# Bell's theorem

Bell's theorem is a theorem that shows that the predictions of quantum mechanics (QM) are counter intuitive, touching upon several fundamental philosophical issues related to modern physics. It is the most famous legacy of the late physicist John S. Bell. Bell's theorem is a no-go theorem, loosely stating that:

 “ No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. ”

## Overview Illustration of Bell test for spin 1/2 particles. Source produces spin singlet pair, one particle sent to Alice another to Bob. Each performs one of the two spin measurements.

As in the situation explored in the EPR paradox, Bell considered an experiment in which a source produces pairs of correlated particles. For example, a pair of particles with correlated spins is created; one particle is sent to Alice and the other to Bob. The experimental arrangement differs from the EPR arrangement in that a measurement is performed on both particles of a pair. On each trial, each of the observers independently chooses between various detector settings and then performs an independent measurement on the particle arriving at his position. Hence, Bell's theorem can be tested by coincidence measurements in which the correlation is measured between two independently chosen observables of the particles. (Note: although the correlated property used here is the particle's spin, it could alternatively be any correlated "quantum state" that encodes exactly one quantum bit.)

When Alice and Bob measure the spin of the particles along the same axis (but in opposite directions), they get identical results 100% of the time. But when Bob measures at orthogonal (right) angles to Alice's measurements, they get identical results only 50% of the time. In terms of mathematics, the two measurements have a correlation of 1, or perfect correlation when read the same way; when read at right angles, they have a correlation of 0; no correlation. (A correlation of −1 would indicate getting opposite results for each measurement.)

Same axis: pair 1 pair 2 pair 3 pair 4 ...n
Alice, 0°: + + ...
Bob, 180°: + + ...
Correlation: ( +1 +1 +1 +1 ...)/n = +1
(100% identical)
Orthogonal axes: pair 1 pair 2 pair 3 pair 4 ...n
Alice, 0°: + + ...
Bob, 90°: + + ...
Correlation: ( −1 +1 +1 −1 ...)/n = 0.0
(50% identical)

So far, the results can be explained by positing local hidden variables — each pair of particles may have been sent out with instructions on how to behave when measured in the two axes (either '+' or '−' for each axis). Clearly, if the source only sends out particles whose instructions are identical for each axis, then when Alice and Bob measure on the same axis, they are bound to get identical results, either (+,+) or (−,−); but (if all four possible pairings of + and − instructions are generated equally) when they measure on perpendicular axes they will see zero correlation.

Now, consider that Alice or Bob can rotate their apparatus relative to each other by any amount at any time before measuring the particles, even after the particles leave the source. If local hidden variables determine the outcome of such measurements, they must encode at the time of leaving the source a result for every possible eventual direction of measurement, not just for the results in one particular axis.

Bob begins this experiment with his apparatus rotated by 45 degrees. We call Alice's axes a and a', and Bob's rotated axes b and b'. Alice and Bob then record the directions they measured the particles in, and the results they got. At the end, they will compare and tally up their results, scoring +1 for each time they got the same result and −1 for an opposite result - except that if Alice measured in a and Bob measured in b', they will score +1 for an opposite result and −1 for the same result.

Using that scoring system, any possible combination of hidden variables would produce an expected average score of at most +0.5. (For example, see the table below, where the most correlated values of the hidden variables have an average correlation of +0.5, i.e. 75% identical. The unusual "scoring system" ensures that maximum average expected correlation is +0.5 for any possible system that relies on local hidden variables.)

Classical model: highly correlated variables less correlated variables
Hidden variable for 0° (a): + + + + + + + +
Hidden variable for 45° (b): + + + + + + + +
Hidden variable for 90° (a'): + + + + + + + +
Hidden variable for 135° (b'): + + + + + + + +
Correlation score:
If measured on a-b, score: +1 +1 +1 −1 +1 +1 +1 -1 +1 −1 −1 −1 −1 −1 −1 +1
If measured on a' − b, score: +1 +1 −1 +1 +1 +1 −1 +1 −1 −1 −1 +1 +1 −1 −1 −1
If measured on a'-b', score: +1 −1 +1 +1 +1 −1 +1 +1 −1 +1 −1 −1 −1 −1 +1 −1
If measured on a − b', score: −1 +1 +1 +1 −1 +1 +1 +1 −1 −1 +1 −1 −1 +1 −1 −1
Expected average score: +0.5 +0.5 +0.5 +0.5 +0.5 +0.5 +0.5 +0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5

Bell's Theorem shows that if the particles behave as predicted by quantum mechanics, Alice and Bob can score higher than the classical hidden variable prediction of +0.5 correlation; if the apparatuses are rotated at 45° to each other, quantum mechanics predicts that the expected average score is 0.71.

(Quantum prediction in detail: When observations at an angle of θ are made on two entangled particles, the predicted correlation is cosθ. The correlation is equal to the length of the projection of the particle's vector onto his measurement vector; by trigonometry, cosθ. θ is 45°, and cosθ is $\tfrac{\sqrt{2}}{2}$, for all pairs of axes except (a,b') – where they are 135° and $-\tfrac{\sqrt{2}}{2}$ – but this last is taken in negative in the agreed scoring system, so the overall score is $\tfrac{\sqrt{2}}{2}$; 0.707. In one explanation, the particles behave as if when Alice or Bob makes a measurement, the other particle usually switches to take that direction instantaneously.)

Multiple researchers have performed equivalent experiments using different methods. It appears most of these experiments produce results which agree with the predictions of quantum mechanics, leading to disproof of local-hidden-variable theories and proof of nonlocality. Still not everyone agrees with these findings. There have been two loopholes found in the earlier of these experiments, the detection loophole and the communication loophole with associated experiments to close these loopholes. After all current experimentation it seems these experiments uphold prima facie support for quantum mechanics' predictions of nonlocality.

## Importance of the theorem

Bell's theorem, derived in his seminal 1964 paper titled "On the Einstein Podolsky Rosen paradox" has been called "the most profound in science.". The title of the article refers to the famous paper by Einstein Podolsky and Rosen purporting to prove the incompleteness of quantum mechanics. In his paper Bell started from essentially the same assumptions as did EPR, viz. i) reality (microscopic objects have real properties determining the outcomes of quantum mechanical measurements) and ii) locality (reality is not influenced by measurements simultaneously performed at a large distance). Bell was able to derive from these assumptions an important result, viz. Bell's inequality, violation of which by quantum mechanics implying that at least one of the assumptions must be abandoned if experiment would turn out to satisfy quantum mechanics. In two respects Bell's 1964 paper was a big step forward compared to the EPR paper: i) it considered more general hidden variables than the elements of physical reality of the EPR paper, ii) more importantly, Bell's inequality was liable to be experimentally tested, thus yielding the opportunity to lift the discussion on the completeness of quantum mechanics from metaphysics to physics. Whereas Bell's 1964 paper deals only with deterministic hidden variables theories, Bell's theorem was later generalized to stochastic theories as well, and it was realized that the theorem can even be proven without introducing hidden variables.

After EPR (Einstein–Podolsky–Rosen), quantum mechanics was left in an unsatisfactory position: either it was incomplete, in the sense that it failed to account for some elements of physical reality, or it violated the principle of finite propagation speed of physical effects. In a modified version of the EPR thought experiment, two observers, now commonly referred to as Alice and Bob, perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state. It was equivalent to the conclusion of EPR that once Alice measured spin in one direction (e.g. on the x axis), Bob's measurement in that direction was determined with certainty, with opposite outcome to that of Alice, whereas immediately before Alice's measurement, Bob's outcome was only statistically determined. Thus, either the spin in each direction is an element of physical reality, or the effects travel from Alice to Bob instantly.

In QM, predictions were formulated in terms of probabilities — for example, the probability that an electron might be detected in a particular region of space, or the probability that it would have spin up or down. The idea persisted, however, that the electron in fact has a definite position and spin, and that QM's weakness was its inability to predict those values precisely. The possibility remained that some yet unknown, but more powerful theory, such as a hidden variables theory, might be able to predict those quantities exactly, while at the same time also being in complete agreement with the probabilistic answers given by QM. If a hidden variables theory were correct, the hidden variables were not described by QM, and thus QM would be an incomplete theory.

The desire for a local realist theory was based on two assumptions:

1. Objects have a definite state that determines the values of all other measurable properties, such as position and momentum.
2. Effects of local actions, such as measurements, cannot travel faster than the speed of light (as a result of special relativity). If the observers are sufficiently far apart, a measurement taken by one has no effect on the measurement taken by the other.

In the formalization of local realism used by Bell, the predictions of theory result from the application of classical probability theory to an underlying parameter space. By a simple (but clever) argument based on classical probability, he then showed that correlations between measurements are bounded in a way that is violated by QM.

Bell's theorem seemed to put an end to local realist hopes for QM. Per Bell's theorem, either quantum mechanics or local realism is wrong. Experiments were needed to determine which is correct, but it took many years and many improvements in technology to perform them.

Bell test experiments to date overwhelmingly show that Bell inequalities are violated. These results provide empirical evidence against local realism and in favor of QM. The no-communication theorem proves that the observers cannot use the inequality violations to communicate information to each other faster than the speed of light.

John Bell's paper examines both John von Neumann's 1932 proof of the incompatibility of hidden variables with QM and the seminal paper on the subject by Albert Einstein and his colleagues.

## Bell inequalities

Bell inequalities concern measurements made by observers on pairs of particles that have interacted and then separated. According to quantum mechanics they are entangled while local realism limits the correlation of subsequent measurements of the particles. Different authors subsequently derived inequalities similar to Bell´s original inequality, collectively termed Bell inequalities. All Bell inequalities describe experiments in which the predicted result assuming entanglement differs from that following from local realism. The inequalities assume that each quantum-level object has a well defined state that accounts for all its measurable properties and that distant objects do not exchange information faster than the speed of light. These well defined states are often called hidden variables, the properties that Einstein posited when he stated his famous objection to quantum mechanics: "God does not play dice."

Bell showed that under quantum mechanics, which lacks local hidden variables, the inequalities (the correlation limit) may be violated. Instead, properties of a particle are not clear to verify in quantum mechanics but may be correlated with those of another particle due to quantum entanglement, allowing their state to be well defined only after a measurement is made on either particle. That restriction agrees with the Heisenberg uncertainty principle, a fundamental and inescapable concept in quantum mechanics.

In Bell's work:

 “ Theoretical physicists live in a classical world, looking out into a quantum-mechanical world. The latter we describe only subjectively, in terms of procedures and results in our classical domain. (...) Now nobody knows just where the boundary between the classical and the quantum domain is situated. (...) More plausible to me is that we will find that there is no boundary. The wave functions would prove to be a provisional or incomplete description of the quantum-mechanical part. It is this possibility, of a homogeneous account of the world, which is for me the chief motivation of the study of the so-called "hidden variable" possibility. (...) A second motivation is connected with the statistical character of quantum-mechanical predictions. Once the incompleteness of the wave function description is suspected, it can be conjectured that random statistical fluctuations are determined by the extra "hidden" variables — "hidden" because at this stage we can only conjecture their existence and certainly cannot control them. (...) A third motivation is in the peculiar character of some quantum-mechanical predictions, which seem almost to cry out for a hidden variable interpretation. This is the famous argument of Einstein, Podolsky and Rosen. (...) We will find, in fact, that no local deterministic hidden-variable theory can reproduce all the experimental predictions of quantum mechanics. This opens the possibility of bringing the question into the experimental domain, by trying to approximate as well as possible the idealized situations in which local hidden variables and quantum mechanics cannot agree ”

In probability theory, repeated measurements of system properties can be regarded as repeated sampling of random variables. In Bell's experiment, Alice can choose a detector setting to measure either A(a) or A(a') and Bob can choose a detector setting to measure either B(b) or B(b'). Measurements of Alice and Bob may be somehow correlated with each other, but the Bell inequalities say that if the correlation stems from local random variables, there is a limit to the amount of correlation one might expect to see.

### Original Bell's inequality

The original inequality that Bell derived was: $1 + \operatorname{C}(b, c) \geq |\operatorname{C}(a, b) - \operatorname{C}(a, c)|,$

where C is the "correlation" of the particle pairs and a, b and c settings of the apparatus. This inequality is not used in practice. For one thing, it is true only for genuinely "two-outcome" systems, not for the "three-outcome" ones (with possible outcomes of zero as well as +1 and −1) encountered in real experiments. For another, it applies only to a very restricted set of hidden variable theories, namely those for which the outcomes on both sides of the experiment are always exactly anticorrelated when the analysers are parallel, in agreement with the quantum mechanical prediction.

There is a simple limit of Bell's inequality which has the virtue of being completely intuitive. If the result of three different statistical coin-flips A,B,C have the property that:

1. A and B are the same (both heads or both tails) 99% of the time
2. B and C are the same 99% of the time

then A and C are the same at least 98% of the time. The number of mismatches between A and B (1/100) plus the number of mismatches between B and C (1/100) are together the maximum possible number of mismatches between A and C.

In quantum mechanics, by letting A,B,C be the values of the spin of two entangled particles measured relative to some axis at 0 degrees, θ degrees, and 2θ degrees respectively, the overlap of the wavefunction between the different angles is proportional to $\cos(S\theta)\approx 1-S^2\theta^2/2$. The probability that A and B give the same answer is $1-\varepsilon^2$, where $\varepsilon$ is proportional to θ. This is also the probability that B and C give the same answer. But A and C are the same 1 − (2ε)2 of the time. Choosing the angle so that ε = .1, A and B are 99% correlated, B and C are 99% correlated and A and C are only 96% correlated.

Imagine that two entangled particles in a spin singlet are shot out to two distant locations, and the spins of both are measured in the direction A. The spins are 100% correlated (actually, anti-correlated but for this argument that is equivalent). The same is true if both spins are measured in directions B or C. It is safe to conclude that any hidden variables which determine the A,B, and C measurements in the two particles are 100% correlated and can be used interchangeably.

If A is measured on one particle and B on the other, the correlation between them is 99%. If B is measured on one and C on the other, the correlation is 99%. This allows us to conclude that the hidden variables determining A and B are 99% correlated and B and C are 99% correlated. But if A is measured in one particle and C in the other, the results are only 96% correlated, which is a contradiction. The intuitive formulation is due to David Mermin, while the small-angle limit is emphasized in Bell's original article.

### CHSH inequality

In addition to Bell's original inequality, the form given by John Clauser, Michael Horne, Abner Shimony and R. A. Holt, (the CHSH form) is especially important, as it gives classical limits to the expected correlation for the above experiment conducted by Alice and Bob: $\ (1) \quad \mathbf{C}[A(a), B(b)] + \mathbf{C}[A(a), B(b')] + \mathbf{C}[A(a'), B(b)] - \mathbf{C}[A(a'), B(b')]\leq 2,$

where C denotes correlation.

Correlation of observables X, Y is defined as $\mathbf{C}(X,Y) = \operatorname{E}(X Y).$

This is a non-normalized form of the correlation coefficient considered in statistics (see Quantum correlation).

In order to formulate Bell's theorem, we formalize local realism as follows:

1. There is a probability space Λ and the observed outcomes by both Alice and Bob result by random sampling of the parameter $\lambda \in \Lambda$.
2. The values observed by Alice or Bob are functions of the local detector settings and the hidden parameter only. Thus
• Value observed by Alice with detector setting a is A(a,λ)
• Value observed by Bob with detector setting b is B(b,λ)

Implicit in assumption 1) above, the hidden parameter space Λ has a probability measure ρ and the expectation of a random variable X on Λ with respect to ρ is written $\operatorname{E}(X) = \int_\Lambda X(\lambda) \rho(\lambda) d \lambda$

where for accessibility of notation we assume that the probability measure has a density.

Bell's inequality. The CHSH inequality (1) holds under the hidden variables assumptions above.

For simplicity, let us first assume the observed values are +1 or −1; we remove this assumption in Remark 1 below.

Let $\lambda \in \Lambda$. Then at least one of $B(b, \lambda) + B(b', \lambda), \quad B(b, \lambda) - B(b', \lambda)$

is 0. Thus $A(a, \lambda) \ B(b, \lambda) + A(a, \lambda) \ B(b', \lambda) +A(a', \lambda) \ B(b, \lambda) - A(a', \lambda) \ B(b', \lambda) =$ $= A(a, \lambda) (B(b, \lambda) + B(b', \lambda))+ A(a', \lambda) (B(b, \lambda) - B(b', \lambda)) \quad$ $\leq 2.$

and therefore $\mathbf{C}(A(a), B(b)) + \mathbf{C}(A(a), B(b')) + \mathbf{C}(A(a'), B(b)) - \mathbf{C}(A(a'), B(b')) =$ $= \int_\Lambda A(a, \lambda) \ B(b, \lambda) \rho(\lambda) d \lambda + \int_\Lambda A(a, \lambda) \ B(b', \lambda) \rho(\lambda) d \lambda + \int_\Lambda A(a', \lambda) \ B(b, \lambda) \rho(\lambda) d \lambda - \int_\Lambda A(a', \lambda) \ B(b', \lambda) \rho(\lambda) d \lambda =$ $= \int_\Lambda \bigg\{A(a, \lambda) \ B(b, \lambda) + A(a, \lambda) \ B(b', \lambda) +A(a', \lambda) \ B(b, \lambda) - A(a', \lambda) \ B(b', \lambda)\bigg\} \rho(\lambda) d \lambda =$ $= \int_\Lambda \bigg\{A(a, \lambda) (B(b, \lambda) + B(b', \lambda))+ A(a', \lambda) (B(b, \lambda) - B(b', \lambda)) \bigg\} \rho(\lambda) d \lambda \quad$ $\leq 2.$

Remark 1. The correlation inequality (1) still holds if the variables A(a,λ), B(b,λ) are allowed to take on any real values between −1 and +1. Indeed, the relevant idea is that each summand in the above average is bounded above by 2. This is easily seen to be true in the more general case: $A(a, \lambda) \ B(b, \lambda) + A(a, \lambda) \ B(b', \lambda) + A(a', \lambda) \ B(b, \lambda) - A(a', \lambda) \ B(b', \lambda) =$ $= A(a, \lambda) (B(b, \lambda) + B(b', \lambda)) +A(a', \lambda) (B(b, \lambda) - B(b', \lambda)) \quad$ $\leq \bigg|A(a, \lambda) (B(b, \lambda) + B(b', \lambda)) +A(a', \lambda) (B(b, \lambda) - B(b', \lambda) ) \bigg| \quad$ $\leq \bigg|A(a, \lambda) (B(b, \lambda) + B(b', \lambda))\bigg| +\bigg|A(a', \lambda) (B(b, \lambda) - B(b', \lambda))\bigg|$ $\leq |B(b, \lambda) + B(b', \lambda)| +| B(b, \lambda) - B(b', \lambda)| \leq 2.$

To justify the upper bound 2 asserted in the last inequality, without loss of generality, we can assume that $B(b, \lambda) \geq B(b', \lambda) \geq 0.$

In that case $|B(b, \lambda) + B(b', \lambda)| +| B(b, \lambda) - B(b', \lambda)| =B(b, \lambda) + B(b', \lambda) + B(b, \lambda) - B(b', \lambda) = \quad$ $= 2 B(b, \lambda) \leq 2. \quad$

Remark 2. Though the important component of the hidden parameter λ in Bell's original proof is associated with the source and is shared by Alice and Bob, there may be others that are associated with the separate detectors, these others being independent. This argument was used by Bell in 1971, and again by Clauser and Horne in 1974, to justify a generalisation of the theorem forced on them by the real experiments, in which detectors were never 100% efficient. The derivations were given in terms of the averages of the outcomes over the local detector variables. The formalisation of local realism was thus effectively changed, replacing A and B by averages and retaining the symbol λ but with a slightly different meaning. It was henceforth restricted (in most theoretical work) to mean only those components that were associated with the source.

However, with the extension proved in Remark 1, CHSH inequality still holds even if the instruments themselves contain hidden variables. In that case, averaging over the instrument hidden variables gives new variables: $\overline{A}(a, \lambda), \quad \overline{B}(b, \lambda)$

on Λ which still have values in the range [−1, +1] to which we can apply the previous result.

## Bell inequalities are violated by quantum mechanical predictions

In the usual quantum mechanical formalism, the observables X and Y are represented as self-adjoint operators on a Hilbert space. To compute the correlation, assume that X and Y are represented by matrices in a finite dimensional space and that X and Y commute; this special case suffices for our purposes below. The von Neumann measurement postulate states: a series of measurements of an observable X on a series of identical systems in state φ produces a distribution of real values. By the assumption that observables are finite matrices, this distribution is discrete. The probability of observing λ is non-zero if and only if λ is an eigenvalue of the matrix X and moreover the probability is $\|\operatorname{E}_X(\lambda) \phi\|^2$

where EX (λ) is the projector corresponding to the eigenvalue λ. The system state immediately after the measurement is $\|\operatorname{E}_X(\lambda) \phi\|^{-1} \operatorname{E}_X(\lambda) \phi.$

From this, we can show that the correlation of commuting observables X and Y in a pure state ψ is $\langle X Y \rangle = \langle X Y \psi \mid \psi \rangle.$

We apply this fact in the context of the EPR paradox. The measurements performed by Alice and Bob are spin measurements on electrons. Alice can choose between two detector settings labelled a and a′; these settings correspond to measurement of spin along the z or the x axis. Bob can choose between two detector settings labelled b and b′; these correspond to measurement of spin along the z′ or x′ axis, where the x′ – z′ coordinate system is rotated 45° relative to the xz coordinate system. The spin observables are represented by the 2 × 2 self-adjoint matrices: $S_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} , S_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}.$

These are the Pauli spin matrices normalized so that the corresponding eigenvalues are +1, −1. As is customary, we denote the eigenvectors of Sx by $\left|+x\right\rang, \quad \left|-x\right\rang.$

Let φ be the spin singlet state for a pair of electrons discussed in the EPR paradox. This is a specially constructed state described by the following vector in the tensor product $\left|\phi\right\rang = \frac{1}{\sqrt{2}} \bigg(\left|+x\right\rang \otimes \left|-x\right\rang - \left|-x\right\rang \otimes \left|+x\right\rang \bigg).$

Now let us apply the CHSH formalism to the measurements that can be performed by Alice and Bob. Illustration of Bell test for spin 1/2 particles. Source produces spin singlet pairs, one particle of each pair is sent to Alice and the other to Bob. Each performs one of the two spin measurements. $A(a) = S_z \otimes I$ $A(a') = S_x \otimes I$ $B(b) = - \frac{1}{\sqrt{2}} \ I \otimes (S_z + S_x)$ $B(b') = \frac{1}{\sqrt{2}} \ I \otimes (S_z - S_x).$

The operators B(b'), B(b) correspond to Bob's spin measurements along x′ and z′. Note that the A operators commute with the B operators, so we can apply our calculation for the correlation. In this case, we can show that the CHSH inequality fails. In fact, a straightforward calculation shows that $\langle A(a) B(b) \rangle = \langle A(a') B(b) \rangle =\langle A(a') B(b') \rangle = \frac{1}{\sqrt{2}},$

and $\langle A(a) B(b') \rangle = - \frac{1}{\sqrt{2}}.$

so that $\langle A(a) B(b) \rangle + \langle A(a') B(b') \rangle + \langle A(a') B(b) \rangle - \langle A(a) B(b') \rangle = \frac{4}{\sqrt{2}} = 2 \sqrt{2} > 2.$

Bell's Theorem: If the quantum mechanical formalism is correct, then the system consisting of a pair of entangled electrons cannot satisfy the principle of local realism. Note that $2 \sqrt{2}$ is indeed the upper bound for quantum mechanics called Tsirelson's bound. The operators giving this maximal value are always isomorphic to the Pauli matrices.

## Practical experiments testing Bell's theorem Scheme 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 (a or b) can be set by the experimenter. Emerging signals from each channel are detected and coincidences of four types (++, −−, +− and −+) counted by the coincidence monitor.

Experimental tests can determine whether the Bell inequalities required by local realism hold up to the empirical evidence.

Bell's inequalities are tested by "coincidence counts" from a Bell test experiment such as the optical one shown in the diagram. Pairs of particles are emitted as a result of a quantum process, analysed with respect to some key property such as polarisation direction, then detected. The setting (orientations) of the analysers are selected by the experimenter.

Bell test experiments to date overwhelmingly violate Bell's inequality. Indeed, a table of Bell test experiments performed prior to 1986 is given in 4.5 of Redhead, 1987. Of the thirteen experiments listed, only two reached results contradictory to quantum mechanics; moreover, according to the same source, when the experiments were repeated, "the discrepancies with QM could not be reproduced".

Nevertheless, the issue is not conclusively settled. According to Shimony's 2004 Stanford Encyclopedia overview article:

Most of the dozens of experiments performed so far have favored Quantum Mechanics, but not decisively because of the 'detection loopholes' or the 'communication loophole.' The latter has been nearly decisively blocked by a recent experiment and there is a good prospect for blocking the former.

To explore the 'detection loophole', one must distinguish the classes of homogeneous and inhomogeneous Bell inequality.

The standard assumption in Quantum Optics is that "all photons of given frequency, direction and polarization are identical" so that photodetectors treat all incident photons on an equal basis. Such a fair sampling assumption generally goes unacknowledged, yet it effectively limits the range of local theories to those which conceive of the light field as corpuscular. The assumption excludes a large family of local realist theories, in particular, Max Planck's description. We must remember the cautionary words of Albert Einstein shortly before he died: "Nowadays every Tom, Dick and Harry ('jeder Kerl' in German original) thinks he knows what a photon is, but he is mistaken".

Objective physical properties for Bell’s analysis (local realist theories) include the wave amplitude of a light signal. Those who maintain the concept of duality, or simply of light being a wave, recognize the possibility or actuality that the emitted atomic light signals have a range of amplitudes and, furthermore, that the amplitudes are modified when the signal passes through analyzing devices such as polarizers and beam splitters. It follows that not all signals have the same detection probability.

### Two classes of Bell inequalities

The fair sampling problem was faced openly in the 1970s. In early designs of their 1973 experiment, Freedman and Clauser used fair sampling in the form of the Clauser-Horne-Shimony-Holt (CHSH) hypothesis. However, shortly afterwards Clauser and Horne made the important distinction between inhomogeneous (IBI) and homogeneous (HBI) Bell inequalities. Testing an IBI requires that we compare certain coincidence rates in two separated detectors with the singles rates of the two detectors. Nobody needed to perform the experiment, because singles rates with all detectors in the 1970s were at least ten times all the coincidence rates. So, taking into account this low detector efficiency, the QM prediction actually satisfied the IBI. To arrive at an experimental design in which the QM prediction violates IBI we require detectors whose efficiency exceeds 82% for singlet states, but have very low dark rate and short dead and resolving times. This is well above the 30% achievable so Shimony’s optimism in the Stanford Encyclopedia, quoted in the preceding section, appears over-stated.

### Practical challenges

Because detectors don't detect a large fraction of all photons, Clauser and Horne recognized that testing Bell's inequality requires some extra assumptions. They introduced the No Enhancement Hypothesis (NEH):

a light signal, originating in an atomic cascade for example, has a certain probability of activating a detector. Then, if a polarizer is interposed between the cascade and the detector, the detection probability cannot increase.

Given this assumption, there is a Bell inequality between the coincidence rates with polarizers and coincidence rates without polarizers.

The experiment was performed by Freedman and Clauser, who found that the Bell's inequality was violated. So the no-enhancement hypothesis cannot be true in a local hidden variables model. The Freedman-Clauser experiment reveals that local hidden variables imply the new phenomenon of signal enhancement:

In the total set of signals from an atomic cascade there is a subset whose detection probability increases as a result of passing through a linear polarizer.

This is perhaps not surprising, as it is known that adding noise to data can, in the presence of a threshold, help reveal hidden signals (this property is known as stochastic resonance). One cannot conclude that this is the only local-realist alternative to Quantum Optics, but it does show that the word loophole is biased. Moreover, the analysis leads us to recognize that the Bell-inequality experiments, rather than showing a breakdown of realism or locality, are capable of revealing important new phenomena.

## Theoretical challenges

Some advocates of the hidden variables idea believe that experiments have ruled out local hidden variables. They are ready to give up locality, explaining the violation of Bell's inequality by means of a "non-local" hidden variable theory, in which the particles exchange information about their states. This is the basis of the Bohm interpretation of quantum mechanics, which requires that all particles in the universe be able to instantaneously exchange information with all others. A recent experiment ruled out a large class of non-Bohmian "non-local" hidden variable theories.

If the hidden variables can communicate with each other faster than light, Bell's inequality can easily be violated. Once one particle is measured, it can communicate the necessary correlations to the other particle. Since in relativity the notion of simultaneity is not absolute, this is unattractive. One idea is to replace instantaneous communication with a process which travels backwards in time along the past light cone. This is the idea behind a transactional interpretation of quantum mechanics, which interprets the statistical emergence of a quantum history as a gradual coming to agreement between histories that go both forward and backward in time.

Recent controversial work by Joy Christian claims that a deterministic, arguably local, and realistic theory can violate Bell's inequalities if the observables are chosen to be non-commuting numbers rather than commuting numbers as Bell had assumed. Christian claims that in this way the statistical predictions of quantum mechanics can be exactly reproduced. The controversy around his work concerns his noncommutative averaging procedure, in which the averages of products of variables at distant sites depend on the order in which they appear in an averaging integral. To many, this looks like nonlocal correlations, although Christian defines locality so that this type of thing is allowed. In his work, Christian builds up a CM view of the Bell's experiment that respects the rotational entanglement of physical reality, which is included in the QM view by construction, as this property of reality manifests itself clearly in the spin of particles, but it is not usually taken into account in the classical realm. Upon building this classical view, Christian suggests that in essence, it is this property of reality that results in the increased values of Bell's inequalities and as a result a local, realistic theory can be constructed. Moreover, Christian suggests a completely macro-object experiment, consisting of thousands of metal spheres, could recreate the results of the usual experiments.

The quantum mechanical wavefunction can also provide a local realistic description, if the wavefunction values are interpreted as the fundamental quantities that describe reality. Such an approach is called a many-worlds interpretation of quantum mechanics. In this controversial view, two distant observers both split into superpositions when measuring a spin. The Bell inequality violations are no longer counterintuitive, because it is not clear which copy of the observer B observer A will see when going to compare notes. If reality includes all the different outcomes, locality in physical space (not outcome space) places no restrictions on how the split observers can meet up.

This implies that there is a subtle assumption in the argument that realism is incompatible with quantum mechanics and locality. The assumption, in its weakest form, is called counterfactual definiteness. This states that if the result of an experiment are always observed to be definite, there is a quantity which determines what the outcome would have been even if you don't do the experiment.

Many worlds interpretations are not only counterfactually indefinite, they are factually indefinite. The results of all experiments, even ones that have been performed, are not uniquely determined.

## Final remarks

The phenomenon of quantum entanglement that is behind violation of Bell's inequality is just one element of quantum physics which cannot be represented by any classical picture of physics; other non-classical elements are complementarity and wavefunction collapse. The problem of interpretation of quantum mechanics is intended to provide a satisfactory picture of these non-classical elements of quantum physics.

The EPR paper "pinpointed" the unusual properties of the entangled states, e.g. the above-mentioned singlet state, which is the foundation for present-day applications of quantum physics, such as quantum cryptography. This strange non-locality was originally supposed to be a Reductio ad absurdum, because the standard interpretation could easily do away with action-at-a-distance by simply assigning to each particle definite spin-states. Bell's theorem showed that the "entangledness" prediction of quantum mechanics have a degree of non-locality that cannot be explained away by any local theory.

In well-defined Bell experiments (see the paragraph on "test experiments") one can now falsify either quantum mechanics or Einstein's quasi-classical assumptions : presently many experiments of this kind have been performed, and the experimental results support quantum mechanics, though some believe that detectors give a biased sample of photons, so that until nearly every photon pair generated is observed there will be loopholes.

What is powerful about Bell's theorem is that it doesn't come from any particular physical theory. What makes Bell's theorem unique and has marked it as one of the most important advances in science is that it relies only on the general properties of quantum mechanics. No physical theory which assumes a deterministic variable inside the particle that determines the outcome, can account for the experimental results, only assuming that this variable cannot acausally change other variables far away.

The following are intended for general audiences.

• Amir D. Aczel, Entanglement: The greatest mystery in physics (Four Walls Eight Windows, New York, 2001).
• A. Afriat and F. Selleri, The Einstein, Podolsky and Rosen Paradox (Plenum Press, New York and London, 1999)
• J. Baggott, The Meaning of Quantum Theory (Oxford University Press, 1992)
• N. David Mermin, "Is the moon there when nobody looks? Reality and the quantum theory", in Physics Today, April 1985, pp. 38–47.
• Louisa Gilder, The Age of Entanglement: When Quantum Physics Was Reborn (New York: Alfred A. Knopf, 2008)
• Brian Greene, The Fabric of the Cosmos (Vintage, 2004, ISBN 0-375-72720-5)
• Nick Herbert, Quantum Reality: Beyond the New Physics (Anchor, 1987, ISBN 0-385-23569-0)
• D. Wick, The infamous boundary: seven decades of controversy in quantum physics (Birkhauser, Boston 1995)
• R. Anton Wilson, Prometheus Rising (New Falcon Publications, 1997, ISBN 1-56184-056-4)
• Gary Zukav "The Dancing Wu Li Masters" (Perennial Classics, 2001, ISBN 0-06-095968-1)