CHSH inequality
In physics, the CHSH inequality can be used in the proof of Bell's theorem, which states that certain consequences of entanglement in quantum mechanics can not be reproduced by local hidden variable theories. Experimental verification of violation of the inequalities is seen as experimental confirmation that nature cannot be described by local hidden variables theories. CHSH stands for John Clauser, Michael Horne, Abner Shimony, and Richard Holt, who described it in a much-cited paper published in 1969. They derived the CHSH inequality, which, as with John Bell's original inequality, is a constraint on the  statistics of "coincidences" in a Bell test which is necessarily true if there exist underlying local hidden variables (local realism). This constraint can, on the other hand, be infringed by 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 correlations 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 proved 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 Aspect'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 formula_1 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 formula_2 are recorded. The experimental estimate for formula_1 is then calculated as: Once all the 's have been estimated, an experimental estimate of "S" () 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. 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 implication that the quantum correlation 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: where "A" and "B" are the average values of 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, Taking absolute values of both sides, and applying the triangle inequality to the right-hand side, we obtain We use the fact that formula_7 and formula_8 are both non-negative to rewrite the right-hand side of this as By (), this must be less than or equal to which, using the fact that the integral of ρ(λ) is  1, is equal to which is equal to formula_12. Putting this together with the left-hand side, we have: which means that the left-hand side is less than or equal to both formula_14 and formula_15. That is: from which we obtain (by the triangle inequality 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: 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. 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 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 formula_19 respectively. At the start of the game, Charlie chooses bits formula_20 uniformly at random , and then sends formula_21 to Alice and formula_22 to Bob. Alice and Bob must then each respond to Charlie with bits formula_23 respectively. Now, once Alice and Bob send their responses back to Charlie, Charlie tests if formula_24. 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 formula_25 depends only on formula_21, and similarly for Bob. This means that Alice and Bob are forbidden from directly communicating with each another 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" classical strategy has success probability at most 75% (where the probability is taken over Charlie's uniformly random choice of formula_27). By a deterministic strategy, we mean a pair of functions formula_28, where formula_29 is a function determining Alice's response as a function of the message she receives from Charlie, and formula_30 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 formula_31). It can be verified that for each of those 8 strategies there is always at least one out of the four possible input pairs formula_32 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 formula_33, 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: formula_34. This state is commonly referred to as the EPR pair, and can be equivalently written as formula_35. 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 formula_21 from Charlie, if formula_37 she will measure her qubit in the basis formula_38, and then respond with 0 if the measurement outcome is formula_39, and 1 if it is formula_40. Otherwise, if formula_41 she will measure her qubit in  the basis formula_42, and respond with 0 if the measurement outcome is formula_43, and 1 if it is formula_44. When Bob receives his bit formula_22 from Charlie, if formula_46 he will measure his qubit in the basis formula_47 where formula_48, and formula_49. He then responds with 0 if the result is formula_50, and 1 if it is formula_51. Otherwise, if formula_52, he will measure his qubit in the basis formula_53 where formula_54, and formula_55. In this case, he responds with 0 if the result is formula_56, and 1 if it is formula_57. 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 formula_58, and then take the average. We analyze the case where formula_59 here: In this case the winning response pairs are formula_60 and formula_61. On input formula_59, we know that Alice will measure in the basis formula_38, and Bob will measure in the basis formula_47. Then the probability that they both output 0 is the same as the probability that their measurements yield formula_65 respectively, so precisely formula_66. Similarly, the probability that they both output 1 is exactly formula_67. So the probability that  either of these successful outcomes happens is formula_68. 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 formula_68, so overall the average win probability for a randomly chosen input is formula_68. Since formula_71, 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 formula_72 where The optimal quantum strategy described above can be recast in this notation as follows: formula_81 is the EPR pair  formula_82, the observable formula_83 (corresponding to Alice measuring in the formula_84 basis), the observable formula_85 (corresponding to Alice measuring in the formula_86 basis), where formula_87 and formula_88 are Pauli matrices. The observables formula_89 and formula_90 (corresponding to each of Bob's choice of basis to measure in). We will denote the success probability of a strategy formula_91 in the CHSH game by formula_92, and we define the "bias" of the strategy formula_91 as formula_94, which is the difference between the winning and losing probabilities of formula_91. In particular, we have formula_96 The bias of the quantum strategy described above is  formula_97. Tsirelson's inequality and CHSH rigidity. Tsirelson's inequality, discovered by Boris Tsirelson in 1980, states that for any quantum strategy formula_91 for the CHSH game, the bias formula_99. Equivalently, it states that success probability formula_100 for any quantum strategy formula_91 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 formula_68, 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: Theorem (Exact CHSH rigidity). Let formula_103 be a quantum strategy for the CHSH game where formula_104 such that formula_105. Then there exist isometries formula_106 and formula_107 where formula_108 are isomorphic to formula_109 such that letting formula_110 we have formula_111 where formula_112 denotes the EPR pair and formula_113 denotes some pure state , and formula_114 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 formula_115) for Alice and Bob such that their shared state formula_116 factors into the tensor of an EPR pair formula_117 and an additional auxiliary state formula_118. Furthermore, Alice and Bob's observables formula_119 and formula_120 behave, up to unitary transformations, like the formula_88 and formula_87 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 formula_91 such that  formula_124 for some formula_125, then there exist isometries under which the strategy formula_91 is formula_127-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.