Bell's Theorem

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 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: " {Link without Title} 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 Variable s. 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: : $(1) \quad \mathbf{C}(A(a), B(b)) + \mathbf{C}(A(a), B(b')) + \mathbf{C}(A(a'), B(b)) - \mathbf{C}(A(a'), B(b'))\leq 2,$ where C denotes correlation. STATEMENT OF BELL'S THEOREM 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 . 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 $ho$ and the expectation of a random variable ''X'' on $\Lambda$ with respect to $ho$ is written : $\operatorname{E}(X) = \int_\Lambda X(\lambda) ho(\lambda) d \lambda$ 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 : $\mathbf{C}(A(a), B(b)) + \mathbf{C}(A(a), B(b')) + \mathbf{C}(A(a'), B(b)) - \mathbf{C}(A(a'), B(b')) =$ : $= \int_\Lambda A(a, \lambda) \ B(b, \lambda) ho(\lambda) d \lambda + \int_\Lambda A(a, \lambda) \ B(b', \lambda) ho(\lambda) d \lambda + \int_\Lambda A(a', \lambda) \ B(b, \lambda) ho(\lambda) d \lambda - \int_\Lambda A(a', \lambda) \ B(b', \lambda) ho(\lambda) d \lambda =$ : $= \int_\Lambda \bigg\{A(a, \lambda) \ B(b, \lambda) + A(a, \lambda) \ B(b', \lambda) +A(a', \lambda) \ B(b, \lambda) - A(a', \lambda) \ B(b', \lambda)\bigg\} ho(\lambda) d \lambda =$ : $= \int_\Lambda \bigg\{A(a, \lambda) (B(b, \lambda) + B(b', \lambda))+ A(a', \lambda) (B(b, \lambda) - B(b', \lambda)) \bigg\} ho(\lambda) d \lambda \quad$ : $\leq 2.$ 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: : $A(a, \lambda) \ B(b, \lambda) + A(a, \lambda) \ B(b', \lambda) + A(a', \lambda) \ B(b, \lambda) - A(a', \lambda) \ B(b', \lambda) =$ : $= A(a, \lambda) (B(b, \lambda) + B(b', \lambda)) +A(a', \lambda) (B(b, \lambda) - B(b', \lambda)) \quad$
	:<math> 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 </math>

B(b') = rac{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) 
angle =  \langle A(a) B(b') 
angle =\langle A(a') B(b) 
angle = rac{1}{\sqrt{2}},

and

:

\langle A(a') B(b') 
angle = - rac{1}{\sqrt{2}}.

so that

:

\langle A(a) B(b) 
angle + \langle A(a) B(b') 
angle + \langle A(a') B(b) 
angle - \langle A(a') B(b') 
angle = rac{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".

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:

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Bell's Theorem