Stern–Gerlach experiment

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https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment 

Stern–Gerlach experiment: Silver atoms travelling through an inhomogeneous magnetic field, and being deflected up or down depending on their spin; (1) furnace, (2) beam of silver atoms, (3) inhomogeneous magnetic field, (4) classically expected result, (5) observed result

In quantum physics, the Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent through a spatially-varying magnetic field, which deflected them before they struck a detector screen, such as a glass slide. Particles with non-zero magnetic moment were deflected, owing to the magnetic field gradient, from a straight path. The screen revealed discrete points of accumulation, rather than a continuous distribution, owing to their quantized spin. Historically, this experiment was decisive in convincing physicists of the reality of angular-momentum quantization in all atomic-scale systems.

After its conception by Otto Stern in 1921, the experiment was first successfully conducted with Walther Gerlach in early 1922.

Description

See also: Spin quantum number

The Stern–Gerlach experiment involves sending silver atoms through an inhomogeneous magnetic field and observing their deflection. Silver atoms were evaporated using an electric furnace in a vacuum. Using thin slits, the atoms were guided into a flat beam and the beam sent through an inhomogeneous magnetic field before colliding with a metallic plate. The laws of classical physics predict that the collection of condensed silver atoms on the plate should form a thin solid line in the same shape as the original beam. However, the inhomogeneous magnetic field caused the beam to split in two separate directions, creating two lines on the metallic plate.

The results show that particles possess an intrinsic angular momentum that is closely analogous to the angular momentum of a classically spinning object, but that takes only certain quantized values. Another important result is that only one component of a particle's spin can be measured at one time, meaning that the measurement of the spin along the z-axis destroys information about a particle's spin along the x and y axis.

The experiment is normally conducted using electrically neutral particles such as silver atoms. This avoids the large deflection in the path of a charged particle moving through a magnetic field and allows spin-dependent effects to dominate.

If the particle is treated as a classical spinning magnetic dipole, it will precess in a magnetic field because of the torque that the magnetic field exerts on the dipole (see torque-induced precession). If it moves through a homogeneous magnetic field, the forces exerted on opposite ends of the dipole cancel each other out and the trajectory of the particle is unaffected. However, if the magnetic field is inhomogeneous then the force on one end of the dipole will be slightly greater than the opposing force on the other end, so that there is a net force which deflects the particle's trajectory. If the particles were classical spinning objects, one would expect the distribution of their spin angular momentum vectors to be random and continuous. Each particle would be deflected by an amount proportional to the dot product of its magnetic moment with the external field gradient, producing some density distribution on the detector screen. Instead, the particles passing through the Stern–Gerlach apparatus are deflected either up or down by a specific amount. This was a measurement of the quantum observable now known as spin angular momentum, which demonstrated possible outcomes of a measurement where the observable has a discrete set of values or point spectrum.

Although some discrete quantum phenomena, such as atomic spectra, were observed much earlier, the Stern–Gerlach experiment allowed scientists to directly observe separation between discrete quantum states for the first time.

Theoretically, quantum angular momentum of any kind has a discrete spectrum, which is sometimes briefly expressed as "angular momentum is quantized".

Experiment using particles with +1/2 or −1/2 spin

If the experiment is conducted using charged particles like electrons, there will be a Lorentz force that tends to bend the trajectory in a circle. This force can be cancelled by an electric field of appropriate magnitude oriented transverse to the charged particle's path.

Spin values for fermions

Electrons are spin-1/2 particles. These have only two possible spin angular momentum values measured along any axis, ${\displaystyle +\frac{\hslash }{2}}$ or ${\displaystyle -\frac{\hslash }{2}}$, a purely quantum mechanical phenomenon. Because its value is always the same, it is regarded as an intrinsic property of electrons, and is sometimes known as "intrinsic angular momentum" (to distinguish it from orbital angular momentum, which can vary and depends on the presence of other particles). If one measures the spin along a vertical axis, electrons are described as "spin up" or "spin down", based on the magnetic moment pointing up or down, respectively.

To mathematically describe the experiment with spin-1/2 particles, it is easiest to use Dirac's bra–ket notation. As the particles pass through the Stern–Gerlach device, they are deflected either up or down, and observed by the detector which resolves to either spin up or spin down. These are described by the angular momentum quantum number ${\displaystyle j}$, which can take on one of the two possible allowed values, either +1/2 or -1/2. The act of observing (measuring) the momentum along the ${\displaystyle z}$ axis corresponds to the ${\displaystyle z}$-axis angular momentum operator, often denoted ${\displaystyle J_z}$. In mathematical terms, the initial state of the particles is

${\displaystyle |\psi ⟩=c_1|{\psi }_{j=+\frac{1}{2}}⟩+c_2|{\psi }_{j=-\frac{1}{2}}⟩}$

where constants ${\displaystyle c_1}$ and ${\displaystyle c_2}$ are complex numbers. This initial state spin can point in any direction. The squares of the absolute values ${\displaystyle |c_1{|}^2}$ and ${\displaystyle |c_2{|}^2}$ are respectively the probabilities for a system in the state ${\displaystyle |\psi ⟩}$ to be found in ${\displaystyle |{\psi }_{j=+\frac{1}{2}}⟩}$ and ${\displaystyle |{\psi }_{j=-\frac{1}{2}}⟩}$ after the measurement along ${\displaystyle z}$ axis is made. The constants ${\displaystyle c_1}$ and ${\displaystyle c_2}$ must also be normalized in order that the probability of finding either one of the values be unity, that is we must ensure that ${\displaystyle |c_1{|}^2+|c_2{|}^2=1}$. However, this information is not sufficient to determine the values of ${\displaystyle c_1}$ and ${\displaystyle c_2}$, because they are complex numbers. Therefore, the measurement yields only the squared magnitudes of the constants, which are interpreted as probabilities.

Sequential experiments

If we link multiple Stern–Gerlach apparatuses (the rectangles containing S-G), we can clearly see that they do not act as simple selectors, i.e. filtering out particles with one of the states (pre-existing to the measurement) and blocking the others. Instead they alter the state by observing it (as in light polarization). In the figure below, x and z name the directions of the (inhomogenous) magnetic field, with the x-z-plane being orthogonal to the particle beam. In the three S-G systems shown below, the cross-hatched squares denote the blocking of a given output, i.e. each of the S-G systems with a blocker allows only particles with one of two states to enter the next S-G apparatus in the sequence.

Exp. 1 - Notice that no z- neutrons are detected at the second S-G analyzer

Experiment 1

The top illustration shows that when a second, identical, S-G apparatus is placed at the exit of the first apparatus, only z+ is seen in the output of the second apparatus. This result is expected since all particles at this point are expected to have z+ spin, as only the z+ beam from the first apparatus entered the second apparatus.

Exp. 2 - The z-spin is known, now measuring the x-spin.

Experiment 2

The middle system shows what happens when a different S-G apparatus is placed at the exit of the z+ beam resulting of the first apparatus, the second apparatus measuring the deflection of the beams on the x axis instead of the z axis. The second apparatus produces x+ and x- outputs. Now classically we would expect to have one beam with the x characteristic oriented + and the z characteristic oriented +, and another with the x characteristic oriented - and the z characteristic oriented +.

Exp. 3 - Neutrons thought to have only z+ spin are measured again, finding that the z-spin has been 'reset'.

Experiment 3

The bottom system contradicts that expectation. The output of the third apparatus which measures the deflection on the z axis again shows an output of z- as well as z+. Given that the input to the second S-G apparatus consisted only of z+, it can be inferred that a S-G apparatus must be altering the states of the particles that pass through it. This experiment can be interpreted to exhibit the uncertainty principle: since the angular momentum cannot be measured on two perpendicular directions at the same time, the measurement of the angular momentum on the x direction destroys the previous determination of the angular momentum in the z direction. That's why the third apparatus measures renewed z+ and z- beams like the x measurement really made a clean slate of the z+ output.

History

A plaque at the Frankfurt institute commemorating the experiment

The Stern–Gerlach experiment was conceived by Otto Stern in 1921 and performed by him and Walther Gerlach in Frankfurt in 1922. At the time of the experiment, the most prevalent model for describing the atom was the Bohr-Sommerfeld model, which described electrons as going around the positively charged nucleus only in certain discrete atomic orbitals or energy levels. Since the electron was quantized to be only in certain positions in space, the separation into distinct orbits was referred to as space quantization. The Stern–Gerlach experiment was meant to test the Bohr–Sommerfeld hypothesis that the direction of the angular momentum of a silver atom is quantized.

The experiment was first performed with an electromagnet that allowed the non-uniform magnetic field to be turned on gradually from a null value. When the field was null, the silver atoms were deposited as a single band on the detecting glass slide. When the field was made stronger, the middle of the band began to widen and eventually to split into two, so that the glass-slide image looked like a lip-print, with an opening in the middle, and closure at either end. In the middle, where the magnetic field was strong enough to split the beam into two, statistically half of the silver atoms had been deflected by the non-uniformity of the field.

Note that the experiment was performed several years before George Uhlenbeck and Samuel Goudsmit formulated their hypothesis about the existence of electron spin in 1925. Even though the result of the Stern−Gerlach experiment has later turned out to be in agreement with the predictions of quantum mechanics for a spin-1/2 particle, the experimental result was also consistent with the Bohr–Sommerfeld theory.

In 1927, T.E. Phipps and J.B. Taylor reproduced the effect using hydrogen atoms in their ground state, thereby eliminating any doubts that may have been caused by the use of silver atoms. However, in 1926 the non-relativistic scalar Schrödinger equation had incorrectly predicted the magnetic moment of hydrogen to be zero in its ground state. To correct this problem Wolfgang Pauli considered a spin-1/2 version of the Schrödinger equation using the 3 Pauli matrices which now bear his name, which was later shown by Paul Dirac in 1928 to be a consequence of his relativistic Dirac equation.

In the early 1930s Stern, together with Otto Robert Frisch and Immanuel Estermann improved the molecular beam apparatus sufficiently to measure the magnetic moment of the proton, a value nearly 2000 times smaller than the electron moment. In 1931, theoretical analysis by Gregory Breit and Isidor Isaac Rabi showed that this apparatus could be used to measure nuclear spin whenever the electronic configuration of the atom was known. The concept was applied by Rabi and Victor W. Cohen in 1934 to determine the ${\displaystyle 3/2}$ spin of sodium atoms.

In 1938 Rabi and coworkers inserted an oscillating magnetic field element into their apparatus, inventing nuclear magnetic resonance spectroscopy. By tuning the frequency of the oscillator to the frequency of the nuclear precessions they could selectively tune into each quantum level of the material under study. Rabi was awarded the Nobel Prize in 1944 for this work.

Importance

The Stern–Gerlach experiment was the first direct evidence of angular-momentum quantization in quantum mechanics, and it strongly influenced later developments in modern physics:

• In the decade that followed, scientists showed using similar techniques, that the nuclei of some atoms also have quantized angular momentum. It is the interaction of this nuclear angular momentum with the spin of the electron that is responsible for the hyperfine structure of the spectroscopic lines.
• Norman F. Ramsey later modified the Rabi apparatus to improve its sensitivity (using the separated oscillatory field method). In the early sixties, Ramsey, H. Mark Goldenberg, and Daniel Kleppner used a Stern–Gerlach system to produce a beam of polarized hydrogen as the source of energy for the hydrogen maser. This led to developing an extremely stable clock based on a hydrogen maser. From 1967 until 2019, the second was defined based on 9,192,631,770 Hz hyperfine transition of a cesium-133 atom; the atomic clock which is used to set this standard is an application of Ramsey's work.
• The Stern–Gerlach experiment has become a prototype for quantum measurement, demonstrating the observation of a discrete value (eigenvalue) of a physical property, previously assumed to be continuous. Entering the Stern–Gerlach magnet, the direction of the silver atom's magnetic moment is indefinite, but when the atom is registered at the screen, it is observed to be at either one spot or the other, and this outcome cannot be predicted in advance. Because the experiment illustrates the character of quantum measurements, The Feynman Lectures on Physics use idealized Stern–Gerlach apparatuses to explain the basic mathematics of quantum theory.

Empathy gap

From Wikipedia, the free encyclopedia

For cognitive bias, see Hot-cold empathy gap.

An empathy gap, sometimes referred to as an empathy bias, is a breakdown or reduction in empathy (the ability to recognize, understand, and share another's thoughts and feelings) where it might otherwise be expected to occur. Empathy gaps may occur due to a failure in the process of empathizing or as a consequence of stable personality characteristics, and may reflect either a lack of ability or motivation to empathize.

Empathy gaps can be interpersonal (toward others) or intrapersonal (toward the self, e.g. when predicting one's own future preferences). A great deal of social psychological research has focused on intergroup empathy gaps, their underlying psychological and neural mechanisms, and their implications for downstream behavior (e.g. prejudice toward outgroup members).

Classification

Cognitive empathy gaps

Failures in cognitive empathy (also referred to as perspective-taking) may sometimes result from a lack of ability. For example, young children often engage in failures of perspective-taking (e.g., on false belief tasks) due to underdeveloped social cognitive abilities. Neurodivergent individuals often face difficulties inferring others' emotional and cognitive states, though the double empathy problem proposes that the problem is mutual, occurring as well in non-neurodivergent individuals' struggle to understand and relate to neurodivergent people. Failures in cognitive empathy may also result from cognitive biases that impair one's ability to understand another's perspective (for example, see the related concept of naive realism.)

One's ability to perspective-take may be limited by one's current emotional state. For example, behavioral economics research has described a number of failures in empathy that occur due to emotional influences on perspective-taking when people make social predictions. People may either fail to accurately predict one's own preferences and decisions (intrapersonal empathy gaps), or to consider how others' preferences might differ from one's own (interpersonal empathy gaps). For example, people not owning a certain good underestimate their attachment to that good were they to own it.

In other circumstances, failures in cognitive empathy may occur due to a lack of motivation. For example, people are less likely to take the perspective of outgroup members with whom they disagree.

Affective empathy gaps

Further information: Hot-cold empathy gap

Affective (i.e. emotional) empathy gaps may describe instances in which an observer and target do not experience similar emotions, or when an observer does not experience anticipated emotional responses toward a target, such as sympathy and compassion.

Certain affective empathy gaps may be driven by a limited ability to share another's emotions. For example, psychopathy is characterized by impairments in emotional empathy.

Individuals may be motivated to avoid empathizing with others' emotions due to the emotional costs of doing so. For example, according to C. D. Batson's model of empathy, empathizing with others may either result in empathic concern (i.e. feelings of warmth and concern for another) or personal distress (i.e. when another's distress causes distress for the self). A trait-level tendency to experience personal distress (vs. empathic concern) may motivate individuals to avoid situations which would require them to empathize with others, and indeed predicts reduced helping behavior.

Notable examples

Intergroup empathy gaps

Humans are less likely to help outgroup members in need, as compared to ingroup members. People are also less likely to value outgroup members' lives as highly as those of ingroup members. These effects are indicative of an ingroup empathy bias, in which people empathize more with ingroup (vs. outgroup) members.

Intergroup empathy gaps are often affective or cognitive in nature, but also extend to other domains such as pain. For example, a great deal of research has demonstrated that people show reduced responses (e.g. neural activity) when observing outgroup (vs. ingroup) members in pain. These effects may occur for real-world social groups such as members of different races. In one study utilizing a minimal groups paradigm (in which groups are randomly assigned, ostensibly based on an arbitrary distinction), individuals also judged the perceived pain of ingroup members to be more painful than that of outgroup members.

Intergroup schadenfreude

Perhaps the most well-known "counter-empathic" emotion—i.e., an emotion that reflects an empathy gap for the target—is schadenfreude, or the experience of pleasure when observing or learning about another's suffering or misfortune. Schadenfreude frequently occurs in intergroup contexts. In fact, the two factors that most strongly predict schadenfreude are identification with one's group and the presence of competition between groups in conflict. Competition may be explicit; for example, one study found that soccer fans were less likely to help an injured stranger wearing a rival team shirt than someone wearing an ingroup team shirt. However, schadenfreude may also be directed toward members of groups associated with high-status, competitive stereotypes. These findings correspond with the stereotype content model, which proposes that such groups elicit envy, thereby precipitating schadenfreude.

Occupational burnout

Stress related to the experience of empathy may cause empathic distress fatigue and occupational burnout, particularly among those in the medical profession. Expressing empathy is an important component of patient-centered care, and can be expressed through behaviors such as concern, attentiveness, sharing emotions, vulnerability, understanding, dialogue, reflection, and authenticity. However, expressing empathy can be cognitively and emotionally demanding for providers. Physicians who lack proper support may experience depression and burnout, particularly in the face of the extended or frequent experiences of personal distress.

Forecasting failures

Within the domain of social psychology, "empathy gaps" typically describe breakdowns in empathy toward others (interpersonal empathy gaps). However, research in behavioral economics has also identified a number of intrapersonal empathy gaps (i.e. toward one's self). For example, "hot-cold empathy gaps" describe a breakdown in empathy for one's future self—specifically, a failure to anticipate how one's future affective states will affect one's preferences. Such failures can negatively impact decision-making, particularly in regards to health outcomes. Hot-cold empathy gaps are related to the psychological concepts of affective forecasting and temporal discounting.

Psychological factors

Mentalizing processes

Both affective and cognitive empathy gaps can occur due to a breakdown in the process of mentalizing others' states. For example, breakdowns in mentalizing may include but are not limited to:

• Mind attribution: People may fail to take another's perspective due to a failure to attribute a mind or agency to that person. Behavioral research has found that individuals are less likely to assign mental states to outgroup compared to ingroup members.
• Episodic simulation: People may find it difficult to empathize with others if they struggle (due to a lack of ability or motivation) to episodically simulate others' mental states—i.e. to imagine events from others' lives which occur at a specific time and place. The ability to engage in episodic simulation is predictive of greater affective empathy and prosocial behavior towards others.

Neural evidence also supports the key role of mentalizing in supporting empathic responses, particularly in an intergroup context. For example, a meta-analysis of neuroimaging studies of intergroup social cognition found that thinking about ingroup members (in comparison to outgroup members) was more frequently related to brain regions known to underlie mentalizing.

Gender norms

Further information: Gender empathy gap

Gender differences in the experience of empathy have been a subject of debate. In particular, scientists have sought to determine whether observed gender differences in empathy are due to variance in ability, motivation, or both between men and women. Research to date raises the possibility that gender norms regarding the experience and expression of empathy may decrease men's willingness to empathize with others, and therefore their tendency to engage in empathy.

A number of studies, primarily utilizing self-report, have found gender differences in men's and women's empathy. A 1977 review of nine studies found women to be more empathic than men on average. A 1983 review found a similar result, although differences in scores were stronger for self-report, as compared to observational, measures. In recent decades, a number of studies utilizing self-reported empathy have shown gender differences in empathy. According to the results of a nationally representative survey, men reported less willingness to give money or volunteer time to a poverty relief organization as compared to women, a finding mediated by men's lower self-reported feelings of empathic concern toward others.

However, more recent work has found little evidence that gender differences in self-reported empathy are related to neurophysiological measures (hemodynamic responses and pupil dilation). This finding raises the possibility that self-reported empathy may not be driven by biological differences in responses, but rather gender differences in willingness to report empathy. Specifically, women may be more likely to report experiencing empathy because it is more gender-normative for women than men. In support of this idea, a study found that manipulating the perceived gender normativity of empathy eliminated gender differences in men and women's self-reported empathy. Specifically, assigning male and female participants to read a narrative describing fictitious neurological research evidence which claimed that males score higher on measures of empathy eliminated the gender gap in self-reported empathy.

Trait differences

Psychological research has identified a number of trait differences associated with reduced empathic responses, including but not limited to:

• Social dominance orientation: Individuals high in social dominance orientation (SDO; i.e., those who endorse inequality and hierarchy between groups), are more likely to be high in prejudice and have less empathic concern for outgroup members. In addition to predicting greater intergroup empathy bias, high SDO scores correlated with greater counter-empathy (i.e. schadenfreude) toward outgroup targets, including Asian and Black targets (compared to ingroup White targets) when group boundaries were previously made salient, as well as toward competitive outgroup members (compared to ingroup members) in a novel group setting.
• Reduced importance of social ideals and relationships: Reduced familial and religious importance also appear to be predictive of reduced empathic responses. In a sample of adults aged 18 to 35 (N = 722), family importance was positively associated with affective empathy and perspective taking, particularly among non-Hispanic whites. Religious importance was significantly related to affective empathy, especially among Black, Indigenous, and/or People of Color.
• Conservative political orientation: In an analysis of data from the 2004 American General Social Survey, researchers found conservatives to have lower levels of empathy as compared to liberals, but only among individuals with low (vs. high) levels of religiosity.
• Higher social class: Some studies have found that people from upper-class backgrounds are less likely to experience feelings of compassion or to engage in empathetic behaviors, such as helping others. Education may play a role in this, wealthy and low-income students often attend different schools and do not get a chance to interact with one another. There is growing evidence to suggest that greater economic inequality is linked with lower empathy among the wealthy.

Neural mechanisms

Neural simulation

According to the perception–action-model of empathy, perception–action-coupling (i.e., the vicarious activation of the neural system for action during the perception of action) allows humans to understand others' actions, intentions, and emotions. According to this theory, when a "subject" individual observes an "object" individual, the object's physical movements and facial expressions activate corresponding neural mechanisms in the subject. That is, by neurally simulating the object's observed states, the subject also experiences these states, the basis of empathy.

The mirror neuron system has been proposed as a neural mechanism supporting perception-action coupling and empathy, although such claims remain a subject of scientific debate. Although the exact (if any) role of mirror neurons in supporting empathy is unclear, evidence suggests that neural simulation (i.e., recreating neural states associated with a process observed in another) may generally support a variety of psychological processes in humans, including disgust, pain, touch, and facial expressions.

Reduced neural simulation of responses to suffering may account in part for observed empathy gaps, particularly in an intergroup context. This possibility is supported by research demonstrating that people show reduced neural activity when they witness ethnic outgroup (vs. ingroup) members in physical or emotional pain. In one study, Chinese and Causian participants viewed videos of Chinese and Causasian targets, who displayed neutral facial expressions as they received either painful or non-painful stimulation to their cheeks. Witnessing racial ingroup faces receive painful stimulation increased activity in the dorsal anterior cingulate cortex and anterior insula (two regions which generally activate during the experience of pain.) However, these responses were diminished toward outgroup members in pain. These results replicated among White-Italian and Black-African participants. Additionally, EEG work has shown reduced neural simulation of movement (in primary motor cortex) for outgroup members, compared to in-group members. This effect was magnified by prejudice and toward disliked groups (i.e. South-Asians, Blacks, and East Asians).

Oxytocin

A great deal of social neuroscience research has been conducted to investigate the social functions of the hormone oxytocin, including its role in empathy. Generally speaking, oxytocin is associated with cooperation between individuals (in both humans and non-human animals). However, these effects interact with group membership in intergroup settings: oxytocin is associated with increased bonding with ingroup, but not outgroup, members, and may thereby contribute to ingroup favoritism and intergroup empathy bias. However, in one study of Israelis and Palestinians, intranasal oxytocin administration improved opposing partisans' empathy for outgroup members by increasing the salience of their pain.

In addition to temporary changes in oxytocin levels, the influence of oxytocin on empathic responses may also be influenced by an oxytocin receptor gene polymorphism, such that certain individuals may differ in the extent to which oxytocin promotes ingroup favoritism.

Specific neural correlates

A number of studies have been conducted to identify the neural regions implicated in intergroup empathy biases. This work has highlighted candidate regions supporting psychological processes such as mentalizing for ingroup members, deindividuation of outgroup members, and the pleasure associated with the experience of schadenfreude.

Role of dmPFC

A meta-analysis of 50 fMRI studies of intergroup social cognition found more consistent activation in dorsomedial prefrontal cortex (dmPFC) during ingroup (vs. outgroup) social cognition. dmPFC has previously been linked to the ability to infer others' mental states, which suggests that individuals may be more likely to engage in mentalizing for ingroup (as compared to outgroup) members. dmPFC activity has also been linked to prosocial behavior; thus, dmPFC's association with cognition about ingroup members suggests a potential neurocognitive mechanism underlying ingroup favoritism.

Role of anterior insula

Activation patterns in the anterior insula (AI) have been observed when thinking about both ingroup and outgroup members. For example, greater activity in the anterior insula has been observed when participants view ingroup members on a sports team receiving pain, compared to outgroup members receiving pain. In contrast, the meta-analysis referenced previously found that anterior insula activation was more reliably related to social cognition about outgroup members.

These seemingly divergent results may be due in part to functional differences between anatomic subregions of the anterior insula. Meta-analyses have identified two distinct subregions of the anterior insula: ventral AI, which is linked to emotional and visceral experiences (e.g. subjective arousal); and dorsal AI, which has been associated with exogenous attention processes such as attention orientation, salience detection, and task performance monitoring. Therefore, anterior insula activation may occur more often when thinking about outgroup members because doing is more attentionally demanding than thinking about ingroup members.

Lateralization of function within the anterior insula may also help account for divergent results, due to differences in connectivity between left and right AI. The right anterior insula has greater connectivity with regions supporting attentional orientation and arousal (e.g. postcentral gyrus and supramarginal gyrus), compared to the left anterior insula, which has greater connectivity with regions involved in perspective-taking and cognitive motor control (e.g. dmPFC and superior frontal gyrus). The previously referenced meta-analysis found right lateralization of anterior insula for outgroup compared to ingroup processing. These findings raise the possibility that when thinking about outgroup members, individuals may use their attention to focus on targets' salient outgroup status, as opposed to thinking about the outgroup member as an individual. In contrast, the meta-analysis found left lateralization of anterior insula activity for thinking about ingroup compared to outgroup members. This finding suggests that left anterior insula may help support perspective-taking and mentalizing about ingroup members, and thinking about them in an individuated way. However, these possibilities are speculative and lateralization may vary due to characteristics such as age, gender, and other individual differences, which should be accounted for in future research.

Role of ventral striatum

A number of fMRI studies have attempted to identify the neural activation patterns underlying the experience of intergroup schadenfreude, particularly toward outgroup members in pain. These studies have found increased activation in the ventral striatum, a region related to reward processing and pleasure.

Consequences

Helping behavior

Breakdowns in empathy may reduce helping behavior, a phenomenon illustrated by the identifiable victim effect. Specifically, humans are less likely to assist others who are not identifiable on an individual level. A related concept is psychological distance—that is, we are less likely to help those who feel more psychologically distant from us.

Reduced empathy for outgroup members is associated with a reduction in willingness to entertain another's points of view, the likelihood of ignoring a customer's complaints, the likelihood of helping others during a natural disaster, and the chance that one opposes social programs designed to benefit disadvantaged individuals.

Prejudice

Empathy gaps may contribute to prejudicial attitudes and behavior. However, training people in perspective-taking, for example by providing instructions about how to take an outgroup member's perspective, has been shown to increase intergroup helping and the recognition of group disparities. Perspective-taking interventions are more likely to be effective when a multicultural approach is used (i.e., an approach that appreciates intergroup differences), as opposed to a "colorblind" approach (e.g. an approach that attempts to emphasize a shared group identity).

Quantum pseudo-telepathy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quantum_pseudo-telepathy

Quantum pseudo-telepathy describes the use of quantum entanglement to eliminate the need for classical communications. A nonlocal game is said to display quantum pseudo-telepathy if players who can use entanglement can win it with certainty while players without it can not. The prefix pseudo refers to the fact that quantum pseudo-telepathy does not involve the exchange of information between any parties. Instead, quantum pseudo-telepathy removes the need for parties to exchange information in some circumstances.

Quantum pseudo-telepathy is generally used as a thought experiment to demonstrate the non-local characteristics of quantum mechanics. However, quantum pseudo-telepathy is a real-world phenomenon which can be verified experimentally. It is thus an especially striking example of an experimental confirmation of Bell inequality violations.

The magic square game

When attempting to construct a 3×3 table filled with the numbers +1 and −1, such that each row has an even number of negative entries and each column an odd number of negative entries, a conflict is bound to emerge.

A simple magic square game demonstrating nonclassical correlations was introduced by P. K. Aravind based on a series of papers by N. David Mermin and Asher Peres and Adán Cabello that developed simplifying demonstrations of Bell's theorem. The game has been reformulated to demonstrate quantum pseudo-telepathy.

Game rules

This is a cooperative game featuring two players, Alice and Bob, and a referee. The referee asks Alice to fill in one row, and Bob one column, of a 3×3 table with plus and minus signs. Their answers must respect the following constraints: Alice's row must contain an even number of minus signs, Bob's column must contain an odd number of minus signs, and they both must assign the same sign to the cell where the row and column intersects. If they manage to do so, they win—otherwise they lose.

Alice and Bob are allowed to elaborate a strategy together, but crucially are not allowed to communicate after they know which row and column they will need to fill in (as otherwise the game would be trivial).

Classical strategy

It is easy to see that if Alice and Bob can come up with a classical strategy where they always win, they can represent it as a 3×3 table encoding their answers. But this is not possible, as the number of minus signs in this hypothetical table would need to be even and odd at the same time: every row must contain an even number of minus signs, making the total number of minus signs even, and every column must contain an odd number of minus signs, making the total number of minus signs odd.

With a bit further analysis one can see that the best possible classical strategy can be represented by a table where each cell now contains both Alice and Bob's answers, that may differ. It is possible to make their answers equal in 8 out of 9 cells, while respecting the parity of Alice's rows and Bob's columns. This implies that if the referee asks for a row and column whose intersection is one of the cells where their answers match they win, and otherwise they lose. Under the usual assumption that the referee asks for them uniformly at random, the best classical winning probability is 8/9.

Pseudo-telepathic strategies

Use of quantum pseudo-telepathy would enable Alice and Bob to win the game 100% of the time without any communication once the game has begun.

This requires Alice and Bob to possess two pairs of particles with entangled states. These particles must have been prepared before the start of the game. One particle of each pair is held by Alice and the other by Bob, so they each have two particles. When Alice and Bob learn which column and row they must fill, each uses that information to select which measurements they should make to their particles. The result of the measurements will appear to each of them to be random (and the observed partial probability distribution of either particle will be independent of the measurement performed by the other party), so no real "communication" takes place.

However, the process of measuring the particles imposes sufficient structure on the joint probability distribution of the results of the measurement such that if Alice and Bob choose their actions based on the results of their measurement, then there will exist a set of strategies and measurements allowing the game to be won with probability 1.

Note that Alice and Bob could be light years apart from one another, and the entangled particles will still enable them to coordinate their actions sufficiently well to win the game with certainty.

Each round of this game uses up one entangled state. Playing N rounds requires that N entangled states (2N independent Bell pairs, see below) be shared in advance. This is because each round needs 2-bits of information to be measured (the third entry is determined by the first two, so measuring it isn't necessary), which destroys the entanglement. There is no way to reuse old measurements from earlier games.

The trick is for Alice and Bob to share an entangled quantum state and to use specific measurements on their components of the entangled state to derive the table entries. A suitable correlated state consists of a pair of entangled Bell states:

${\displaystyle |\phi ⟩=\frac{1}{\sqrt{2}}({|+⟩}_a\otimes {|+⟩}_b+{|-⟩}_a\otimes {|-⟩}_b)\otimes \frac{1}{\sqrt{2}}({|+⟩}_c\otimes {|+⟩}_d+{|-⟩}_c\otimes {|-⟩}_d)}$

here ${\displaystyle |+⟩}$ and ${\displaystyle |-⟩}$ are eigenstates of the Pauli operator S_x with eigenvalues +1 and −1, respectively, whilst the subscripts a, b, c, and d identify the components of each Bell state, with a and c going to Alice, and b and d going to Bob. The symbol ${\displaystyle \otimes }$ represents a tensor product.

Observables for these components can be written as products of the Pauli matrices:

${\displaystyle X=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],Y=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right],Z=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]}$

Products of these Pauli spin operators can be used to fill the 3×3 table such that each row and each column contains a mutually commuting set of observables with eigenvalues +1 and −1, and with the product of the observables in each row being the identity operator, and the product of observables in each column equating to minus the identity operator. This is a so-called Mermin–Peres magic square. It is shown in below table.

${\displaystyle +I\otimes Z}$	${\displaystyle +Z\otimes I}$	${\displaystyle +Z\otimes Z}$
${\displaystyle +X\otimes I}$	${\displaystyle +I\otimes X}$	${\displaystyle +X\otimes X}$
${\displaystyle -X\otimes Z}$	${\displaystyle -Z\otimes X}$	${\displaystyle +Y\otimes Y}$

Effectively, while it is not possible to construct a 3×3 table with entries +1 and −1 such that the product of the elements in each row equals +1 and the product of elements in each column equals −1, it is possible to do so with the richer algebraic structure based on spin matrices.

The play proceeds by having each player make one measurement on their part of the entangled state per round of play. Each of Alice's measurements will give her the values for a row, and each of Bob's measurements will give him the values for a column. It is possible to do that because all observables in a given row or column commute, so there exists a basis in which they can be measured simultaneously. For Alice's first row she needs to measure both her particles in the ${\displaystyle Z}$ basis, for the second row she needs to measure them in the ${\displaystyle X}$ basis, and for the third row she needs to measure them in an entangled basis. For Bob's first column he needs to measure his first particle in the ${\displaystyle X}$ basis and the second in the ${\displaystyle Z}$ basis, for second column he needs to measure his first particle in the ${\displaystyle Z}$ basis and the second in the ${\displaystyle X}$ basis, and for his third column he needs to measure both his particles in a different entangled basis, the Bell basis. As long as the table above is used, the measurement results are guaranteed to always multiply out to +1 for Alice along her row, and −1 for Bob down his column. Of course, each completely new round requires a new entangled state, as different rows and columns are not compatible with each other.

Current research

It has been demonstrated that the above-described game is the simplest two-player game of its type in which quantum pseudo-telepathy allows a win with probability one. Other games in which quantum pseudo-telepathy occurs have been studied, including larger magic square games, graph colouring games giving rise to the notion of quantum chromatic number, and multiplayer games involving more than two participants.

In July 2022 a study reported the experimental demonstration of quantum pseudotelepathy via playing the nonlocal version of Mermin-Peres magic square game.

Greenberger–Horne–Zeilinger game

The Greenberger–Horne–Zeilinger (GHZ) game is another example of quantum pseudo-telepathy. Classically, the game has 0.75 winning probability. However, with a quantum strategy, the players can achieve a winning probability of 1, meaning they always win.

In the game there are three players, Alice, Bob, and Carol playing against a referee. The referee poses a binary question to each player (either ${\displaystyle 0}$ or ${\displaystyle 1}$). The three players each respond with an answer again in the form of either ${\displaystyle 0}$ or ${\displaystyle 1}$. Therefore, when the game is played the three questions of the referee x, y, z are drawn from the 4 options ${\displaystyle \{(0,0,0),(1,1,0),(1,0,1),(0,1,1)\}}$. For example, if question triple ${\displaystyle (0,1,1)}$ is chosen, then Alice receives bit 0, Bob receives bit 1, and Carol receives bit 1 from the referee. Based on the question bit received, Alice, Bob, and Carol each respond with an answer a, b, c, also in the form of 0 or 1. The players can formulate a strategy together prior to the start of the game. However, no communication is allowed during the game itself.

The players win if ${\displaystyle {\mathrm{m}\mathrm{o}\mathrm{d}}_2(a+b+c)=x\vee y\vee z}$, where ${\displaystyle \vee }$ indicates OR condition and ${\displaystyle mo{d}_2}$ indicates summation of answers modulo 2. In other words, the sum of three answers has to be even if ${\displaystyle x=y=z=0}$. Otherwise, the sum of answers has to be odd.

Winning condition of GHZ game

${\displaystyle x}$	${\displaystyle y}$	${\displaystyle z}$	${\displaystyle a\oplus b\oplus c}$
0	0	0	0 mod 2
1	1	0	1 mod 2
1	0	1	1 mod 2
0	1	1	1 mod 2

Classical strategy

Classically, Alice, Bob, and Carol can employ a deterministic strategy that always end up with odd sum (e.g. Alice always output 1. Bob and Carol always output 0). The players win 75% of the time and only lose if the questions are ${\displaystyle (0,0,0)}$.

This is the best classical strategy: only 3 out of 4 winning conditions can be satisfied simultaneously. Let ${\displaystyle a_0,a_1}$ be Alice's response to question 0 and 1 respectively, ${\displaystyle b_0,b_1}$ be Bob's response to question 0, 1, and ${\displaystyle c_0,c_1}$ be Carol's response to question 0, 1. We can write all constraints that satisfy winning conditions as $${\displaystyle \begin{array}{rl}& a_0+b_0+c_0=0\mathrm{mod}2\\ & a_1+b_1+c_0=1\mathrm{mod}2\\ & a_1+b_0+c_1=1\mathrm{mod}2\\ & a_0+b_1+c_1=1\mathrm{mod}2\end{array}}$$

Suppose that there is a classical strategy that satisfies all four winning conditions, all four conditions hold true. Through observation, each term appears twice on the left hand side. Hence, the left side sum = 0 mod 2. However, the right side sum = 1 mod 2. The contradiction shows that all four winning conditions cannot be simultaneously satisfied.

Quantum strategy

When Alice, Bob, and Carol decide to adopt a quantum strategy they share a tripartite entangled state $|\psi ⟩=\frac{1}{\sqrt{2}}(|000⟩+|111⟩)$, known as the GHZ state.

If question 0 is received, the player makes a measurement in the X basis $\{|+⟩,|-⟩\}$. If question 1 is received, the player makes a measurement in the Y basis $\left\{\frac{1}{\sqrt{2}}(|0⟩+i|1⟩),\frac{1}{\sqrt{2}}(|0⟩-i|1⟩)\right\}$. In both cases, the players give answer 0 if the result of the measurement is the first state of the pair, and answer 1 if the result is the second state of the pair. With this strategy the players win the game with probability 1.