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) 54-56

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 A are said to be complementary if [A,B] 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 = 2P-1,b = 2Q-1 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
C = a(a+b)+b(b-a).
Then since a2 = b2 = a2 = b2 = 1 is follows that [1]
C2 = 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 C2 8 which implies

|R|2 = |r(C)|2 r(C2) 8
We thus obtain [1]-[3]

Proposition 1. In quantum theory |R| 22.

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 G, where

G = a(a+b)+b(b-a)
and    denotes the average with respect to the joint distribution. Since the functions a,b,a,b commute, eq. (1) here becomes G2 = 4 and thus
|R|2 = |G|2 G2 = 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
r([P,Q][P,Q])
(2)
is positive, perhaps by interchanging the roles of P and Q, which results in a change of sign of (2). Then
r(C2) = 4+16||[P,Q][P,Q]||.
It follows that
||C2|| = 4+16||[P,Q][P,Q]||.
Thus
||C|| = b,
where
b = 2{1+4||[P,Q][P,Q]||}1/2.
(3)
We may then find a state r on A such that
|r(C)| = b.
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] 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 bP, where

bP = 2{1+4||P([P,Q][P,Q])||}1/2
If P is faithful, bP = 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 bP = b for all non-trivial representations.

We now apply the above results to projections in independent sub-algebras of A.

Definition. Let A1,A2 be C* sub-algebras of A. Then A1,A2 are said to be independent if

(i)
A1 and A2 commute: [A,B] = 0 for all A A1,B A2;
(ii)
the Schlieder property holds: If A A1,B A2 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] 0 and [P,Q] 0 then [P,Q][P,Q] 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 non-commutative 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 non-commutative von Neumann algebras A1,A2 in A, there are projections P,Q A1,P,Q A2 such that b = 22.

That is, Bell's inequality may be violated by the maximum amount 22 by some projections P,Q A1; P,Q A2 and some state on A. A similar conclusion holds for vector states in a faithful representation of A.

Lemma. If M is a non-commutative 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] 0. Let T = rs(1-r). Then T 0. Let T = u|T| be the polar decomposition of T and define x = ufu, y = uuf. By the way T is constructed it follows that x is orthogonal to y and u2 = 0. Let z denote the projection x+y. Define the self-adjoint operators A = u+uf, B = i-1(u-uf). Then A2 = B2 = z and [A,B] = 2i-1(x-y) 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 space-time an associated von Neumann algebra A(R) satisfying

(i)
R1 R2A(R1) A(R2);
(ii)
the algebra of observables A = C* algebra generated by {RA(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 R1 is space-like separated from R2 then A(R1) commutes with A(R2);
(v)
for any R, A = {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 light-cone;
(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 R1 is strictly space-like separated from R2 if there is a positive number e such that R1 is space-like separated from R2e, where R2e denotes the set of points within a distance e of R2. The uniqueness of the vacuum and the spectral condition imply by a theorem of Schlieder [7] that A(R1) and A(R2) are independent when R1 is strictly space-like separated from R2.

We see by proposition 4 that given P,Q A(R), P,Q A(R2) where R1 and R2 are strictly space-like separated, the violation of Bell's inequality depends only on ||[P,Q]|| ||[P,Q]|| and not on the distance between R1 and R2. By proposition 5, supb for projections in A(R1) and A(R2) is 22 independent of the separation of R1 and R2. Notice furthermore that since A is simple, in any non-trivial representation of A there are projections in A(R) and A(R2) and vector states which violate Bell's inequality arbitrarily closely to 22.

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 R1 and R2 [3]:

|r(C)| D1+D2,
where
D1
=
|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
D2 = |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
D2 |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 R1 is space-like separated from R2(t) for all t, |t| T. Here R2(t) is the time translation of the region R2 by the amount t. Thus |r(C)| 2+4e-mT. Notice that T is proportional to the distance between R1 and R2.

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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