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.

CHSH stands for John Clauser, Michael Horne, Abner Shimony, and Richard Holt, who described it in a much-cited paper published in 1969.

In practice, the inequality is routinely violated by modern experiments in quantum mechanics.

Clauser et al.'s 1969[1] 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.

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.

exceeds 2 for systems prepared in suitable entangled states and the appropriate choice of measurement settings (see below).

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.

[7]In practice most actual experiments have used light rather than the electrons that Bell originally had in mind.

Coincidences (simultaneous detections) are recorded, the results being categorised as '++', '+−', '−+' or '−−' and corresponding counts accumulated.

The settings a = 0°, a′ = 45°, b = 22.5°, and b′ = 67.5° 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.

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.

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 following is based on page 37 of Bell's Speakable and Unspeakable,[4] the main change being to use the symbol ‘E’ instead of ‘P’ for the expected value of the quantum correlation.

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.

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.

to violate the CHSH inequality, expressed by the maximum attainable polynomial Smax defined in Eq.

The numerical values of the basis vectors, when found, can be directly translated to the configuration of the projective measurements.

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).

[16] 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.

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.

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%.

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.

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

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.

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

An approximate or quantitative version of CHSH rigidity was obtained by McKague, et al.[19] who proved that if you have a quantum strategy

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,[21][22] certifiable randomness expansion,[23] and device-independent cryptography.