We prove that for any two commuting von Neumann algebras of infinite type, the open set of Bell correlated states for the two algebras is norm dense. We then apply this result to algebraic quantum field theory—where all local algebras are of infinite type—in order to show that for any two spacelike separated regions, there is an open dense set of field states that dictate Bell correlations between the regions. We also show that any vector state cyclic for one of a pair of commuting non-Abelian von Neumann algebras is entangled (i.e., nonseparable) across the algebras—from which it follows that every field state with bounded energy is entangled across any two spacelike separated regions.
REFERENCES
1.
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge U.P., Cambridge, 1987).
2.
J. F.
Clauser
, M. A.
Horne
, A.
Shimony
, and R. A.
Holt
, “Proposed experiment to test local hidden-variable theories
,” Phys. Rev. Lett.
26
, 880
–884
(1969
).3.
S. J.
Summers
, “On the independence of local algebras in quantum field theory
,” Rev. Math. Phys.
2
, 201
–247
(1990
).4.
L. J.
Landau
, “On the violation of Bell’s inequality in quantum theory
,” Phys. Lett. A
120
, 54
–56
(1987
).5.
S. J.
Summers
and R. F.
Werner
, “Maximal violation of Bell’s inequalities is generic in quantum field theory
,” Commun. Math. Phys.
110
, 247
–259
(1987
).6.
S. J.
Summers
and R. F.
Werner
, “Maximal violation of Bell’s inequalities for algebras of observables in tangent spacetime regions
,” Ann. Inst. Henri Poincaré
49
, 215
–243
(1988
).7.
S. J.
Summers
and R. F.
Werner
, “On Bell’s inequalities and algebraic invariants
,” Lett. Math. Phys.
33
, 321
–334
(1995
).8.
S. J. Summers, “Bell’s inequalities and algebraic structure,” in Operator Algebras and Quantum Field Theory, edited by S. Doplicher, R. Longo, J. E. Roberts, and L. Zsido (International, Cambridge, MA, 1997).
9.
M. L. G.
Redhead
, “More ado about nothing
,” Found. Phys.
25
, 123
–137
(1995
).10.
L. J.
Landau
, “On the non-classical structure of the vacuum
,” Phys. Lett. A
123
, 115
–118
(1987
).11.
R. F.
Werner
, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model
,” Phys. Rev. A
40
, 4277
–4281
(1989
).12.
R. Clifton, H. Halvorson, and A. Kent, “Non-local correlations are generic in infinite-dimensional bipartite systems,” to appear in Phys. Rev. A.
13.
R. Kadison and J. Ringrose, Fundamentals of the Theory of Operator Algebras (American Mathematical Society, Providence, RI, 1997).
14.
S.
Popescu
, “Bell’s inequalities and density matrices: Revealing ‘hidden’ nonlocality
,” Phys. Rev. Lett.
74
, 2619
–2622
(1995
).15.
R.
Clifton
and H.
Halvorson
, “Bipartite mixed states of infinite-dimensional systems are generically nonseparable
,” Phys. Rev. A
61
, 012108
(2000
).16.
G. G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Wiley, New York, 1972), p. 101.
17.
R. Haag, Local Quantum Physics (Springer, New York, 1992), p. 125.
18.
G. F.
Dell’Antonio
, “On the limit of sequences of normal states
,” Commun. Pure Appl. Math.
20
, 413
–429
(1967
).19.
J.
Dixmier
and O.
Maréchal
, “Vecteurs totalisateurs d’une algèbre de von Neumann
,” Commun. Math. Phys.
22
, 44
–50
(1971
).20.
K.
Zyczkowski
, P.
Horodecki
, A.
Sanpera
, and M.
Lewenstein
, “Volume of the set of separable states
,” Phys. Rev. A
58
, 883
–892
(1998
).21.
H.-J. Borchers, Translation Group and Particle Representations in Quantum Field Theory (Springer, New York, 1996).
22.
J.
Dimock
, “Algebras of local observables on a manifold
,” Commun. Math. Phys.
77
, 219
–228
(1980
).23.
S.
Doplicher
, R.
Haag
, and J. E.
Roberts
, “Fields, observables and gauge transformations I, II
,” Commun. Math. Phys.
13
, 1
–23
(1969
);24.
D.
Buchholz
and K.
Fredenhagen
, “Locality and the structure of particle states
,” Commun. Math. Phys.
84
, 1
–54
(1982
).25.
S. S. Horuzhy, Introduction to Algebraic Quantum Field Theory (Kluwer, Dordrecht, 1988).
26.
S.
Schlieder
, “Einige Bemerkungen über Projektionsoperatoren
,” Commun. Math. Phys.
13
, 216
–225
(1969
).27.
B. S.
Kay
and R. M.
Wald
, “Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon
,” Phys. Rep.
207
, 49
–136
(1991
).28.
R.
Verch
, “Continuity of symplectically adjoint maps and the algebraic structure of Hadamard vacuum representations for quantum fields on curved spacetime
,” Rev. Math. Phys.
9
, 635
–674
(1997
).
This content is only available via PDF.
© 2000 American Institute of Physics.
2000
American Institute of Physics
You do not currently have access to this content.