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Experimental consequences of stochastic noncontextual hidden variable theories
0O83-6656/93$24.0O @1993PergamonPressLtd
Viataa in Astronomy, Vol.37,pp. 317-320,1993 Printedin GreatBritain.Allfightsreserved.
EXPERIMENTAL CONSEQUENCES OF STOCHASTIC NONCONTEXTUAL HIDDEN VARIABLE THEORIES S. M. R o y and Virendra Singh Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Bombay 400 005, India
ABSTRACT We formulate stochastic noncontextukl hidden variable theories. In such theories the hidden state A specifies, not definite values, but expectation values A(A), ..- of observables A, ..and noncontextuality means that A(A) is independent of which other commuting observables commuting with A are measured together with A. We derive Bell inequalities which show that such theories conflict with quantum theory and propose a two photon experiment to test them.
In quantum theory, the expectation value of an observable A is unaffected by the previous (or simultaneous) measurement of any set of mutually commuting observables (Ghiraxdi et al. 1980) commuting with A. This 'statistical noncontextuality' provides the inspiration for the 'noncontextuality' hypothesis in hidden variable theories or deeper level theories. In a determiaistic noncontextual hidden vaxiabletheory, the value of a~q observable A is independent of which particular set of mutually commuting observables commuting with A are measured together with A. The Gleason and Kochen-Specker theorems (Gieason 1957; Kochen and Specker 1967; Bell 1966) prove the impossibility of such ,~ hidden variable theory for quantum mechanics. A particnlaxly interesting case of 'noncontextuality' arises when observables A and B commute due to spacelike separation, and the noncontextuality hypothesis becomes the Einstein locality hypothesis (Einstein et al. 1935). In this case Bell's theorem (Bell 1964; Clauser et al. 1969; Bell 1971) proves a stronger result, viz. the impossibility, not only of deterministic but ,dso of stochastic local hidden variable theory for quantum mechanics. The purpose of the present work is twofold. (1) We formulate noncontextual stochastic hidden variable theories and prove that they contradict quantum mechanical predictions. (2) We propose a two photon experiment to test inequalities on photon polarization correlations implied by noncontextual stochastic hidden variable theories and violated by quantum theory. The inequalities do not follow from locality. Evidently, noncontextuality is more stringent than locaJitv because it aDvlies to a ~reater vaxietv of ohvsieal situations.
S.M. Roy and V. Singh
318
Stochastic n o n c o n t e x t ual h i d d e n variable theories. Let At(at), .- -, A' (a,) be dynaanical variables corresponding to different degrees of freedom of a given physical system, the settings
Ai(ai) being denoted by a i. In quantum theory the dynamical variables are represented by observables A~(a;), with
of the apparatus measuring
[A,(a,), A, (a~)] = 0.
(a)
In a hidden variable theory., the state of the system may be characterized by variables ~ (which may include the quantum state vector as well), with p(A) being their probability distribution obeying i
~)>0, For given ~, the dynamical variables have
/dX~)=l
(2)
expectation valnes At(,X,at), a~(,X,a~), ..., A,(~,a.),
noncontextpality implying that A; depends only on ~ and a;, but not on a i with j ~ i. We assume that by the very definition of the A~, JA;I _< I (e.g. A~ = +1 for transmission through a polarizer and -1 for nontransmission), and hence that
IA,O,~,)I ~_ 1, Vi
(3)
Further, in complete analogy to Bell's definition of local stochastic theories (Bell 1971) we define noncontextual stochastic theories to be those in which ArAb(A, ax, a~) = At(A, at) A2(A, a2). The n-variable correlation function thus has the representation,
/
~'(~,~,,...,~.) =
d,X p(.X)
fi.4,(,X,a,).
(4)
i----1
The corresponding quantum correlation function in a state ~ is
[P(at, a~,..., a,,)]¢ = (Ol ]-I A,(a:)[O).
(5)
i_'t
T h e o r e m . There exist quantum systems for which some experimentally verifiable quantum predictions cannot be reproduced by any noncontextual stochastic hidden varibales model; i.e. there exist quantum states ~ such that
P(a~,,,,, ..-, a.) # [P(,~,, a,,.-.,a.)],~
(6)
On the other hand it is obvious that the representation for n-variable correlation functions assumed for stochastic noncontextual hidden variable theories necessarily holds in any classical theory. Hence the noncontextuality inequalities derived below yield a quantitative measure of the amount by which ouantum theory must violate any classical theory.
Experimental Consequences
319
P r o o f of t h e t h e o r e m for two suin 1/~ uarti©les. Let ~x and J~ be the Pauli spin opera-
tots for two particles. Then
[~1. ~1~2. ~2,~1. ~ 2
(~)
~2] = o,
if ~t, ~2, bt, b'2 are unit vectors obeying ~i .1_ hi, i.e., gt" gt = a2" b2 ffi 0. Let us denote a ffi {~x, g2}, b ffi {b',,g2}, and A(a) ffi 8x" 8x 82" ~2; then A(a) has eigenvalues 4-1, and [A(a), A(b)] = 0. Hence, in a stochastic noncontextual hidden variable theory we must have the correlation function P(a, b) obeying (4) with (A(~, a)) 2 _< 1. This leads to Bell's inequalities,
(8)
IP(a, b) - P(a, b')l + IP(a', b) + P(a', b')l _< 2,
provided that ~, .1. bl, ~ ± b~'~, ~ _L hi, ~ _1. bt'~for i = 1 and 2. On the otherhaud quantum mechanics gives in the singiet state ~,
(P(a,b))q.Mi = (~[A(a)A(b)[~) = (~t × bt)" (~2 × ~).
(9)
The orthogonality conditions on a,b are obeyed if we choose bx, b~, ~2 and ~2 along the negative z-axis, ~ ffi b'~ ffi ~2 = ~ = (0, 0 , - 1), and the remaining vectors in the a~-9 plane. In particulax, the choice ~x = (1,0,0), ~'x = (0,-1,0), b2 = ( - l / x / 2 , 1/x/2,0), br~ = (1/x/~, 1/Vf2,0), leads to
I
-
I+ I
+
I
which violates the noncontextuality inequality by a factor V/2. This proves the announced theorem. For experimental purposes, a two photon version of the violation of noncontextuaiity by quantum mechanics might be more practical to test for the two photon state, ~ --
(I~)I~)
- I~)I~))/v r~, where Is) and I~) denote photon states plane polarized along m and y axes
respectively. R.eplacing ~ by the 3 x 3 photon spin operator E which eqnals ~ for • and 9 components of photon spin wave function, we obtain as before Eq. (9), which violates the Bell inequalities following from noncontextuality by a factor V~, for the choice of a.b, a',b f given already. The experiment (Fig. I) will provide a test of the quantum superposition principle against the classical idea of noncontextual realism in a situation where locality is not the issue. Another striking consequence of the ideas developed here is that for a single high spin particle, quantum theory implies exponential violation of classical behaviour. This will be reported separately.
S. M. Roy and V. Singh
320
DETECTORS
DETECTORS
(~ s~'+!
(?,--,
-<-IKI"'I
I-~/
P.o,o°~
P.o,o°
'~
'
'"
• ISz'--I ' "Is..,)
(~s~ -I SLITS TWO
CHANNEL
ELLIPTIC
POLARIZERS
FIG.I Schematic diagram of experimental apparatus to test the noncontextuality inequality. The source S emits two photons in the polarization state (l~)ly) - ly)lx))/V~, along - z and +z-axis. The left photon goes throngh an elliptic polarizer measuring ~x" '~1 = rl = +1, and then through another elfiptic polarizer measuring ~31 • bx = sl - 4-1. The right photon similarly goes through two channel elliptic polarizers measuring ~2 • ~2 --- r2 = 4-1, and ~ . b2 = s2 = 4-1. Here, at _L bx and ~, J. ~2. Detectors on the left and right &re wired to measure 2' = 16 coincidence counts N,,,,,,2,~, and hence P(a, b) =
'~
,.,.r.,s.,.s.,.,,v,,.... ,,,/z.,,'v,,,,,,,,,.
f l #1 ~t't jB]I
References Bell, J.S. (1964) Physics (Long Island Cityj N.Y.) I, 195. Bell, J.S. (1966) Re~. Mod. Phys. 38, 447. Bell, J.S. (1971) in "Foundations of Q~antum Mechanics", International School of Physics, "Enrico Fermi", Course IL (Academic, New York), p. 171. Clanser, J.P., Home, M.A., Schimony, A. and Holt, R.A. (1969) Phys. Re~. Left. 26, 880. Einstein, A., Podolsky, B. and Rosen, N. (1935) Phys. Re~. 47, 777. Ghir&rdi, G.C., Rimini, A. and Weber, T. (1980) Lett. Nuo~o Cimento 27, 293. Gleason, A.M. (1957) J. Math. Mech. 6, 885. Kochen, S. and Specker, I~.P. (1967) J. Math. Mech. 17, 59.
Viataa in Astronomy, Vol.37,pp. 317-320,1993 Printedin GreatBritain.Allfightsreserved.
EXPERIMENTAL CONSEQUENCES OF STOCHASTIC NONCONTEXTUAL HIDDEN VARIABLE THEORIES S. M. R o y and Virendra Singh Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Bombay 400 005, India
ABSTRACT We formulate stochastic noncontextukl hidden variable theories. In such theories the hidden state A specifies, not definite values, but expectation values A(A), ..- of observables A, ..and noncontextuality means that A(A) is independent of which other commuting observables commuting with A are measured together with A. We derive Bell inequalities which show that such theories conflict with quantum theory and propose a two photon experiment to test them.
In quantum theory, the expectation value of an observable A is unaffected by the previous (or simultaneous) measurement of any set of mutually commuting observables (Ghiraxdi et al. 1980) commuting with A. This 'statistical noncontextuality' provides the inspiration for the 'noncontextuality' hypothesis in hidden variable theories or deeper level theories. In a determiaistic noncontextual hidden vaxiabletheory, the value of a~q observable A is independent of which particular set of mutually commuting observables commuting with A are measured together with A. The Gleason and Kochen-Specker theorems (Gieason 1957; Kochen and Specker 1967; Bell 1966) prove the impossibility of such ,~ hidden variable theory for quantum mechanics. A particnlaxly interesting case of 'noncontextuality' arises when observables A and B commute due to spacelike separation, and the noncontextuality hypothesis becomes the Einstein locality hypothesis (Einstein et al. 1935). In this case Bell's theorem (Bell 1964; Clauser et al. 1969; Bell 1971) proves a stronger result, viz. the impossibility, not only of deterministic but ,dso of stochastic local hidden variable theory for quantum mechanics. The purpose of the present work is twofold. (1) We formulate noncontextual stochastic hidden variable theories and prove that they contradict quantum mechanical predictions. (2) We propose a two photon experiment to test inequalities on photon polarization correlations implied by noncontextual stochastic hidden variable theories and violated by quantum theory. The inequalities do not follow from locality. Evidently, noncontextuality is more stringent than locaJitv because it aDvlies to a ~reater vaxietv of ohvsieal situations.
S.M. Roy and V. Singh
318
Stochastic n o n c o n t e x t ual h i d d e n variable theories. Let At(at), .- -, A' (a,) be dynaanical variables corresponding to different degrees of freedom of a given physical system, the settings
Ai(ai) being denoted by a i. In quantum theory the dynamical variables are represented by observables A~(a;), with
of the apparatus measuring
[A,(a,), A, (a~)] = 0.
(a)
In a hidden variable theory., the state of the system may be characterized by variables ~ (which may include the quantum state vector as well), with p(A) being their probability distribution obeying i
~)>0, For given ~, the dynamical variables have
/dX~)=l
(2)
expectation valnes At(,X,at), a~(,X,a~), ..., A,(~,a.),
noncontextpality implying that A; depends only on ~ and a;, but not on a i with j ~ i. We assume that by the very definition of the A~, JA;I _< I (e.g. A~ = +1 for transmission through a polarizer and -1 for nontransmission), and hence that
IA,O,~,)I ~_ 1, Vi
(3)
Further, in complete analogy to Bell's definition of local stochastic theories (Bell 1971) we define noncontextual stochastic theories to be those in which ArAb(A, ax, a~) = At(A, at) A2(A, a2). The n-variable correlation function thus has the representation,
/
~'(~,~,,...,~.) =
d,X p(.X)
fi.4,(,X,a,).
(4)
i----1
The corresponding quantum correlation function in a state ~ is
[P(at, a~,..., a,,)]¢ = (Ol ]-I A,(a:)[O).
(5)
i_'t
T h e o r e m . There exist quantum systems for which some experimentally verifiable quantum predictions cannot be reproduced by any noncontextual stochastic hidden varibales model; i.e. there exist quantum states ~ such that
P(a~,,,,, ..-, a.) # [P(,~,, a,,.-.,a.)],~
(6)
On the other hand it is obvious that the representation for n-variable correlation functions assumed for stochastic noncontextual hidden variable theories necessarily holds in any classical theory. Hence the noncontextuality inequalities derived below yield a quantitative measure of the amount by which ouantum theory must violate any classical theory.
Experimental Consequences
319
P r o o f of t h e t h e o r e m for two suin 1/~ uarti©les. Let ~x and J~ be the Pauli spin opera-
tots for two particles. Then
[~1. ~1~2. ~2,~1. ~ 2
(~)
~2] = o,
if ~t, ~2, bt, b'2 are unit vectors obeying ~i .1_ hi, i.e., gt" gt = a2" b2 ffi 0. Let us denote a ffi {~x, g2}, b ffi {b',,g2}, and A(a) ffi 8x" 8x 82" ~2; then A(a) has eigenvalues 4-1, and [A(a), A(b)] = 0. Hence, in a stochastic noncontextual hidden variable theory we must have the correlation function P(a, b) obeying (4) with (A(~, a)) 2 _< 1. This leads to Bell's inequalities,
(8)
IP(a, b) - P(a, b')l + IP(a', b) + P(a', b')l _< 2,
provided that ~, .1. bl, ~ ± b~'~, ~ _L hi, ~ _1. bt'~for i = 1 and 2. On the otherhaud quantum mechanics gives in the singiet state ~,
(P(a,b))q.Mi = (~[A(a)A(b)[~) = (~t × bt)" (~2 × ~).
(9)
The orthogonality conditions on a,b are obeyed if we choose bx, b~, ~2 and ~2 along the negative z-axis, ~ ffi b'~ ffi ~2 = ~ = (0, 0 , - 1), and the remaining vectors in the a~-9 plane. In particulax, the choice ~x = (1,0,0), ~'x = (0,-1,0), b2 = ( - l / x / 2 , 1/x/2,0), br~ = (1/x/~, 1/Vf2,0), leads to
I
-
I+ I
+
I
which violates the noncontextuality inequality by a factor V/2. This proves the announced theorem. For experimental purposes, a two photon version of the violation of noncontextuaiity by quantum mechanics might be more practical to test for the two photon state, ~ --
(I~)I~)
- I~)I~))/v r~, where Is) and I~) denote photon states plane polarized along m and y axes
respectively. R.eplacing ~ by the 3 x 3 photon spin operator E which eqnals ~ for • and 9 components of photon spin wave function, we obtain as before Eq. (9), which violates the Bell inequalities following from noncontextuality by a factor V~, for the choice of a.b, a',b f given already. The experiment (Fig. I) will provide a test of the quantum superposition principle against the classical idea of noncontextual realism in a situation where locality is not the issue. Another striking consequence of the ideas developed here is that for a single high spin particle, quantum theory implies exponential violation of classical behaviour. This will be reported separately.
S. M. Roy and V. Singh
320
DETECTORS
DETECTORS
(~ s~'+!
(?,--,
-<-IKI"'I
I-~/
P.o,o°~
P.o,o°
'~
'
'"
• ISz'--I ' "Is..,)
(~s~ -I SLITS TWO
CHANNEL
ELLIPTIC
POLARIZERS
FIG.I Schematic diagram of experimental apparatus to test the noncontextuality inequality. The source S emits two photons in the polarization state (l~)ly) - ly)lx))/V~, along - z and +z-axis. The left photon goes throngh an elliptic polarizer measuring ~x" '~1 = rl = +1, and then through another elfiptic polarizer measuring ~31 • bx = sl - 4-1. The right photon similarly goes through two channel elliptic polarizers measuring ~2 • ~2 --- r2 = 4-1, and ~ . b2 = s2 = 4-1. Here, at _L bx and ~, J. ~2. Detectors on the left and right &re wired to measure 2' = 16 coincidence counts N,,,,,,2,~, and hence P(a, b) =
'~
,.,.r.,s.,.s.,.,,v,,.... ,,,/z.,,'v,,,,,,,,,.
f l #1 ~t't jB]I
References Bell, J.S. (1964) Physics (Long Island Cityj N.Y.) I, 195. Bell, J.S. (1966) Re~. Mod. Phys. 38, 447. Bell, J.S. (1971) in "Foundations of Q~antum Mechanics", International School of Physics, "Enrico Fermi", Course IL (Academic, New York), p. 171. Clanser, J.P., Home, M.A., Schimony, A. and Holt, R.A. (1969) Phys. Re~. Left. 26, 880. Einstein, A., Podolsky, B. and Rosen, N. (1935) Phys. Re~. 47, 777. Ghir&rdi, G.C., Rimini, A. and Weber, T. (1980) Lett. Nuo~o Cimento 27, 293. Gleason, A.M. (1957) J. Math. Mech. 6, 885. Kochen, S. and Specker, I~.P. (1967) J. Math. Mech. 17, 59.