Bell's theorem

The inequality that Bell derived can be written as:[2]

${\displaystyle \mid C_{h}(a,b)-C_{h}(a,c)\mid \leq 1+C_{h}(b,c),}$

where a, b and c refer to three arbitrary settings of the two analysers. This inequality is however restricted in its application to the rather special case in which the outcomes on both sides of the experiment are always exactly anticorrelated whenever the analysers are parallel. The advantage of restricting attention to this special case is the resulting simplicity of the derivation. In experimental work, the inequality is not very useful because it is hard, if not impossible, to create perfect anti-correlation.

This simple form has an intuitive explanation, however. It is equivalent to the following elementary result from probability theory. Consider three (highly correlated, and possibly biased) coin-flips X, Y, and Z, with the property that:

1. X and Y give the same outcome (both heads or both tails) 99% of the time
2. Y and Z also give the same outcome 99% of the time,

then X and Z must also yield the same outcome at least 98% of the time. The number of mismatches between X and Y (1/100) plus the number of mismatches between Y and Z (1/100) are together the maximum possible number of mismatches between X and Z (a simple Boole–Fréchet inequality).

Imagine a pair of particles that can be measured at distant locations. Suppose that the measurement devices have settings, which are angles—e.g., the devices measure something called spin in some direction. The experimenter chooses the directions, one for each particle, separately. Suppose the measurement outcome is binary (e.g., spin up, spin down). Suppose the two particles are perfectly anti-correlated—in the sense that whenever both measured in the same direction, one gets identically opposite outcomes, when both measured in opposite directions they always give the same outcome. The only way to imagine how this works is that both particles leave their common source with, somehow, the outcomes they will deliver when measured in any possible direction. (How else could particle 1 know how to deliver the same answer as particle 2 when measured in the same direction? They don't know in advance how they are going to be measured...). The measurement on particle 2 (after switching its sign) can be thought of as telling us what the same measurement on particle 1 would have given.

Start with one setting exactly opposite to the other. All the pairs of particles give the same outcome (each pair is either both spin up or both spin down). Now shift Alice's setting by one degree relative to Bob's. They are now one degree off being exactly opposite to one another. A small fraction of the pairs, say f, now give different outcomes. If instead we had left Alice's setting unchanged but shifted Bob's by one degree (in the opposite direction), then again a fraction f of the pairs of particles turns out to give different outcomes. Finally consider what happens when both shifts are implemented at the same time: the two settings are now exactly two degrees away from being opposite to one another. By the mismatch argument, the chance of a mismatch at two degrees can't be more than twice the chance of a mismatch at one degree: it cannot be more than 2f.

Compare this with the predictions from quantum mechanics for the singlet state. For a small angle θ, measured in radians, the chance of a different outcome is approximately ${\displaystyle f_{1}=\theta ^{2}/4}$ as explained by small-angle approximation. At two times this small angle, the chance of a mismatch is therefore about 4 times larger, since ${\displaystyle f_{2}=(2\theta )^{2}/4=2^{2}\theta ^{2}/4\approx 4f_{1}}$. But we just argued that it cannot be more than 2 times as large.

This intuitive formulation is due to David Mermin. The small-angle limit is discussed in Bell's original article, and therefore goes right back to the origin of the Bell inequalities.

Generalizing Bell's original inequality,[2] John Clauser, Michael Horne, Abner Shimony and R. A. Holt introduced the CHSH inequality,[22] which puts classical limits on the set of four correlations in Alice and Bob's experiment, without any assumption of perfect correlations (or anti-correlations) at equal settings

${\displaystyle (1)\quad C_{h}(a,b)+C_{h}(a,b')+C_{h}(a',b)-C_{h}(a',b')\leq 2.}$

Making the special choice ${\displaystyle a'=b+\pi }$, denoting ${\displaystyle b'=c}$, and assuming perfect anti-correlation at equal settings, perfect correlation at opposite settings, therefore ${\displaystyle \rho (a,a+\pi )=1}$ and ${\displaystyle \rho (b,a+\pi )=-\rho (b,a)}$, the CHSH inequality reduces to the original Bell inequality. Nowadays, (1) is also often simply called "the Bell inequality", but sometimes more completely "the Bell-CHSH inequality".

With abbreviated notation

${\displaystyle A=A(a,\lambda ),A'=A(a',\lambda ),B=B(b,\lambda ),B'=B(b',\lambda ),}$

the CHSH inequality can be derived as follows. Each of the four quantities is ${\displaystyle \pm 1}$ and each depends on ${\displaystyle \lambda }$. It follows that for any ${\displaystyle \lambda \in \Lambda }$, one of ${\displaystyle B+B'}$ and ${\displaystyle B-B'}$ is zero, and the other is ${\displaystyle \pm 2}$. From this it follows that

${\displaystyle AB+AB'+A'B-A'B'=A(B+B')+A'(B-B')\leq 2,}$

and therefore

${\displaystyle {\begin{aligned}C_{h}(a,b)+C_{h}(a,b')+C_{h}(a',b)-C_{h}(a',b')&=\int _{\Lambda }ABp\,d\lambda +\int _{\Lambda }AB'p\,d\lambda +\int _{\Lambda }A'Bp\,d\lambda -\int _{\Lambda }A'B'p\,d\lambda \\&=\int _{\Lambda }(AB+AB'+A'B-A'B')p\,d\lambda \\&=\int _{\Lambda }(A(B+B')+A'(B-B'))p\,d\lambda \leq 2.\end{aligned}}}$

At the heart of this derivation is a simple algebraic inequality concerning four variables, ${\displaystyle A,A',B,B'}$, which take the values ${\displaystyle \pm 1}$ only:

${\displaystyle AB+AB'+A'B-A'B'=A(B+B')+A'(B-B')\leq 2.}$

The CHSH inequality is seen to depend only on the following three key features of a local hidden variables theory: (1) realism: alongside of the outcomes of actually performed measurements, the outcomes of potentially performed measurements also exist at the same time; (2) locality, the outcomes of measurements on Alice's particle don't depend on which measurement Bob chooses to perform on the other particle; (3) freedom: Alice and Bob can indeed choose freely which measurements to perform.

The realism assumption is actually somewhat idealistic, and Bell's theorem only proves non-locality with respect to variables that only exist for metaphysical reasons. However, before the discovery of quantum mechanics, both realism and locality were completely uncontroversial features of physical theories.

The measurements performed by Alice and Bob are spin measurements on electrons. Alice can choose between two detector settings labeled ${\displaystyle a}$ and ${\displaystyle a'}$; these settings correspond to measurement of spin along the ${\displaystyle z}$ or the ${\displaystyle x}$ axis. Bob can choose between two detector settings labeled ${\displaystyle b}$ and ${\displaystyle b'}$; these correspond to measurement of spin along the ${\displaystyle z'}$ or ${\displaystyle x'}$ axis, where the ${\displaystyle x'-z'}$ coordinate system is rotated 135° relative to the ${\displaystyle x-z}$ coordinate system. The spin observables are represented by the 2 × 2 self-adjoint matrices:

${\displaystyle S_{x}={\begin{bmatrix}0&1\\1&0\end{bmatrix}},\quad S_{z}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}$

These are the Pauli spin matrices, which are known to have eigenvalues equal to ${\displaystyle \pm 1}$. As is customary, we will use bra–ket notation to denote the eigenvectors of ${\displaystyle S_{z}}$ as ${\displaystyle |0\rangle ,|1\rangle }$, where

$${\displaystyle |0\rangle \equiv {\begin{pmatrix}1\\0\end{pmatrix}},\qquad |1\rangle \equiv {\begin{pmatrix}0\\1\end{pmatrix}}.}$$

Consider now the single state ${\displaystyle |\Phi ^{-}\rangle }$ defined as

$${\displaystyle |\Phi ^{-}\rangle \equiv {\frac {1}{\sqrt {2}}}\left(|0,1\rangle -|1,0\rangle \right),}$$

where we used the shortened notation ${\displaystyle |0,1\rangle \equiv |0\rangle \otimes |1\rangle ,|1,0\rangle \equiv |1\rangle \otimes |0\rangle .}$

According to quantum mechanics, the choice of measurements is encoded into the choice of Hermitian operators applied to this state. In particular, consider the following operators:

$${\displaystyle {\begin{aligned}A(a)&=S_{z}\otimes I\\A(a')&=S_{x}\otimes I\\B(b)&={\frac {-1}{\sqrt {2}}}\ I\otimes (S_{z}+S_{x})\\B(b')&={\frac {1}{\sqrt {2}}}\ I\otimes (S_{z}-S_{x}),\end{aligned}}}$$

where ${\displaystyle A(a),A(a')}$ represent two measurement choices of Alice, and ${\displaystyle B(b),B(b')}$ two measurement choices of Bob.

To obtain the expectation value given by a given measurement choice of Alice and Bob, one has to compute the expectation value of the corresponding pair of operators (for example, ${\displaystyle A(a)B(b)}$ if the inputs are chosen to be ${\displaystyle a,b}$) over the shared state ${\displaystyle |\Phi ^{-}\rangle }$.

For example, the expectation value ${\displaystyle \langle A(a)B(b)\rangle }$ corresponding to Alice choosing the measurement setting ${\displaystyle a}$ and Bob choosing the measurement setting ${\displaystyle b}$ is computed as

$${\displaystyle \langle A(a)B(b)\rangle \equiv \langle \Phi ^{-}|\left({\frac {-1}{\sqrt {2}}}S_{z}\otimes (S_{x}+S_{z})\right)|\Phi ^{-}\rangle =-{\frac {1}{2}}\langle \Phi ^{-}|{\Big [}|0\rangle \otimes (|0\rangle -|1\rangle )+|1\rangle \otimes (|1\rangle +|0\rangle ){\Big ]}={\frac {1}{\sqrt {2}}}.}$$

Similar computations are used to obtain

$${\displaystyle {\begin{aligned}\left\langle A(a)B(b)\right\rangle =\left\langle A(a')B(b)\right\rangle =\left\langle A(a')B(b')\right\rangle &={\frac {1}{\sqrt {2}}}\\\left\langle A(a)B(b')\right\rangle &=-{\frac {1}{\sqrt {2}}}.\end{aligned}}}$$

It follows that the value of ${\displaystyle S}$ given by this particular experimental arrangement is

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

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 ${\displaystyle 2{\sqrt {2}}}$ 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.[23]

The fair sampling problem was faced openly in the 1970s. In early designs of their 1973 experiment, Freedman and Clauser[24] used fair sampling in the form of the Clauser–Horne–Shimony–Holt (CHSH[22]) hypothesis. However, shortly afterwards Clauser and Horne[20] 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 82.8% for singlet states,[25] but have very low dark rate and short dead and resolving times. However, Eberhard (1976) discovered that with a variant of the Clauser-Horne inequality, and using less than maximally entangled states, only 66.67% detection efficiency was required. This was achieved in 2015 by two successful “loophole-free” Bell-type experiments, in Vienna (Giustina er al.) and at NIST in Boulder, Colorado (Shalm te al.) [references to be added].

Because, at that time, even the best detectors didn't detect a large fraction of all photons, Clauser and Horne[20] recognized that testing Bell's inequality required 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,[24] who found that the Bell's inequality was violated. So the no-enhancement hypothesis cannot be true in a local hidden variables model.

While early experiments used atomic cascades, later experiments have used parametric down-conversion, following a suggestion by Reid and Walls,[26] giving improved generation and detection properties. As a result, recent experiments with photons no longer have to suffer from the detection loophole. This made the photon the first experimental system for which all main experimental loopholes were surmounted, although at first only in separate experiments. From 2015, experimentalists were able to surmount all the main experimental loopholes simultaneously; see Bell test experiments.

Most 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 2007 experiment ruled out a large class of non-Bohmian non-local hidden variable theories.[27]

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 that 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.[28]

The Many-Worlds interpretation is local and deterministic, as it consists of the unitary part of quantum mechanics without collapse. It can generate correlations that violate a Bell inequality because it doesn't satisfy the implicit assumption that Bell made that measurements have a single outcome. In fact, Bell's theorem can be proven in the Many-Worlds framework from the assumption that a measurement has a single outcome. Therefore a violation of a Bell inequality can be interpreted as a demonstration that measurements have multiple outcomes.[29]

The explanation it provides for the Bell correlations is that when Alice and Bob make their measurements, they split into local branches. From the point of view of each copy of Alice, there are multiple copies of Bob experiencing different results, so Bob cannot have a definite result, and the same is true from the point of view of each copy of Bob. They will obtain a mutually well-defined result only when their future light cones overlap. At this point we can say that the Bell correlation starts existing, but it was produced by a purely local mechanism. Therefore the violation of a Bell inequality cannot be interpreted as a proof of non-locality.[30]

Bell himself summarized one of the possible ways to address the theorem, superdeterminism, in a 1985 BBC Radio interview:

There is a way to escape the inference of superluminal speeds and spooky action at a distance. But it involves absolute determinism in the universe, the complete absence of free will. Suppose the world is super-deterministic, with not just inanimate nature running on behind-the-scenes clockwork, but with our behavior, including our belief that we are free to choose to do one experiment rather than another, absolutely predetermined, including the 'decision' by the experimenter to carry out one set of measurements rather than another, the difficulty disappears. There is no need for a faster-than-light signal to tell particle A what measurement has been carried out on particle B, because the universe, including particle A, already 'knows' what that measurement, and its outcome, will be.[31]

A few advocates of deterministic models have not given up on local hidden variables. For example, Gerard 't Hooft has argued that the aforementioned superdeterminism loophole cannot be dismissed.[32] For a hidden-variable theory, if Bell's conditions are correct, the results that agree with quantum mechanical theory appear to indicate superluminal (faster-than-light) effects, in contradiction to relativistic physics.

There have also been repeated claims that Bell's arguments are irrelevant because they depend on hidden assumptions that, in fact, are questionable. For example, E. T. Jaynes[33] argued in 1989 that there are two hidden assumptions in Bell's theorem that limit its generality. According to Jaynes:

1. Bell interpreted conditional probability P(X | Y) as a causal influence, i.e. Y exerted a causal influence on X in reality. This interpretation is a misunderstanding of probability theory. As Jaynes shows,[33] "one cannot even reason correctly in so simple a problem as drawing two balls from Bernoulli's Urn, if he interprets probabilities in this way."
2. Bell's inequality does not apply to some possible hidden variable theories. It only applies to a certain class of local hidden variable theories. In fact, it might have just missed the kind of hidden variable theories that Einstein is most interested in.

Richard D. Gill claimed that Jaynes misunderstood Bell's analysis. Gill points out that in the same conference volume in which Jaynes argues against Bell, Jaynes confesses to being extremely impressed by a short proof by Steve Gull presented at the same conference, that the singlet correlations could not be reproduced by a computer simulation of a local hidden variables theory.[34] According to Jaynes (writing nearly 30 years after Bell's landmark contributions), it would probably take us another 30 years to fully appreciate Gull's stunning result.

In 2006 a flurry of activity about implications for determinism arose with John Horton Conway and Simon B. Kochen's free will theorem,[35] which stated "the response of a spin 1 particle to a triple experiment is free—that is to say, is not a function of properties of that part of the universe that is earlier than this response with respect to any given inertial frame."[36] This theorem raised awareness of a tension between determinism fully governing an experiment (on the one hand) and Alice and Bob being free to choose any settings they like for their observations (on the other).[37][38] The philosopher David Hodgson supports this theorem as showing that determinism is unscientific, thereby leaving the door open for our own free will.[39]