Bell's theorem

Bell's theorem is the most famous legacy of the late John Bell. It is notable for showing that the predictions of quantum mechanics (QM) differ from those of intuition. It is simple and elegant, and touches upon 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". The Einstein Podolsky Rosen paradox (EPR paradox) assumes local realism, the intuitive notion that particle attributes have definite values independent of the act of observation and that physical effects have a finite propagation speed. Bell showed that local realism leads to a requirement for certain types of phenomena that are not present in quantum mechanics. This requirement is called Bell's inequality.

Different authors subsequently derived similar inequalities, collectively termed Bell inequalities, that also assume local realism. That is, they 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 properties are often called hidden variables, the properties that Einstein posited when he stated his famous objection to quantum mechanics: "^* does not play dice."

The inequalities concern measurements made by observers (often called Alice and Bob) on entangled pairs of particles that have interacted and then separated. Hidden variable assumptions limit the correlation of subsequent measurements of the particles. Bell discovered that under quantum mechanics this correlation limit may be violated. Quantum mechanics lacks local hidden variables associated with individual particles, and so the inequalities do not apply to it. Instead, it predicts correlation due to quantum entanglement of the particles, allowing their state to be well defined only after a measurement is made on either particle. That restriction agrees with the Heisenberg uncertainty principle, one of the most fundamental concepts in quantum mechanics.

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 the inequalities of Bell's theorem are violated. This provides empirical evidence against local realism and demonstrates 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 the observers cannot use the inequality violations to communicate information to each other 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.

Importance of the theorem

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 one direction (e.g. on the x axis), Bob's measurement in that direction was determined with certainty, whereas immediately before Alice's measurement, Bob's outcome was only statistically determined. Thus, either the spin in each direction is not 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. 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 two ideas: first, that objects have a definite state that 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 measurements by Alice and Bob to be somehow correlated with each other: the random variables are assumed not to be independent, but linked in some way. Nonetheless, there is a limit to the amount of correlation one might expect to see. This is what the Bell inequality expresses.

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

\leq 2,

where C denotes correlation.

Statement of Bell's theorem

In this article correlation of observables X, Y is defined as

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

This is 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:

There is a probability space $\Lambda$ and the observed outcomes by both Alice and Bob result by random sampling of the parameter $\lambda \in \Lambda$ .
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 $A(a,\lambda)$
Value observed by Bob with detector setting b is $B(b,\lambda)$

Implicit in assumption 1) above, the hidden parameter space $\Lambda$ has a probability measure $\rho$ and the expectation of a random variable X on $\Lambda$ with respect to $\rho$ 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 $\lambda \in \Lambda$ . 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 $A(a,\lambda)$ , $B(b,\lambda)$ 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))\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 $\lambda$ 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 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 $\lambda$ 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)

\Lambda

which still have values in the range +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 $\phi$ produces a distribution of real values in which the probability of observing $\lambda$ is

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

(where $E_X(\lambda)$ is the eigenspace corresponding to $\lambda$ ) 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 $\psi$ 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 45^o relative to the x – z 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 S_x by

\left|+x\right\rang, \quad \left|-x\right\rang.

Let $\phi$ 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).

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.

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. Hence many worlds can adhere to both the properties of philosophical realism and the principle of locality and not violate Bell's conditions -- the only interpretation that can do this.

Original Bell's 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.

Notable quotes

Heinz Pagels, in The Cosmic Code, writes:

References

A. Aspect et al., Experimental Tests of Realistic Local Theories via Bell's Theorem, Phys. Rev. Lett. 47, 460 (1981)
A. Aspect et al., Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities, Phys. Rev. Lett. 49, 91 (1982).
A. Aspect et al., Experimental Test of Bell's Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett. 49, 1804 (1982).
A. Aspect and P. Grangier, About resonant scattering and other hypothetical effects in the Orsay atomic-cascade experiment tests of Bell inequalities: a discussion and some new experimental data, Lettere al Nuovo Cimento 43, 345 (1985)
J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 195 (1964)
J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys. 38, 447 (1966)
J. S. Bell, Introduction to the hidden variable question, Proceedings of the International School of Physics 'Enrico Fermi', Course IL, Foundations of Quantum Mechanics (1971) 171-81
J. S. Bell, Bertlmann’s socks and the nature of reality, Journal de Physique, Colloque C2, suppl. au numero 3, Tome 42 (1981) pp C2 41-61
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press 1987) collection of Bell's papers, including all of the above.
J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Proposed experiment to test local hidden-variable theories, Physical Review Letters 23, 880-884 (1969).
J. F. Clauser and M. A. Horne, Experimental consequences of objective local theories, Physical Review D, 10, 526-35 (1974)
J. F. Clauser and A. Shimony, Bell's theorem: experimental tests and implications, Reports on Progress in Physics 41, 1881 (1978)
S. J. Freedman and J. F. Clauser, Experimental test of local hidden-variable theories, Phys. Rev. Lett. 28, 938 (1972)
E. S. Fry, T. Walther and S. Li, Proposal for a loophole-free test of the Bell inequalities, Phys. Rev. A 52, 4381 (1995)
E. S. Fry, and T. Walther, Atom based tests of the Bell Inequalities - the legacy of John Bell continues, pp 103-117 of Quantum ^*speakables, R.A. Bertlmann and A. Zeilinger (eds.) (Springer, Berlin-Heidelberg-New York, 2002)
R. B. Griffiths, Consistent Quantum Theory', Cambridge University Press (2002).
L. Hardy, Nonlocality for 2 particles without inequalities for almost all entangled states. Physical Review Letters 71 (11) 1665-1668 (1993)
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000)
P. Pearle, Hidden-Variable Example Based upon Data Rejection, Physical Review D 2, 1418-25 (1970)
A. Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht, 1993.
M. Redhead, Incompleteness, Nonlocality and Realism, Clarendon Press (1987)
B. C. van Frassen, Quantum Mechanics, Clarendon Press, 1991.
M.A. Rowe, D. Kielpinski, V. Meyer, C.A. Sackett, W.M. Itano, C. Monroe, and D.J. Wineland, "Experimental violation of Bell's inequalities with efficient detection",(Nature, 409, 791-794, 2001).

External links

An explanation of Bell's Theorem, based on N. D. Mermin's article, "Bringing Home the Atomic World: Quantum Mysteries for Anybody," Am. J. of Phys. 49 (10), 940 (October 1981)
Article on Bell's Theorem by Shimony in the Stanford Encyclopedia of Philosophy, (2004). Includes a useful list of references, including general reading.

Quantum information science | Quantum measurement | Physics theorems

Importance of the theorem

Bell's thought experiment

Statement of Bell's theorem

Comparison to quantum mechanical prediction

Bell test experiments

Implications of violation of Bell's inequality

Original Bell's inequality

Notable quotes

See also

Further reading

References

External links

Gourt Home::Gourt :: Bell's theorem

Importance of the theorem

Bell's thought experiment

Statement of Bell's theorem

Comparison to quantum mechanical prediction

Bell test experiments

Implications of violation of Bell's inequality

Original Bell's inequality

Notable quotes

See also

Further reading

References

External links