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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the CHSH inequality can be used in the proof of Bell's theorem, which states that certain consequences of entanglement in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
can not be reproduced by local hidden-variable theories. Experimental verification of the inequality being violated is seen as
confirmation In Christian denominations that practice infant baptism, confirmation is seen as the sealing of the covenant created in baptism. Those being confirmed are known as confirmands. For adults, it is an affirmation of belief. It involves laying on ...
that nature cannot be described by such theories. CHSH stands for
John Clauser John Francis Clauser (; born December 1, 1942) is an American theoretical and experimental physicist known for contributions to the foundations of quantum mechanics, in particular the Clauser–Horne–Shimony–Holt inequality. Clauser was a ...
, Michael Horne,
Abner Shimony Abner Eliezer Shimony (; March 10, 1928 – August 8, 2015) was an American physicist and philosopher. He specialized in quantum theory and philosophy of science. As a physicist, he concentrated on the interaction between relativity theory and qu ...
, and Richard Holt, who described it in a much-cited paper published in 1969. They derived the CHSH inequality, which, as with
John Stewart Bell John Stewart Bell FRS (28 July 1928 – 1 October 1990) was a physicist from Northern Ireland and the originator of Bell's theorem, an important theorem in quantum physics regarding hidden-variable theories. In 2022, the Nobel Prize in Phy ...
's original inequality, is a constraint on the statistical occurrence of "coincidences" in a
Bell test A Bell test, also known as Bell inequality test or Bell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics in relation to Albert Einstein's concept of local realism. Named for John Stewart Bell, the e ...
which is necessarily true if there exist underlying local hidden variables, an assumption that is sometimes termed local realism. It is in fact the case that the inequality is routinely violated by modern experiments in quantum mechanics.


Statement

The usual form of the CHSH inequality is where ''a'' and ''a''′ are detector settings on side A, ''b'' and ''b''′ on side B, the four combinations being tested in separate subexperiments. The terms ''E''(''a'', ''b'') etc. are the
quantum correlation In quantum mechanics, quantum correlation is the expected value of the product of the alternative outcomes. In other words, it is the expected change in physical characteristics as one quantum system passes through an interaction site. In John Be ...
s of the particle pairs, where the quantum correlation is defined to be the expectation value of the product of the "outcomes" of the experiment, i.e. the statistical average of ''A''(''a'')·''B''(''b''), where ''A'' and ''B'' are the separate outcomes, using the coding +1 for the '+' channel and −1 for the '−' channel. Clauser et al.'s 1969 derivation was oriented towards the use of "two-channel" detectors, and indeed it is for these that it is generally used, but under their method the only possible outcomes were +1 and −1. In order to adapt to real situations, which at the time meant the use of polarised light and single-channel polarisers, they had to interpret '−' as meaning "non-detection in the '+' channel", i.e. either '−' or nothing. They did not in the original article discuss how the two-channel inequality could be applied in real experiments with real imperfect detectors, though it was later provedJ. S. Bell, in ''Foundations of Quantum Mechanics'', Proceedings of the International School of Physics “Enrico Fermi”, Course XLIX, B. d’Espagnat (Ed.) (Academic, New York, 1971), p. 171 and Appendix B. Pages 171-81 are reproduced as Ch. 4 of J. S. Bell, ''Speakable and Unspeakable in Quantum Mechanics'' (Cambridge University Press 1987) that the inequality itself was equally valid. The occurrence of zero outcomes, though, means it is no longer so obvious how the values of ''E'' are to be estimated from the experimental data. The mathematical formalism of quantum mechanics predicts a maximum value for S of 2 ( Tsirelson's bound), which is greater than 2, and CHSH violations are therefore predicted by the theory of quantum mechanics.


Experiments

Many Bell tests conducted subsequent to
Alain Aspect Alain Aspect (; born 15 June 1947) is a French physicist noted for his experimental work on quantum entanglement. Aspect was awarded the 2022 Nobel Prize in Physics, jointly with John Clauser and Anton Zeilinger, "for experiments with entangl ...
's second experiment in 1982 have used the CHSH inequality, estimating the terms using (3) and assuming fair sampling. Some dramatic violations of the inequality have been reported. In practice most actual experiments have used light rather than the electrons that Bell originally had in mind. The property of interest is, in the best known experiments, the polarisation direction, though other properties can be used. The diagram shows a typical optical experiment. Coincidences (simultaneous detections) are recorded, the results being categorised as '++', '+−', '−+' or '−−' and corresponding counts accumulated. Four separate subexperiments are conducted, corresponding to the four terms E(a, b) in the test statistic ''S'' (, above). The settings , , , and are generally in practice chosen — the "Bell test angles" — these being the ones for which the quantum mechanical formula gives the greatest violation of the inequality. For each selected value of ''a'' and ''b'', the numbers of coincidences in each category \left\ are recorded. The experimental estimate for E(a, b) is then calculated as: Once all the 's have been estimated, an experimental estimate of ''S'' (Eq. ) can be found. If it is numerically greater than 2 it has infringed the CHSH inequality and the experiment is declared to have supported the quantum mechanics prediction and ruled out all local hidden-variable theories. The CHSH paper lists many preconditions (or "reasonable and/or presumable assumptions") to derive the simplified theorem and formula. For example, for the method to be valid, it has to be assumed that the detected pairs are a fair sample of those emitted. In actual experiments, detectors are never 100% efficient, so that only a sample of the emitted pairs are detected. A subtle, related requirement is that the hidden variables do not influence or determine detection probability in a way that would lead to different samples at each arm of the experiment. Various labs have been entangled and violated the CHSH inequality with
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
pairs,
beryllium Beryllium is a chemical element with the symbol Be and atomic number 4. It is a steel-gray, strong, lightweight and brittle alkaline earth metal. It is a divalent element that occurs naturally only in combination with other elements to form mi ...
ion pairs,
ytterbium Ytterbium is a chemical element with the symbol Yb and atomic number 70. It is a metal, the fourteenth and penultimate element in the lanthanide series, which is the basis of the relative stability of its +2 oxidation state. However, like the othe ...
ion pairs,
rubidium Rubidium is the chemical element with the symbol Rb and atomic number 37. It is a very soft, whitish-grey solid in the alkali metal group, similar to potassium and caesium. Rubidium is the first alkali metal in the group to have a density higher ...
atom pairs, whole rubidium-atom cloud pairs, nitrogen vacancies in
diamonds Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, b ...
, and
Josephson phase In physics, the Josephson effect is a phenomenon that occurs when two superconductors are placed in proximity, with some barrier or restriction between them. It is an example of a macroscopic quantum phenomenon, where the effects of quantum mec ...
qubits In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
.


Derivation

The original 1969 derivation will not be given here since it is not easy to follow and involves the assumption that the outcomes are all +1 or −1, never zero. Bell's 1971 derivation is more general. He effectively assumes the "Objective Local Theory" later used by Clauser and Horne. It is assumed that any hidden variables associated with the detectors themselves are independent on the two sides and can be averaged out from the start. Another derivation of interest is given in Clauser and Horne's 1974 paper, in which they start from the CH74 inequality. It would appear from both these later derivations that the only assumptions really needed for the inequality itself (as opposed to the method of estimation of the test statistic) are that the distribution of the possible states of the source remains constant and the detectors on the two sides act independently.


Bell's 1971 derivation

The following is based on page 37 of Bell's ''Speakable and Unspeakable'', the main change being to use the symbol ‘''E''’ instead of ‘''P''’ for the expected value of the quantum correlation. This avoids any suggestion that the
quantum correlation In quantum mechanics, quantum correlation is the expected value of the product of the alternative outcomes. In other words, it is the expected change in physical characteristics as one quantum system passes through an interaction site. In John Be ...
is itself a probability. We start with the standard assumption of independence of the two sides, enabling us to obtain the joint probabilities of pairs of outcomes by multiplying the separate probabilities, for any selected value of the "hidden variable" λ. λ is assumed to be drawn from a fixed distribution of possible states of the source, the probability of the source being in the state λ for any particular trial being given by the density function ρ(λ), the integral of which over the complete hidden variable space is 1. We thus assume we can write: E(a, b) = \int \underline(a, \lambda) \underline(b, \lambda) \rho(\lambda) d\lambda where ''A'' and ''B'' are the outcomes. Since the possible values of ''A'' and ''B'' are −1, 0 and +1, it follows that: Then, if ''a'', ''a''′, ''b'' and ''b''′ are alternative settings for the detectors, \begin &E(a, b) - E\left(a, b'\right) \\ = &\int \left \underline(a, \lambda) \underline(b, \lambda) - \underline(a, \lambda) \underline\left(b', \lambda\right) \right\rho(\lambda) d\lambda \\ = &\int \left[ \underline(a, \lambda) \underline(b, \lambda) - \underline(a, \lambda) \underline\left(b', \lambda\right) \pm \underline(a, \lambda) \underline(b, \lambda) \underline\left(a', \lambda\right) \underline\left(b', \lambda\right) \mp \underline(a, \lambda) \underline(b, \lambda) \underline\left(a', \lambda\right) \underline\left(b', \lambda\right) \right] \rho(\lambda) d\lambda\\ = &\int \underline(a, \lambda) \underline(b, \lambda) \left[1 \pm \underline\left(a', \lambda\right) \underline\left(b', \lambda\right) \right] \rho(\lambda) d\lambda - \int \underline(a, \lambda) \underline\left(b', \lambda\right) \left \pm \underline\left(a', \lambda\right) \underline(b, \lambda) \right\rho(\lambda) d\lambda \end Taking absolute values of both sides, and applying the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
to the right-hand side, we obtain :\left, E(a, b) - E\left(a, b'\right) \ \leq \left, \int \underline(a, \lambda) \underline(b, \lambda) \left 1 \pm \underline\left(a', \lambda\right) \underline\left(b', \lambda\right) \right\rho(\lambda) d\lambda \ + \left, \int \underline(a, \lambda) \underline\left(b', \lambda\right) \left \pm \underline\left(a', \lambda\right) \underline(b, \lambda) \right\rho(\lambda) d\lambda \ We use the fact that \left 1 \pm \underline\left(a', \lambda\right) \underline\left(b', \lambda\right) \right\rho(\lambda) and \left 1 \pm \underline\left(a', \lambda\right) \underline(b, \lambda) \right\rho(\lambda) are both non-negative to rewrite the right-hand side of this as \int \left, \underline(a, \lambda) \underline(b, \lambda) \ \left, \left 1 \pm \underline\left(a', \lambda\right) \underline\left(b', \lambda\right)\right\rho(\lambda) d\lambda \ + \int \left, \underline(a, \lambda) \underline(b', \lambda) \ \left, \left 1 \pm \underline\left(a', \lambda\right) \underline(b, \lambda) \right\rho(\lambda) d\lambda \ By (), this must be less than or equal to \int \left 1 \pm \underline\left(a', \lambda\right) \underline\left(b', \lambda\right) \right\rho(\lambda) d\lambda + \int \left 1 \pm \underline\left(a', \lambda\right) \underline(b, \lambda) \right\rho(\lambda) d\lambda which, using the fact that the integral of is 1, is equal to 2 \pm \left \int \underline\left(a', \lambda\right) \underline\left(b', \lambda\right) \rho(\lambda) d\lambda + \int \underline\left(a', \lambda\right) \underline(b, \lambda) \rho(\lambda) d\lambda \right/math> which is equal to 2 \pm \left E\left(a', b'\right) + E\left(a', b\right) \right/math>. Putting this together with the left-hand side, we have: \left, E(a, b) - E\left(a, b'\right) \ \; \leq 2 \; \pm \left E\left(a', b'\right) + E\left(a', b\right) \right/math> which means that the left-hand side is less than or equal to both 2 + \left E\left(a', b'\right) + E\left(a', b\right) \right/math> and 2 - \left E\left(a', b'\right) + E\left(a', b\right) \right/math>. That is: \left, E(a, b) - E\left(a, b'\right) \ \; \leq \; 2 - \left, E\left(a', b'\right) + E\left(a', b\right) \ from which we obtain 2 \;\geq\; \left, E(a, b) - E\left(a, b'\right) \ + \left, E\left(a', b'\right) + E\left(a', b\right) \ \;\geq\; \left, E(a, b) - E\left(a, b'\right) + E\left(a', b'\right) + E\left(a', b\right) \ (by the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
again), which is the CHSH inequality.


Derivation from Clauser and Horne's 1974 inequality

In their 1974 paper, Clauser and Horne show that the CHSH inequality can be derived from the CH74 one. As they tell us, in a two-channel experiment the CH74 single-channel test is still applicable and provides four sets of inequalities governing the probabilities ''p'' of coincidences. Working from the inhomogeneous version of the inequality, we can write: - 1 \; \leq \; p_(a, b) - p_(a, b') + p_(a', b) + p_(a', b') - p_(a') - p_(b) \; \leq \; 0 where ''j'' and ''k'' are each '+' or '−', indicating which detectors are being considered. To obtain the CHSH test statistic ''S'' (), all that is needed is to multiply the inequalities for which ''j'' is different from ''k'' by −1 and add these to the inequalities for which ''j'' and ''k'' are the same.


Optimal violation by a general quantum state

In experimental practice, the two particles are not an ideal
EPR pair The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form o ...
. There is a necessary and sufficient condition for a two-
qubit In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
density matrix In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
\rho to violate the CHSH inequality, expressed by the maximum attainable polynomial ''S''max defined in . This is important in entanglement-based
quantum key distribution Quantum key distribution (QKD) is a secure communication method which implements a cryptographic protocol involving components of quantum mechanics. It enables two parties to produce a shared random secret key known only to them, which can then be ...
, where the secret key rate depends on the degree of measurement correlations. Let us introduce a 3×3 real matrix T_ with elements t_ = \operatorname rho\cdot(\sigma_i \otimes \sigma_j)/math>, where \sigma_1, \sigma_2, \sigma_3 are the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
. Then we find the eigensystem of the real symmetric matrix U_\rho = T_\rho^\text T_\rho, U_\rho \boldsymbol_i = \lambda_i \boldsymbol_i, \quad , \boldsymbol_i, = 1, \quad i=1,2,3, where the indices are sorted by \lambda_1 \geq \lambda_2 \geq \lambda_3. Then, the maximal CHSH polynomial is determined by the two greatest eigenvalues, S_\text(\rho) = 2\sqrt.


Optimal measurement bases

There exists an optimal configuration of the measurement bases ''a, a', b, b for a given \rho that yields ''S''max with at least one free parameter. The projective measurement that yields either +1 or −1 for two orthogonal states , \alpha\rangle, , \alpha^\perp\rangle respectively, can be expressed by an operator \Alpha = , \alpha\rangle\langle\alpha, - , \alpha^\perp\rangle\langle\alpha^\perp, . The choice of this measurement basis can be parametrized by a real unit vector \boldsymbol \in \mathbb^3, , \boldsymbol, =1 and the Pauli vector \boldsymbol by expressing \Alpha = \boldsymbol \cdot \boldsymbol. Then, the expected correlation in bases ''a, b'' is E(a,b) = \operatorname rho(\boldsymbol \cdot \boldsymbol)\otimes(\boldsymbol \cdot \boldsymbol)= \boldsymbol^\text T_\rho \boldsymbol. The numerical values of the basis vectors, when found, can be directly translated to the configuration of the projective measurements. The optimal set of bases for the state \rho is found by taking the two greatest eigenvalues \lambda_ and the corresponding eigenvectors \boldsymbol_ of U_\rho, and finding the auxiliary unit vectors \begin \boldsymbol &= \boldsymbol_1 \cos\varphi + \boldsymbol_2 \sin\varphi, \\ \boldsymbol' &= \boldsymbol_1 \sin\varphi - \boldsymbol_2 \cos\varphi, \end where \varphi is a free parameter. We also calculate the acute angle \theta = \operatorname \sqrt to obtain the bases that maximize , \begin \boldsymbol &= T_\rho \boldsymbol' / , T_\rho \boldsymbol', , \\ \boldsymbol' &= T_\rho \boldsymbol / , T_\rho \boldsymbol, , \\ \boldsymbol &= \boldsymbol \cos\theta + \boldsymbol' \sin\theta, \\ \boldsymbol' &= \boldsymbol \cos\theta - \boldsymbol' \sin\theta. \end In entanglement-based
quantum key distribution Quantum key distribution (QKD) is a secure communication method which implements a cryptographic protocol involving components of quantum mechanics. It enables two parties to produce a shared random secret key known only to them, which can then be ...
, there is another measurement basis used to communicate the secret key (\boldsymbol_0 assuming Alice uses the side A). The bases \boldsymbol_0, \boldsymbol then need to minimize the quantum bit error rate ''Q'', which is the probability of obtaining different measurement outcomes (+1 on one particle and −1 on the other). The corresponding bases are \begin \boldsymbol_0 &= T_\rho \boldsymbol_1 / , T_\rho \boldsymbol_1, , \\ \boldsymbol &= \boldsymbol_1. \end The CHSH polynomial ''S'' needs to be maximized as well, which together with the bases above creates the constraint \varphi = \pi/4.


CHSH game

The CHSH game is a thought experiment involving two parties separated at a great distance (far enough to preclude classical communication at the speed of light), each of whom has access to one half of an entangled two-qubit pair. Analysis of this game shows that no classical
local hidden-variable theory In the interpretation of quantum mechanics, a local hidden-variable theory is a hidden-variable theory that satisfies the condition of being consistent with local realism. This includes all types of the theory that attempt to account for the proba ...
can explain the correlations that can result from entanglement. Since this game is indeed physically realizable, this gives strong evidence that classical physics is fundamentally incapable of explaining certain quantum phenomena, at least in a "local" fashion. In the CHSH game, there are two cooperating players, Alice and Bob, and a referee, Charlie. These agents will be abbreviated A, B, C respectively. At the start of the game, Charlie chooses bits x,y \in \ uniformly at random, and then sends x to Alice and y to Bob. Alice and Bob must then each respond to Charlie with bits a,b \in \ respectively. Now, once Alice and Bob send their responses back to Charlie, Charlie tests if a \oplus b = x \land y. If this equality holds, then Alice and Bob win, and if not then they lose. It is also required that Alice and Bob's responses can only depend on the bits they see: so Alice's response a depends only on x, and similarly for Bob. This means that Alice and Bob are forbidden from directly communicating with each other about the values of the bits sent to them by Charlie. However, Alice and Bob are allowed to decide on a common ''strategy'' before the game begins. In the following sections, it is shown that if Alice and Bob use only classical strategies involving their local information (and potentially some random coin tosses), it is impossible for them to win with a probability higher than 75%. However, if Alice and Bob are allowed to share a single entangled qubit pair, then there exists a strategy which allows Alice and Bob to succeed with a probability of ~85%.


Optimal classical strategy

We first establish that any ''
deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
'' classical strategy has success probability at most 75% (where the probability is taken over Charlie's uniformly random choice of x,y). By a deterministic strategy, we mean a pair of functions f_A, f_B: \ \mapsto \, where f_A is a function determining Alice's response as a function of the message she receives from Charlie, and f_B is a function determining Bob's response based on what he receives. To prove that any deterministic strategy fails at least 25% of the time, we can simply consider all possible pairs of strategies for Alice and Bob, of which there are at most 8 (for each party, there are 4 functions \ \mapsto \). It can be verified that for each of those 8 strategies there is always at least one out of the four possible input pairs (x, y) \in \^2 which makes the strategy fail. For example, in the strategy where both players always answer 0, we have that Alice and Bob win in all cases except for when x=y=1, so using this strategy their win probability is exactly 75%. Now, consider the case of randomized classical strategies, where Alice and Bob have access to ''correlated'' random numbers. They can be produced by jointly flipping a coin several times before the game has started and Alice and Bob are still allowed to communicate. The output they give at each round is then a function of both Charlie's message and the outcome of the corresponding coin flip. Such a strategy can be viewed as a probability distribution over deterministic strategies, and thus its success probability is a weighted sum over the success probabilities of the deterministic strategies. But since every deterministic strategy has a success probability of at most 75%, this weighted sum cannot exceed 75% either.


Optimal quantum strategy

Now, imagine that Alice and Bob each possess one qubit of the following 2-qubit entangled state: \Phi = \frac(, 00\rangle + , 11\rangle). This state is commonly referred to as the
EPR pair The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form o ...
, and can be equivalently written as \Phi = \frac(, ++\rangle + , --\rangle). Alice and Bob will use this entangled pair in their strategy as described below. The optimality of this strategy follows from Tsirelson's bound. When Alice receives her bit x from Charlie, if x=0 she will measure her qubit in the basis , 0\rangle, , 1\rangle, and then respond with 0 if the measurement outcome is , 0\rangle, and 1 if it is , 1\rangle. Otherwise, if x=1 she will measure her qubit in the basis , +\rangle, , -\rangle, and respond with 0 if the measurement outcome is , +\rangle, and 1 if it is , -\rangle. When Bob receives his bit y from Charlie, if y=0 he will measure his qubit in the basis , a_0\rangle, , a_1\rangle where , a_0\rangle = \left(\cos\frac\right), 0\rangle + \left(\sin\frac\right), 1\rangle, and , a_1\rangle = \left(-\sin\frac\right), 0\rangle + \left(\cos\frac\right), 1\rangle. He then responds with 0 if the result is , a_0\rangle, and 1 if it is , a_1\rangle. Otherwise, if y=1, he will measure his qubit in the basis , b_0\rangle, , b_1\rangle where , b_0\rangle = \left(\cos\frac\right), 0\rangle - \left(\sin\frac\right), 1\rangle, and , b_1\rangle = (\sin\frac), 0\rangle + \left(\cos\frac\right), 1\rangle. In this case, he responds with 0 if the result is , b_0\rangle, and 1 if it is , b_1\rangle. To analyze the success probability, it suffices to analyze the probability that they output a winning value pair on each of the four possible inputs (x,y), and then take the average. We analyze the case where x=y=0 here: In this case the winning response pairs are a = b = 0 and a=b=1. On input x=y=0, we know that Alice will measure in the basis , 0\rangle, , 1\rangle, and Bob will measure in the basis , a_0\rangle, , a_1\rangle. Then the probability that they both output 0 is the same as the probability that their measurements yield , 0\rangle, , a_0\rangle respectively, so precisely , (\langle 0, \otimes \langle a_0, ) , \Phi \rangle, ^2 = \frac\cos^2\left(\frac\right). Similarly, the probability that they both output 1 is exactly , (\langle 1, \otimes \langle a_1, ) , \Phi \rangle, ^2 = \frac\cos^2\left(\frac\right). So the probability that either of these successful outcomes happens is \cos^2\left(\frac\right). In the case of the 3 other possible input pairs, essentially identical analysis shows that Alice and Bob will have the same win probability of \cos^2\left(\frac\right), so overall the average win probability for a randomly chosen input is \cos^2\left(\frac\right). Since \cos^2\left(\frac\right) \approx 85\%, this is strictly better than what was possible in the classical case.


Modeling general quantum strategies

An arbitrary quantum strategy for the CHSH game can be modeled as a triple \mathcal = \left(, \psi\rangle, (A_, A_1), (B_0, B_1)\right) where * , \psi\rangle \in \mathbb^d \otimes \mathbb^d is a bipartite state for some d, * A_ and A_ are Alice's observables each corresponding to receiving x\in\ from the referee, and * B_ and B_ are Bob's observables each corresponding to receiving y\in\ from the referee. The optimal quantum strategy described above can be recast in this notation as follows: , \psi\rangle \in\mathbb^2\otimes\mathbb^2 is the EPR pair , \psi\rangle = \frac(, 00\rangle + , 11\rangle), the observable A_0 = Z (corresponding to Alice measuring in the \ basis), the observable A_1 = X (corresponding to Alice measuring in the \ basis), where X and Z are
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
. The observables B_ = \frac(X+Z) and B_ = \frac(Z-X) (corresponding to each of Bob's choice of basis to measure in). We will denote the success probability of a strategy \mathcal in the CHSH game by \omega^*_(\mathcal), and we define the ''bias'' of the strategy \mathcal as \beta^*_(\mathcal) := 2\omega^*_(\mathcal) - 1, which is the difference between the winning and losing probabilities of \mathcal. In particular, we have \beta^*_(\mathcal) = \frac \sum_ (-1)^ \cdot \langle \psi , A_x\otimes B_y , \psi\rangle. The bias of the quantum strategy described above is \frac.


Tsirelson's inequality and CHSH rigidity

Tsirelson's inequality, discovered by
Boris Tsirelson Boris Semyonovich Tsirelson (May 4, 1950 – January 21, 2020) ( he, בוריס סמיונוביץ' צירלסון, russian: Борис Семёнович Цирельсон) was a Russian–Israeli mathematician and Professor of Mathematics ...
in 1980, states that for any quantum strategy \mathcal for the CHSH game, the bias \beta^*_(\mathcal) \leq \frac. Equivalently, it states that success probability \omega^*_(\mathcal) \leq \cos^2\left(\frac\right) = \frac + \frac for any quantum strategy \mathcal for the CHSH game. In particular, this implies the optimality of the quantum strategy described above for the CHSH game. Tsirelsen's inequality establishes that the maximum success probability of ''any'' quantum strategy is \cos^2\left(\frac\right), and we saw that this maximum success probability is achieved by the quantum strategy described above. In fact, any quantum strategy that achieves this maximum success probability must be isomorphic (in a precise sense) to the canonical quantum strategy described above; this property is called the ''rigidity'' of the CHSH game, first attributed to Summers and Werner. More formally, we have the following result: Informally, the above theorem states that given an arbitrary optimal strategy for the CHSH game, there exists a local change-of-basis (given by the isometries V, W) for Alice and Bob such that their shared state , \psi\rangle factors into the tensor of an EPR pair , \Phi\rangle and an additional auxiliary state , \phi\rangle. Furthermore, Alice and Bob's observables (A_0, A_1) and (B_0, B_1) behave, up to unitary transformations, like the Z and X observables on their respective qubits from the EPR pair. An ''approximate'' or ''quantitative'' version of CHSH rigidity was obtained by McKague, et al. who proved that if you have a quantum strategy \mathcal such that \omega_(\mathcal) = \cos^2\left(\frac\right) - \epsilon for some \epsilon > 0, then there exist isometries under which the strategy \mathcal is O(\sqrt)-close to the canonical quantum strategy. Representation-theoretic proofs of approximate rigity are also known.


Applications

Note that the CHSH game can be viewed as a ''test'' for quantum entanglement and quantum measurements, and that the rigidity of the CHSH game lets us test for a ''specific'' entanglement as well as ''specific'' quantum measurements. This in turn can be leveraged to test or even verify entire quantum computations—in particular, the rigidity of CHSH games has been harnessed to construct protocols for verifiable quantum delegation, certifiable randomness expansion, and device-independent cryptography.


See also

*
Correlation does not imply causation The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. The id ...
* Leggett–Garg inequality *
Quantum game theory Quantum game theory is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways: # Superposed initial states, #Quantum entanglement of initial states, #Superposition of strategies to b ...


References

{{Quantum mechanics topics Quantum measurement Inequalities