﻿

# Quantum mechanical Bell test prediction

In physics, the quantum mechanical Bell test prediction is the prediction that quantum mechanics would give for the correlation probabilities for a set of measurements performed on a quantum entangled state. An important outcome of this prediction is that it violates the Bell inequality, which, as a result, has serious implications for the interpretation of quantum mechanics.

The following is based on section 2 of the Stanford Encyclopedia of Philosophy article written by Abner Shimony, one of the authors of the original Clauser, Horne, Shimony and Holt article (1969) after which the CHSH Bell test is named (Shimony, 2004).

himony's derivation of the quantum mechanical prediction

Let the system consist of a pair of photons A and B propagating respectively in the "z" and −"z" directions. The two kets |"x">j and |"y">j constitute a polarization basis for photon j (j =A, B), the former representing (in Dirac's notation) a state in which the photon A is linearly polarized in the "x"-direction and the latter a state in which it is linearly polarized in the "y"-direction. For the two-photon system the four product kets |"x">A |"x">B, |"x">A |"y">B, |"y">A |"x">B, and |"y">A |"y">B constitute a polarization basis. Each two-photon polarization state can be expressed as a linear combination of these four basis states with complex coefficients. Of particular interest are the entangled quantum states, which in no way can be expressed as |&phi;>A|&xi;>B, with |&phi;> and |&xi;> single-photon states, an example being

: (1) |&Phi;> = (1/&radic;2) [ |"x">A |"x">B + |"y">A |"y">B ] , which has the useful property of being invariant under rotation of the "x" and "y" axes in the plane perpendicular to "z". Neither photon A nor photon B is in a definite polarization state when the pair is in the state |&Phi;>, but their potentialities (in the terminology of Heisenberg 1958) are correlated: if by measurement or some other process the potentiality of photon A to be polarized along the "x"-direction or along the "y"-direction is actualized, then the same will be true of photon B, and conversely.

Suppose now that photons A and B impinge respectively on the faces of birefringent crystal polarization analyzers, with the entrance face of each analyzer perpendicular to "z". Each analyzer has the property of separating light incident upon its face into two outgoing non-parallel rays, the ordinary ray and the extraordinary ray. The transmission axis of the analyzer is a direction with the property that a photon polarized along it will emerge in the ordinary ray (with certainty if the crystals are assumed to be ideal), while a photon polarized in a direction perpendicular to "z" and to the transmission axis will emerge in the extraordinary ray. (See diagram.) The crystals are also idealized by assuming that no incident photon is absorbed, but each emerges in either the '+' or the '−' channel. Quantum mechanics provides an algorithm for computing the probabilities that photons A and B will emerge from these idealized analyzers in specified rays, as functions of the orientations "a" and "b" of the analyzers, "a" being the angle between the transmission axis of the 'A' analyzer and an arbitrary fixed direction in the "x&ndash;y" plane, and "b" having the analogous meaning for B:

: (2a) prob&Phi; (j, k |"a", "b") = | <&Phi;|&theta;j>A |&xi;k>B |2.

Here j is a quantum number associated with the ray into which photon A emerges, taking values +1 or −1 depending on which channel it emerges from, while k is the analogous quantum number for photon B; and | <&Phi;|&theta;j>A |&xi;k>B | is the ket representing the quantum state of photons A and B with the respective quantum numbers j and k. Calculation of the probabilities of interest from Eq. (2a) can be simplified by using the invariance noted after Eq. (1) and rewriting |&Phi;> as

: (3) |&Phi;> = (1/&radic;2) [ |&theta;1>A |&theta;1>B + |&theta;−1>A |&theta;−1>B ] .

Eq. (3) results from Eq. (1) by substituting the transmission axis of the A analyzer for "x" and the direction perpendicular to both "z" and this transmission axis for "y".

Since |&theta;−1>A is orthogonal to |&theta;1>A, only the first term of Eq. (3) contributes to the inner product in Eq. (2a) if j = k = 1; and since the inner product of |&theta;1>A with itself is unity because of normalization, Eq. (2a) reduces for j = k = 1 to

: (2b) prob&Phi; (1, 1 |"a", "b") = 1/2 | B<&theta;1 | &phi;1>B |2. Finally, the expression on the right hand side of Eq. (2b) is evaluated by using the law of Malus, which is preserved in the quantum mechanical treatment of polarization states: that the probability for a photon polarized in a direction "n" to pass through an ideal polarization analyzer with axis of transmission "n"&prime; equals the squared cosine of the angle between "n" and "n"&prime;. Hence

: (4a) prob&Phi; (1, 1 |"a", "b") = 1/2 cos2 &sigma; where &sigma; is "b" − "a". Likewise,

: (4b) prob&Phi; (−1, −1 |"a", "b") = 1/2 cos2 &sigma;, and

: (4c) prob&Phi; (1, −1|"a", "b") = prob&Phi; (−1, 1|"a", "b") = 1/2 sin2 &sigma;.

The expectation value of the product of the results j and k of the polarization analyzes of photons A and B by their respective analyzers is

: (5) "E"&Phi; ("a", "b") = prob&Phi; (1, 1 |"a", "b") + prob&Phi; (−1, −1 |"a", "b") − prob&Phi; (1, −1 |"a", "b") − prob&Phi; (−1, 1 |"a", "b") ::= cos2 &sigma; − sin2 &sigma; ::= cos 2&sigma;.

The required quantum mechanical predictions are thus 1/2 cos2 &sigma; for coincidence probabilities and cos 2&sigma; for quantum correlations, where &sigma; is the angle between the detectors.

Demonstration of infringement of a Bell inequality

Now choose as the orientation angles of the transmission axes

: (6) "a" = 0, "a"&prime; = &pi;/4, "b" = &pi;/8, "b"&prime; = 3 &pi;/8 .

Then

: (7a) "E"&Phi;("a", "b") = cos 2(&pi;/8) = 0.707,

: (7b) "E"&Phi;("a", "b"&prime;) = cos 2(3&pi;/8) = −0.707,

: (7c) "E"&Phi;("a"&prime;, "b") = cos 2(−&pi;/8) = 0.707, and

: (7d) "E"&Phi;("a"&prime;, "b"&prime;) = cos 2(&pi;/8) = 0.707.

Therefore the quantum mechanical prediction for the CHSH test statistic is

: (8) "S"&Phi; = "E"&Phi;("a", "b") − "E"&Phi;("a", "b"&prime;) + "E"&Phi;("a"&prime;, "b") + "E"&Phi;("a"&prime;, "b"&prime;) = 2.828,

exceeding the CHSH Bell test limit of 2 and thus completing the proof of a version of Bell's Theorem. It is important to note, however, that "all" entangled quantum states yield predictions in violation of the inequality, as Gisin (1991) and Popescu and Rohrlich (1992) have independently demonstrated. Popescu and Rohrlich (1992) also show that the maximum amount of violation is achieved with a quantum state of maximum degree of entanglement, exemplified by |&Phi;> of Eq. (1).

References

* Clauser, J. F., M. A. Horne, A. Shimony and R. A. Holt  , "Proposed experiment to test local hidden-variable theories", Physical Review Letters 23, 880-884
* Gisin, N.  , "Bell's inequality holds for all non-product states", Physics Letters A 154, 201-202 [Note: the title of this paper is erroneous and should be replaced by "Bell's inequality is violated by all non-product states"]
* Popescu, S. and D. Rohrlich  , "Generic quantum nonlocality", Physics Letters A 166, 293-297
* Abner Shimony, [http://plato.stanford.edu/archives/sum2005/entries/bell-theorem/ "Bell's Theorem"] (2005), The Stanford Encyclopedia of Philosophy (Summer 2005 Edition), Edward N. Zalta (ed.)

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Bell test experiments — The Bell test experiments serve to investigate the validity of the entanglement effect in quantum mechanics by using some kind of Bell inequality. John Bell published the first inequality of this kind in his paper On the Einstein Podolsky Rosen… …   Wikipedia

• Loopholes in Bell test experiments — In Bell test experiments, there may be experimental problems that affect the validity of the experimental findings. The term Loopholes is frequently used to denote these problems. See the page on Bell s theorem for the theoretical background to… …   Wikipedia

• Bell's theorem — is a theorem that shows that the predictions of quantum mechanics (QM) are not intuitive, and touches upon fundamental philosophical issues that relate to modern physics. It is the most famous legacy of the late physicist John S. Bell. Bell s… …   Wikipedia

• John Stewart Bell — (June 28 1928 ndash; October 1 1990) was a physicist, and the originator of Bell s Theorem, one of the most important theorems in quantum physics. Life and work He was born in Belfast, Northern Ireland, and graduated in experimental physics at… …   Wikipedia

• Sakurai's Bell inequality — The intention of a Bell inequality is to serve as a test of local realism or local hidden variable theories as against quantum mechanics, applying Bell s theorem, which shows them to be incompatible. Not all the Bell s inequalities that appear in …   Wikipedia

• Interpretation of quantum mechanics — An interpretation of quantum mechanics is a statement which attempts to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has received thorough experimental testing, many of these experiments are open… …   Wikipedia

• Mathematics and Physical Sciences — ▪ 2003 Introduction Mathematics       Mathematics in 2002 was marked by two discoveries in number theory. The first may have practical implications; the second satisfied a 150 year old curiosity.       Computer scientist Manindra Agrawal of the… …   Universalium

• Local hidden variable theory — In quantum mechanics, a local hidden variable theory is one in which distant events are assumed to have no instantaneous (or at least faster than light) effect on local ones. According to the quantum entanglement theory of quantum mechanics, on… …   Wikipedia

• Nobel Prizes — ▪ 2009 Introduction Prize for Peace       The 2008 Nobel Prize for Peace was awarded to Martti Ahtisaari, former president (1994–2000) of Finland, for his work over more than 30 years in settling international disputes, many involving ethnic,… …   Universalium

• physical science — physical scientist. 1. any of the natural sciences dealing with inanimate matter or with energy, as physics, chemistry, and astronomy. 2. these sciences collectively. [1835 45] * * * Introduction       the systematic study of the inorganic world …   Universalium