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.

 

Quantum foundations

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Quantum_foundations

Quantum foundations is a discipline of science that seeks to understand the most counter-intuitive aspects of quantum theory, reformulate it and even propose new generalizations thereof. Contrary to other physical theories, such as general relativity, the defining axioms of quantum theory are quite ad hoc, with no obvious physical intuition. While they lead to the right experimental predictions, they do not come with a mental picture of the world where they fit.

There exist different approaches to resolve this conceptual gap:

• First, one can put quantum physics in contraposition with classical physics: by identifying scenarios, such as Bell experiments, where quantum theory radically deviates from classical predictions, one hopes to gain physical insights on the structure of quantum physics.
• Second, one can attempt to find a re-derivation of the quantum formalism in terms of operational axioms.
• Third, one can search for a full correspondence between the mathematical elements of the quantum framework and physical phenomena: any such correspondence is called an interpretation.
• Fourth, one can renounce quantum theory altogether and propose a different model of the world.

Research in quantum foundations is structured along these roads.

Non-classical features of quantum theory

Quantum nonlocality

Main article: Quantum nonlocality

Two or more separate parties conducting measurements over a quantum state can observe correlations which cannot be explained with any local hidden variable theory. Whether this should be regarded as proving that the physical world itself is "nonlocal" is a topic of debate, but the terminology of "quantum nonlocality" is commonplace. Nonlocality research efforts in quantum foundations focus on determining the exact limits that classical or quantum physics enforces on the correlations observed in a Bell experiment or more complex causal scenarios. This research program has so far provided a generalization of Bell's theorem that allows falsifying all classical theories with a superluminal, yet finite, hidden influence.

Quantum contextuality

Main article: Quantum contextuality

Nonlocality can be understood as an instance of quantum contextuality. A situation is contextual when the value of an observable depends on the context in which it is measured (namely, on which other observables are being measured as well). The original definition of measurement contextuality can be extended to state preparations and even general physical transformations.

Epistemic models for the quantum wave-function

A physical property is epistemic when it represents our knowledge or beliefs on the value of a second, more fundamental feature. The probability of an event to occur is an example of an epistemic property. In contrast, a non-epistemic or ontic variable captures the notion of a “real” property of the system under consideration.

There is an on-going debate on whether the wave-function represents the epistemic state of a yet to be discovered ontic variable or, on the contrary, it is a fundamental entity. Under some physical assumptions, the Pusey–Barrett–Rudolph (PBR) theorem demonstrates the inconsistency of quantum states as epistemic states, in the sense above. Note that, in QBism and Copenhagen-type views, quantum states are still regarded as epistemic, not with respect to some ontic variable, but to one's expectations about future experimental outcomes. The PBR theorem does not exclude such epistemic views on quantum states.

Axiomatic reconstructions

Some of the counter-intuitive aspects of quantum theory, as well as the difficulty to extend it, follow from the fact that its defining axioms lack a physical motivation. An active area of research in quantum foundations is therefore to find alternative formulations of quantum theory which rely on physically compelling principles. Those efforts come in two flavors, depending on the desired level of description of the theory: the so-called Generalized Probabilistic Theories approach and the Black boxes approach.

The framework of generalized probabilistic theories

Main article: Generalized probabilistic theories

Generalized Probabilistic Theories (GPTs) are a general framework to describe the operational features of arbitrary physical theories. Essentially, they provide a statistical description of any experiment combining state preparations, transformations and measurements. The framework of GPTs can accommodate classical and quantum physics, as well as hypothetical non-quantum physical theories which nonetheless possess quantum theory's most remarkable features, such as entanglement or teleportation. Notably, a small set of physically motivated axioms is enough to single out the GPT representation of quantum theory.

L. Hardy introduced the concept of GPT in 2001, in an attempt to re-derive quantum theory from basic physical principles. Although Hardy's work was very influential (see the follow-ups below), one of his axioms was regarded as unsatisfactory: it stipulated that, of all the physical theories compatible with the rest of the axioms, one should choose the simplest one. The work of Dakic and Brukner eliminated this “axiom of simplicity” and provided a reconstruction of quantum theory based on three physical principles. This was followed by the more rigorous reconstruction of Masanes and Müller.

Axioms common to these three reconstructions are:

• The subspace axiom: systems which can store the same amount of information are physically equivalent.
• Local tomography: to characterize the state of a composite system it is enough to conduct measurements at each part.
• Reversibility: for any two extremal states [i.e., states which are not statistical mixtures of other states], there exists a reversible physical transformation that maps one into the other.

An alternative GPT reconstruction proposed by Chiribella, D'Ariano and Perinotti  around the same time is also based on the

• Purification axiom: for any state ${\displaystyle S_A}$ of a physical system A there exists a bipartite physical system ${\displaystyle A-B}$ and an extremal state (or purification) ${\displaystyle T_{AB}}$ such that ${\displaystyle S_A}$ is the restriction of ${\displaystyle T_{AB}}$ to system ${\displaystyle A}$. In addition, any two such purifications ${\displaystyle T_{AB},T_{AB}^{\mathrm{\prime }}}$ of ${\displaystyle S_A}$ can be mapped into one another via a reversible physical transformation on system ${\displaystyle B}$.

The use of purification to characterize quantum theory has been criticized on the grounds that it also applies in the Spekkens toy model.

To the success of the GPT approach, it can be countered that all such works just recover finite dimensional quantum theory. In addition, none of the previous axioms can be experimentally falsified unless the measurement apparatuses are assumed to be tomographically complete.

Categorical quantum mechanics or process theories

Main article: Categorical quantum mechanics

Categorical Quantum Mechanics (CQM) or Process Theories are a general framework to describe physical theories, with an emphasis on processes and their compositions. It was pioneered by Samson Abramsky and Bob Coecke. Besides its influence in quantum foundations, most notably the use of a diagrammatic formalism, CQM also plays an important role in quantum technologies, most notably in the form of ZX-calculus. It also has been used to model theories outside of physics, for example the DisCoCat compositional natural language meaning model.

The framework of black boxes

Main article: Quantum nonlocality

In the black box or device-independent framework, an experiment is regarded as a black box where the experimentalist introduces an input (the type of experiment) and obtains an output (the outcome of the experiment). Experiments conducted by two or more parties in separate labs are hence described by their statistical correlations alone.

From Bell's theorem, we know that classical and quantum physics predict different sets of allowed correlations. It is expected, therefore, that far-from-quantum physical theories should predict correlations beyond the quantum set. In fact, there exist instances of theoretical non-quantum correlations which, a priori, do not seem physically implausible. The aim of device-independent reconstructions is to show that all such supra-quantum examples are precluded by a reasonable physical principle.

The physical principles proposed so far include no-signalling, Non-Trivial Communication Complexity, No-Advantage for Nonlocal computation, Information Causality, Macroscopic Locality, and Local Orthogonality. All these principles limit the set of possible correlations in non-trivial ways. Moreover, they are all device-independent: this means that they can be falsified under the assumption that we can decide if two or more events are space-like separated. The drawback of the device-independent approach is that, even when taken together, all the afore-mentioned physical principles do not suffice to single out the set of quantum correlations. In other words: all such reconstructions are partial.

Interpretations of quantum theory

Main article: Interpretations of quantum mechanics

An interpretation of quantum theory is a correspondence between the elements of its mathematical formalism and physical phenomena. For instance, in the pilot wave theory, the quantum wave function is interpreted as a field that guides the particle trajectory and evolves with it via a system of coupled differential equations. Most interpretations of quantum theory stem from the desire to solve the quantum measurement problem.

Extensions of quantum theory

In an attempt to reconcile quantum and classical physics, or to identify non-classical models with a dynamical causal structure, some modifications of quantum theory have been proposed.

Collapse models

Collapse models posit the existence of natural processes which periodically localize the wave-function. Such theories provide an explanation to the nonexistence of superpositions of macroscopic objects, at the cost of abandoning unitarity and exact energy conservation.

Quantum measure theory

In Sorkin's quantum measure theory (QMT), physical systems are not modeled via unitary rays and Hermitian operators, but through a single matrix-like object, the decoherence functional. The entries of the decoherence functional determine the feasibility to experimentally discriminate between two or more different sets of classical histories, as well as the probabilities of each experimental outcome. In some models of QMT the decoherence functional is further constrained to be positive semidefinite (strong positivity). Even under the assumption of strong positivity, there exist models of QMT which generate stronger-than-quantum Bell correlations.

Acausal quantum processes

The formalism of process matrices starts from the observation that, given the structure of quantum states, the set of feasible quantum operations follows from positivity considerations. Namely, for any linear map from states to probabilities one can find a physical system where this map corresponds to a physical measurement. Likewise, any linear transformation that maps composite states to states corresponds to a valid operation in some physical system. In view of this trend, it is reasonable to postulate that any high-order map from quantum instruments (namely, measurement processes) to probabilities should also be physically realizable. Any such map is termed a process matrix. As shown by Oreshkov et al., some process matrices describe situations where the notion of global causality breaks.

The starting point of this claim is the following mental experiment: two parties, Alice and Bob, enter a building and end up in separate rooms. The rooms have ingoing and outgoing channels from which a quantum system periodically enters and leaves the room. While those systems are in the lab, Alice and Bob are able to interact with them in any way; in particular, they can measure some of their properties.

Since Alice and Bob's interactions can be modeled by quantum instruments, the statistics they observe when they apply one instrument or another are given by a process matrix. As it turns out, there exist process matrices which would guarantee that the measurement statistics collected by Alice and Bob is incompatible with Alice interacting with her system at the same time, before or after Bob, or any convex combination of these three situations. Such processes are called acausal.