CHSH inequality

In physics, the Clauser–Horne–Shimony–Holt '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
and are detector settings on side, and on side, the four combinations being tested in separate subexperiments. The terms 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, where 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 that the value of exceeds 2 for systems prepared in suitable entangled states and the appropriate choice of measurement settings. The maximum violation predicted by quantum mechanics is 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 and assuming fair sampling. Some dramatic violations of the inequality have been reported.Image:Two channel bell test.svg|300px|thumb|right|Schematic of a "two-channel" Bell test
The source S produces pairs of photons, sent in opposite directions. Each photon encounters a two-channel polariser whose orientation can be set by the experimenter. Emerging signals from each channel are detected and coincidences counted by the coincidence monitor CM.
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 are recorded, the results being categorised as '++', '+−', '−+' or '−−' and corresponding counts accumulated.
Four separate subexperiments are conducted, corresponding to the four terms in the test statistic S. 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, the numbers of coincidences in each category are recorded. The experimental estimate for 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 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.
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, the integral of which over the complete hidden variable space is 1. We thus assume we can write:
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
We use the fact that and 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 is 1, is equal to
which is equal to.
Putting this together with the left-hand side, we have:
which means that the left-hand side is less than or equal to both and. That is:
from which we obtain
, 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.

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 S_max defined in. This is important in entanglement-based quantum key distribution, where the secret key rate depends on the degree of measurement correlations.
Let us introduce a 3×3 real matrix with elements, where are the Pauli matrices. Then we find the eigenvalues and eigenvectors of the real symmetric matrix,
where the indices are sorted by. 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' for a given that yields S_max with at least one free parameter.
The projective measurement that yields either +1 or −1 for two orthogonal states respectively, can be expressed by an operator. The choice of this measurement basis can be parametrized by a real unit vector and the Pauli vector by expressing. Then, the expected correlation in bases a, b is
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 is found by taking the two greatest eigenvalues and the corresponding eigenvectors of, and finding the auxiliary unit vectors
where is a free parameter. We also calculate the acute angle
to obtain the bases that maximize,
In entanglement-based quantum key distribution, there is another measurement basis used to communicate the secret key. The bases then need to minimize the quantum bit error rate Q, which is the probability of obtaining different measurement outcomes. The corresponding bases are
The CHSH polynomial S needs to be maximized as well, which together with the bases above creates the constraint.