Introduction

“Bell inequalities” are designed to form the basis of tests for the non-local “quantum entanglement” phenomenon predicted by quantum mechanics (QM) as against “local hidden variable” theories. They place a limit on the correlation, as measured by coincidence counts, between particles that have taken part in an interaction and then separated. The limit can be exceeded if quantum mechanics is correct but not if local realism is obeyed. Over the years, several different ones have been derived, most never used in real experiments. As discussed at length in a seminal report by Clauser and Shimony in 1978[1], they are by no means all equivalent to John Bell’s original (1964[2]) inequality.

The question discussed in the present paper is whether the “best” version is being used in recent “Bell test” experiments. It is suggested here that the choice is being made in ignorance of some of the facts. As a result largely of historical accident, it is not fully realised that the “single channel” experiments that were conducted for the first decade (the 1970s) possess a distinct advantage over the “two-channel” ones that are currently in vogue. They do not depend on the assumption of fair sampling. They are immune to what is variously known as the “fair sampling”, “detection efficiency” or “variable detection probability” “loophole” – a loophole that, as is now well known amongst experts, enables local realist explanations to match the experimental results in optical two-channel tests[3]^,[4]. They also, in point of fact, have other advantages, making them the logical choice for use in optical tests.

The Clauser-Horne-Shimony-Holt[5] (CHSH) two-channel test is briefly discussed, then two different derivations of the single-channel inequality. It emerges that the second of these, that given by Clauser and Horne in their 1974 paper[6], illuminates the true assumptions required for what is here termed the CH74 test, showing that it is considerably more general than was at first thought. The CHSH inequality can be derived from the CH74 one by adding the fair sampling assumption. The “no enhancement” assumption required for CH74 is not needed for CHSH, but is this as important?

A historical note follows, attempting to rationalise the evolution of the theory and practice of Bell test experiments. The physics community at large has not always had access to the information needed for an informed decision. Marshall, Santos and Selleri[7], the leading local realists at the time of Aspect’s famous experiments, argued that there must in fact be some “enhancement”, but I maintain that they were almost certainly wrong, acting on wrong information.

The CHSH Bell inequality

Bell himself had originally thought not of optical tests but of experiments using ions, which were to fly apart with opposite “spins” and each be detected as either “up” or “down”. This had led him to consider two-channel experiments, where the “outcome” of a measurement could be +1 or –1. When it came to testing his idea, though, it was found that ions were not practical. Real “Bell test experiments” have almost all been performed using light, treating it as particle-like photons. Experimenters and theorists had to modify his idea, allowing for the fact that detection was never going to be perfect.

Fig. 1: Scheme of a two-channel Bell test experiment.

Photons from a source S pass through polarisers set at angles a and b, and, in the QM picture, exit either by the ‘+’ or by the ‘–’ channel, to be detected by photomultipliers D+ or D–. If the signals from the two sides reach the coincidence monitor CM within a preset time window, they are registered as a coincidence.

In 1969 the CHSH modification of Bell’s original inequality was published. It applied to two-channel experiments (fig. 1), but the derivation given in 1969 was restricted to the perfect case, with all outcomes ±1, never zero. It seems (see the 1978 report) that even after Bell’s extension of it in 1971 to cover zero outcomes, it was not considered usable unless there were “event-ready” detectors to determine N, the number of pairs reaching the analysers (polarisers). The CHSH test was to remain unused for the next decade. Partly for practical reasons, all experiments in the 1970s used single channels, applying the test in fact recommended in the 1969 paper: the one I term the CH74 test. Aspect in 1982 was the first to apply the CHSH inequality, using it in the second of his three well-known experiments[8].

The CHSH inequality as used by Aspect in 1982 and the majority of subsequent Bell test experiments[9] is:

–2 £ S £ 2, (1)

where

S = E(a, b) – E(a, b¢ ) + E(a¢, b¢) + E(a¢, b¢ ) (2)

and, for each choice a = a or a¢ and b = b or b¢

. (3)

The terms E(a, b) are estimates of the quantum correlation between the two sides (the expected value of the product of the outcomes) for detector settings a and b respectively, a separate set of experimental runs being required for each term. N₊₊, N_+– etc. are coincidence counts. In actual experiments the values of a, a¢, b and b¢ are chosen to be the “Bell test angles”, maximizing the predicted QM violation of (1).

The derivation given in 1969 is confusing, and possibly not even correct. Bell in 1971 tidied it up, dropping his original (quantum-mechanical) assumption that parallel detectors would always produce exactly opposite results and extending it to cover functions A(a, l) and B(b, l) that can represent outcomes of zero as well as ±1 (l is the assumed hidden variable set at the source). It is Bell’s version that is reproduced in Clauser and Shimony’s 1978 report. Neither in 1969 nor in 1971 does the description of the method continue as far as expression (3) above. All concerned were, at this stage in the evolution of the Bell tests, agreed that the inequality should be used only if N were known. Presumably the intention was that, instead of (3) above, the expression

(3¢)

should be used, each of the terms N₊₊/N etc. representing the probability of a given coincidence. The report moves hastily on to discuss the single-channel (CH74) inequality that can be applied in practical situations.

The CH74 inequality

The CH74 (single-channel) inequality, as used in, for example, two of Aspect’s experiments[10], is:

(4)

where

F(a, a¢, b, b¢ ) = N(a, b) – N(a, b¢ ) + N(a¢, b) + N(a¢, b¢ ) – N(a¢, ¥) – N(¥, b),

the symbol ¥ denoting absence of a polariser. Only ‘+’ outcomes are measured (fig. 2) so no suffices are needed on the N’s. Runs conducted with polariser absent on either one or both sides compensate for the information lost by not measuring both ‘+’ and ‘–’ outcomes, though, as will be seen later, the compensation need not be exact (equalities (5) below are not actually necessary).

Clauser et al.’s 1969 derivation

In their 1969 paper, Clauser, Horne, Shimony and Holt give their first derivation of the “CH74” inequality. Aspect in a recent paper[11] presents what is essentially the 1969 derivation only using different notation. The inequality is derived from the CHSH one and requires all the assumptions of that derivation and a rather stronger version of the “fair sampling” assumption[12]. The latter is embodied in the following equations, which are used to replace unmeasured quantities by measured ones in expression (3):

Fig. 2: Setup for “single-channel” Bell test

“Pile of plates” polarisers or the equivalent, with only one output, are used instead of the polarising cube of a two-channel test.

N₊₊(∞,∞) = N₊₊(a, b) + N_+–(a, b) + N_--(a, b) + N_–+(a, b)

N₊₊(a, ∞) = N₊₊(a, b) + N_+–(a, b) (5)

N₊₊(∞, b) = N₊₊(a, b) + N_–+(a, b)

It seems possible that the significance of Clauser and Horne’s later (1974) derivation, working from first principles and with no mention of expression (3), is not generally recognised. It is, in any event, not the one quoted by Aspect. He implies in the above-mentioned paper that the CH74 test is inferior on two counts. Firstly, it uses a “simplified experimental scheme, somewhat different from the ideal one”, and secondly “the probabilities involved in the expression of E(a, b) must be redefined on the ensemble of pairs that would be detected with polarisers removed”. The validity of this procedure requires, he says, Clauser et al.’s 1969 assumption:

“Given that a pair of photons emerges from the polarisers, the probability of their joint detection is independent of the polariser orientations” (or, Aspect adds, of their removal).

Aspect does mention the existence of another proof. He says at the end of section 8.1: “Clauser and Horne [1974]have exhibited another assumption, leading to the same inequalities. The status of these assumptions has been thoroughly discussed in [Clauser and Shimony’s 1978 report].” He makes no mention (and one must presume is unaware) of the fact, stated though perhaps with insufficient emphasis in the 1978 report, that the only assumption actually needed is that of “no enhancement”. As will be seen below, neither knowledge of N nor acceptance of the “fair sampling” assumptions, devised to circumvent the problem when N is not known, are in fact needed.

It is noteworthy that in the 1978 report, though the existence of the 1969 derivation is mentioned (page 1894) it is not discussed.

Clauser and Horne’s 1974 derivation

Now let us turn to Clauser and Horne’s own later exposition. In 1974 they make a complete break with the tradition of dealing with “quantum correlations” and “outcome” functions taking values ±1. They assume that the hidden variable l set at the source determines, in conjunction with the detector setting, the probability of detection, not the actual individual outcome. They refer to their theory as an “Objective Local Theory” (OLT) rather than a “hidden variable” one[13].

I shall quote in full (apart from the few lines relating to the rotationally invariant case) the derivation of an “inhomogenous” version of the test, then briefly outline the practical (homogeneous) version, which follows a similar method. (The inhomogeneous test is impractical since it involves comparing coincidence rates with singles rates, which are, for low efficiencies, much greater.)

The inhomogeneous CH74 inequality

Starting on page 528 we find the following (in their own words apart from equation numbers):

... in this section, we derive a consequence of [the factorability assumption] which is experimentally testable without N being known, and which contradicts the quantum-mechanical predictions.

Let a and a¢ be two orientations of analyser 1, and let b and b¢ be two orientations of analyser 2.

The inequalities

0 £ p₁(l, a) £ 1

0 £ p₁(l, a¢ ) £ 1 (CH1)

0 £ p₂(l, b) £ 1

0 £ p₂(l, b¢ ) £ 1

hold if the probabilities are sensible. These inequalities and the theorem [see below, from Appendix A of original paper] give

– 1 £ p₁(l, a) p₂(l, b) – p₁(l, a) p₂(l, b¢ ) + p₁(l, a¢ ) p₂(l, b)

+ p₁(l, a¢ ) p₂(l, b¢ ) – p₁(l, a¢ ) – p₂(l, b) £ 0 (CH2)

for each l. Multiplication by r(l) [the probability of the source being in state l] and integration over l gives [assuming factorability]

– 1 £ p₁₂(a, b) – p₁₂(a, b¢ ) + p₁₂(a¢, b) + p₁₂(a¢, b¢ ) – p₁(a¢ ) – p₂(b) £ 0 (CH3)

as a necessary constraint on the statistical predictions of any OLT [Objective Local Theory] …

… The upper limits in (CH3) are experimentally testable without N being known. Inequality (CH3) holds perfectly generally for any systems described by OLT. These are new results not previously presented elsewhere. [My italics]

[End of quoted text.]

The CH74 inequality as used in practice

On page 530, we find the derivation of an inequality of similar structure that can be used with real, low-efficiency, detectors. It employs an assumption rather stronger than (CH1). This is the “no enhancement” assumption, which can be expressed mathematically in the form:

0 £ p₁(l, a) £ p₁(l, ¥) £ 1,

0 £ p₂(l, b) £ p₂(l, ¥) £ 1 (CH5)

where ¥ denotes absence of the polariser, p₁(l, ¥) the probability of a count from detector 1 when the polariser is absent and the emission is in state l, and p₂(l, ¥) likewise for detector 2.

Using the same arguments as before, we find (CH3) replaced by:

p₁₂(¥, ¥) £ p₁₂(a, b) – p₁₂(a, b¢ )

+ p₁₂(a