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<ab>+<ab’>+<a’b>—<a’b’>I~2,
where <ab> = <A ®B>
(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.
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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
<ab>
=
<A (a) ®B(fl) > 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
<ab> + <ab’> + <a’b> — <a’b’>
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
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Then, finally, ordinary Stern—Gerlach experiments
performed on these two beams give the correlation
<a2®a2 >,~,,=—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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