On the Violation of Bell's Inequality in Quantum Theory
Lawrence J. Landau
Mathematics Department, King's College London
Strand, London WC2R 2LS
Physics Letters A 120 (1987) 5456
Abstract
Elementary considerations concerning Bell's inequality are presented and applied to the quantum theory of local observables.
1 General results
Let A denote the C^{*} algebra of quantum observables (perhaps the set of all bounded operators on a Hilbert space). Two observables A,B Î A are said to be complementary if [A,B] ¹ 0 and compatible if [A,B] = 0. Let P and Q be complementary projections, P¢ and Q¢ complementary projections, while P is compatible with P¢ and with Q¢ and Q is compatible with P¢ and with Q¢. Define a = 2P1,b = 2Q1 and similarly for a¢, b¢. For any state r on A define R by
R = r(aa¢)+r(ab¢)+r(bb¢)r(ba¢) = r(C), 

where
Then since a^{2} = b^{2} = a^{¢2} = b^{¢2} = 1 is follows that [1]
C^{2} = 4+[a,b][a¢,b¢] = 4+16[P,Q][P¢,Q¢]. 
 (1) 
Eq. (1) is a very useful identity and is the basis for the following analysis.
Since a = b = a¢ = b¢ = 1 it follows that [a,b][a¢,b¢] £ 4 so that C^{2} £ 8 which implies
R^{2} = r(C)^{2} £ r(C^{2}) £ 8 

We thus obtain [1][3]
Proposition 1. In quantum theory R £ 2Ö2.
If the four experiments corresponding to the compatible pairs of projections from P,Q,P¢,Q¢ have a classical description by means of a joint distribution then we may represent a,b,a¢,b¢ as functions a,b,a¢,b¢
taking the values ±1 on the joint distribution space and correspondingly we may express R as áGñ, where
and á ñ denotes the average with respect to the joint distribution. Since the functions a,b,a¢,b¢
commute, eq. (1) here becomes G^{2} = 4 and thus
R^{2} = áGñ^{2} £ áG^{2}ñ = 4 

and so
Proposition 2. (Bell's inequality [1][4].) In a classical theory with joint distributions R £ 2.
If R violates Bell's inequality then no classical description in terms of a joint distribution is possible.
We may strengthen proposition 1 by noting that there exists a state r on A such that
r([P,Q][P¢,Q¢]) = [P,Q][P¢, Q¢]. 

We may suppose
is positive, perhaps by interchanging the roles of P and Q, which results in a change of sign of (2). Then
r(C^{2}) = 4+16[P,Q][P¢,Q¢]. 

It follows that
C^{2} = 4+16[P,Q][P¢,Q¢]. 

Thus
where
b = 2{1+4[P,Q][P¢,Q¢]}^{1/2}. 
 (3) 
We may then find a state r on A such that
Therefore we have
Proposition 3. For given P,Q,P¢,Q¢ there is a state r on A such that R = b (eq. (3)).
Thus the maximal violation of Bell's inequality is determined by [P,Q][P¢,Q¢]. In particular,
Corollary 1. Given P,Q,P¢,Q¢ Bell's inequality is violated for some state on A if and only if [P,Q][P¢,Q¢] ¹ 0. The analysis leading to proposition 3 may be carried out in a given representation of A and gives
Corollary 2. For given P,Q,P¢,Q¢ the supremum of R over all vector states in the representation P of A is b_{P}, where
b_{P} = 2{1+4P([P,Q][P¢,Q¢])}^{1/2} 

If P is faithful, b_{P} = b.
Note that if A is a simple algebra then all nontrivial representations of A are faithful and we have
Corollary 3. If A is simple then b_{P} = b for all nontrivial representations.
We now apply the above results to projections in independent subalgebras of A.
Definition. Let A_{1},A_{2} be C^{*} subalgebras of A. Then A_{1},A_{2} are said to be independent if
 (i)
 A_{1} and A_{2} commute: [A,B] = 0 for all A Î A_{1},B Î A_{2};
 (ii)
 the Schlieder property holds: If A Î A_{1},B Î A_{2} and AB = 0 then A = 0 or B = 0.
Now let us suppose that the algebra generated by P,Q is independent of the algebra generated by P¢,Q¢. If [P,Q] ¹ 0 and [P¢,Q¢] ¹ 0 then [P,Q][P¢,Q¢] ¹ 0 and so by corollary 1 Bell's inequality is violated by some state on A for P,Q,P¢,Q¢. This result can be sharpened using the analysis of Roos [5] which implies [P,Q][P¢,Q¢] = [ P,Q] [P¢,Q¢]. Thus
Proposition 4. If the algebra generated by P,Q is independent of the algebra generated by P¢,Q¢ then
b = 2{1+4[P,Q] [P¢,Q¢]}^{1/2} 

We finally remark that for any projections P,Q, [P,Q] £ 1/2, and for any noncommutative von Neumann algebra M there are projections P,Q such that [P,Q] = 1/2. (See the lemma below.) We conclude
Proposition 5. For any two independent noncommutative von Neumann algebras A_{1},A_{2} in A, there are projections P,Q Î A_{1},P¢,Q¢ Î A_{2} such that b = 2Ö2.
That is, Bell's inequality may be violated by the maximum amount 2Ö2 by some projections P,Q Î A_{1}; P¢,Q¢ Î A_{2} and some state on A. A similar conclusion holds for vector states in a faithful representation of A.
Lemma. If M is a noncommutative von Neumann algebra there are projections P,Q such that [P,Q] = 1/2.
Proof. Let r,s be two projections in M, [r,s] ¹ 0. Let T = rs(1r). Then T ¹ 0. Let T = uT be the polar decomposition of T and define x = u^{f}u, y = uu^{f}. By the way T is constructed it follows that x is orthogonal to y and u^{2} = 0. Let z denote the projection x+y. Define the selfadjoint operators A = u+u^{f}, B = i^{1}(uu^{f}). Then A^{2} = B^{2} = z and [A,B] = 2i^{1}(xy) so that [A,B] = 2. Let p = (A+z)/2 = projection onto the eigenspace of A with eigenvalue 1, q = (B+z)/2 = projection onto the eigenspace of B with eigenvalue 1. Then [p,q] = ^{1}/_{4}[A,B] so [p,q] = 1/2.
2 Application to local algebras of observables
The following framework is typical of local relativistic quantum theory. In a Hilbert space H there is given for each bounded open region R of spacetime an associated von Neumann algebra A(R) satisfying
 (i)
 R_{1} Ì R_{2}ÞA(R_{1}) Ì A(R_{2});
 (ii)
 the algebra of observables A = C^{*} algebra generated by {È_{R}A(R)};
 (iii)
 A is irreducible, i.e. the von Neumann algebra A^{¢¢} generated by A is the set of all bounded operators on H;
 (iv)
 if R_{1} is spacelike separated from R_{2} then A(R_{1}) commutes with A(R_{2});
 (v)
 for any R, A^{¢¢} = {È_{x} A(R+x)}^{¢¢}.
In H there is a unitary representation U(x) of the translation group satisfying
 (vi)
 A(R+x) = U(x)A(R)U(x)^{1};
 (vii)
 the spectrum of U(x) is contained in the closed forward lightcone;
 (viii)
 there is a unique vacuum vector W invariant under translations.
The irreducibility of A implies that the center of A is trivial and by a theorem of Borchers [6] the algebra A is simple.
We say that the region R_{1} is strictly spacelike separated from R_{2} if there is a positive number e such that R_{1} is spacelike separated from R_{2}^{e}, where R_{2}^{e} denotes the set of points within a distance e of R_{2}. The uniqueness of the vacuum and the spectral condition imply by a theorem of Schlieder [7] that A(R_{1}) and A(R_{2}) are independent when R_{1} is strictly spacelike separated from R_{2}.
We see by proposition 4 that given P,Q Î A(R_{)}, P¢,Q¢ Î A(R_{2}) where R_{1} and R_{2} are strictly spacelike separated, the violation of Bell's inequality depends only on [P,Q] [P¢,Q¢] and not on the distance between R_{1} and R_{2}. By proposition 5, supb for projections in A(R_{1}) and A(R_{2}) is 2Ö2 independent of the separation of R_{1} and R_{2}. Notice furthermore that since A is simple, in any nontrivial representation of A there are projections in A(R_{)} and A(R_{2}) and vector states which violate Bell's inequality arbitrarily closely to 2Ö2.
For completeness we present the argument showing that for the vacuum state with a mass gap m, the maximum violation of Bell's inequality falls off exponentially with the separation of R_{1} and R_{2} [3]:
where



r(aa¢)r(a)r(a¢)+r(ab¢) r(a)r(b¢) 
 

r(bb¢)r(b)r(b¢)+r(ba¢) r(b)r(a¢) 

 

and
D_{2} = r(a)[r(a¢)+r(b¢)]+r(b)[r(b¢) r(a¢)]. 

As the numbers r(a),r(b),r(a¢),r(b¢) all lie between 1 and 1 it follows that
D_{2} £ r(a¢)+r(b¢)+r(b¢)r(a¢) £ 2. 

By a result of Fredenhaben [8] if r is the vacuum state and there is a mass gap m > 0,
r(aa¢)r(a)r(a¢) £ e^{mT}, 

where R_{1} is spacelike separated from R_{2}(t) for all t, t £ T. Here R_{2}(t) is the time translation of the region R_{2} by the amount t. Thus r(C) £ 2+4e^{mT}. Notice that T is proportional to the distance between R_{1} and R_{2}.
Acknowledgement
I wish to thank R. Werner for correspondence which led to a sharpening of my original versions of proposition 3 and the lemma.
References
 [1]
 L. Landau, Lett. Math. Phys. 14 (1987) 33.
 [2]
 B.S. Cirel'son, Lett. Math. Phys. 4 (1980) 93.
 [3]
 S.J. Summers and R. Werner, Phys. Lett. A 110 (1985) 257.
 [4]
 J.S. Bell, Physics 1 (1964) 195;
A. Fine, Phys. Rev. Lett. 48 (1982) 291.
 [5]
 H. Roos, Commun. Math. Phys. 16 (1970) 238.
 [6]
 H.J. Borchers, Commun. Math. Phys. 4 (1967) 315.
 [7]
 S. Schlieder, Commun. Math. Phys. 13 (1969) 216.
 [8]
 K. Fredenhagen, Commun. Math. Phys. 97 (1985) 461.
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