experiments in telepathy

Scientists Prove That Telepathic Communication Is Within Reach

An international research team develops a way to say “hello” with your mind

Corinne Iozzio

In a recent experiment, a person in India said “hola” and “ciao” to three other people in France. Today, the Web, smartphones and international calling might make that not seem like an impressive feat, but it was. The greetings were not spoken, typed or texted. The communication in question happened between the brains of a set of study subjects, marking one of the first instances of brain-to-brain communication on record.

The team, whose members come from Barcelona-based research institute Starlab , French firm Axilum Robotics and Harvard Medical School, published its findings earlier this month in the journal PLOS One . Study co-author Alvaro Pascual-Leone, director of the Berenson-Allen Center for Noninvasive Brain Stimulation at Beth Israel Deaconess Medical Center and a neurology professor at Harvard Medical School, hopes this and forthcoming research in the field will one day provide a new communication pathway for patients who might not be able to speak.

“We want to improve the ways people can communicate in the face of limitations—those who might not be able to speak or have sensory impairments,” he says. “Can we work around those limitations and communicate with another person or a computer?”

Pascual-Leone’s experiment was successful—the correspondents neither spoke, nor typed, nor even looked at one another. But he freely concedes that the test was more a proof of concept than anything else, and the technique still has a long way to go. “It’s still very, very early,” he says, “[but] we can show that this is even possible with technology that’s available. It’s the difference between talking on the phone and sending Morse code. To get where we’re going, you need certain steps to be taken first.”

Indeed, the process was drawn out, if not downright inelegant. First, the team had to establish binary-code equivalents of letters; for example “h” is “0-0-1-1-1.” Then, with EEG (electroencephalography) sensors attached to the scalp, the sender moved either his hands or feet to indicate a 1 or a 0. The code then passed to the recipient over email. On the other end, the receiver was blindfolded with a transcranial magnetic stimulation (TMS) system on his head. (TMS is a non-invasive method of stimulating neurons in the brain; it’s most commonly used to treat depression .) The TMS headset stimulated the recipient’s brain, causing him to see quick flashes of light. A flash was equivalent to a “1” and a blank was a “0.” From there, the code was translated back into text. It took about 70 minutes to relay the message.

There is a bit of contention about the degree to which this approach was actually novel. IEEE Spectrum reports that this recent study is quite similar to one conducted at the University of Washington last year. In that study, researchers used the same EEG-to-TMS setup, but rather than pulsed light, stimulated the brain’s motor cortex to subconsciously cause the recipient to strike a key on a keyboard. Pascual-Leone contends, however, that his work is notable because the recipient was conscious of the communication.

Both studies represent only a small step toward engineering telepathy, which might take years—or decades—to perfect. Ultimately, the goal is to remove the computer middleman from the transmission equation and allow direct brain-to-brain communication between people. “We’re still a long way from that,” Pascual-Leone admits, “but in the end, I think it’s a pursuit worthy of the effort.”

Outside of medicine, brain-to-brain communication could find applications across many disciplines. Soldiers, for instance, could use the technology on the battlefield , sending commands and warnings to one another. Civilians might benefit, as well; businesspeople could use it to send cues to partners during negotiations, or pitchers and catchers could avoid sign-stealing during baseball games.

Still, telepathic communication that works like a sort of futuristic walkie-talkie will involve major advances in sensing, emitting and receiving technologies—and perhaps even a slight retraining of the human brain. At the same time, Pascual-Leone cautions that scientists must also keep in mind the ethics of telepathy.

“Could there be potential for sending someone a thought that’s not desirable to them?” he says. “Those kinds of things are theoretically in the realm of possibility."

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Corinne Iozzio | | READ MORE

Corinne Iozzio is a New York–based technology writer and editor. When she’s not fiddling with LEGOs or Nerf blasters, she covers gadgets and emerging tech for various publications, including Popular Science and Scientific American.

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Science News

How hans berger’s quest for telepathy spurred modern brain science.

Instead of finding long-range signals, he invented EEG

Hans Berger, shown, invented the electroencephalogram, or EEG, in an effort to uncover how thoughts could travel between people.

Apic/Getty Images

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By Laura Sanders

July 6, 2021 at 8:00 am

A brush with death led Hans Berger to invent a machine that could eavesdrop on the brain.

In 1893, when he was 19, Berger fell off his horse during maneuvers training with the German military and was nearly trampled. On that same day, his sister, far away, got a bad feeling about Hans. She talked her father into sending a telegram asking if everything was all right.

To young Berger, this eerie timing was no coincidence: It was a case of “spontaneous telepathy,” he later wrote. Hans was convinced that he had transmitted his thoughts of mortal fear to his sister — somehow.

So he decided to study psychiatry, beginning a quest to uncover how thoughts could travel between people. Chasing after a scientific basis for telepathy was a dead end, of course. But in the attempt, Berger ended up making a key contribution to modern medicine and science: He invented the electroencephalogram, or EEG , a device that could read the brain’s electrical activity.

Berger’s machine, first used successfully in 1924, produced a readout of squiggles that represented the electricity created by collections of firing nerve cells in the brain.

To celebrate our upcoming 100th anniversary, we’re launching a series that highlights some of the biggest advances in science over the last century. For more on the past, present and future of neuroscience, visit Century of Science:  Our brains, our futures .

In the century since, the EEG has become an indispensable clinical tool. It can spot seizures , monitor sleep and even help determine brain death . It has also yielded fundamental insights into how the brain works, revealing details about the brain’s activity while at rest, or while crunching numbers or tripping on hallucinogens .

When Berger was young, the idea of paranormal psychic communication didn’t sound as wacko as it does now. “The hangover from the 19th century was this idea of trying to explain cases of telepathy,” says communications expert Caitlin Shure, who wrote her thesis at Columbia University on the concept of brain waves. At that time, scientific societies and serious research initiatives were devoted to probing these occurrences. British physician and author Arthur Conan Doyle, of Sherlock Holmes fame, was a staunch believer. It was, as Shure puts it, “peak telepathy enthusiasm time.”

In a way, this makes sense. Scientific understanding of the world was deepening, along with technological advances in radio. “Why shouldn’t thoughts be able to travel through the universe just like wireless telegraphy?” Shure says.

Berger slogged away to prove how telepathy worked, trying to measure mental activity. He scrutinized blood flow and brain temperature before turning to electrical output. His breakthrough finally came on July 6, 1924, while studying a man with a skull injury called Patient K. Using a vacuum tube amplifier to enhance the electrical signals, Berger was finally able to spot a brain wave.

In 1929, Berger finally published his results, the first of a series of papers with the exact same title, numbered 1 to 14: “Über das Elektrenkephalogramm des Menschen,” or “On the Electroencephalogram of Man.”

The findings “go down like the proverbial lead balloon,” says medical historian and forensic psychiatrist Robert Kaplan of the University of Wollongong in Australia. A more prominent scientist, Nobel laureate Edgar Adrian of the University of Cambridge, was deeply skeptical of Berger’s findings, and repeated the experiments. But Adrian confirmed the results and began to publicize the method and Berger along with it.

The rest of Berger’s story takes a dark turn. In the run up to the second World War, he was ousted from his research position at the University of Jena in Germany and forced into a non-research job at a nursing home. Convinced that he had a fatal heart disease and sick with an infection and depression, Berger died by suicide in 1941 — “a terrible, sad ending to this story,” Kaplan says. The year before, Adrian had nominated Berger for the Nobel Prize in physiology or medicine, but no prize was awarded that year.

Berger wrote late in his life that the waves he discovered couldn’t explain the psychic transference he sought; his waves could not have traveled far enough to reach his sister. But, as Shure points out, echoes of that idea ripple into today’s world, in which we are all instantly and digitally connected. “There’s a way in which these false beliefs, or fantasies, about brain waves or telepathy or thought transference ended up creating that reality,” Shure says. Technology has already begun linking brains wirelessly .

It’s not Berger’s telepathy. But today’s technology is getting us closer to something like it. And at the very least, if you had a near-death experience this morning, your sister would soon know about it.

The National Suicide Prevention Lifeline can be reached at 1-800-273-TALK (8255).

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Brain-to -brain interfaces: the science of telepathy

Research Assistant Professor in Neuroscience, The University of Western Australia

Disclosure statement

Kristyn Bates receives funding from The Raine Medical Research Foundation and The Neurotrauma Research Program (Western Australia).

University of Western Australia provides funding as a founding partner of The Conversation AU.

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Have you ever wondered what it would be like to walk a mile (or 1.6 kilometres) in somebody else’s shoes? Or have you ever tried to send a telepathic message to a partner in transit to “pick up milk on your way home”?

Recent advances in brain-computer interfaces are turning the science fantasy of transmitting thoughts directly from one brain to another into reality.

Studies published in the last two years have reported direct transmission of brain activity between two animals , between two humans and even between a human and a rat . These “brain-to-brain interfaces” (BBIs) allow for direct transmission of brain activity in real time by coupling the brains of two individuals.

So what is the science behind this?

Reading the brainwaves

Brain-to-brain interface is made possible because of the way brain cells communicate with each other. Cell-to-cell communication occurs via a process known as synaptic transmission , where chemical signals are passed between cells resulting in electrical spikes in the receiving cell.

Synaptic transmission forms the basis of all brain activity, including motor control, memory, perception and emotion. Because cells are connected in a network, brain activity produces a synchronised pulse of electrical activity, which is called a “brain wave”.

Brain waves change according to the cognitive processes that the brain is currently working through and are characterised by the time-frequency pattern of the up and down states (oscillations).

For example, there are brainwaves that are characteristic of the different phases of sleep , and patterns characteristic of various states of awareness and consciousness.

Brainwaves are detected using a technique known as electroencephalography ( EEG ), where a swimming-cap like device is worn over the scalp and electrical activity detected via electrodes. The pattern of activity is then recorded and interpreted using computer software.

This kind of brain-machine interface forms the basis of neural prosthesis technology and is used to restore brain function . This may sound far-fetched, but neural prostheses are actually commonplace, just think of the Cochlear implant !

Technical telepathy

The electrical nature of the brain allows not only for sending of signals, but also for the receiving of electrical pulses. These can be delivered in a non-invasive way using a technique called transcranial magnetic stimulation ( TMS ).

A TMS device creates a magnetic field over the scalp, which then causes an electrical current in the brain. When a TMS coil is placed over the motor cortex, the motor pathways can be activated, resulting in movement of a limb, hand or foot, or even a finger or toe.

Scientists are now working on ways to sort through all the noise in brainwaves to uncover specific signals that can then be used to create an artificial communication channel between animals.

The first demonstration of this was in a 2013 study where a pair of rats were connected through a BBI to perform a behavioural task. The connection was reinforced by giving both rats a reward when the receiver rat performed the task correctly.

Hot on the heels of this study was a demonstration that a human could control the tail movements of a rat via BBI.

We now know that BBIs can work between humans too. By combining EEG and TMS, scientists have transmitted the thought of moving a hand from one person to a separate individual, who actually moved their hand. The BBI works best when both participants are conscious cooperators in the experiment. In this case, the subjects were engaged in a computer game.

Thinking at you

The latest advance in human BBIs represents another leap forward. This is where transmission of conscious thought was achieved between two human beings in August last year.

Using a combination of technologies – including EEG, the Internet and TMS – the team of researchers was able to transmit a thought all the way from India to France.

Words were first coded into binary notation (i.e. 1 = “hola”; 0 = “ciao”). Then the resulting EEG signal from the person thinking the 1 or the 0 was transmitted to a robot-driven TMS device positioned over the visual cortex of the receiver’s brain.

In this case, the TMS pulses resulted in the perception of flashes of light for the receiver, who was then able to decode this information into the original words (hola or ciao).

Now that these BBI technologies are becoming a reality, they have a huge potential to impact the way we interact with other humans. And maybe even the way we communicate with animals through direct transmission of thought.

Such technologies have obvious ethical and legal implications, however. So it is important to note that the success of BBIs depends upon the conscious coupling of the subjects.

In this respect, there is a terrific potential for BBIs to one day be integrated into psychotherapies, including cognitive behavioural therapy , learning of motor skills, or even more fantastical situations akin to remote control of robots on distant planets or Vulcan-like mind melds a la Star Trek.

Soon, it might well be possible to really experience walking a mile (or a kilometre) in another person’s shoes.

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• v.1(2); Jul-Dec 2008

Investigating paranormal phenomena: Functional brain imaging of telepathy

Ganesan venkatasubramanian.

National Institute of Mental Health and Neurosciences, Bangalore, India

Peruvumba N Jayakumar

Hongasandra r nagendra.

1 Swami Vivekananda Yoga Anusandhana Samsthana, Vivekananda Yoga Research Foundation, Bangalore, India

Dindagur Nagaraja

Bangalore n gangadhar.

“Telepathy” is defined as “the communication of impressions of any kind from one mind to another, independently of the recognized channels of sense”. Meta-analyses of “ganzfield” studies as well as “card-guessing task” studies provide compelling evidence for the existence of telepathic phenomena. The aim of this study was to elucidate the neural basis of telepathy by examining an individual with this special ability.

Materials and Methods:

Using functional MRI, we examined a famous “mentalist” while he was performing a telepathic task in a 1.5 T scanner. A matched control subject without this special ability was also examined under similar conditions.

The mentalist demonstrated significant activation of the right parahippocampal gyrus after successful performance of a telepathic task. The comparison subject, who did not show any telepathic ability, demonstrated significant activation of the left inferior frontal gyrus.

Conclusions:

The findings of this study are suggestive of a limbic basis for telepathy and warrant further systematic research.

INTRODUCTION

“Telepathy” is defined as “the communication of impressions of any kind from one mind to another, independently of the recognized channels of sense”.[ 1 ] With the help of various rigorous paradigms over the last 70 years, systematic research has lent support to the reality of telepathy.[ 2 ] Meta-analyses of “ganzfield” studies[ 3 ] as well as “card-guessing task”[ 4 ] studies provide compelling evidence for the existence of telepathy. This mysterious phenomenon has implications not only in the cognitive sciences but also in the biological and healing sciences.[ 2 ] It has long been assumed that conscious intention has the capacity to affect living systems across a distance. Intercessory prayers, healing energy, and similar other methods have long been a part of medicine.[ 2 ] Hence, analyzing the underpinnings of telepathy might potentially help in understanding the “distant-healing” phenomena also.

Examining people with extraordinary capabilities involving paranormal phenomena might help in a better understanding of these puzzling entities.[ 5 ] Previous such studies examining people with “special talents”[ 5 , 6 ] yielded significant insights. Similarly, studies have been conducted on people experiencing paranormal phenomena. A functional MRI study on “distant intentionality” (defined as sending thoughts at a distance) examined the brain activation pattern in a recipient of thoughts from healers who espoused some form for connecting or healing at a distance. The recipient demonstrated significant brain activations in the anterior and middle cingulate areas, precuneus, and the frontal regions.[ 7 ] Previous studies[ 8 , 9 ] examining subjects with telepathic ability suggested an association of paranormal phenomena with the right cerebral hemisphere. It has been reported that correlated neural signals may be detected by fMRI in the brains of subjects who are physically and sensorily isolated from each other.[ 10 ] In light of these previous studies, we aimed to examine the functional neuroanatomical correlates of telepathy in Mr. Gerard Senehi, an “expert with telepathic ability (mentalist)” using functional Magnetic Resonance Imaging (fMRI).

MATERIALS AND METHODS

Mr. Gerard Senehi [Mr. GS] (aged 46 years) is well known for his abilities to perform various paranormal tasks such as telekinesis, mind reading, and telepathy ( http://www.experimentalist.com ). Mr. JS, the comparison subject, is a 43 –year-old male, who was aware of various paranormal phenomena including telepathy, but did not have any paranormal abilities to the best of his knowledge. Both the subjects were right-handed[ 11 ] and possessed Master's Degrees. Both the subjects were screened using the General Health Questionnaire[ 12 ] and a comprehensive mental status examination was done to rule out any psychiatric disorder. Neither of them had any history suggestive of substance abuse or dependence, medical or neurological disorders. Neither had any contraindication for MRI. The study procedures were explained to the subjects and written informed consent was obtained. The study protocol was reviewed and approved by the institute's ethics committee.

Telepathy task

One of the investigators (PNJ) drew an image in the presence of other investigators [HRN, BNG, and GVS]. Figures ​ Figures1A 1A and ​ and2A 2A were the images drawn by PNJ for the “mentalist” and the control subject while both were seated in separate rooms. Neither the mentalist [GS] nor the control subject [JS] knew what the image was. The subject was then shifted to the MRI scanner and the investigator (PNJ) was seated in the MRI console room (about 15 feet away). Adequate precautions were taken to avoid sensory leakages by following the guidelines of Hyman and Honorton.[ 13 ] During the scan, the subject was instructed to perform the act of telepathy to think about and identify the probable image that would have been drawn by the investigator during the designated epochs of “activation” and not to engage in this task during the periods of “rest”. The subjects were visually cued (using a mirror attached to the head coil which reflected the cues projected on a screen) by green and red stars to indicate the respective onset of activation and rest epochs. The investigator (PNJ) was also given the same cues and was engaged in transmitting the image to the subject in the MRI scanner during the “activation” periods, stopping during the periods of rest. After the scanning, the subject was asked to draw the image that he was able to obtain by performing telepathy. Figure 1B was the image reproduced by the “mentalist” and Figure 2B was the image reproduced by the control. Both the subjects were scanned on the 3 rd day of the lunar cycle and at the same time of the day (1400 hours IST) separated by a three-month interval.

Image drawn by the investigator (PNJ) for the “mentalist” [Mr. GS]

Image reproduced by the mentalist [Mr. GS] after the telepathic task

Image drawn by the investigator (PNJ) for the control subject [Mr. JS]

Image reproduced by the control subject [Mr. JS] after the telepathic task

Imaging procedures

MRI was done with 1.5 Tesla Magnetom ‘vision’ scanner. First, a T 1 -weighted three-dimensional Magnetization Prepared Rapid Acquisition Gradient Echo sequence was performed (TR = 9.7 msec; TE = 4 msec; nutation angle = 12°; FOV = 250 mm; slice thickness 1 mm; NEX = 1; matrix = 200 × 256; 160 sagittal slices). After obtaining the anatomical MR images, echo-planar images (EPI) were obtained. They consisted of 112 functional acquisitions, with each acquisition consisting of 16 slices (slice thickness = 8 mm without any interslice gap) in the axial plane covering the entire brain. The parameters for a multishot EPI sequence using Blood Oxygen Level Dependent (BOLD) contrast were as follows: repetition time = 4000 msec; echo time = 76 msec; flip angle = 90°; FOV = 250 mm; matrix 128 × 128. The acquisitions were grouped in blocks of eight, yielding 14 blocks. The condition for successive blocks alternated between “rest” and the “telepathic” task, starting with “rest”. This “rest-telepathy” paradigm yielded seven sets of “rest” and “telepathy”.

Image analysis

The fMRI analysis was performed using Statistical Parametric Mapping-2 (SPM2)( http://www.fil.ion.ucl.ac.uk/spm ). The EPI images were realigned and corrected for slice timing variations. The images were then normalized[ 14 ] to the Montreal Neurological Institute (MNI) space.[ 15 ] Finally, the images were smoothened with a gaussian kernel of 6 mm full-width, half-maximum.

SPM2 combines the General Linear Model and Gaussian field theory to draw statistical inferences from BOLD response data regarding deviations from the null hypothesis in three-dimensional brain space.[ 16 ] The images were analyzed using a block design paradigm with a canonical hemodynamic response function. The epochs of rest were subtracted from the epochs of the telepathic task performance. The voxel-wise analysis produced a statistical parametric map of brain activation associated with the telepathic task in the MNI space. Significance corrections for multiple comparisons were performed using a False Discovery Rate (FDR) correction[ 17 ] ( P < 0.05). The coordinates of significant areas of activation were transformed from MNI space[ 15 ] into the stereotactic space of Talairach and Tournoux[ 18 ] using nonlinear transform.[ 19 ] The brain regions were localized from the Talairach and Tournoux co-ordinates using automated software.[ 20 ]

The image [ Figure 1B ] reproduced by the “mentalist” showed striking similarity to the original image drawn by the investigator (PNJ) whereas the one reproduced by the control subject [ Figure 2B ] did not. The mentalist showed significant activation involving the right parahippocampal gyrus [Number of voxels = 160; Talairach and Tournoux co-ordinates of peak activation: ‘x’ = 32, ‘y’ = -41, ‘z’ = -6; T = 4.88; P (uncorrected) < 0.001; FDR-corrected P = 0.018] [ Figure 3 ] whereas the control subject showed significant activation involving the left inferior frontal gyrus [number of voxels = 363; Talairach and Tournoux co-ordinates of peak activation: ‘x’ = -42, ‘y’ = 25, ‘z’ = -8; T = 4.21; P (uncorrected) < 0.001; FDR-corrected P = 0.037] [ Figure 4 ].

Right Parahippocampal Gyrus Activation in the subject with telepathic ability [Mr. GS], while performing a successful telepathic task. On the left hand side, the activation is superimposed on a glass brain and on the right hand side, the activation [yellow] is superimposed on a structural MR image

Left Inferior Frontal Gyrus Activation in the control subject without any telepathic ability [Mr. JS], while performing an unsuccessful telepathic task. On the left hand side, the activation is superimposed on a glass brain and on the right hand side, the activation [yellow] is superimposed on a structural MR image

To our knowledge, this is the first fMRI study to examine the brain correlates of telepathy. Previous studies have employed other functional brain mapping techniques such as Single Photon Emission Computed Tomography (SPECT)[ 8 ] and electroencephalography (EEG) and MRI[ 9 ] to investigate paranormal phenomena in selected individuals. In our study, telepathy was associated with significant activation of the right parahippocampal gyrus; whereas the control subject without telepathic ability, activated the left inferior frontal gyrus under similar task conditions.

A previous study[ 9 ] on Mr. Ingo Swann (who had the special ability of remote-viewing) showed that the proportions of unusual 7-Hz EEG spike and slow wave activity over the occipital lobes per trial had a correlation with the ratings of response accuracy. Neuropsychological and MRI analyses suggested a differential structural and functional organization within the parieto-occipital region of Mr. Swann's right hemisphere.

Another SPECT study[ 8 ] examined Mr. Sean Harribance, who routinely experienced “flashes of images” of objects that were hidden and of accurate personal information concerning people with whom he was not familiar. The “extrasensory” processes in Mr. Harribance correlated quantitatively with morphological and functional changes involving the right parietotemporal cortices (or its thalamic inputs) and hippocampal formation.

Together, these two studies suggest that paranormal phenomena might have a relationship with the right cerebral hemisphere, especially the right posterior cortical and hippocampal regions. The parahippocampal region is very closely linked to the hippocampus, both structurally and functionally.[ 21 ] So, the current study findings also support the association between the right hippocampal system and paranormal phenomena.

In our study, the control subject activated his left inferior frontal gyrus during his unsuccessful telepathic task performance; this brain area is implicated in the “Theory of Mind [ToM]”.[ 22 ] The attribution of mental states, such as desires, intentions, and beliefs, to others has been referred to as ToM.[ 23 ] Empathy, conceptually related to ToM, is described as the ability to infer and share the emotional experiences of another.[ 24 ] An earlier study reported that psychic mind readers had greater cognitive empathy than individuals without these abilities.[ 5 ] Importantly, hippocampal brain regions are important for empathy.[ 25 ] Thus, our observations derive indirect support from this earlier study.[ 5 ]

Superior empathizing abilities have been hypothesized to be important for both telepathy[ 5 ] as well as for distant intentionality.[ 7 ] Interestingly, the cuneus (a brain region associated with empathy[ 26 ]) has been reported to be linked with distant intentionality.[ 7 ] Also, in our study, the hippocampal region (associated with empathy[ 25 ]) is implicated in telepathy. These observations support the hypothesized link between empathy and special abilities. It is possible that people with telepathy or distant healing abilities might possess the ability to activate differentially specific brain regions (in localization, e.g , anterior vs posterior brain regions or in lateralization, e.g , right vs left brain) related to the empathy circuit in comparison to individuals without these abilities.

On the contrary, empathy deficits[ 27 ] and cuneus[ 28 ] and parahippocampal abnormalities[ 29 ] and anomalous right hemisphere overactivation[ 30 ] have been reported in schizophrenia. Most of these “left-hemisphere dominance failure” findings have been conceptualized as being “abnormal” in their tendency to increase a person's proclivity towards psychosis. Paradoxically, evolutionary theories on psychosis propose an alternative possibility that some of these traits might be of crucial utility.[ 31 ] It has been proposed that this dominance failure (and consequent right hemisphere overactivation) facilitates the emergence of paranormal and delusion-like ideas by way of right hemispheric associative processing characteristics, i.e. , coarse rather than focused semantic activation. Interestingly, the ability to detect subtle magnetic field energies might underlie paranormal phenomena.[ 32 ] Moreover, magnetic field abnormalities have been described to be the underlying basis for psychotic symptoms.[ 33 , 34 ] However, it is yet to be examined whether a conglomeration of these features ( i.e. , reduced left hemispheric dominance, paranormal beliefs) are also indicative of an inherent advantage towards acquiring “special” abilities in some people (of course, with enhancement towards psychosis in others) possibly due to an enhanced tendency to perceive subtle geomagnetic energy alterations.

Ours is probably the first fMRI study to examine the neuroanatomical correlates of telepathy. fMRI offers methodological advantages of nonradioactive and noninvasive real-time imaging of the brain. We have employed a well-researched and validated image analysis paradigm with optimal correction for false positive results. Our study methodology strictly adhered to the guidelines for research on paranormal phenomena proposed by Hyman and Honorton.[ 11 ] These include rigorous precautions against sensory leakage, extensive security procedures to prevent malpractices, full documentation of all experimental procedures and equipment, and complete specifications about statistical analyses.

Nonetheless, one has to be cautious while interpreting the study findings due to the following limitations: i) ideally, it would have been methodologically more rigorous if Mr. Gerard had replicated the successful telepathic task with similar brain activation during another session of fMRI on a different occasion. As Mr. Gerard had reported some inexplicable discomfort in the few days following the fMRI, this could not be done; Ii) examination of just one control subject is another limiting factor.

CONCLUSIONS

In summary, this study's findings are suggestive of an association between telepathy and the right parahippocampal gyrus. The methodological rigor, isolated and robust brain activation with telepathy, and established theoretical relevance of this brain region with reference to paranormal phenomena highlight the need for further studies using advanced fusion imaging techniques (simultaneous fMRI, EEG, and magnetoencephalography) to examine telepathy.

Acknowledgments

We sincerely thank Mr. Gerard Senehi and the comparison subject [Mr. JS] for consenting to be subjects for this study.

Persona 4 Golden

Persona 4 Golden Quests guide: complete quest walkthroughs and rewards

Like any good RPG - and trust us, this is a good one indeed -  Persona 4 Golden  has a whole lot of  side quests  with  unique rewards . On this page, we list  every quest  - and even have a short  walkthrough  for each one.

That makes this a long page, as there are 69 side quests in P4G (nice). The side quests provide a range of rewards - many give useful battle items, while some give money, raise your social stats and unlock new weapons and books . A handful also unlock new costumes for characters, too.

Side quests are great to complete in Persona 4, as generally speaking they don't use up your valuable limited in-game days - they're free ways to make some gains in some way or another. Some of the rewards are genuinely pretty sweet, too, though the solutions for the quests - many of which are simple dungeon loot fetch quests - can be a little esoteric - which is what our mini walkthroughs are here to help with.

Persona 4 Golden Quests guide: side quest walkthroughs for every mission

Side Quests in Persona 4 Golden take on different forms, but there's generally a few different templates. Some will ask you to grab an item from inside a dungeon, either from a chest as a drop from a specific enemy.

Others will ask you quiz questions (a bit like with the Persona 4 Golden school answers ), or get you to puzzle something out around the combat-free overworld of Inaba. Some quests will require access to the Okina City and Shichiri Beach areas, optionally unlocked through bike riding once you get a scooter .

Most quests don't have time limits, but a few do have strict cut-off points. We make clear which below. Some quests also lead into others directly, meaning there's a strict order of progression - again, that's listed below. 

Anyway, let's get to the quests and walkthroughs:

• Date: Available from 4/18 
• Location: Recieve from 'Funky Student' at the Classroom Building 3F
• Reward: 3x Chest Keys
• Solution: Quiz questions. Answer 1 is "Group A", Answer 2 is "Top 6 Flag Colors".
• Date: Available from 4/25 
• Location: Male Student in Classroom 2-2, Yasogami High
• Reward: 1x Goho-m
• Solution:  Go to the school rooftop and speak to the girl; choose the second option. Do this three times. The girl won't be there on rainy days.
• Date: Available from 5/1
• Location: Man at the Samegawa Flood Plain
• Reward: 4000 Yen
• Solution: An item found inside the Twisted Shoping District inside the TV; search the table in the middle of the room.
• Date: From 5/2
• Location: Classroom 2/2 from a Female Student 
• Reward: 3x Chest Key
• Solution: The item you need is dropped by the Avenger Knight enemy in Yukiko's Castle, common around floor 7.
• ​Location: Shady Student in the Classroom Building 2F
• Reward: 3x Dokudami Tea
• Solution: This item drops from the Magical Magus enemies in the Yukiko's Castle dungeon, around floor 7.
• Location: A Boy in Practice Building 1F
• Reward: 1x Olympic Tape
• Solution: Required item is a drop from Heat Balance in Yukiko's Castle dungeon, around floor 7.
• Date: Quest 7 from 5/6, each other quest in order after the previous is completed
• Location: The Shrine
• Reward: Hermit S-link Rank Up
• Solution: Quests 7, 8, 9, 10, 11, 12, 13, 14 and 15 are all ones you complete as part of the Hermit social link. We've got a separate guide with the solutions to each Fox quest for the Hermit s-link , so check that out.
• Date: 5/18 onwards
• Location: From a student in Practice Building 2F at school 
• Reward: 5x Royal Jelly
• Solution: Get a Fitting Board, which is a drop from the Laughing Table enemies on Yukiko's Castle level 6 & 7.
• Date: 5/18 
• Location: Ms. Sofue, Classroom Building 2F
• Reward: 2x Pulsating Stone
• Solution: You need a Suspicious Pole. Get this from the Trance Twins enemy found around Yukiko's Castle 4F.
• Date: 5/23 on
• Location: Avid Reader NPC in the South Shopping District
• Reward: The Gentle Way book
• Solution: Hand over a Peach Seed, commonly found in unlocked chests in any dungeon.
• Prerequisites: Scooter and access to Okina required
• Location: Male NPC in front of the Theatre at Okina City
• Reward: 5000 Yen
• Solution: The quest giver wants you to give out his tissues to everyone in the immediate area. Do this and return to him.
• Prerequisites: Must have completed Quest 18
• Location: From the Avid Reader in the South Shopping District Again
• Reward: The Punk Way Book
• Solution: Feed the cat on the Samegawa Flood Plain
• Date: 6/3, after Quest 20 is complete
• Location: Samegawa Flood Plain, the cat
• Reward: 1x Heal Jelly
• Solution: Feed the Cat.
• Date: 6/3, after Quest 21 is complete
• Location: Samegawa Flood Plain, the cat
• Reward: 1x Soma
• Solution: Feed the cat 7 times.
• Location: Old Woman NPC on the Samegawa Flood Plain
• Reward: Knowledge & Courage stats raised
• Solution: The item you need is dropped by the Bribed Fuzz enemy in the first 2 floors of the Steamy Bathhouse dungeon.
• Location: Northern Shopping District, from an old man 
• Reward: 15,000 Yen
• Solution: Item required is dropped by the Selfish Basalt enemy around the Steamy Bathhouse dungeon 6F.
• Prerequisites: you must have started Gardening
• Location:  Fussy Housewife in Junes
• Reward: Sharp Shovel (Protagonist Weapon)
• As well as being a weapon for the protagonist, you can use the Sharp Shovel elsewhere. Go to the Samegawa Flood Plain Riverbank at night, talk to the dog there and then use the shovel to dig - you'll get the Bone, a weapon for Yosuke.
• Prerequisites: Complete Quest 4
• Location: Timid Female Student in Classroom 2-2
• Reward: 3x Ointment
• Solution: The item you need drops from the Tranquil Idol enemy - around Steamy Bathhouse 7F.
• Prerequisites: Must have completed Quest 26
• Location: Classroom 2-2, from the same female student 
• Reward: 5x Chest Key
• Solution: This item drops from the Liberatring Idol enemy in the Marukyu Striptease dungeon, between floors 5 and 10.
• Prerequisites: Must have completed Quest 16
• Location: Student in the Practice Building 2F 
• Reward: 2x Snuff Soul;
• Solution: The Reflecting Board item you need drops from the Crying Table found around the 10th floor of the Steamy Bathhouse dungeon.
• Date: From 6/9 on
• Location: From a woman in the South Shopping District
• Reward: 18,000 Yen
• Solution: The item you want drops from the Grave Beetle, around the 10th floor of the Steamy Bathhouse.
• Cut-off: Must be completed by 11/17
• Location: A gir4l in the South Shopping District
• Reward: 5x Goho-m
• Solution: Find the girl's twin. She's in the Samegawa Flood Plain, at the Gazebo.
• Location: Male Student in Classroom Building 3F
• Reward: Bamboo Broom (Protagonist Weapon)
• The Lai Katana unlocks at the main weapon shop after you sell 6 Golden Cloth materials; you can get these from Phantom Mage enemies in Yukiko's Castle, around Floor 6 and 7.
• To get the Cleaning Uniform, travel to Okina City on your scooter. The Cleaning Uniform is one of the Persona 4 character costumes you can buy, for 14,800 Yen.
• Location: Daidara in the Weapon Store
• Reward: Inaba Trout (Yosuke Weapon)
• Solution: Catch this fish in the fishing mini-game, or buy it from the shopping TV channel on 6/5.
• Prerequisites: Must have completed Quest 6
• Location: Practice Building 1F from a Male Student
• Reward: 2x Uplifting Radio
• Solution: This item is a drop from the Silver Dice enemies found on floors 8, 9 and 10 of the Marukyu Striptease dungeon.
• ​​​​​​​Date: 7/3
• Prerequisites: Must have completed Quest 20
• Location: Avid Reader NPC in the Shopping District South, once again 
• Reward: Guide to Pests book, 5000 Yen
• Solution: To get the Hard Boots, sell the weapon store 5 Thick Hide materials to unlock the boots, then buy them. The Thick Hide drops from the Dancing Hand enemies in the Steamy Bathhouse floors 7 and 8.
• ​​​​​​​Date: 7/13
• Cut-off: Complete before 11/17
• Location: A young girl at the Samegawa Flood Plain Riverbank 
• Reward: 3x Value Medicines
• Solution: Get a Flower Brooch, which drops from the Soul Dancer on the first few floors of Marukyu Striptease
• ​​​​​​​Date: 8/22
• Prerequisites: Must have completed Side Quest 35
• Reward: 3x Macca Leaf
• Solution: The same girl as before now wants a Leaf Pochette. This drops from the Blind Cupid enemy found in the Void Quest dungeon - try floors 2, 3 and 4. 
• ​​​​​​​Date: 7/16
• Location: From a woman found in Okina City, accessed via the Scooter.
• Reward: 3x Royal Jelly
• Solution: You need the Gentleman's Tux armor, which is found in a golden (locked) chest in the Yukiko's Castle dungeon.
• ​​​​​​​Date: 7/25
• Location: From Daidara in the weapon shop
• Reward: Grilled Corn (Yosuke Weapon)
• Solution: The quest name is pretty clear - buy Barrier Corn seedlings, and then plant and harvest them in your garden.
• Location:  Samegawa Flood Plain, from an old man
• Reward: 30,000 Yen
• Solution: The item you need is a drop from the Amenti Raven enemy, in the Void Quest dungeon - try floor 4 or floor 2.
• ​​​​​​​Date: 8/9
• Prerequisites: Complete Quest 34
• Location: Avid Reader NPC in the South Shopping District 
• Reward: Riddlemaina book & 10,000 Yen
• Solution: You'll need the Fashionable Dishes item; this drops from the Sky Balance enemies in Marukyu Striptease around floor 9 or 10. These are materials, not key items, meaning if you're not careful you can sell them to the weapon store and then thus have to go and farm more.
• ​​​​​​​Date: 9/2
• Location: Okina City, a girl in front of the theatre  
• Solution: After getting the quest, go back to your school and to the Classroom Building 2F. Talk to your history teacher and ask about the movie - then relay that information back to the questgiver.
• Prerequisites: Must have completed Quest 1
• Location: Funky Student in the Classroom Building 3F
• Reward: 1x Snuff Soul
• Solution: Questions again - the answers are "Group A", then "Human Motion", then "Group B" and finally "Indefinite Articles". 
• ​​​​​​​Date: 9/5
• Prerequisites: Complete Quest 30
• Location: A girl in the Southern Shopping District.  
• Reward: 5x Dokudami Tea
• Solution: Go back to the other twin at the Flood Plain and then report back.
• ​​​​​​​Date: 9/20
• Prerequisites: Must have completed Quest 17
• Location:  Ms. Soufe in the Classroom Building 2F
• Reward: 2x Mysterious Scarab
• Solution: You need to get the Culurium item; get it as a drop from the Steel Machine enemy found on level 10 of the Void Quest dungeon.
• ​​​​​​​Date: 9/26
• Prerequisites: Complete Quest 40
• Location: Once again the Avid Reader NPC in the South Shopping District 
• Reward: Who Am I? book
• Solution: Another quiz. The anwers are "Judo medalists' names", "How one should live as a punk", "2 pages per test" and finally "Me". 
• ​​​​​​​Date: 9/28
• Prerequisites: Clear Quest 28
• Location: The same desk-obsessed student in the Practice Building 2F 
• Reward: 1x Bead
• Solution: You now need 3x Proof of Passion items; they drop from the Furious Gigas enemies in level B4 of the Secret Lab.
• ​​​​​​​Date: 10-12
• Prerequisites: Access to Okina City
• Location: A woman in Okina
• Reward: Disco Fan (Yukiko Weapon)
• To get the Power Rocks, go to the Secret Lab - floors 4-7 have the Power Castle enemies that drop them.
• Note: Quest 51 requires Fine Coal from the same enemy. These cannot be done at the same time. If Quest 51 is accepted, the item for Quest 47 will not drop.
• Location: Ms. Nakayama in the Classroom Building 3F
• Reward: 35,000 Yen
• Solution: The item you require drops from the Constancy Relic enemy in the Secret Lab's first few floors.
• Location:  The Principal, Classroom Building 3F
• Reward: Understanding and Diligence stats raised
• Solution: The item you need drops from the Wicked Turret enemies, commonly found on levels B7 and B8 of the Secret Lab
• Location: Classroom Building 1F, from a female student
• Reward: 3x Physical Mirror
• Solution: Required item drops from, the Mach Wheel enemy - try the Secret Lab B5 and B6. 
• Prerequisites: Cleared Quest 24
• Location: North Shopping District, from an old man
• Reward: 40,000 Yen
• Note: Quest 47 requires Power Rocks from the same enemy. These cannot be done at the same time. If Quest 51 is accepted, the item for Quest 47 will not drop.
• Prerequisites: Quest 28 Completed
• Cut-off: Clear before 11/17
• Location: Samegawa Riverbank, with the twins again 
• Reward: Chain Bead
• Solution: The twin wants a Branch Headband item; this drops from the Elegant Mother enemy in the Secret Lab's B8.
• Date: 10/31
• Prerequisites: Complete Quest 52
• Location: Samegawa Riverbank with the twin again 
• Reward: 3x Super Sonic
• Solution: Head to the shopping district to speak to the other twin, then back to the questgiver twin, then back to the shopping district twin once more.
• Location:  From a Woman in Okina City, reached oin the scooter
• Reward: Haikara Shirt (Armor)
• First, go to the Northern side of the Shopping District in Inaba. The dog is in front of Souzai Daigaku - but first go to the store and buy 3 Steak Skewers. 
• With your 3 Steak Skewers in hand, talk to the dog. Select "Mika-chan", then feed it the Skewer.
• Keep coming back daily to feed the Skewers until the dog returns home.
• Date: 11/12
• Prerequisites: Cleared Quest 47
• Location: Funky Student in Classroom Building 3F 
• Reward: 1x Chewing Soul
• Solution: Another quiz. Answer "Group B", "The way they're drawn", "Promethium" and finally "Need". 
• Date: 11/22
• Prerequisites: Quest 1 Cleared
• Location: Classroom 2-2 from a male student
• Reward: 1x Mokoi Doll
• Solution: The student needs an Animal Guide item, dropped from the Prime Magus enemy in the first few floors of the Heaven dungeon.
• Prerequisites: Quest 46 Cleared
• Location: Practice Building 2F, back with the desk student who treats you like their own personal IKEA
• Reward: 3x Fire Signal
• Solution: This time, she needs Classy Lumber. This is a loot drop from the Angry Table, found between floors 5 and 7 of the Heaven dungeon.
• Prerequisites: Complete Quest 33
• Location:  Boy in the Practice Building 1F
• Reward: 1x Spirit Radio
• Solution: The item you want is a drop from the Revelation Pesce enemy - found in Heaven 7F and 8F.
• Prerequisites: Complete Quest 39
• Location: Old man at the Samegawa Flood Plain
• Reward: 45,000 Yen
• Solution: The item you're after is a drop from defeated Phantom Lord enemies. Find them in the Heaven dungeon, floors 3, 4, 5 and 6.
• Location: Daidara in the weapon store
• Reward: Guardian (Kanji Weapon)
• So if you haven't caught one before this quest pops and held onto it, you'll need one now. To catch it, get an Inaba Jewel Beetle bait from the bug-catching mini-game and get to fishing. It may take quite a few attempts for it to appear.
• Prerequisites: Complete Quest 54
• Location:  From the same woman in Okina City as the previous pet-finding quest
• Reward: Haikara Shirt
• Solution: This time, there's a cat at the Samegawa Riverbank. You'll need to feed it a fish every day, daily, for five days. It has to be a different fish each day, too.
• Prerequisites: Complete Quest 44
• Location:  From Ms. Sofue in the Classroom Building 2F
• Reward: 1x Silver Tray
• Solution: You need an Orichalcum. This drops from the Solemn Machine enemy - found in the Magatsu Mandala dungeon's 6th floor.
• Prerequisites: Clear Quest 48
• Location: From Ms. Nakayama in the Classroom Building 3F
• Reward: 65,000 Yen
• Solution: Grab the Golden Chains you need - 3 of them - from the Minotaur II enemy found on the 6th floor of Magatsu Mandala. 
• Prerequisites: Complete Quest 55
• Location:  From Funky Student, Classroom Building 3F
• Reward: 3x Soul Food
• Solution: Another quiz, with only one answer: Uruguay.
• Location: from Ms. Kashiwagi, Classroom Building 1F
• Reward: Coronet Armor (for Naoto)
• Solution: A little pop quiz. Pick the "Deoxyribonucleric Acid" and "Rabbit". 
• Prerequisites: Cleared Quest 65
• Location: Another from Ms. Kashiwagi, Classroom Building 1F
• Reward: Fighter Armor (for Chie)
• First, get 8 Bloody Hides from the Jotun of Blood enemies on Level 8 of Heaven. 
• Sell these materials to unlock the Vidar's Boots in the weapon store. 
• With Vidar's Boots, talk to the Artisan Apprentice NPC in the Practice Building 2F. Trade them for the Animal Slippers.
• Take the slippers back to Kashiwagi.
• Prerequisites: Cleared Quest 66
• Location:  Another from Ms. Kashiwagi in the Classroom Building 1F
• Reward: Magical Armor (for Yukiko) and Animal Paw (Teddie weapon)
• 8 Magatsu Xandrite - these drop rarely when the Gold Hand enemies are defeated in dungeons. They're more common in later dungeons, if you need to farm.
• 1 Mondo Stone - to get this, talk to a lady in white who appears most (but not all) nights in the Northern part of the Shopping District, near the Shrine entrance. You'll need to trade her a Samegawa Guardian, which is a rare fish you hopefully caught through a few earlier quests.
• Take these to the Shiroku Pub (the weapon shop at night) to get the Animal Paw Weapon for Teddie. Show it to Kashiwagi for a reward.
• Prerequisites: Quest 67 Completed
• Location: Ms. Kashiwagi, Classroom Building 1F 
• Reward: School Swimsuit Costume
• Solution: Another pop quiz - the answers are "Raster", "Latin" and finally "Rome". 
• Location: The Principal, Classroom Building 3F
• Reward: Victory Fan (Yukiko Weapon)
• Solution: Another quiz - the answers are "Wind", "Upper, middle and lower", "7" and "Mameluke"

The Time When the U.S. Conducted Telepathic Experiments at Fort Meade

Our government’s dip into the paranormal is a matter of public record..

Looking across government buildings in Fort Meade, Maryland. (Photo: Brooks Kraft/Getty Images)

Conspiracy theories about secret government investigations into seemingly-science fictional phenomena are a dime a dozen. What’s rarer is actual, documented proof of governmental tinkerings with these forces.

But for nearly 20 years beginning in the 1970s, a secret army unit, working in conjunction with the Defense Intelligence Agency, conducted research and experiments in which they attempted to test the existence and application of so-called “psychoenergetics” for use in military operations. And it’s all a matter of public record .

With the $20 million Stargate Project—a collective name for a series of programs with codenames like GRILL FLAME and SUN STREAK—the U.S. government was training an army of telepaths. Or, at least, they were trying to.

These programs, based out of a base in Fort Meade, Maryland, primarily focused on “remote viewing,” the practice of using extrasensory perception to gain information about locations that, because of distance or other impediment, can’t be seen by the human eye. In some cases, this included using remote viewing for the purposes of precognition– seeing into the future.

The Defense Intelligence Agency headquarters in 1988, when the Stargate Project was still ongoing. (Photo: Public Domain )

In typical Stargate experiments, psychics personnel were asked to do things like “access and describe” locations such as the U.S. Library of Congress , a distant lighthouse , and Stonehenge . In many cases, they were believed to be successful.

Reportedly, Stargate boasted as many as 22 of these psychics, comprising both civilian and military personnel, over the course of its operations. Many had been recruited by by Lieutenant. Frederick Holmes “Skip” Atwater, who was an aide to Major General Albert Stubblebine, and claimed to have experienced his own psychic insights since his early childhood.

It wasn’t all just experimentation either, either. According to retired U.S. Army officer Joseph McMeonagle, who worked extensively with Project Stargate from its early days until 1984, the team’s psychics sometimes assisted with actual intelligence operations when all other methods failed. Supposedly, their methods were used to locating Brigadier General Joseph L. Dozier, who had been kidnapped in Italy in 1981.

Other assignments included ascertaining the whereabouts of Saddam Hussein during the Gulf War, gathering information about Russian submarine capabilities, and searching for plutonium in North Korea.

Night settles on Fort Meade. (Photo: Bossi/CC BY-SA 2.0 )

While many inside the project considered it a success, and believed that their findings provided proof of the effectiveness of psychic techniques for military uses, these claims were later refuted. After the project was transferred to the CIA in 1995, the CIA convened a panel with the American Institutes for Research, who issued a report saying that Stargate had been an expensive failure, and cited sloppy methodology as the reason for most of its supposed positive findings.

The Stargate Project was shuttered and declassified in September of that that year. By that time, there were only three remaining psychics, one of whom reportedly had a shockingly unscientific method of divination: tarot cards.

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Parapsychology

Mental telepathy is real, scientists show mind-to-mind communication over the internet..

Posted March 6, 2015 | Reviewed by Jessica Schrader

• Mental telepathy is the process of transferring thoughts from one mind to another.
• Research in 2014 scientifically validated the demonstration of mind-to-mind communication.
• Someday, instead of texting, speaking, or looking into a camera to communicate, we may simply “think” our message.

Mental telepathy , the process of transferring thoughts from one mind to another, has traditionally occupied the realms of either science fiction or the paranormal, both of which are outside of mainstream science.

Research in 2014 has changed all that, with a scientifically validated demonstration of mind-to-mind communication.

Neuroscientist Carlos Grau of the University of Barcelona and colleagues set up a clever experiment in which signals picked up by an Electroencephalagraph EEG from subjects in India were transmitted over the internet as email messages to other subjects in France, whose scalps had been fitted with Trans Cranial Magnetic (TMS) stimulators.

TMS devices, which have been used to treat anxiety and depression , electrically stimulate neural activity in the brain through intact scalps using strong magnetic fields. In this experiment, TMS stimulators were placed over the occipital (visual) cortex at the back of the brain, creating a perceived flash of light, called a phosphene, through neural activations in the visual cortex.

The subjects in India were trained to generate an EEG signal representing either a one or a zero using a biofeedback monitor. A one was generated when subjects imagined moving a hand, while a zero was produced when subjects imagined moving a foot. These ones and zeros were then emailed from India to France, and routed to one of two TMS devices mounted on subjects’ scalps. Ones were routed to a TMS electrode that caused a phosphene to be perceived, while zeros were routed to a different TMS device whose activity produced no phosphenes.

This figure illustrates how the mind to mind communication path worked.

Think of this as a kind of neurological Morse code. The researchers encoded a series of ones and zeros into words such as “hola” and “ciao,” showing that simple linguistic communication was possible.

The bottom line is that the experiment more or less worked. Error rates of transmission of ones and zeros varied between 1% and 11%, well below what would be expected by random noise.

Why should we care?

Well, one way of viewing this demonstration is that it is as historic as Alexander Graham Bell saying, “Watson, come here, I want to see you:” the first-ever voice communication over the telephone.

In Grau et al’s experiments, we may be seeing the birth of a revolutionary means of communication that will transform our world in the way the telegraph, telephone, or television did. Someday, instead of texting, speaking or looking into a camera to communicate, we may simply “think” our message, saving all the bother of typing or speaking, or even getting out of the shower.

Finally, if mind-to-mind communication proves practical, it’s possible that we will learn to communicate subtle ideas and nuances that text, speech and facial expressions cannot, forever changing the way humans relate to each other by amplifying and enriching the depth of communication.

Words sometimes fail us. Thoughts may not!

http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0105225 .

Eric Haseltine, Ph.D ., is a neuroscientist and the author of Long Fuse, Big Bang.

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• Published: 22 August 2024

Beating one bit of communication with and without quantum pseudo-telepathy

• István Márton   ORCID: orcid.org/0000-0001-7024-8245 1 ,
• Erika Bene   ORCID: orcid.org/0000-0002-8073-7525 1 ,
• Péter Diviánszky 1 , 2 &
• Tamás Vértesi   ORCID: orcid.org/0000-0003-4437-9414 1  

npj Quantum Information volume  10 , Article number:  79 ( 2024 ) Cite this article

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• Information theory and computation
• Quantum information
• Quantum mechanics

According to Bell’s theorem, certain entangled states cannot be simulated classically using local hidden variables (LHV). Suppose that we can augment LHV by some amount of classical communication. The question then arises as to how many bits are needed to simulate entangled states? There is very strong evidence that a single bit of communication is powerful enough to simulate projective measurements on any two-qubit entangled state. However, the problem of simulating measurements on higher-dimensional systems remains largely unexplored. In this study, we present Bell-like scenarios, even with three inputs per party, in which bipartite correlations resulting from measurements on higher-dimensional states cannot be simulated with a single bit of communication. We consider the case where the communication direction is fixed and the case where it is bidirectional. To this end, we introduce constructions based on parallel repetition of pseudo-telepathy games and an original algorithm based on branch-and-bound technique to compute the one-bit classical bound. Two copies of emblematic Bell expressions, such as the Magic square pseudo-telepathy game, prove to be particularly powerful, requiring a 16 × 16 state to beat the bidirectional one-bit classical bound, and look a promising candidate for implementation on an optical platform.

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Introduction.

Certain mutipartite quantum correlations cannot be simulated by local hidden variables (LHV), also known as shared random variables. This forms the core of Bell’s theorem 1 , 2 . When a quantum correlation cannot be simulated by LHV models, it is referred to as Bell nonlocal 3 . One question that arises is: What resources are needed on top of LHVs to simulate quantum correlations? The amount of classical communication that must be supplied is an obvious resource 4 , 5 , 6 , 7 , 8 . In particular, the question can be made quantitative: at least how many bits of classical communication are required to reproduce Bell nonlocal correlations arising from any number of measurements on a given d  ×  d quantum state. It is worth noting that Bell nonlocal correlations have also found a crucial role in applied physics, e.g., they can be used for the device-independent certification of the correct functioning of quantum key distribution 9 , random number generators 10 and other devices (see e.g., ref. 11 for a thorough review of the field). However, there are less strict frameworks that include partially characterized devices, and the classical communication costs of these protocols have also been studied (see, for example, references for bipartite systems 12 , 13 , 14 , 15 and for single systems 6 , 16 ).

Historically, the communication cost of simulating maximally entangled states has been addressed first. Following initial results 5 , 8 , it has been proven that correlations arising from projective measurements on a maximally entangled state of two qubits can be simulated with LHV augmented by one bit of classical communication (the so-called one-bit classical model) 17 .

There have also been results on the simulation of partially entangled qubits. Research has shown that projective measurements on all partially entangled two-qubit states can be classically simulated with at most two bits of communication 17 . More recently, the two-bit simulation result has been extended to arbitrary POVM measurements 16 . However, Gisin has posed the question of whether a single bit is sufficient to simulate all two-qubit states 18 . This problem can be approached systematically, since the one-bit classical resources are contained in a Bell-type polytope 19 , 20 , 21 . However, the size of the one-bit classical polytope grows rapidly with the number of inputs and outputs. Currently, the largest completely characterized one-bit classical polytope has three binary-outcome measurements for one party and two binary-outcome measurements for the other party 20 . No quantum violation has been found, even for three inputs per party on both sides 22 . In a recent study by Renner and Quintino 21 , the problem was approached from a different angle, and developed a one-bit classical protocol that perfectly simulates an arbitrary number of projective measurements performed on weakly entangled two-qubit states. Moreover, a recent numerical study based on neural networks by Sidayaja et al. 23 has gathered strong evidence that projective measurements on all entangled two-qubit states can be simulated with a one-bit classical model. See also the recent perspective by Tavakoli 24 .

Our strategy is as follows. We identify Bell-like inequalities that are satisfied by all LHV models supplemented with one bit of communication (i.e., one-bit classical models) and then we search for a quantum violation of these inequalities. As the two-qubit scenario has been widely studied without violating the one-bit classical model 21 , 23 , here we turn to higher-dimensional bipartite systems. What are the perspectives for solving this problem? On the one hand, complexity arguments show that one bit of communication is not sufficient to classically simulate all bipartite quantum correlations 5 . On the other hand, it is an open problem to identify such Bell-type scenarios with a modest number of inputs and outputs (see, e.g., refs. 23 , 24 ). For example, correlations resulting from two-output measurements on arbitrary high-dimensional maximally entangled states can be classically simulated using only two bits of communication 25 . In fact, this bound is tight since 4 × 4 dimensional maximally entangled quantum states cannot be simulated with a single bit of communication 26 . However, the proof requires an infinite number of measurement inputs.

In the present study, we use several techniques to prove that no classical model with one bit of communication can simulate a finite number of measurements performed on higher-dimensional bipartite quantum systems. On the one hand, we rely on pseudo-telepathy games 27 and apply them in a multi-copy scenario 28 . On the other hand, we make use of so-called Platonic Bell-like inequalities 29 , which belong to the class of correlation-type Bell inequalities constructed from the vertices of higher-dimensional regular polyhedra. The elegant structure of these expressions allows us to find the exact quantum maximum and to set nontrivial upper bounds on the one-bit score.

Overview of the results on Bell-type constructions

We present several examples of two-qudit systems that beat the one-bit and the c -bit classical bound with a finite and typically modest number of measurement inputs and outputs. Our examples are based on four bipartite Bell inequalities as building blocks for our Bell-type constructions: the CHSH inequality 2 , the Magic square game 27 , 30 , 31 , the family of CGLMP inequalities 32 , and Platonic Bell inequalities 29 , 33 , 34 . These are discussed in detail further below in the Results section.

Table 1 collects the Bell-type constructions presented in this paper. We show the setup that involves the input cardinality ( m A and m B ), output cardinality ( o A and o B ), and the d × d quantum state. Furthermore, we provide the one-bit classical bound ( L 1bit), the quantum value ( Q ) of Bell-type inequalities and D P  =  m A m B o A o B . We use D P as a measure of the complexity of a given Bell-type construction. Note that D P  > 24 is a lower bound to break the one-bit classical bound using bipartite quantum states. This is due to the fact that the nontrivial Bell-type scenarios ( m A , m B , o A , o B ) that have at most dimension D P  = 24 are given by the scenarios (2, 3, 2, 2), (3, 2, 2, 2), (2, 2, 3, 2), and (2, 2, 2, 3), and all of them can be simulated classically with one bit of bidirectional communication (i.e., communication either from Alice to Bob or vice versa) as shown in ref. 22 .

It should be noted that the values L 1bit presented in Table 1 are the result of rigorous calculations except for the cases of CHSH ⊗ 4 and the Platonic inequalities E7 and FR10, for which L 1bit are obtained from heuristics. However, in the section on Platonic inequalities the upper bounds of 565 and 806 are given on the one-bit bound, which are still below the respective quantum bounds of 567 and 810. Table 1 reveals that the smallest D P of 12,544 for the bidirectional case is given by the \({[{{\rm{Magic}}}^{\otimes 2}]}_{{\rm{s}}}\) inequality. The inequality features seven inputs and sixteen outputs. We propose it as a candidate for experimentally violating the one-bit classical bound. On the other hand, if we aim to violate quantumly the fixed-directional (e.g., from Alice to Bob) one-bit bound, then the most suitable candidate appears to be the \({[{{\rm{Magic}}}^{\otimes 2}]}_{{\rm{a}}}\) inequality, which has D P  = 5376. The inequality has seven inputs on Alice’s side and three inputs on Bob’s side, with sixteen outputs per measurement on each side.

As we can see in Table 1 , all the values of D P are much higher than 24, demonstrating the power of a single bit of classical communication, or alternatively, our lack of success in finding constructions that exceed the one-bit bound with lower complexity of the input-output scenario. We propose it as an open problem to shrink the gap between 24 and 5376 in the fixed-directional problem and the gap between 24 and 12544 in the bidirectional problem.

In addition to the one-bit scenario (i.e., c  = 1), we provide Bell-like inequalities augmented by c bits of one-way classical communication for c ≥2. These examples are based on multiple copies of the CHSH expression, as well as a truncated version of multiple copies of the CGLMP d inequalities. We examine the scaling of the input and output cardinality and the Hilbert space dimension of the bipartite quantum systems to beat the one-way c -bit classical bound of the aforementioned inequalities. Crucially, we find a Bell inequality with c bits of one-way classical communication that has 2 c  + 1 inputs and can be violated by quantum systems of sufficiently high dimensions. Note that this is a minimal scenario in terms of the number of inputs; otherwise, the c -bit inequality cannot be violated.

The rest of the Results section is structured as follows. In the following subsection, we provide the notation, define the Bell-like scenario augmented by one bit and also by c  ≥ 2 bits of communication, and demonstrate the usefulness of multiple copies of the quantum CHSH game in this problem. The next subsection considers the double Magic square game and its various truncated versions. It also provides the one-bit classical bound of the Bell-type inequality associated with the game. In particular, a quantum violation of the one-bit classical bound of a 7-input ( m A  =  m B  = 7) and 16-output ( o A  =  o B  = 16) Bell-type inequality is shown. The next subsection examines the CGLMP d inequalities and their truncated versions and demonstrates quantum violation of the one-bit bound of a three-input, 283 2 -output Bell-like inequality. In the following subsection, we investigate a different construction based on the so-called Platonic Bell inequalities that belong to a class of correlation-type Bell inequalities. We compute the one-bit classical bound for such Bell-type inequalities with 63 (and also 90) inputs numerically and provide an analytical upper bound as well. We then find a quantum violation of this bound using 8 × 8 and 32 × 32 dimensional maximally entangled quantum states, respectively.

The power of multiple copies of the CHSH expression

We establish notation and introduce the concept of Bell-like inequalities that are valid for all correlations that can be simulated classically with a single bit of communication. Consider a scenario where two non-communicating parties, named Alice and Bob, produce outcomes (alternatively outputs) a ∈ {0, …, o A − 1} and b ∈ {0, …, o B − 1} for settings (alternatively inputs) x ∈ {0, …, m A − 1} and y ∈ {0, …, m B − 1}. In such a scenario, a generic bipartite Bell inequality can be expressed as

where P ( a b ∣ x y ) represents the conditional probabilities and we assume that S a b x y   ≥ 0. Writing a Bell inequality in this form can also be viewed as a bipartite Bell nonlocal game 35 . The local bound L that appears on the right-hand side of Eq. ( 1 ) is the maximal value of the Bell expression \({\mathcal{B}}\) when the probabilities P ( a b ∣ x y ) admit an LHV model. In this case, P ( a b ∣ x y ) can be explained using a common past history and local operations by Alice and Bob, and can be written as follows:

Here λ represents a local variable, q ( λ ) denotes a probability distribution, and P A and P B refer to Alice’s and Bob’s respective marginals.

Let us now allow one bit of classical communication, say from Alice to Bob, in addition to LHV operations. See panel a in Fig. 1 . In this fixed-directional case, the protocol follows these steps. First, Alice and Bob receive their inputs x and y . Then Alice is allowed to send one bit of classical message, l  = 0, 1, to Bob. Afterward, Alice and Bob produce the respective outputs a and b . In this way, Alice and Bob can simulate all P ( a b ∣ x y ) that satisfy:

where the marginal of Bob ( P B ) also depends on the value of the classical bit l  =  l ( x , λ ), l being either 0 or 1. The maximum on \({\mathcal{B}}\) in Eq. ( 1 ) achieved by these strategies is referred to as L 1bit or the one-bit bound. That is

where the maximum is taken over all P ( a b ∣ x y ) in the form of ( 3 ).

a the classical model with one extra bit of communication. After receiving the input settings x and y , Alice sends Bob a classical binary message l ∈ 0, 1. Then Alice and Bob give the outputs a and b as a function of the information available to each party. This results in P 1bit ( a , b ∣ x , y ). b The quantum Bell setup that produces the probability distribution P Q ( a b ∣ x y ). By evaluating the Bell functional B in Eq. ( 1 ), we aim to violate the one-bit bound: L 1bit <  Q .

When probabilities are obtained from quantum mechanics instead, the maximum value of \({\mathcal{B}}\) in Eq. ( 1 ) is called the Tsirelson bound 36 . In such a case,

where ρ is a density matrix on the space \({{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d}\) , and A and B are d -dimensional projection matrices. These matrices add up to the identity, ∑ a A a ∣ x  = ∑ b B b ∣ y  =  I d , and fulfill \({A}_{a| x}^{2}={A}_{a| x}\) and \({B}_{b| y}^{2}={B}_{b| y}\) for all x and y , where the number of outcomes a (or b ) is not necessarily equal to the dimension d . We note that the Tsirelson bound of all Bell expressions considered in this paper can be obtained by such projection operators.

We now discuss the multi-copy CHSH scenario. Let us write the CHSH Bell inequality 2 in the following form (see e.g., refs. 28 , 32 ):

where x , y and a , b are assumed to have values of 0 and 1. More compactly the above inequality reads \({\rm{CHSH}}=\mathop{\sum }\nolimits_{b,x,y = 0}^{1}P(a=xy\oplus b,b| x,y)\le 3\) . On the right-hand side, L  = 3 is the local bound that can be achieved by suitable local deterministic strategies. An appropriate strategy is for Alice to output a  = 1 for x  = 0, 1 while Bob outputs b  = 1 for y  = 0, 1. Therefore, these correlations within the local set ( 2 ) can be expressed as P L ( a b ∣ x y ) =  δ a ,1 δ b ,1 for any x , y .

On the other hand, L 1bit(CHSH) = 4 can be achieved if Alice sends l  =  x to Bob, with Alice returning a  = 1 for x  = 0, 1 and Bob returning b  = 1 for y  = 0 and b  = (1 − l ) for y  = 1. Note that 4 is also the algebraic bound that can be achieved with P ( a b ∣ x y ) respecting only positivity.

In the quantum case, using a two-qubit maximally entangled state and mutually unbiased measurements, the following statistics can be obtained

in Eq. ( 5 ). By substituting these values into the CHSH inequality (6), one obtains \(Q({\rm{CHSH}})=2+\sqrt{2}\) .

When given n instances of a Bell nonlocal game \({\mathcal{B}}\) , a straightforward way is to play them in parallel. For example, when presented with two copies ( n  = 2) and \({\mathcal{B}}={\rm{CHSH}}\) , the resulting double CHSH expression 28 (see also refs. 37 , 38 ) is:

where CHSH A , B acts on the first copy, while \({{\rm{CHSH}}}_{A^{\prime} ,B^{\prime} }\) acts on the second copy of the space of input-output variables. The Bell expression is written explicitly for n  = 2 as follows

where P ( a b ∣ x y ) =  P ( a 1 , a 2 , b 1 , b 2 ∣ x 1 , x 2 , y 1 , y 2 ). The formula for n  ≥ 2 copies reads as

Although computing the Tsirelson bound of \({{\mathcal{B}}}^{\otimes n}\) for a generic \({\mathcal{B}}\) is difficult, we can often obtain a good enough lower bound by playing each instance of \({\mathcal{B}}\) independently with the optimal quantum strategy for the single copy case. In the case of n copies, we then have

In general, for the quantum maximum of the n -copy Bell functional \({\mathcal{B}}\) we only obtain a lower bound: \(Q({{\mathcal{B}}}^{\otimes n})\ge Q{({\mathcal{B}})}^{n}\) . However, in the particular case of the n -copy CHSH expression, the Tsirelson bound saturates the lower bound 39 , and we have

We now ask for the local bound of CHSH ⊗ n . An obvious lower bound is L  ≥ 3 n , which can be achieved by using independent classical deterministic strategies between individual copies. However, exploiting joint strategies enables better performance. In this case, Alice’s output a i depends not only on input x i , but also on input \({x}_{i^{\prime} }\) , where \(i^{\prime} \,\ne\, i\) . For two and three copies, we have the respective bounds L (CHSH ⊗ 2 ) = 10 (see ref. 28 ) and L (CHSH ⊗ 3 ) = 31. The latter value was obtained independently by S. Aaronson and B. Toner (see the footnotes in ref. 39 ). However, only empirical values can be found in the literature for n  > 3 40 . On the other hand, the following upper bound was found in 2014 by A. Ambainis 41 , building upon ref. 42 :

which holds for any number of copies n  ≥ 1.

We next discuss the calculation of the one-bit bound for the multi-copy CHSH scenario. We show analytically that there are a certain number of n copies for which quantum correlations exceed the one-bit classical bound, L 1bit. Two ingredients are required for the proof. The first corresponds to the bound in Eq. ( 13 ). The second one is the relation

which applies to any bipartite Bell expression \({\mathcal{B}}\) written in the form ( 1 ) defined by positive coefficients S a b x y . Note that any bipartite Bell inequality can be brought into this form, e.g., by adding an appropriate constant to every pair of settings x , y and rescaling the local bound L . To show ( 14 ), let us observe that \(L({\mathcal{B}})\ge L({\sum }_{abxy}{S}_{abxy}P(ab| xy))\) , where the summation is over all a , b , y and \(x\in X^{\prime}\) where \(X^{\prime} \subseteq \{0,\ldots ,{m}_{A}-1\}\) . Then we have the one-bit bound from Alice to Bob

where maximization is carried out over all bipartitions X 1 ∪ X 2  = {0, …, m A − 1}. Note that the same bound applies if Bob is the one sending a single bit to Alice. So the upper bound on the one-bit bound in ( 14 ) applies in both directions.

By choosing \({\mathcal{B}}={{\rm{CHSH}}}^{\otimes 2}\) and combining the two relations (( 13 ), ( 14 )), the following upper bound is reached:

On the other hand, the Tsirelson bound of the n -copy CHSH expression is given by Eq. ( 12 ). Applying Eq. ( 16 ) gives the number of copies n  = 13 at which the quantum value Q exceeds the one-bit bound of L 1bit. This calculation relied on applying analytical upper bounds. However, is n  = 13 tight in our one-bit problem? To decide, we used a heuristic search to calculate L 1bit(CHSH ⊗ n ) for small integers n . The numerical computation strongly suggests that n  = 4 is the critical value at which L 1bit(CHSH ⊗ n ) <  Q (CHSH ⊗ n ). In this Bell-type scenario, each party has m  = 2 n  = 16 measurement inputs, o  = 16 measurement outputs, and (16 × 16)-dimensional states. See Table 2 . The entries without stars in the table are obtained by an exact enumeration, while those marked with stars (*) are based on (reliable) heuristics. In particular, the heuristics we used for the one-bit classical bound is a modified version of the see-saw procedure, where iteration is also performed for the optimal strategies for the l ( x ) = 0, 1 bit message, as used, e.g., in refs. 40 , 43 . An implementation of the procedure for two outcomes can be found in ref. 44 . Note that, since the CHSH ⊗ n expression is symmetric for the exchange of parties, all the above findings apply in both directions, that is, in the bidirectional case. It is interesting to note that in Table 2 the L 1bit bound for n  = 3 is equal to 40, whereas the quantum maximum is only slightly smaller, 39.79898. Can we modify the CHSH Bell functional to obtain a case where L 1bit becomes smaller than the quantum maximum for n  = 3? If such a Bell functional exists, this would considerably reduce the Hilbert space dimension required to break the one-bit barrier (namely, from 16 × 16 to 8 × 8). Note that for any k  = (1, …, n − 1), it is straightforward to establish a lower bound of L 1bit(CHSH ⊗ n ) ≥  L 1bit(CHSH ⊗ k ) ×  L (CHSH ⊗ n − k ). For the special case of n  = 3 and k  = 1, this bound is tight (40 = 4 × 10). However, L (CHSH ⊗ 2 ) = 10 > 3 2 signifies that the multiplicative nature of the local bound for the parallel repetition of the classical CHSH game does not hold true even for n  = 2 copies. This makes the bound L 1bit(CHSH ⊗ 3 ) so large that it cannot be beaten quantumly for three copies ( n  = 3).

We next generalize the one-bit result of the n -copy CHSH scenario to the communication of l  = (2 c )-level classical messages, where c is the number of bits. It should be noted that we are considering only a fixed amount of one-way classical communication between the two parties, but our findings can be extended to a fixed amount of two-way communication as well. The set of possible classical protocols that use at most c bits of communication is the subject of the field of communication complexity 45 . We inquire about the number of n copies of CHSH expressions required to surpass the one-way c -bit classical bound with the quantum value ( 12 ). In particular, we give an upper bound on the setup parameters, including the number of inputs m , outputs o , and the dimension d per party needed to beat the c -bit classical bound.

Let us generalize the results obtained for one bit to c bits. First, we have

The proof is analogous to that for the c  = 1 bit case ( 14 ). However, in this case the number of possible partitions of the set of Alice’s input X is 2 c , which explains the prefactor in ( 17 ). Specifically for \({\mathcal{B}}={{\rm{CHSH}}}^{\otimes n}\) and having applied the upper bound ( 13 ) to a given c and n , we obtain

Therefore, our condition to exceed the L c bit bound is:

which we solve for n to arrive at the upper bound for the n crit ( c ) value:

Since for the n -copy CHSH expression, the number of inputs, outputs, and dimensionality of the component space are the same (i.e., 2 n ), we obtain the following upper bounds:

Note that a lower bound of m crit ( c ) > 2 c follows from Alice simply communicating her own input to Bob using an l  = (2 c )-level classical message.

We now improve on the n -copy CHSH scenario. In particular, regarding the one-bit classical bound, we ask about the possibility of finding more economical Bell-like inequalities augmented by one bit of communication that can be violated quantumly with fewer inputs, outputs and dimensions. Regarding the c -bit classical bound, we will beat the n -copy CHSH inequality in all of the aforementioned bounds ( 21 ). Nonetheless, we cannot improve all three parameters at the same time. Is it still possible to find tighter upper bounds for m crit ( c ), o crit ( c ) or d crit ( c ) for c  > 1 by using other Bell-like inequalities? We will present an optimal solution to m crit ( c ) based on a truncated version of the 2 c copies of the CGLMP d inequality. A tight lower bound of m crit ( c ) > 2 c can be obtained by using a (possibly huge) D  ×  D quantum state, where \(D={{d}^{2}}^{c}\) and d is large enough (see the subsection on CGLMP inequalities for details).

In the following three subsections, we will examine the double Magic square game, the double CGLMP d inequalities, and a Platonic Bell-type inequality. All of these examples are shown to beat the one-bit classical bound quantumly.

Two copies of the Magic square and pseudo-telepathy games

We will calculate the one-bit bound on the double Magic square game. First, however, let us give a brief description of the single-copy Magic square game 27 , 30 , 31 . This is a nonlocal game for which the Tsirelson bound achieves the algebraic maximum. This game is within the class of quantum pseudo-telepathy games 46 . Each party has three inputs and four outputs, that is, m A  =  m B  = 3 and o A  =  o B  = 4. The Bell functional “Magic” with a local bound of 8 is written as follows:

where the parties produce three bits each, which are represented as a  = ( a 0 a 1 a 2 ) and b  = ( b 0 b 1 b 2 ). Furthermore, it is assumed that the following conditions hold: \({a}_{0}\oplus {a}_{1}\oplus {a}_{2}=0\,{\rm{mod}}\,2\) and \({b}_{0}\oplus {b}_{1}\oplus {b}_{2}=1\,{\rm{mod}}\,2\) . Due to this parity constraint, the third output becomes unnecessary and each party has only four outputs: a  =  a 0 a 1 ∈ {00, 01, 10, 11} and similarly b  =  b 0 b 1 ∈ {00, 01, 10, 11}. In the game terminology, the existence of a winning quantum strategy means that quantum physics violates this inequality up to the maximal algebraic value of 9. This violation can be obtained with a 4 × 4 dimensional maximally entangled state. The optimal measurements, on the other hand, correspond to 4-outcome POVMs with rank-1 projection operators 27 .

It is known that L (Magic) = 8, and L 1bit(Magic) =  Q (Magic) = 9 (see ref. 47 ). Therefore, there is no advantage in using quantum strategies over the optimal one-bit classical protocol. How does the picture change if we use n copies of the Magic square game? Table 3 summarizes the different bounds for the n -qubit Magic square game up to n  = 3. The quantum value Q  = 9 n defines the exact Tsirelson bound for any value of n due to the pseudo-telepathy property of the game. As we can see, two copies are sufficient to beat the one-bit bound ( L 1bit = 75) quantumly ( Q  = 81). The value of 75 has been verified using a branch-and-bound algorithm 48 adapted from calculating the local bound 43 , 49 to the one-bit bound 44 . We use this algorithm (see ref. 44 for the implementation using parallel computing on CPUs) to obtain the value of L 1bit(Magic ⊗ 2 ) = 75 in 11 s on our workstation. In addition to the computer-assisted proof above, in the Methods section, we provide a fully analytic proof that L 1bit is strictly less than 81. Namely, we prove that L 1bit(Magic ⊗ 2 ) ≤ 80, which conclusively proves that the maximum quantum value 81 of the double Magic square game cannot be obtained classically with one bit of communication.

Below we provide the one-bit classical bound for the truncated double Magic square game. The Magic ⊗ 2 inequality consists of 9 inputs and 16 outputs. Let us now remove two inputs from the input set X  =  Y  = {0, 1, 2} 2 . Specifically, we remove {21, 22} from both sides and obtain X  =  Y  = {00, 01, 02, 10, 11, 12, 20}. By doing so, we obtain a Bell inequality with 7 inputs and 16 outputs, which we call \({[{{\rm{Magic}}}^{\otimes 2}]}_{{\rm{s}}}\) . This game still remains a pseudo-telepathy game, since the algebraic bound matches the quantum bound, \(Q({[{{\rm{Magic}}}^{\otimes 2}]}_{{\rm{s}}})=49\) , and the local bound is \(L({[{{\rm{Magic}}}^{\otimes 2}]}_{{\rm{s}}})=44\) due to an exhaustive search performed by enumerating all deterministic strategies. However, \(L1{\rm{bit}}({[{{\rm{Magic}}}^{\otimes 2}]}_{{\rm{s}}})=48\) . The proof is analogous to that of the double Magic square game in the Methods section. We observe that for an arbitrarily partitioning of the 7-element set \(X(x,x' )=\{00,01,02,10,11,12,20\}\) into two disjoint subsets X 1 and X 2 , one of the subsets will necessarily contain either X L  or  X R (defined by Eq. ( 32 )). The remaining part of the proof is similar to that of double Magic square game. Notice that due to the symmetry of the Bell functional \({[{{\rm{Magic}}}^{\otimes 2}]}_{{\rm{s}}}\) , the quantum value of 49 also exceeds the bidirectional one-bit classical bound.

Now consider the case where the communication direction is fixed, and Alice is allowed to communicate one bit to Bob. This is called the asymmetrically truncated double Magic square game. Once again, we begin with the double Magic square game, but this time, inputs on Alice and Bob respective sides are as follows

This provides us with a Bell functional having seven inputs on the Alice side and three inputs on Bob’s side, with 16 outputs per measurement on both sides. The Bell functional is denoted as \({[{{\rm{Magic}}}^{\otimes 2}]}_{a}\) . In this case, the proof follows the same line of reasoning as that used to prove the one-bit classical bound \({[{{\rm{Magic}}}^{\otimes 2}]}_{{\rm{s}}}\) . Here, we give different bounds for the expression \({[{{\rm{Magic}}}^{\otimes 2}]}_{{\rm{a}}}\) , namely, L 1bit = 20 (where communication is fixed directional), Q  = 21 and L  = 18. Note that a lower bound to the input cardinality in the fixed directional one-bit scenario involves three inputs on Alice’s side and two inputs on Bob’s side. The question that arises is whether such a Bell-like inequality with quantum violation exists. The answer is affirmative. To this end, in the sequel we will recap another family of bipartite Bell inequalities known as the CGLMP d family 32 .

We next consider the one-bit classical bound for two copies of generic pseudo-telepathy games. We observed that by considering a pseudo-telepathy game, specifically the Magic square game, and using two copies of them, we can beat the one-bit classical bound by allowing quantum resources. We ask if it is a generic property of bipartite pseudo-telepathy games. This turns out to be the case. The proof is given in the Methods section.

What is known about (a single copy of) two-party pseudo-telepathy games? Several of them have been discussed in the literature 35 , 46 . According to our proof in the Methods section, the one-bit classical bound can also be violated quantumly with any two-copy pseudo-telepathy game. Let us examine the so-called Impossible coloring game 50 , 51 and for a modern formulation, see ref. 35 . In this game, Alice has three outputs ( o A  = 3), and Bob has two outputs ( o B  = 2). The shared state is a 3 × 3 maximally entangled state. If two copies of the Impossible coloring game are played, the output cardinalities are squared, resulting in o A  = 9, o B  = 4, and a state space of 9 × 9. Note that this inequality is not symmetric for party exchange. Since pseudo-telepathy is a property which is symmetric for party exchange, the one-bit classical bound for the double Impossible coloring game is violated quantumly in both directions. Although the dimensionality of the problem (i.e., d  = 9) is less than that of the double Magic square game (which is d  = 16), it requires many more inputs. It is worth noting that this is the smallest output cardinality and dimensionality of the state space that a two-copy bipartite pseudo-telepathy game can have 35 . It should also be noted that there is a correlation Bell inequality with one bit of communication, which is more economical in terms of dimensionality. In the subsection on Platonic inequalities, we present the construction for d  = 8 with 63 inputs and two outputs per party. There is also a construction in the literature based on correlation-type Bell inequalities for 4 × 4 systems, which however uses an infinite number of inputs 26 .

Multiple copies of the CGLMP inequality

In this subsection, our aim is to violate quantumly the one-bit and c -bit bounds with minimum input cardinality using two or more copies of the family of CGLMP inequalities. First we discuss the one-bit bound for the double CGLMP inequalities. The family of CGLMP inequalities, which was introduced in ref. 32 , forms a one-parameter family of bipartite Bell inequalities. Each party has two inputs labeled as 0 and 1, and d  ≥ 2 outputs per party, which are labeled from 0 to d − 1. Just like in the CHSH and Magic square games, we will use them as basic building blocks of the multi-copy inequalities. For d  = 2, they reduce to the CHSH inequality (in equivalent form) and are tight for all d 52 . For d  = 3, the original three-output CGLMP inequality is recovered 53 . We use the form of the inequalities presented in ref. 54 . With a slight modification to the notation in ref. 54 , we have

where P ( A x  <  B y ) = ∑ a < b P ( a b ∣ x y ), and L (CGLMP d ) = 3 for any d  ≥ 2. Since all the coefficients in the inequality ( 24 ) are positive, this form of CGLMP d can be interpreted as a Bell nonlocal game.

The quantum value Q (CGLMP d ) up to d  = 10 6 has been computed by Zohren and Gill 55 , which is believed to be the maximum quantum value, that is, the Tsirelson bound of the inequality for any d . The optimal conjectured bipartite quantum state for CGLMP d has dimensions d  ×  d . That is, the dimension of the Hilbert space d is equal to the number of outcomes d . Each POVM element corresponding to the optimal solution is a rank-1 projector.

In the second column of Table 4 , we reproduce the quantum values up to d  = 10. For any d  ≥ 2, the algebraic maximum of the single-copy inequality is 4 and the local bound is 3. It has been proven in ref. 55 that Q (CGLMP d ) tends to 4 when d → ∞ . As a result, in the limit of d → ∞ , it provides us with a pseudo-telepathy game. However, as a two-input Bell inequality, the one-bit bound always saturates the algebraic bound, meaning L 1bit(CGLMP d ) = 4. This value is greater than Q (CGLMP d ) for any finite d  ≥ 2.

Let us now consider playing two or more instances of the CGLMP d game in parallel. Table 4 (third column) presents the quantum values for the double CGLMP inequalities, where the lower bound \(Q({{\rm{CGLMP}}}_{d}^{\otimes 2})\ge Q{({{\rm{CGLMP}}}_{d})}^{2}\) is used. When d  = 2, this formula defines the exact Tsirelson bound since CGLMP 2 is equivalent to CHSH, which is a type of XOR game 39 . The one-bit bound \(L1{\rm{bit}}({{\rm{CGLMP}}}_{d}^{\otimes 2})=12\) in the last column is verified by the branch-and-bound algorithm up to d  = 10. We conjecture that this is the exact bound for any d  ≥ 2. We also computed the local bound up to d  = 10 and obtained \(L({{\rm{CGLMP}}}_{d}^{\otimes 2})=10\) . According to the results in Table 4 , \(Q({{\rm{CGLMP}}}_{8}^{\otimes 2})\, >\, L1{\rm{bit}}({{\rm{CGLMP}}}_{8}^{\otimes 2})\) , therefore we have an example of a four-input, 64-output Bell-like inequality with one bit of communication that can be violated with a 64 × 64 quantum state.

To achieve Q (CGLMP d ) in Table 4 , non-maximally entangled states of two d -dimensional quantum systems are required, except for d  = 2 53 . Let us denote the quantum value by \(\tilde{Q}({{\rm{CGLMP}}}_{d})\) that can be attained with conjectured optimal measurements and d  ×  d maximally entangled states 32 , 54 . The values obtained are \(\tilde{Q}({{\rm{CGLMP}}}_{31})=3.6345\) and \(\tilde{Q}({{\rm{CGLMP}}}_{31}^{\otimes 2})\ge \tilde{Q}{({{\rm{CGLMP}}}_{31})}^{2}=12.0031\) . Thus, a (961 × 961)-dimensional maximally entangled state allows us to exceed the conjectured value of \(L1{\rm{bit}}({{\rm{CGLMP}}}_{31}^{\otimes 2})=12\) .

In this respect, the so-called SATWAP inequalities 56 look promising to further reduce the dimensionality threshold corresponding to maximally entangled states, since these inequalities are based on the CGLMP inequalities, but tailored to maximally entangled states. At this point, the following question arises: What is the minimum number of inputs required to exceed the one-bit bound of a Bell-like inequality with quantum systems? The previous example consists of four inputs. Is there any three-input Bell inequality with one bit of communication violated by quantum systems? We next provide such a construction. That is, we compute the bidirectional and then the fixed directional one-bit classical bounds for the truncated double CGLMP inequalities.

Consider the double CGLMP d inequality discussed above, where we label the four inputs by \(X(x,x' )={\{0,1\}}^{2}\) and \(Y(y,y' )={\{0,1\}}^{2}\) on the respective sides of Alice and Bob. To obtain the settings, let us remove setting (1, 0) from both X and Y :

Let us denote this three-input inequality by \({[{{\rm{CGLMP}}}_{d}^{\otimes 2}]}_{{\rm{s}}}\) . Note that the algebraic maximum is 9 for any d  ≥ 2. This value can be attained quantumly in the limiting case of d → ∞ . On the other hand, it can be proven that \(L1{\rm{bit}}({[{{\rm{CGLMP}}}_{d}^{\otimes 2}]}_{{\rm{s}}})\) = 8 for any d  ≥ 2. The proof follows the same line of reasoning as the proof of \(L1{\rm{bit}}({[{{\rm{Magic}}}^{\otimes 2}]}_{{\rm{s}}})=48\) in the subsection on Magic square games. In particular, we show that for any bipartition of the three-element set \(X(x,x^{\prime} )=\{00,01,11\}\) , the two-setting CGLMP d will appear on one of the partitions. This cannot be played perfectly using local strategies, hence the bound has to be smaller than the algebraic maximum of 9. As all deterministic one-bit strategies produce an integer value, an upper bound for \(L1{\rm{bit}}({[{{\rm{CGLMP}}}_{d}^{\otimes 2}]}_{{\rm{s}}})\) is 8, which is tight since it can be achieved with a specific one-bit classical strategy. On the other hand, the conjectured local bound is \(L({[{{\rm{CGLMP}}}_{d}^{\otimes 2}]}_{{\rm{s}}})=7\) , which we verified up to d  = 20.

We expect to exceed the bound of \(L1{\rm{bit}}({[{{\rm{CGLMP}}}_{d}^{\otimes 2}]}_{s})=8\) with a potentially large but finite value of d . Why is that? This is due to the fact that \({[{{\rm{CGLMP}}}_{d}^{\otimes 2}]}_{{\rm{s}}}\) can be played perfectly when d is infinite, and its quantum value tends to 9 as d becomes large. For this reason, there must be a threshold value for which \(Q({[{{\rm{CGLMP}}}_{d}^{\otimes 2}]}_{{\rm{s}}})\) exceeds the one-bit bound of 8. Indeed, using the specific settings stated in ref. 55 and the same quantum states, we obtain \(Q({[{{\rm{CGLMP}}}_{d}^{\otimes 2}]}_{{\rm{s}}})=8.0002059\) for d = 283, exceeding the one-bit bound 8. Hence, we can conclude that a (283 2  × 283 2 )-dimensional quantum state with well-chosen measurements violates the three-input and 283 2 output Bell inequality with one bit of communication.

Below is the computation of the fixed directional one-bit classical bound for the truncated double CGLMP inequalities. We begin with the four-setting X  =  Y  = {0,1} 2 double CGLMP inequalities by keeping the settings \(X(x,x' )=\{00,01,11\}\) and \(Y(y,y' )=\{00,11\}\) . We call this expression as \({[{{\rm{CGLMP}}}_{d}^{\otimes 2}]}_{{\rm{a}}}\) . We allow for a single bit of communication from Alice to Bob. Here, we find that the one-bit classical bound is 5, and we know that for large d the quantum value converges to 6. The L 1bit = 5 value is argued similarly to the proof presented above for the bidirectional one-bit bound of the truncated double CGLMP inequalities, while the conjectured local bound is L  = 4, which we verified up to d  = 20.

What is the threshold parameter d at which the quantum value exceeds the one-bit classical bound? It turns out that \(Q({[{{\rm{CGLMP}}}_{d}^{\otimes 2}]}_{{\rm{a}}})=5.0005455617\) for d  = 38. Therefore, if we consider one bit of classical communication in a fixed direction from Alice to Bob, there is a Bell-like inequality augmented by one bit of communication with three inputs on Alice’s side and two inputs on Bob’s side that can be violated by a (38 2  × 38 2 )-dimensional quantum state.

Here we generalize the construction described above in the case of the bidirectional one-bit bound of the truncated double CGLMP inequalities to the c -bit bound of l  = 2 c copies of the CGLMP inequalities. Note that here the communication of c bits can be either from Alice to Bob or from Bob to Alice, due to the symmetry of the games considered. We begin with l  = 2 c copies of the CGLMP d expression, and keep the following ( l  + 1) inputs on the respective sides of Alice and Bob:

Let us allow c bits of classical communication from Alice to Bob. With this message, we can make any l  = 2 c -partition of the ( l  + 1)-element input set on Alice’s side. This will lead to one of the partitions having two inputs. As all l  + 1 strings in ( 26 ) are different, there will be at least one index, say i , where these two strings differ. Let us select the same two strings on Bob’s side as well. Let us coarse-grain on all the indices, except for index i , on both Alice’s and Bob’s side. In this way, we obtain a CGLMP d inequality that cannot be played perfectly with only local resources. Therefore, the c -bit bound of the truncated l -copy CGLMP d inequality with c bits of communication from Alice to Bob, is at most L c bit = ( l +1) 2 − 1, given that the algebraic maximum is ( l  + 1) 2 . Since the quantum bound of this truncated single copy CGLMP d inequality approaches the algebraic maximum when d → ∞ , there must be a critical d for any l  = 2 c , where the quantum bound exceeds L c bit. Note, however, that this critical d can be quite large even for moderate c . As we have already found above, for c  = 1 the critical d is 283. It is worth noting that due to the symmetry of the inequality, our findings regarding exceeding the c -bit bound quantumly are still valid when Bob communicates c classical bits to Alice. For a given c number of bits, the number of inputs per party is 2 c  + 1 defining a minimal scenario, and both parties have \(o={{d}^{2}}^{c}\) outputs per measurement. Similar results have been obtained by Maxwell and Chitambar 20 regarding the input cardinality for the one-way communication cost of simulating no-signaling distributions. However, unlike our case, binary outputs could be chosen. It remains an open question of how to reduce the dimension and number of outputs by considering alternative Bell inequalities with c bits of communication, where the number of settings is the minimum value of 2 c  + 1.

Platonic correlation-type Bell inequalities with one bit of communication

We investigate a construction other than pseudo-telepathy games, belonging to the class of Platonic correlation-type Bell inequalities, which is defined as follows. Consider an m -vertex solid in Euclidean dimension n , where the unit vectors pointing towards the vertices of this solid are denoted by \({\{{{\boldsymbol{V}}}_{i}\}}_{i}\) with i = 1, …, m . We suppose that the columns of the matrix, with elements \({V}_{ij}={({{\boldsymbol{V}}}_{i})}_{j}\) , are orthogonal to each other and have the same norm. This property applies to all Platonic and Archimedean solids 34 . We shall define the coefficients of the m -input two-output correlation Bell inequality

by M x , y  =  V x ⋅ V y , where L denotes the local bound, and E x y  =  P (00 ∣ x y ) +  P (11 ∣ x y ) − P (01 ∣ x y ) − P (10 ∣ x y ) is the two-party correlation between inputs x and y . The maximum quantum value of the Bell inequality ( 27 ) is Q (Plato) =  m 2 / D , which defines the Tsirelson bound of the Bell inequality 33 , 34 . All such Bell inequalities are referred to as Platonic Bell inequalities.

One such Platonic construction follows from halving the 126 minimal vectors in the E 7 lattice (see the database 57 ). It results in 63 unit vectors \({\{{{\boldsymbol{V}}}_{i}\}}_{i}\) in dimension 7, which obey the aforementioned semi-orthogonality property. Therefore, the maximum quantum value of this Bell expression is given by Q (Plato E7 ) =  m 2 / n = 567. On the other hand, the exact value of the local bound L (Plato E7 ) = 399 is obtained through a branch-and-bound search over all local deterministic strategies (see ref. 43 for a description and refs. 49 , 58 for a CPU- and GPU-based implementation of the algorithm, respectively). The ratio of Q / L  = 567/399 = 1.421052 exceeds the ratio of the maximum quantum violation of the CHSH inequality, which is \(\sqrt{2}\) . On the contrary, calculating the L 1bit bound is more demanding, and we cannot use the branch-and-bound algorithm in ref. 44 . For this purpose, we use a heuristic see-saw-type algorithm similar to the algorithms described in ref. 40 , 43 . This iteratively searches within the set of deterministic one-bit strategies. In this way, we find the best possible lower bound of 563 to L 1bit(Plato E7 ), which is smaller than Q (Plato E7 ) = 567. See the directory 1bit2out in ref. 44 for all the details of the implementation. In Fig. 2 , panel a shows the 63 × 63 Bell matrix and panel b shows the one-bit strategies giving L 1bit(Plato E7 ) = 563, which is conjectured to be optimal.

a , b The 63 × 63 dimensional E7 Platonic construction. c , d The 90 × 90 dimensional FR10 Platonic construction. The rows (columns) of each subplot are labeled by the inputs x ( y ). On the left ( a , c ) panels: the witness matrix M  = ( M x y ). On the right ( b , d ) panels: the correlation pattern E  = ( E x y ) of the optimal one-bit strategy. Each row x is labeled by the communicated bit c  = 0, 1. For c  = 0 ( c  = 1) the entry E x y   = − 1 is marked in black (gray). For E x y  = +1 it is marked white.

Similarly to the 63-setting Platonic inequality, we can construct a 90-setting Platonic inequality. In this case, the inequality is constructed from the 90 Fishburn-Reeds vectors 59 at dimension k  = 10 without diagonal modification. The local bound of the 90 × 90 Bell inequality is 570 and the quantum maximum is 810 59 . On the other hand, the one-bit bound due to a heuristic see-saw search gives 805. See the directory 1bit2out in 44 for all the details. The panels c and d of Fig. 2 show the case Plato FR10 . The coefficients of the Platonic expression are shown in c, whereas the corresponding one-bit strategies are shown in d.

In fact, we can establish an upper bound of 565 on L 1bit(Plato E7 ), conclusively proving that the quantum value Q (Plato E7  = 567) exceeds the one-bit classical bound. Due to Tsirelson’s work 60 (see also ref. 61 ) then it is possible to construct 63 two-outcome projective measurements corresponding to traceless observables in dimension d  = 2 ⌊ n /2 ⌋  = 8 for n  = 7, together with an 8 × 8 maximally entangled state, which breaks the one-bit barrier (analogously in the FR10 case the local dimension is d  = 32). The analytical upper bound of 565 on the one-bit classical bound can be seen from the following observation.

Observation 1

The relation

is valid for any Platonic Bell inequality.

The local bound (on the right) and an upper bound to the one-bit classical maximum (on the left) are defined by their respective formulas as follows

where the maximum is taken over all a x ∈ ± 1 and

where the maximum is taken over all a x ∈ ± 1 and over all bipartitions X 1 ∪ X 2 = {1, …, m }. Above \({{\boldsymbol{V}}}_{i}\in {{\mathbb{R}}}^{n},\,i=1,\ldots ,m\) are the construction vectors that define the Platonic Bell coefficients as M x y = V x ⋅ V y . One can show the validity of Obs. 1 using geometric arguments based on formulas (( 29 ),( 30 )). See the Methods section for the proof of these formulas and of Obs. 1. Let us now apply Obs. 1 to the Plato E7 Bell expression to obtain an upper bound of \(L1{\rm{bit}}\le \sqrt{2}\times 399 \,< \,565\) . As Plato E7 is constructed to be symmetric for party exchange, the one-bit classical bound is the same for both communication directions. Therefore, we conclude that the maximum quantum value of the Platonic Bell inequality Plato E7 above cannot be achieved with a bidirectional one-bit classical communication model. In this case, the dimension of the full probability space is D P  = 63 2  × 2 2  = 15876, which is larger than D P  = 7 2  × 16 2  = 12544 corresponding to the truncated double Magic square game.

We used diverse techniques to prove that a classical model with one bit of classical communication cannot simulate measurements performed on higher-dimensional bipartite quantum systems. Table 1 highlights our main findings for the different constructions. We defined a hardness measure for a one-bit classical simulation by the dimension of the full probability space of the bipartite correlations D P . This is the dimension that quantum correlations require to refute classical models with one bit of communication. We place the upper bound for the value of D P at 12,544. However, this number is quite far from the best lower bound of D P  > 24. This suggests that there is still much room for further improvement. We leave it as an open problem to reduce the gap described above. However, it is possible that our attempts to find a smaller upper bound for D P failed, as its true value could be closer to our upper bound of 12,544. If that is the case, we can argue how surprisingly powerful LHV models plus a single bit of classical communication are when the goal is to simulate bipartite quantum correlations. Note that all the above measurement constructions giving maximum violations are projection valued. On the other hand, the lowest D P value in this paper is achieved with maximally entangled states. It is then interesting to ask whether non-projective measurements or non-maximally entangled states can reduce the lowest upper bound value of 12,544 for D P reported in this paper.

From an experimental point of view, it is also crucial to find violations of the one-bit bound using the smallest possible dimensional bipartite states. In this regard, it is known that double pseudo-telepathy games require at least 9 × 9 dimensional states. Furthermore, our best construction with a finite number of inputs in terms of dimensionality is based on a Platonic Bell inequality and involves 8 × 8 dimensional states. On the other hand, there is strong evidence that all 2 × 2 quantum states can be simulated classically with a single bit of communication. The question arises whether one can rule out one-bit classical simulation with a component space dimension less than 8 (and possibly a modest number of inputs) by considering other Bell-like constructions. Nevertheless, given recent progress in the generation of photonic hyperentangled states, the double Magic square and the CHSH ⊗ 4 games (although both with a higher dimensionality of 16 × 16) look like promising candidates for experimental implementation.

We note that in a more recent work 62 , Sidajaya and Scarani present a fixed-directional one-bit inequality in scenario (5, 2, 5, 5) that reduces the complexity of our most economical scenario (7, 3, 16, 16). To this end, the authors truncated Bell-like inequalities, a method also used in our paper. However, the truncated double Magic square scenario (7, 7, 16, 16), which corresponds to our lowest D P value in the bidirectional case, could not be improved in ref. 62 .

Analytical proof on the one-bit bound for the double Magic square game

Below we derive an analytical upper bound on the one-bit classical bound for the double Magic square game. We prove that L 1bit(Magic ⊗ 2 ) ≤ 80, which is strictly less than the algebraic bound of 81, which is the quantum maximum. This proves conclusively that playing the double Magic square game quantumly cannot be simulated classically with a single bit of communication. But notice that the exact one-bit bound is the smaller value of 75, which is given by the branch-and-bound algorithm implemented in ref. 44 .

Denote the set of nine inputs on Alice’s side as \(X(x,x' )={\{0,1,2\}}^{2}\) , and use the same set of inputs on Bob’s side \(Y(y,y' )={\{0,1,2\}}^{2}\) . In order to prove that the one-bit classical bound for Magic ⊗ 2 is less than the algebraic bound, we make use of the definition ( 3 ) for the one-bit classical set. We have to consider all possible bipartitions of X according to the classical message l  = 0, 1, add up the local bound for each partition, and choose the partition with the highest sum to obtain the L 1bit bound. Since we are only concerned whether L 1bit can attain the algebraic maximum or not, it is enough to show that the local bound of one of the partitions cannot attain the algebraic maximum that corresponds to that particular partition. In this context, it is useful to introduce a coarse graining of the joint probability distribution \(P(aa' bb' | xx' yy' )\) for a given \(x' \in \{0,1,2\}\) on Alice’s side and \(y' \in \{0,1,2\}\) on Bob’s side:

where summation is over all \(a' ={\{0,1\}}^{2}\) and \(b' ={\{0,1\}}^{2}\) outputs. Let us observe that a probability distribution P ( a b ∣ x y ) in Eq. ( 31 ) that corresponds to the algebraic maximum of a single-copy pseudo-telepathy game does not allow the original two-copy distribution \(P(aa' bb' | xx' yy' )\) to be achieved via a local strategy ( 2 ). Otherwise, it would be possible to obtain a nonlocal distribution from local operations, which would contradict ( 2 ). Let us divide the set X into two arbitrary subsets named X 1 and X 2 , i.e., X  =  X 1 ∪ X 2 . Let us then use the following notation to represent two specific subsets of X :

where x i and \(x_i'\) can take values in {0, 1, 2}. Through the grouping of X into X 1 and X 2 we see that one of them will contain either X L or X R or both. It is worth noting that Bob’s nine-input set \(Y(y,y' )={\{0,1,2\}}^{2}\) has not been partitioned. Consequently, if either the subset X 1 or X 2 contains X L ( X R ), a coarse-grained distribution P ( a b ∣ x y ) on the first copy ( \(P(a' b' | x' y' )\) on the second copy) will certainly correspond to the algebraic maximum of the Magic square game. Since the probability distribution corresponding to Q (Magic) = 9 cannot be obtained with local strategies (where L (Magic) = 8), an upper bound of L 1bit(Magic ⊗ 2 ) < 81 is established. Note that the one-bit classical bound can take only positive integers, resulting in a maximum upper bound of 80. □

Using more detailed analytical arguments, it is possible to show that L 1bit(Magic ⊗ 2 ) ≤ 75. As the bound can be attained with a specific one-bit strategy, the bound is strict, meaning L 1bit(Magic ⊗ 2 ) = 75.

One-bit bound for two copies of generic pseudo-telepathy games

We prove that it is a generic property of bipartite pseudo-telepathy games that using two copies of them, we can break the one-bit classical barrier by allowing quantum resources. This is because for any bipartition X 1 and X 2 of Alice’s set, \(X(x,x' )={\{0,...,{m}_{A}-1\}}^{2}\) , one of the bipartitions will involve \({X}_{L}=\{(0,x'_{0}),...,({m}_{A}-1,x'_{{m}_{A}-1})\}\) or \({X}_{R}=\{({x}_{0},0),...,({x}_{{m}_{A}-1},{m}_{A}-1)\}\) , or even both, which can be subsequently coarse-grained to the original single-copy pseudo-telepathy game. This property is shown below.

Consider partition X 1 . There are two options. Either it includes X L or it does not. If it does, we have completed the proof. Let us assume that X 1 does not include X L . In this case, there is a missing entry e ∉ {0, …, m A − 1} on copy x . Therefore, X 2 must contain e in the same copy x . Hence, all pairs {( e , 0), ( e , 1), …, ( e , m A − 1)} are included in X 2 . Consequently, X 2 contains X R , where x i = e . □

Local and one-bit bounds for Platonic correlation-type Bell inequalities

Recall that we define the m × m Bell matrix M by the construction vectors V i ∈ R n for i = (1, …, m ) as follows M x , y = V x ⋅ V y . However, in contrast to the definition of the Platonic properties in the main text, in the proof below we do not need to assume that V i has unit norm, nor do we assume the semi-orthogonal property of the matrix M . We now give two lemmas. The first lemma gives, in terms of its construction vectors, the local bound of the correlation Bell expression Plato.

The same as Eq. ( 29 ) in the main text.

We invoke the definition of the matrix M and of the local bound L :

The next lemma gives, in terms of its local bound and its construction vectors, an upper bound on the one-bit bound of the correlation Bell expression Plato.

The same as Eq. ( 30 ) in the main text.

where the maximum is taken over all a x ∈ ± 1 and over all bipartitions X 1 ∪ X 2 = {1, …, m }.

First we adapt the definition of L 1bit in Eq. ( 4 ) for correlation-type Bell inequalities:

where the maximum is over all \({a}_{x}=\pm 1,\,{b}_{y}^{(1)}=\pm 1,\,{b}_{y}^{(2)}=\pm 1\) and X 1 ∪ X 2 = {1, …, m }. Keeping in mind that M x , y = V x ⋅ V y and using the Cauchy-Schwarz inequality we obtain the upper bound

where the maximum is over all a x = ± 1, b y = ± 1 and X 1 ∪ X 2 = {1, …, m }. Notice from Lemma 1 that \(\sqrt{L({\rm{Plato}})}={\max }_{\{{b}_{y}\}\in {\pm 1}^{m}}\left\vert {\sum }_{y}{b}_{y}{{\boldsymbol{V}}}_{y}\right\vert\) , and we get our desired result. □

Now we are ready to prove the following upper bound.

holds true for any Platonic correlation Bell inequality. This is the same as Obs. 1 in the main text.

Invoking Lemmas 1 and 2 , we establish the upper bound

where the maximum in the denominator is taken over all a x  = ± 1, and the maximum in the numerator is taken over all a x   = ±1 and bipartition X 1 ∪ X 2 = {1, …, m }. Let \(\{{a}_{x}^{* }\}\) along with the bipartition \({X}_{1}^{* }\cup {X}_{2}^{* }\) be maximizing the numerator. Let \({{\boldsymbol{A}}}_{1}={\sum }_{x\in {X}_{1}^{* }}{a}_{x}^{* }{{\boldsymbol{V}}}_{x}\) and \({{\boldsymbol{A}}}_{2}={\sum }_{x\in {X}_{2}^{* }}{a}_{x}^{* }{{\boldsymbol{V}}}_{x}\) . Note that we can always choose \(\{{a}_{x}^{* }\}\) in such a way that the overlap A 1 ⋅ A 2 is positive. Indeed, we simply reverse the sign of \({a}_{x}^{* }\) for all x ∈ X 1 if the overlap is negative. This has no effect on the numerator. On the other hand, if we use the specific values of \({a}_{x}^{* }\) instead maximizing over all a x = ± 1, we will have \(\max \left\vert {\sum }_{x}{a}_{x}{{\boldsymbol{V}}}_{x}\right\vert \ge \left\vert {\sum }_{x}{a}_{x}^{* }{{\boldsymbol{V}}}_{x}\right\vert\) in the denominator. This gives an upper bound on the right-hand side of ( 39 ) and we obtain

where the last relation is due to the fact that for any two vectors a and b such that a ⋅ b  ≥ 0 and \(\left\vert {\boldsymbol{a}}+{\boldsymbol{b}}\right\vert > 0\) , it holds that \(\left\vert {\boldsymbol{a}}\right\vert +\left\vert {\boldsymbol{b}}\right\vert \le \sqrt{2}\left\vert {\boldsymbol{a}}+{\boldsymbol{b}}\right\vert\) . □

Code availability

The computer codes used to find the classical one-bit bound on the Bell-like inequalities presented in the article are available as Haskell and Matlab codes at https://github.com/istvanmarton/Lcom .

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Acknowledgements

We thank Antonio Acín and Jonatan Bohr Brask for inspiring conversations and Csaba Hruska for providing technical assistance in the GPU computations. We acknowledge the support of the EU (QuantERA eDICT, CHIST-ERA MoDIC) and the National Research, Development and Innovation Office NKFIH (No. 2019-2.1.7-ERA-NET-2020-00003, No. 2023-1.2.1-ERA_NET-2023-00009 and No. K145927).

Open access funding provided by HUN-REN Institute for Nuclear Research.

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Márton, I., Bene, E., Diviánszky, P. et al. Beating one bit of communication with and without quantum pseudo-telepathy. npj Quantum Inf 10 , 79 (2024). https://doi.org/10.1038/s41534-024-00874-1

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Stanley krippner.

Stanley Krippner, Ph.D., Affiliated Distinguished Professor at the California Institute for Integral Studies, is a Fellow in five APA divisions, and past-president of two divisions (30 and 32). Formerly, he was director of the Kent State University Child Study Center, Kent OH, and the Maimonides Medical Center Dream Research Laboratory, in Brooklyn NY. He is co-author of Extraordinary Dreams (SUNY, 2002), The Mythic Path, 3rd ed. (Energy Psychology Press, 2006), Haunted by Combat: Understanding PTSD in War Veterans (Greenwood, 2007), The Voice of Rolling Thunder (Bear/Inner Traditions, 2012), and Understanding Suicide's Allure (Praeger, 2021), and co-editor of The Psychological Impact of War on Civilians: An International Perspective (Greenwood, 2003), Varieties of Anomalous Experience: Examining the Scientific Evidence (APA, 2000), The Shamanic Powers of Rolling Thunder (Bear/Inner Traditions, 2016), Integrated Health Care for the Traumatized (Roman & Littlefield, 2019), and Holistic Treatment in Mental Health (McFarland, 2020).

Stanley has conducted workshops and seminars in Argentina, Brazil, Canada, China, Colombia, Cuba, Cyprus, Ecuador, Finland, France, Germany, Great Britain, Italy, Japan, Lithuania, Mexico, the Netherlands, Panama, the Philippines, Portugal, Puerto Rico, Russia, South Africa, Spain, Sweden, Venezuela, and at several congresses of the Interamerican Psychological Association. He is an advisory board member for the International School for Psychotherapy, Counseling, and Group Leadership (St. Petersburg) and the Czech Unitaria (Prague). He holds faculty appointments at the Universidade Holistica Internacional (Brasilia) and the Instituto de Medicina y Tecnologia Avanzada de la Conducta (Ciudad Juarez). He has given invited addresses for the Chinese Academy of Sciences, the Russian Academy of Pedagogical Sciences, and the School for Diplomatic Studies, Montevideo, Uruguay. He is a Fellow of the Society for the Scientific Study of Religion, the Society for the Scientific Study of Sexuality, and the Society for Psychological Science. In 2002 he was the recipient of the American Psychological Association's Award for Distinguished Contributions to the Advancement of International Psychology.

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Can human beings communicate by thought alone? For more than a century the issue of telepathy has divided the scientific community, and even today it still sparks bitter controversy among top academics. 

Since the 1970s, parapsychologists at leading universities and research institutes around the world have risked the derision of sceptical colleagues by putting the various claims for telepathy to the test in dozens of rigorous scientific studies. The results and their implications are dividing even the researchers who uncovered them.

Some researchers say the results constitute compelling evidence that telepathy is genuine. Other parapsychologists believe the field is on the brink of collapse, having tried to produce definitive scientific proof and failed. Sceptics and advocates alike do concur on one issue, however: that the most impressive evidence so far has come from the so-called 'ganzfeld' experiments , a German term that means 'whole field'. Reports of telepathic experiences had by people during meditation led parapsychologists to suspect that telepathy might involve 'signals' passing between people that were so faint that they were usually swamped by normal brain activity. In this case, such signals might be more easily detected by those experiencing meditation-like tranquillity in a relaxing 'whole field' of light, sound and warmth .

The ganzfeld experiment tries to recreate these conditions with participants sitting in soft reclining chairs in a sealed room, listening to relaxing sounds while their eyes are covered with special filters letting in only soft pink light. In early ganzfeld experiments, the telepathy test involved identification of a picture chosen from a random selection of four taken from a large image bank. The idea was that a person acting as a 'sender' would attempt to beam the image over to the ' receiver ' relaxing in the sealed room.

Once the session was over, this person was asked to identify which of the four images had been used. Random guessing would give a hit-rate of 25 per cent; if telepathy is real, however, the hit-rate would be higher. In 1982, the results from the first ganzfeld studies were analysed by one of its pioneers, the American parapsychologist Charles Honorton. They pointed to typical hit-rates of better than 30 per cent - a small effect, but one which statistical tests suggested could not be put down to chance.

The implication was that the ganzfeld method had revealed real evidence for telepathy. But there was a crucial flaw in this argument - one routinely overlooked in more conventional areas of science. Just because chance had been ruled out as an explanation did not prove telepathy must exist; there were many other ways of getting positive results. These ranged from ' sensory leakage ' - where clues about the pictures accidentally reach the receiver - to outright fraud . In response, the researchers issued a review of all the ganzfeld studies done up to 1985 to show that 80 per cent had found statistically significant evidence. However, they also agreed that there were still too many problems in the experiments which could lead to positive results, and they drew up a list demanding new standards for future research.

After this, many researchers switched to autoganzfeld tests - an automated variant of the technique which used computers to perform many of the key tasks such as the random selection of images. By minimising human involvement , the idea was to minimise the risk of flawed results. In 1987, results from hundreds of autoganzfeld tests were studied by Honorton in a ' meta-analysis ', a statistical technique for finding the overall results from a set of studies. Though less compelling than before, the outcome was still impressive.

Yet some parapsychologists remain disturbed by the lack of consistency between individual ganzfeld studies. Defenders of telepathy point out that demanding impressive evidence from every study ignores one basic statistical fact: it takes large samples to detect small effects. If, as current results suggest, telepathy produces hit-rates only marginally above the 25 per cent expected by chance, it's unlikely to be detected by a typical ganzfeld study involving around 40 people: the group is just not big enough . Only when many studies are combined in a meta-analysis will the faint signal of telepathy really become apparent. And that is what researchers do seem to be finding.

What they are certainly not finding, however, is any change in attitude of mainstream scientists: most still totally reject the very idea of telepathy. The problem stems at least in part from the lack of any plausible mechanism for telepathy.

Various theories have been put forward, many focusing on esoteric ideas from theoretical physics. They include 'quantum entanglement', in which events affecting one group of atoms instantly affect another group, no matter how far apart they may be. While physicists have demonstrated entanglement with specially prepared atoms, no-one knows if it also exists between atoms making up human minds. Answering such questions would transform parapsychology. This has prompted some researchers to argue that the future lies not in collecting more evidence for telepathy, but in probing possible mechanisms. Some work has begun already, with researchers trying to identify people who are particularly successful in autoganzfeld trials. Early results show that creative and artistic people do much better than average : in one study at the University of Edinburgh, musicians achieved a hit-rate of 56 per cent. Perhaps more tests like these will eventually give the researchers the evidence they are seeking and strengthen the case for the existence of telepathy.

Questions 1-4

Complete each sentence with the correct ending, A-G , below.

Write the correct letter, A-G , in boxes 1-4  on your answer sheet. 

A the discovery of a mechanism for telepathy.

B the need to create a suitable environment for telepathy.

C their claims of a high success rate.

D a solution to the problem posed by random guessing.

E the significance of the ganzfeld experiments.

F a more careful selection of subjects.

G a need to keep altering conditions.

1     Researchers with differing attitudes towards telepathy agree on A B C D E F G Answer: E      Locate

2     Reports of experiences during meditation indicated A B C D E F G Answer: B      Locate

3     Attitudes to parapsychology would alter drastically with A B C D E F G Answer: A      Locate

4    Recent autoganzfeld trials suggest that success rates will improve with A B C D E F G Answer: F      Locate

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Write your answers in boxes 5-14  on your answer sheet.

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