# Bell's theorem

Bell's theorem is the most famous legacy of the late John Bell. It is famous for drawing an important line in the sand between quantum mechanics (QM) and the world as we know it intuitively. It is simple and elegant, and at the same time touches upon many of the fundamental philosophical issues that relate to modern physics. In its simplest form, Bell's theorem states:

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

This theorem has even been called "the most profound in science" (Stapp, 1975). Bell's seminal 1965 paper was entitled "On the Einstein Podolsky Rosen paradox". He showed that the assumption of local realism - that particle attributes have definite values independent of the act of observation and that physical effects have a finite propagation speed - leads to a requirement for certain types of phenomena which is not present in quantum mechanics. This requirement is called "Bell's inequality". Similar inequalities have subsequently been derived by different authors which are collectively termed "Bell inequalities". They all make the same assumptions about local realism -- that a quantum-level object has a well defined state which accounts for all its measurable properties and that distant objects do not exchange information faster than the speed of light. These well defined properties are often called hidden variables.

The inequalities concern measurements made by remotely located observers (often called Alice and Bob) on entangled pairs of particles that have interacted and then separated. Assuming hidden variables, they lead to strict limits on the possible values of the correlation of subsequent measurements that can be obtained from the particle pairs. Bell discovered that these limits are outside the predictions of quantum mechanics in special cases. Quantum mechanics does not assume the existence of local hidden variables associated with individual particles, and so the inequalities do not apply to it. The QM predicted correlation is due to quantum entanglement of the pair, with the idea that their state is not determined until the point at which a measurement is made on one or the other. This idea is fully in accordance with the Heisenberg uncertainty principle, one of the most fundamental concepts in quantum mechanics.

If one accepts Bell's theorem: either quantum mechanics is wrong, or local realism is wrong. They cannot both be correct. So which is right? To answer that scientifically, experiments needed to be performed. It took many years and many improvements in technology to get the answer.

Bell test experiments to date overwhelmingly show that the inequalities of Bell's theorem are violated. This provides empirical evidence against local realism and demonstrating that some of the "spooky action at a distance" suggested by the famous Einstein Podolsky Rosen (EPR) thought experiment do in fact occur. They are also taken as positive evidence in favor of QM. The principle of special relativity is saved by the no-communication theorem, which proves that it is impossible for Alice to communicate information to Bob (or vice versa) faster than the speed of light.

John Bell's papers examined both John von Neumann's 1932 proof of the incompatibility of hidden variables with QM and Albert Einstein and his colleagues' seminal 1935 paper on the subject.

## Why Bell's theorem is important

After EPR, quantum mechanics was left in the unsatisfactory position that it was either 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 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 a conclusion of EPR, that once Alice measured spin in the x direction, Bob's measurement in the x direction was determined with certainty, whereas immediately before Alice's measurement Bob's outcome was only statistically determined. Thus either spin in the x direction is not an element of physical reality or 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. However, there still remained the idea that the electron had a definite position and spin, and that QM's failing was its inability to predict those values precisely. The possibility remained that some yet unknown, but more powerful theory, such as a "Hidden Variable" theory, might be able to predict these 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 ideas about "how the real world works": first that objects have a definite state which determines the values of all other measurable properties such as position and momentum and second, that (as a result of special relativity) effects of local actions such as measurements cannot travel faster than the speed of light. In the formalization of local realism used by Bell, the predictions of a 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 seal the fate of those that had local realist hopes for QM.

## Bell's thought experiment

Bell considered a setup in which two observers, Alice and Bob, perform independent measurements on a system S prepared in some fixed state. Each observer has a detector with which to make measurements. On each trial, Alice and Bob can independently choose between various detector settings. Alice can choose a detector setting a to obtain a measurement A(a) and Bob can choose a detector setting b to measure B(b). After repeated trials Alice and Bob collect statistics on their measurements and correlate the results.

There are two key assumptions in Bell's analysis: (1) each measurement reveals an objective physical property of the system (2) a measurement taken by one observer has no effect on the measurement taken by the other.

In the language of probability theory, repeated measurements of system properties can be regarded as repeated sampling of random variables. One might expect that measurements by Alice and Bob to be somehow correlated with each other: the random variables are assumed to not be independent, but linked in some way. Nonetheless, there is a limit to the amount of correlation one might expect to see. The Bell inequality expresses that maximum amount of correlation one can expect.

A version of the Bell inequality appropriate for this example is given by Clauser, Horne, Shimony and Holt, and is called the CHSH form:

$(1) \quad \mathbf{C}(A(a), B(b)) + \mathbf{C}(A(a), B(b')) + \mathbf{C}(A(a'), B(b)) - \mathbf{C}(A(a'), B(b'))\leq 2,$

where C denotes correlation.

## Statement of Bell's theorem

$\mathbf{C}(X,Y) = \operatorname{E}(X Y).$

This is non-normalized form of the correlation coefficient considered in statistics.

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 λ ∈ Λ.
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 theorem. 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 λ ∈ Λ. 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, +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:

$A(a, \lambda) \ B(b, \lambda) + A(a, \lambda) \ B(b', \lambda) + A(a', \lambda) \ B(b, \lambda) - A(a', \lambda) \ B(b', \lambda) =$
$= A(a, \lambda) (B(b, \lambda) + B(b', \lambda)) +A(a', \lambda) (B(b, \lambda) - B(b', \lambda)) \quad$
$\leq \bigg|A(a, \lambda) (B(b, \lambda) + B(b', \lambda)) +A(a', \lambda) (B(b, \lambda) - B(b', \lambda) ) \bigg| \quad$
$\leq \bigg|A(a, \lambda) (B(b, \lambda) + B(b', \lambda), \lambda)\bigg| +\bigg|A(a', \lambda) (B(b, \lambda) - B(b', \lambda), \lambda)\bigg|$
$\leq |B(b, \lambda) + B(b', \lambda)| +| B(b, \lambda) - B(b', \lambda)| \leq 2.$

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 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 detector 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.

## Comparison to quantum mechanical prediction

To apply Bell's theorem we will show that quantum mechanics makes a prediction that violates a "Bell inequality" in the setup considered in the EPR thought experiment. In order to do this, we first need to show how to compute correlations of quantum mechanical observables.

In the usual quantum mechanical formalism, observables X, Y are represented as self-adjoint operators on a Hilbert space. To compute the correlation, assume that X, Y are represented by matrices in a finite dimensional space and that X, Y commute; this special case suffices for our purposes below. We then use the von Neumann measurement postulate: a series of measurements of an observable X on a series of identical systems in state φ produces a distribution of real values in which the probability of observing λ is

$\|\operatorname{E}_X(\lambda) \phi\|^2$

(where EX(λ) is the eigenspace corresponding to λ) and 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, 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 for an electron. 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 45o 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.
File:Bells-thm.png
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.
$A(a) = S_z \otimes I$
$A(a') = S_x \otimes I$
$B(b) = - \frac{1}{\sqrt{2}} \ I \otimes (S_z + S_x)$
$B(b') = \frac{1}{\sqrt{2}} \ I \otimes (S_z - S_x).$

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

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

Thus, 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, it's called Tsirelson's bound. The operators giving this maximal value are always isomorphic to the Pauli matrices.

The next sections consider experimental tests to see whether the Bell inequalities required by local realism hold up to the empirical evidence.

## Bell test experiments

Main article: Bell test experiments.

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 suggest that Bell's inequality is violated. 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".
File:Bell-test-photon-analyer.png
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.

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."

## Implications of violation of Bell's inequality

The phenomenon of quantum entanglement that is implied by 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.

Some advocates of the hidden variables idea prefer to accept the opinion that experiments have ruled out local hidden variables. They are ready to give up locality (and probably also causality), 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. It is, however, considered by most to be unconvincing, requiring, for example, that all particles in the universe be able to instantaneously exchange information with all others.

Finally, one subtle assumption of the Bell inequalities is counterfactual definiteness. The derivation refers to several objective properties that cannot all be measured for any given particle, since the act of taking the measurement changes the state. Under local realism the difficulty is readily overcome, so long as we can assume that the source is stable, producing the same statistical distribution of states for all the subexperiments. If this assumption is felt to be unjustifiable, though, one can argue that Bell's inequality is unproven. In the Everett many-worlds interpretation, the assumption of counterfactual definiteness is abandoned, this interpretation assuming that the universe branches into many different observers, each of whom measures a different observation.

## Bell's original inequality

The original inequality that Bell derived (Bell, 1964) 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.

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.
• Brian Greene, The Fabric of the Cosmos (Vintage, 2004, ISBN 0-375-72720-5)
• D. Wick, The infamous boundary: seven decades of controversy in quantum physics (Birkhauser, Boston 1995)

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