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Bell's inequalities for quantum mechanics
PhysicsLettersA 167 (1992) 6-10 North-Holland
PHYSICS LETTERS A
Bell's inequalities for quantum mechanics H.E. AndAs Department of Physics, Universityof Oslo, P.O. Box 1048,Blindern, N-0316Oslo 3, Norway Received4 October 1991;revisedmanuscript received 13 May 1992;acceptedfor publication 13 May 1992 Communicatedby J.P. Vigier
Inequalities correspondingto the generalizedBell inequalities of local realism are derived for the quantum case. The extremal values permittedby these inequalitiesexceedthose allowedby the generalizedBellinequalities. Quantum predictionsfor systems of two spin-½particlesprepared as mixturesdo not violate Bell's inequalities.
With his famous paper establishing Bell's inequalities, Bell [ 1 ] showed that quantum theory is incompatible with realism in the form expressed by local hidden variable theories. Later, several authors have confirmed and sharpened his results, providing inequalities by more general approaches [ 2-10 ]. However, a characterization of the set of quantum states giving predictions contradicting those of local realism has only rarely been attempted [ 1 1-13 ] and should not be considered exhaustive. We will therefore in the following examine the situation more closely, first concentrating on the derivation of inequalities, corresponding to the generalized Bell-tyoe inequalities, for the quantum case. Let {~iul/z= 1, 2 .... } and { ~ I v = 1, 2, ...} be sets ofdichotomic variables pertaining to each of the two particles forming a two-particle system prepared in a state p. Define the quantity ~ - - - ~ , cu~a~,®/~, c ~ e ~ and let L - Ep ( ~ ) = Y~~ c~Ep (~, ® 6~ ) be the expectation value of this quantity in the state p. Then the following generalized version of Bell's theorem applies. Theorem 1. (i) If the state p allows a description of its properties by classical probability concepts, then ILcul <~Mo, with Mo=max(Y,~ c~,~,~/~), ~ (r/u) = + 1 V/z(o). (ii) I f p is a state as described by quantum mechanics, then 1/2
+ ,u'V' o' ~ o
Iz' ~ , u
where the upper limit can be reached by pure states only. Proof. Garuccio et al. [9] have proven (i) for so-called "Einstein-local" theories, i.e. theories satisfying a probabilistic reality criterion and a corresponding separability principle specified in ref. [ 9 ]. A theorem providing partially equivalent results for the case c~,~= + 1, essentially stating that the existence of a joint probability distribution for the variables ~ ® 6~ implies inequalities of type (i), is due to Suppes and Zanotti [ 10 ]. (ii) Using Ep (ti~,® 6~) = Tr (p~, ® 6~), one obtains ILQM l = ~ cuoEp(Cl,~®~v)= E ¢uu Tr(/~iu ® / ~ ) = T r ( / ~ C ) , ,uo
(1)
gv
where/~ is the density operator of the state. Let now dr, /~ and fi be represented by their square-matrix representations a t, b. and p. These representations are Hermitian square matrices, constituting a linear space .g. 6
0375-9601/92/$ 05.00 © 1992 ElsevierSciencePublishers B.V. All fights reserved.
Volume 167, number l
PHYSICSLETTERSA
6 July 1992
Equally, the trace of the matrix products of ~¢/satisfy the conditions for a scalar product on this space, i.e. (x, y) - T r ( x y ) for x, ye.¢/. This definition allows the introduction of a scalar product norm N2(x) =Tr(x2), implying that the unity norm be given by N2( 1 ) = T r ( 1 ). The Cauchy-Schwarz inequality then applies, giving 1
1
ILQM Iz= N 2 ( I ® l ) ITr(pC)124 T - - ~
Tr(p2) T r ( C 2)
(2)
for the appropriately normalized L. By a~ = 1 =b~ for dichotomic variables, C = Y~,~c~,~a.®b~ entrains T r ( C 2 ) = ~ c ~ Tr2(1)+ uv
+
Y. c~,~c~,~,Tr(b~b~,)+ .uuu'
~.
F. cF,~c.,~Tr(aua.,) vl.z#'
c.~c~,,~,Tr(a~a~,) Tr(b~b~,).
(3)
,u' ¢ I t , v ' ~ : v
Applying again the Cauchy-Schwarz inequality to the case of two matrices x and y representing dichotomic variables, i.e. ITr(xy) 12~Tr(x 2) Tr(y 2) =Tr2( 1 ), one arrives at
Tr(C2)=ITr(C2)I<~Tr2(1) Yc.'2~+~.
~,~ ~c~,.c~,~, ITr(b~b~,)l+ ..'~" ~c.~c.,~ ITr(a~a.,)l u' ~ u
+
~
lt' v~ ,u
Ic~,.c.,~,I lTr(a~,a~,,)llTr(b~b~,)l
la u ,u ' v '
(; ¢ .
c..c.. ÷
. ~c#,c~,.~-t-~lc,,~l
~.,.~ Ic,,,~,l).
(4)
I1' ~11
For the density matrix p, we have Tr(p 2) ~ 1 ,
(5)
of which the upper limit can be reached only by pure states. Now, by substituting the appropriate quantities in eq. (2) by eqs. (4) and (5), we arrive at the desired result. Q.E.D. The generalized inequalities can be illustrated by two examples. Taking
CB =al ®bl +al ®b2 q-a2®b2 -a2®bl , and
Cr. =al ®bl +al ®b2 d-al ®b3 -al ®b4 -a2 ®bl +aE®b2 +aE®b3 -a2®b4 + 2a3 ® b2
-
a3 ® ba + a3 ® b4 + a4 ® b3 + a4 ® b4,
we obtain Bell's [ 1 ] and Kempermann's [ 10,14] inequality in the case where (i) applies. For CB one gets (i) ILcLI ~<2 (Bell's ineq~lity), (ii) ILQMI ~<2v/2. For CK we have (i) ILcLI ~<6 (Kempermann's inequality), (ii) ILQMI ~2x/33. In the last case the classical inequality is violated by 92%, while for the Bell case the violation is 41%. It is realized from these considerations that Kempermann's inequality considerably sharpens the conflict between quantum mechanics and a classical description. This is also recognized by the fact that Kempermann's inequality is a superinequality, i.e. Mo < ½Y~,~ Icj,~I, which means that it provides restrictions on a theory that cannot be provided by any Bell inequality [ 10,14 ]. From theorem 1 an important lemma follows, first noted by Landau [ 7 ].
V o l u m e 167, n u m b e r 1
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6 July 1992
Lemma 1. Let C=a~ ®b~ +a~ ®b2 +a2®b2 -a2®bl. Then ILQMI ~
(6)
If [a~, a2] =0 or [b~, b2] =0, eq. (6) is expressed as Tr(C 2) = 4 Tr2( 1 ) .
(7)
Substituting the appropriate quantities in (2) by eqs. (7) and (5) then gives ILQMI ~<2. Q.E.D. Further interesting aspects can be studied by specifying to a system of two spin-½ particles. A general density matrix describing the spin of one (single) of these particles is given by
p1=1(1+ ~ r i a ' ) ,
~i r2<~l,
(8)
where a~, i= 1, 2, 3 are the standard Pauli matrices and 1 is the 2 × 2 identity matrix [ 15]. The upper limit ~ r 2 = 1 can be reached by pure states only. The general density matrix of the two-particle system can then be given by ) ,i] , p = 1 [ 1®1+ ( ~ r,a] ® 1 + 1 ® ( ~ sJ)all + Etnma~1 ®am
~i r2+ E S2+ E t2m~< 3.
mn
j
(9)
nm
Again, the upper limit is obtainable for pure states only. In this formalism, the density matrix for the pure state is found by p V= V, where V is the vector representation of the state in Hilbert space. For e.g. the singlet s state }J=0) = ( 1/x/~) ( Iu d ) z - Idu)z) we obtain po = ¼( 1® 1 - ~-11ol -I ~ O 1 -- ~1 ~'2 t¢~ ~¢~'t~ ~II2 _ _ 6 3 ® 6 3 1 II ). Using P~ =- ½( 1 + ~r1"n) and Q ~ - ½( 1 + ~rn" m), the operators projecting the spins of particles I and II along the axes n and m respectively, a mixture of these spin projections, representing a mixed state of the two-particle system, is expressed as
W=aP~_ ®Q~ + flP~ ®Q=_+ ?P~ ®Q~ +2P ~_ ®Q_=.
(10)
Here a, r, ? and 2 are the probabilities of observing the spins of the two particles as up up, up down, down up and down down the n and m axes respectively. Consequently, a, r, ?, 2>/0 and o r + r + 7 + 2 = I. Equation (10) can be written in the form (9) as W= ~ [1®1 + (ot+fl-7-2)a~®l + ( c t - r + ~ , - 2 ) l ® a ~ + (a-fl-~,+2)aI®a~] , ak=o'k= ~ k~ai. i
(11) This formalism may now be used to prove the following theorem.
Theorem 2. For a two-particle spin-½ system prepared as a mixture W,
[ L Q M [ ~ M 0.
Proof. By writing it in the form ( 11 ), it is recognized that Wcan be transformed into a diagonal matrix W d by an appropriate unitary transformation U= UI@ U n, i.e. Wa= UI@ UnW( U I) *@ ( U I1) *. Since the trace of a matrix is conserved under unitary transformations, we obtain for the appropriately normalized L:
V o l u m e 167, n u m b e r
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PHYSICS
LETTERS
A
6 July 1992
1 1 ILoM I -- N( 1® 1~ ITr(WC) I - Tr ( 1 ) ITr( WdC d) I
, Tr(
-- 4Tr(1)
cup[a(ad ® b vd + a .Id au®bv d {::1+au®a,,, d Ild b~+ d a nId a ud® u m- - l I d iuk vd ~!
+ B ( a u ®db u . . l _c1- - X d ~ d , o , t . d
~ d K:~ --lid L d
--Id - d ,¢x --IId/..d x
--On ¢d.'u~Oo--U'U~O m Up--O n fd'u~O m Up)
+7(a d® b ~d - o . Id au®bp d d d lid d d Ild d +a'u®a., b p - a nId au®o., b~) d Id d d ) + 2 ( a d, ® b pd - a . Id a ud ® b pd - a ud® a , ,lid b~+o. au®a,,lid bp)]
=
-
8
C~p{a [ (adl,
d d +bp~) d + a'u~:) (bp,, +
+ (adl, +au~2)(bp, d ~_b~:2) + (ad, '
d
(adll
d d +b~) -a'u~) (bp,,
d
d
+ ~ [ (ad,, +ads:) (bdl, +bd:~) + (ad,, -ad~D (bd,, + b¢:,)
+7[
+am2)(bp _ b ~ ) _ ( a d . -a'u~) d (bp. d -b~2) ] (a'u,, +b~:~)_(ad -a'u2~) d d +bp~2) d d +ap::)(bd d (bp,,
d d d +(a'u.+a'u~)(bp,,_bd _~_ ~ [ ( a d l ,
)_(ad
d d +bd~) +amO (bp,,
-
d d -a'u,~)(bp,,-b~:~) ]
(ad,, -a'u,,)(bv,, ,1 d +bd2~)
- (a.,, +a'u~:) (bp,, -bp~:) + (a'u,, - a "u~) ( b p,. - b p~2) d d +4,8au,,bp~ d d d d - 81 ~ c'u~(4ota'u,,bp,, ~+47am2bp, , + 42ad~,bd20
<.~ Ol Z ¢.upa'uHd
+B
C'upa'u.d
,up
+7
+2
c'upa'u~2bp22aa
,
(12)
"UV
since or, p, 7, 2>_-0. For dichotomic observables x, Tr(x:) =x~:l +x~2 + 2 ]x~2 [:=Tr(1 ) =2, implying Ix.I Thus, and V i,j. Since I Y'up C'upad~bd~jjI is linear in the two independent variables ad. and bd., the maximum for this quantity will be found on the boundary of the variables, i.e. lad,, I = x / ~ = Ib~jj I. Furthermore, the maximum depends only on the coefficients c'up, so that one can perform the substitutions a~--,x/~'u and b¢,~x/~np, ~'u (qp) = + 1. Thus,
c'upad,,b¢, ~<2max ~ c'up¢'uqp - 2 M o . Introducing eq. (13) into eq. (12) and recalling that a + f l + ~ + 2 = +2)Mo =Mo. Q.E.D.
(13) 1, we finally obtain ILQ~[~< ( a + f l +
For the special case ~ = fil ®/~ + d~ ®/~: + t~z ®/~2 - fi~ ®/~, i.e. Bell's inequality, a corresponding theorem has been proven by Capasso et al. [ 11 ]. Again, this theorem has an interesting lemma.
9
Volume 167, number 1
PHYSICS LETTERS A
6 July 1992
Lemma 2. F o r p r o d u c t states, i.e. factorizable states represented by density matrices o f the form P = e~ ®Q~,
ILQM[ ~
Proof. The l e m m a is p r o v e n by observing that P is a special case o f W with either a,/~, 7 or 2 equal to unity. Q.E.D. A recent p r o o f o f a t h e o r e m by G i s i n [ 13 ] implicitly contains the p r o o f o f l e m m a 2 for the Bell case. T h e o r e m 2 a n d its l e m m a thus establish the c o m p a t i b i l i t y between local realism and certain aspects o f the q u a n t u m description as related to the generalized Bell theorem. In this respect it represents an extension o f the works by Capasso et al. [ 11 ] a n d G i s i n [ 13 ]. It has been shown that the vector-space structure typical for the q u a n t u m description leads to inequalities that are i n c o m p a t i b l e with the a s s u m p t i o n s o f local realism. However, this does not hold for the case o f twop a n i c l e spin-½ systems p r e p a r e d as mixtures or in p r o d u c t states, indicating that the q u a n t u m description o f multi-particle systems possesses an inherent non-locality ( " e n t a n g l e m e n t " [ 13 ] ) with no classical analogy. A b r o a d e r discussion o f this a n d related issues bearing on their implications for the conceptual foundations o f q u a n t u m mechanics has been given a separate presentation [ 16 ]. I would like to t h a n k K. G j ~ t t e r u d for helpful discussions and for his reading o f the manuscript. Equally, s u p p o r t by the N o r w e g i a n Research Council for Science a n d the Humanities, reference no. 420.88/004, is gratefully acknowledged.
References [ 1] J.S. Bell, Physics 1 (1964) 195. [2] J. Clauser, M. Home, A. Shimony and R. Holt, Phys. Rev. Lett. 23 (1969) 880. [ 3 ] J.F. Clauser and M.A. Home, Phys. Rev. D 10 (1974) 526. [4 ] S.M. Roy and V. Singh, J. Phys. A 11 ( 1978 ) L 167. [ 5 ] A. Fine, Phys. Rev. Len. 48 (1982) 291. [6] E. Santos, Phys. Lctt. A 115 (1986) 363. [7] L.J. Landau, Phys. Lett. A 120 (1987) 54. [ 8 ] S.P. Gudder, Quantum probability (Academic Press, New York, 1988 ). [9] A. Garuccio, V.L. Lcpore and F. Selleri, Found. Phys. 20 (1990) 1173. [ 10] P. Suppes and M. Zanotti, Found. Phys. Lett. 4 ( 1991 ) 101. [ 11 ] V. Capasso, D. Fortunato and F. Selleri, Int. J. Theor. Phys. 7 (1973) 319. [ 12 ] J. Baez, Lctt. Math. Phys. 13 (1987) 135. [ 13 ] N. Gisin, Phys. Lctt. A 154 ( 1991 ) 201. [ 14] F. Selleri and A. van der Merwe, eds., Quantum paradoxes and physical reality (Kluwer, Dordrecht, 1990) pp. 290-302. [ 15 ] E.G. Beltrametti and G. Cassinelli, The logic of quantum mechanics (Addison-Wesley, Reading, MA, 1981 ). [ 16 ] H.E. And,Jtsand O.K. Gjetterud, Department of Physics Report 91-22, University of Oslo ( 1991 ), submitted to Found. Phys.
10
PHYSICS LETTERS A
Bell's inequalities for quantum mechanics H.E. AndAs Department of Physics, Universityof Oslo, P.O. Box 1048,Blindern, N-0316Oslo 3, Norway Received4 October 1991;revisedmanuscript received 13 May 1992;acceptedfor publication 13 May 1992 Communicatedby J.P. Vigier
Inequalities correspondingto the generalizedBell inequalities of local realism are derived for the quantum case. The extremal values permittedby these inequalitiesexceedthose allowedby the generalizedBellinequalities. Quantum predictionsfor systems of two spin-½particlesprepared as mixturesdo not violate Bell's inequalities.
With his famous paper establishing Bell's inequalities, Bell [ 1 ] showed that quantum theory is incompatible with realism in the form expressed by local hidden variable theories. Later, several authors have confirmed and sharpened his results, providing inequalities by more general approaches [ 2-10 ]. However, a characterization of the set of quantum states giving predictions contradicting those of local realism has only rarely been attempted [ 1 1-13 ] and should not be considered exhaustive. We will therefore in the following examine the situation more closely, first concentrating on the derivation of inequalities, corresponding to the generalized Bell-tyoe inequalities, for the quantum case. Let {~iul/z= 1, 2 .... } and { ~ I v = 1, 2, ...} be sets ofdichotomic variables pertaining to each of the two particles forming a two-particle system prepared in a state p. Define the quantity ~ - - - ~ , cu~a~,®/~, c ~ e ~ and let L - Ep ( ~ ) = Y~~ c~Ep (~, ® 6~ ) be the expectation value of this quantity in the state p. Then the following generalized version of Bell's theorem applies. Theorem 1. (i) If the state p allows a description of its properties by classical probability concepts, then ILcul <~Mo, with Mo=max(Y,~ c~,~,~/~), ~ (r/u) = + 1 V/z(o). (ii) I f p is a state as described by quantum mechanics, then 1/2
+ ,u'V' o' ~ o
Iz' ~ , u
where the upper limit can be reached by pure states only. Proof. Garuccio et al. [9] have proven (i) for so-called "Einstein-local" theories, i.e. theories satisfying a probabilistic reality criterion and a corresponding separability principle specified in ref. [ 9 ]. A theorem providing partially equivalent results for the case c~,~= + 1, essentially stating that the existence of a joint probability distribution for the variables ~ ® 6~ implies inequalities of type (i), is due to Suppes and Zanotti [ 10 ]. (ii) Using Ep (ti~,® 6~) = Tr (p~, ® 6~), one obtains ILQM l = ~ cuoEp(Cl,~®~v)= E ¢uu Tr(/~iu ® / ~ ) = T r ( / ~ C ) , ,uo
(1)
gv
where/~ is the density operator of the state. Let now dr, /~ and fi be represented by their square-matrix representations a t, b. and p. These representations are Hermitian square matrices, constituting a linear space .g. 6
0375-9601/92/$ 05.00 © 1992 ElsevierSciencePublishers B.V. All fights reserved.
Volume 167, number l
PHYSICSLETTERSA
6 July 1992
Equally, the trace of the matrix products of ~¢/satisfy the conditions for a scalar product on this space, i.e. (x, y) - T r ( x y ) for x, ye.¢/. This definition allows the introduction of a scalar product norm N2(x) =Tr(x2), implying that the unity norm be given by N2( 1 ) = T r ( 1 ). The Cauchy-Schwarz inequality then applies, giving 1
1
ILQM Iz= N 2 ( I ® l ) ITr(pC)124 T - - ~
Tr(p2) T r ( C 2)
(2)
for the appropriately normalized L. By a~ = 1 =b~ for dichotomic variables, C = Y~,~c~,~a.®b~ entrains T r ( C 2 ) = ~ c ~ Tr2(1)+ uv
+
Y. c~,~c~,~,Tr(b~b~,)+ .uuu'
~.
F. cF,~c.,~Tr(aua.,) vl.z#'
c.~c~,,~,Tr(a~a~,) Tr(b~b~,).
(3)
,u' ¢ I t , v ' ~ : v
Applying again the Cauchy-Schwarz inequality to the case of two matrices x and y representing dichotomic variables, i.e. ITr(xy) 12~Tr(x 2) Tr(y 2) =Tr2( 1 ), one arrives at
Tr(C2)=ITr(C2)I<~Tr2(1) Yc.'2~+~.
~,~ ~c~,.c~,~, ITr(b~b~,)l+ ..'~" ~c.~c.,~ ITr(a~a.,)l u' ~ u
+
~
lt' v~ ,u
Ic~,.c.,~,I lTr(a~,a~,,)llTr(b~b~,)l
la u ,u ' v '
(; ¢ .
c..c.. ÷
. ~c#,c~,.~-t-~lc,,~l
~.,.~ Ic,,,~,l).
(4)
I1' ~11
For the density matrix p, we have Tr(p 2) ~ 1 ,
(5)
of which the upper limit can be reached only by pure states. Now, by substituting the appropriate quantities in eq. (2) by eqs. (4) and (5), we arrive at the desired result. Q.E.D. The generalized inequalities can be illustrated by two examples. Taking
CB =al ®bl +al ®b2 q-a2®b2 -a2®bl , and
Cr. =al ®bl +al ®b2 d-al ®b3 -al ®b4 -a2 ®bl +aE®b2 +aE®b3 -a2®b4 + 2a3 ® b2
-
a3 ® ba + a3 ® b4 + a4 ® b3 + a4 ® b4,
we obtain Bell's [ 1 ] and Kempermann's [ 10,14] inequality in the case where (i) applies. For CB one gets (i) ILcLI ~<2 (Bell's ineq~lity), (ii) ILQMI ~<2v/2. For CK we have (i) ILcLI ~<6 (Kempermann's inequality), (ii) ILQMI ~2x/33. In the last case the classical inequality is violated by 92%, while for the Bell case the violation is 41%. It is realized from these considerations that Kempermann's inequality considerably sharpens the conflict between quantum mechanics and a classical description. This is also recognized by the fact that Kempermann's inequality is a superinequality, i.e. Mo < ½Y~,~ Icj,~I, which means that it provides restrictions on a theory that cannot be provided by any Bell inequality [ 10,14 ]. From theorem 1 an important lemma follows, first noted by Landau [ 7 ].
V o l u m e 167, n u m b e r 1
PHYSICS LETTERS A
6 July 1992
Lemma 1. Let C=a~ ®b~ +a~ ®b2 +a2®b2 -a2®bl. Then ILQMI ~
(6)
If [a~, a2] =0 or [b~, b2] =0, eq. (6) is expressed as Tr(C 2) = 4 Tr2( 1 ) .
(7)
Substituting the appropriate quantities in (2) by eqs. (7) and (5) then gives ILQMI ~<2. Q.E.D. Further interesting aspects can be studied by specifying to a system of two spin-½ particles. A general density matrix describing the spin of one (single) of these particles is given by
p1=1(1+ ~ r i a ' ) ,
~i r2<~l,
(8)
where a~, i= 1, 2, 3 are the standard Pauli matrices and 1 is the 2 × 2 identity matrix [ 15]. The upper limit ~ r 2 = 1 can be reached by pure states only. The general density matrix of the two-particle system can then be given by ) ,i] , p = 1 [ 1®1+ ( ~ r,a] ® 1 + 1 ® ( ~ sJ)all + Etnma~1 ®am
~i r2+ E S2+ E t2m~< 3.
mn
j
(9)
nm
Again, the upper limit is obtainable for pure states only. In this formalism, the density matrix for the pure state is found by p V= V, where V is the vector representation of the state in Hilbert space. For e.g. the singlet s state }J=0) = ( 1/x/~) ( Iu d ) z - Idu)z) we obtain po = ¼( 1® 1 - ~-11ol -I ~ O 1 -- ~1 ~'2 t¢~ ~¢~'t~ ~II2 _ _ 6 3 ® 6 3 1 II ). Using P~ =- ½( 1 + ~r1"n) and Q ~ - ½( 1 + ~rn" m), the operators projecting the spins of particles I and II along the axes n and m respectively, a mixture of these spin projections, representing a mixed state of the two-particle system, is expressed as
W=aP~_ ®Q~ + flP~ ®Q=_+ ?P~ ®Q~ +2P ~_ ®Q_=.
(10)
Here a, r, ? and 2 are the probabilities of observing the spins of the two particles as up up, up down, down up and down down the n and m axes respectively. Consequently, a, r, ?, 2>/0 and o r + r + 7 + 2 = I. Equation (10) can be written in the form (9) as W= ~ [1®1 + (ot+fl-7-2)a~®l + ( c t - r + ~ , - 2 ) l ® a ~ + (a-fl-~,+2)aI®a~] , ak=o'k= ~ k~ai. i
(11) This formalism may now be used to prove the following theorem.
Theorem 2. For a two-particle spin-½ system prepared as a mixture W,
[ L Q M [ ~ M 0.
Proof. By writing it in the form ( 11 ), it is recognized that Wcan be transformed into a diagonal matrix W d by an appropriate unitary transformation U= UI@ U n, i.e. Wa= UI@ UnW( U I) *@ ( U I1) *. Since the trace of a matrix is conserved under unitary transformations, we obtain for the appropriately normalized L:
V o l u m e 167, n u m b e r
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PHYSICS
LETTERS
A
6 July 1992
1 1 ILoM I -- N( 1® 1~ ITr(WC) I - Tr ( 1 ) ITr( WdC d) I
, Tr(
-- 4Tr(1)
cup[a(ad ® b vd + a .Id au®bv d {::1+au®a,,, d Ild b~+ d a nId a ud® u m- - l I d iuk vd ~!
+ B ( a u ®db u . . l _c1- - X d ~ d , o , t . d
~ d K:~ --lid L d
--Id - d ,¢x --IId/..d x
--On ¢d.'u~Oo--U'U~O m Up--O n fd'u~O m Up)
+7(a d® b ~d - o . Id au®bp d d d lid d d Ild d +a'u®a., b p - a nId au®o., b~) d Id d d ) + 2 ( a d, ® b pd - a . Id a ud ® b pd - a ud® a , ,lid b~+o. au®a,,lid bp)]
=
-
8
C~p{a [ (adl,
d d +bp~) d + a'u~:) (bp,, +
+ (adl, +au~2)(bp, d ~_b~:2) + (ad, '
d
(adll
d d +b~) -a'u~) (bp,,
d
d
+ ~ [ (ad,, +ads:) (bdl, +bd:~) + (ad,, -ad~D (bd,, + b¢:,)
+7[
+am2)(bp _ b ~ ) _ ( a d . -a'u~) d (bp. d -b~2) ] (a'u,, +b~:~)_(ad -a'u2~) d d +bp~2) d d +ap::)(bd d (bp,,
d d d +(a'u.+a'u~)(bp,,_bd _~_ ~ [ ( a d l ,
)_(ad
d d +bd~) +amO (bp,,
-
d d -a'u,~)(bp,,-b~:~) ]
(ad,, -a'u,,)(bv,, ,1 d +bd2~)
- (a.,, +a'u~:) (bp,, -bp~:) + (a'u,, - a "u~) ( b p,. - b p~2) d d +4,8au,,bp~ d d d d - 81 ~ c'u~(4ota'u,,bp,, ~+47am2bp, , + 42ad~,bd20
<.~ Ol Z ¢.upa'uHd
+B
C'upa'u.d
,up
+7
+2
c'upa'u~2bp22aa
,
(12)
"UV
since or, p, 7, 2>_-0. For dichotomic observables x, Tr(x:) =x~:l +x~2 + 2 ]x~2 [:=Tr(1 ) =2, implying Ix.I Thus, and V i,j. Since I Y'up C'upad~bd~jjI is linear in the two independent variables ad. and bd., the maximum for this quantity will be found on the boundary of the variables, i.e. lad,, I = x / ~ = Ib~jj I. Furthermore, the maximum depends only on the coefficients c'up, so that one can perform the substitutions a~--,x/~'u and b¢,~x/~np, ~'u (qp) = + 1. Thus,
c'upad,,b¢, ~<2max ~ c'up¢'uqp - 2 M o . Introducing eq. (13) into eq. (12) and recalling that a + f l + ~ + 2 = +2)Mo =Mo. Q.E.D.
(13) 1, we finally obtain ILQ~[~< ( a + f l +
For the special case ~ = fil ®/~ + d~ ®/~: + t~z ®/~2 - fi~ ®/~, i.e. Bell's inequality, a corresponding theorem has been proven by Capasso et al. [ 11 ]. Again, this theorem has an interesting lemma.
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Volume 167, number 1
PHYSICS LETTERS A
6 July 1992
Lemma 2. F o r p r o d u c t states, i.e. factorizable states represented by density matrices o f the form P = e~ ®Q~,
ILQM[ ~
Proof. The l e m m a is p r o v e n by observing that P is a special case o f W with either a,/~, 7 or 2 equal to unity. Q.E.D. A recent p r o o f o f a t h e o r e m by G i s i n [ 13 ] implicitly contains the p r o o f o f l e m m a 2 for the Bell case. T h e o r e m 2 a n d its l e m m a thus establish the c o m p a t i b i l i t y between local realism and certain aspects o f the q u a n t u m description as related to the generalized Bell theorem. In this respect it represents an extension o f the works by Capasso et al. [ 11 ] a n d G i s i n [ 13 ]. It has been shown that the vector-space structure typical for the q u a n t u m description leads to inequalities that are i n c o m p a t i b l e with the a s s u m p t i o n s o f local realism. However, this does not hold for the case o f twop a n i c l e spin-½ systems p r e p a r e d as mixtures or in p r o d u c t states, indicating that the q u a n t u m description o f multi-particle systems possesses an inherent non-locality ( " e n t a n g l e m e n t " [ 13 ] ) with no classical analogy. A b r o a d e r discussion o f this a n d related issues bearing on their implications for the conceptual foundations o f q u a n t u m mechanics has been given a separate presentation [ 16 ]. I would like to t h a n k K. G j ~ t t e r u d for helpful discussions and for his reading o f the manuscript. Equally, s u p p o r t by the N o r w e g i a n Research Council for Science a n d the Humanities, reference no. 420.88/004, is gratefully acknowledged.
References [ 1] J.S. Bell, Physics 1 (1964) 195. [2] J. Clauser, M. Home, A. Shimony and R. Holt, Phys. Rev. Lett. 23 (1969) 880. [ 3 ] J.F. Clauser and M.A. Home, Phys. Rev. D 10 (1974) 526. [4 ] S.M. Roy and V. Singh, J. Phys. A 11 ( 1978 ) L 167. [ 5 ] A. Fine, Phys. Rev. Len. 48 (1982) 291. [6] E. Santos, Phys. Lctt. A 115 (1986) 363. [7] L.J. Landau, Phys. Lett. A 120 (1987) 54. [ 8 ] S.P. Gudder, Quantum probability (Academic Press, New York, 1988 ). [9] A. Garuccio, V.L. Lcpore and F. Selleri, Found. Phys. 20 (1990) 1173. [ 10] P. Suppes and M. Zanotti, Found. Phys. Lett. 4 ( 1991 ) 101. [ 11 ] V. Capasso, D. Fortunato and F. Selleri, Int. J. Theor. Phys. 7 (1973) 319. [ 12 ] J. Baez, Lctt. Math. Phys. 13 (1987) 135. [ 13 ] N. Gisin, Phys. Lctt. A 154 ( 1991 ) 201. [ 14] F. Selleri and A. van der Merwe, eds., Quantum paradoxes and physical reality (Kluwer, Dordrecht, 1990) pp. 290-302. [ 15 ] E.G. Beltrametti and G. Cassinelli, The logic of quantum mechanics (Addison-Wesley, Reading, MA, 1981 ). [ 16 ] H.E. And,Jtsand O.K. Gjetterud, Department of Physics Report 91-22, University of Oslo ( 1991 ), submitted to Found. Phys.
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