Maximal violation of Bell's inequality for arbitrarily large spin

Physics LettersA 162 (1992) 15—17 North-Holland

PHYSICS LETTERS A

Maximal violation of Bell’s inequality for arbitrarily large spin N. Gisin Group ofApplied Physics, University of Geneva, 1211 Geneva 4, Switzerland

and A. Peres Department ofPhysics, Technion — Israel Institute ofTechnology, 32 000 Haifa, Israel Received 4 November 1991; accepted for publication 2 December 1991 Communicated by J.P. Vigier

For any nonfactorable state of two quantum systems, it is possible to find pairs ofobservables whose correlations violate Bell’s inequality. In the case oftwo particles of spin j prepared in a singlet state, the violation of Bell’s inequality remains maximal for arbitrarily largej. It is thus seen that large quantum numbers are no guarantee ofclassical behaviour.

The violation of Bell’s inequality by quantum theoryis the most radical departure of quantum physics from classical local realism. Early proofs of violation [1,2] involved pairs of spin ~ particles in a singlet state, or polarization components of correlated photons, which have similar algebraic properties. These proofs were later generalized by Mermin et al. [3—5]to pairs of spin j particles, but the magnitude of the violation rapidly decreased for large 3. In this article, we show that for any nonfactorable state of two quantum systems, it is possible to find pairs of observables whose correlations violate Bell’s inequality. The proof presented here is simpler, and gives stronger correlations, than the one previously published by one ofus [61. In the special case oftwo particles of spin j in a singlet state, the violation of Bell’s inequality is as large as for spin which gives the maximal attainable value [7,81. This improves a recent result [9] where the violation was asymptotically constant for j—* oo, but it was only 24%, instead of 41% in the present work. Finally, we describe a conceptual experiment which can substantiate this result. ~,

Theorem. Let ~ If ~vis not factorable, there exist observables A®B, with eigenvalues ±1,

whose correlations violate Bell’s inequality [2] I++—I~2, where =

(1)

,,.

Proof Any w can be written as a Schmidt biorthogonal sum ~ c~Ø1®O~

(2)

where {Ø~}and {O,} are orthonormal bases in and ~ respectively. It is possible to choose their phases so that all the c• are real and nonnegative, and to label them so that c1 ~ c2 ~ ~ 0. We can now restrict .~

...

our attention to the N-dimensional subspaces of ~ and ,W~which correspond to nonvanishing c~.A nonfactorable state is one for which N> 1. With orthonormal bases defined as above, let f’. and 1~be block-diagonal matrices, in which each block is an ordinary Pauli matrix, o~and a~,respectively. That is, the only nonvanishing elements ofT’~ and P~are (F~) 1~)2n..l,2~= ( 2~,2fl_, = I (3) ,

and (f’~)~~ = (— l)’~ .

0375-9601 /92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

(4) 15

Volume 162, number 1

PHYSICS LETtERS A

However, if N is odd — which slightly complicates the proof—we shall take (f’Z)NN=Oinsteadofl,and define still another matrix, 17, whose only nonvanishing element is 11NN= 1. If N is even, 17 is the null matrix. Furthermore, it is convenient to define a number y by y=c~ (odd N), ‘=O (even N). (5) With the above notations, let us define observables

A(a)=f’~sin a+[’~cos a+H,

(6)

and B(/3) =1’ sin /3+T’~cos /3+17.

(7)

The eigenvalues of A (a) and B(/3) are ±1, and the correlation of these observables is

=

w

=(l—y) cos a cos fl+Ksin a sin fl+y,

(9)

or A or B, are not obtainable from f’~by a rotation in our physical P’ space (except for the trivial case J= ~). However, an equivalent procedure is to perform separate unitary rotations, for each beam, in the subspace spanned by Ørn and 0—rn (and likewise

—

—

(10)

which contradicts eq. (1). Q.E.D. Corollary. In the special case ofa pair ofspinj partides in a singlet state, we have [3] 2 Vi, (11) c,=(2j+l)’’ and therefore K= 1 if 21+ 1 is even. The right hand side of eq. (10) then is 2~h,which is the maximal violation of Bell’s inequality allowed by Cirel’son’s theorem [7,8]. If however 2j is even, one term of the sum (9) has no partner and K is only 2j/ (21+ 1). The right hand side of eq. (10) then becomes 2(2~hf+ 1)1(21+ 1), which tends to 2~.J~(1 0.1464/f), for largej. (For j= 1, the result is 2.552. It is an open question whether a higher result could be obtained by means of a different set of operators.) —

Experimental verification. Finally, let us outline a 16

order of indices is j, —j, j — 1, 1 —j, etc.). Moreover, these states are correlated: the two particles will never be found in beams with different m2 The next problem is to measure the matrices A and B (or their relevant submatrices). Here, the difficulty is that only T’~can easily be measured by an ordinary Stern—Gerlach experiment. The matrices 1~,

is always positive for a nonfactorable state. In particular, if we choose a = 0, a’ = It/2, and /3= /3’ = tan—’ [K! (1 y)], we obtain + + — 2+K2] 112~2y =2 [(1 —y)

conceptual experiment aimed at verifying the correlation (8). We take for granted [3—5] that it is possible to prepare two spin j particles in a singlet state. Let these particles have not only a magnetic moment (that is, an interaction energy 1zB2J2) but also an electric quadrupole moment (an interaction energy proportional to EZJZ~). The two particles, flying away from each other, first pass through parallel but inhomogeneous electric fields. This is an electrostatic analogue of the Stern—Gerlach experiment, which produces beams with Im~I=1~j— 1, ~ (or 0). The state of each one of these beams lies in the subspace of ~ or ,$~which corresponds to one of the 2x2 blocks in the A and B matrices (here, the

(8)

where K= 2 (c1 c2 + c3 c4 +...)

27 January 1992

by Orn and 0_rn). This can be donebylettingeach one form field given B~.Thism2 produces an energy difof themagnetic beams, with pass through a uniference E=2um 2B~between the two components (±m~)of the beam. Then, an rf pulse, with the ap~,

propriate frequency w=E/h, can generate transitions (Rabi oscillations): 0m~0’rn=0rncosa+0_,,,sma, 0_,n~0_,n=0_rnc0sa—0ms’b0~,

(12) (13)

and 0rn~0’rn0m

0_rn

cos

/30_rn

0~.rn= 0_rn COS

sin /3,

/3+ Orn sin

(14) /3,

(15)

where the rotation angles a and /3 are proportional to the intensities of the rf pulses and are independently controlledstate by the two distant to observers. The entangled corresponding the mth pair of beams thus evolves from y/rn = (Orn ® 0_rn + Ø~ ~n 0 0rn) /,,/~,

(16)

Volume 162, number 1

PHYSICS LETTERS A

to (17)

Wn = (0’rn®0’~m+0’..rn®0~n)/~/~ =[(Orn®0_rn+0_rn®0rn)

cos(a—fl)

— (Orn ®Orn — 0_rn 00_rn)

sin (a /3)] —

/,.J~.

27 January 1992

Work by AP was supported by the Gerard Swope Fund and by the Fund for Encouragement of Research at Technion.

References

(18) Then, finally, ordinary Stern—Gerlach experiments performed on these two beams give the correlation ,~,,=—dos 2(a—/3) ,

(19)

which leads to the familiar maximal violation of inequality (1). The fact that particles with arbitrarily large spin can maximally violate Bell’s inequality corroborates the view that classical properties do not automatically emerge for “large” quantum systems, whatever “large” may mean: assemblies of many subsystems, as in refs. [10] and [11], or states with large quantum numbers, as in the present work.

[1] J.S. Bell, Physics 1 (1964)195. [2] J,F. Clauser, M.A. Home, A. Shimony and R.A. Holt, Phys. Rev. Lett. 23 (1969) 880. [3] N.D. Mermin, Phys. Rev. D 22 (1980) 356. [4]N.D. Mermin and G.M. Schwarz, Found. Phys. 12 (1982) 101. [5] A. Garg and N.D. Mermin, Phys. Rev. Lett. 49 (1982) 901, 1294. [6] N. Gisin, Phys. Lett. A 154 (1991) 201. [7]B.S. Cirel’son, Lett. Math. Phys. 4 (1980) 93. [8] L.J. Landau, Phys. Lett. A 120 (1987) 54. [9] A. Peres, Finite violation of a Bell inequality for arbitrarily largespin, Phys. Rev. A, submitted for publication. [10] N.D. Mermin, Phys. Rev. Lett. 65 (1990) 1838. [11] C. Pagonis, M.L.C. Redhead and R.K. Clifton, Phys. Lett. A 155 (1991) 441.

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