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Wikipedia / CHSH inequality

In physics, the Clauser–Horne–Shimony–Holt (CHSH) inequality can be used in the proof of Bell's theorem, which states that certain consequences of entanglement in quantum mechanics cannot be reproduced by local hidden-variable theories. Experimental verification of the inequality being violated is seen as confirmation that nature cannot be described by such 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 Stewart Bell's original inequality, is a constraint—on the statistical occurrence of "coincidences" in a Bell test—which is necessarily true if an underlying local hidden-variable theory exists. In practice, the inequality is routinely violated by modern experiments in quantum mechanics.

Statement

The usual form of the CHSH inequality is where

etc. are the quantum correlations 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.

(Tsirelson's bound) and can be obtained from a maximal entangled Bell state.

Experiments

Many Bell tests conducted subsequent to Alain 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.

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.

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.

The CHSH inequality has been violated with photon pairs, beryllium ion pairs, ytterbium ion pairs, rubidium atom pairs, whole rubidium-atom cloud pairs, nitrogen vacancies in diamonds, and Josephson phase qubits.

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.

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 is itself a probability.

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,

Taking absolute values of both sides, and applying the triangle inequality to the right-hand side, we obtain

By ( which, using the fact that the integral of .

(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.

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. There is a necessary and sufficient condition for a two-qubit density matrix to violate the CHSH inequality, expressed by the maximum attainable polynomial Smax defined in . This is important in entanglement-based quantum key distribution, where the secret key rate depends on the degree of measurement correlations.

are the Pauli matrices. Then we find the eigenvalues and eigenvectors. Then, the maximal CHSH polynomial is determined by the two greatest eigenvalues,

Optimal measurement bases

There exists an optimal configuration of the measurement bases a, a', b, b that yields Smax with at least one free parameter.

and the Pauli vector. Then, the expected correlation in bases a, b The numerical values of the basis vectors, when found, can be directly translated to the configuration of the projective measurements.

to obtain the bases that maximize

In entanglement-based quantum key distribution 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 The CHSH polynomial S.

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.

, where ∧ denotes a logical AND operation and ⊕ denotes a logical XOR operation. If this equality holds, then Alice and Bob win, and if not then they lose.

, 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, 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

, commonly referred to as an EPR pair. Alice and Bob will use this entangled pair in their strategy as described below. The optimality of this strategy then follows from Tsirelson's bound.

otherwise.

.

The following table shows how the game is played. The states are arranged in the order that puts each state between the two most similar. They could correspond, for example, to photons polarized at angles of 0°, 22.5°, 45°, ... 180° (with 180° and 0° being the same state)..

, this is strictly better than what was possible in the classical case.

Modeling general quantum strategies

where

  • ,

  • are Alice's observables from the referee, and

  • from the referee.

are Pauli matrices, and we define the bias.

.

Tsirelson's inequality and CHSH rigidity

Tsirelson's inequality, discovered by Boris Tsirelson in 1980, for the CHSH game. In particular, this implies the optimality of the quantum strategy described above for the CHSH game.

Tsirelson's inequality establishes that the maximum success probability of any, 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:

observables on their respective qubits from the EPR pair. An approximate or quantitative version of CHSH rigidity was obtained by McKague, et al.-close to the canonical quantum strategy. Representation-theoretic proofs of approximate rigidity 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

References