The Hughston–Jozsa–Wootters theorem shows that any finite ensemble of quantum states can be prepared “at a distance,” and it has been used to demonstrate the insecurity of all bit commitment protocols based on finite quantum systems without superselection rules. In this paper, we prove a generalized HJW theorem for arbitrary ensembles of states on a -algebra. We then use this result to demonstrate the insecurity of bit commitment protocols based on infinite quantum systems, and quantum systems with Abelian superselection rules.
REFERENCES
1.
Alfsen
, E.
, Compact Convex Sets and Boundary Integrals (Springer, New York, 1971
).2.
Bennett
, C.
and Brassard
, G.
, “Quantum cryptography: Public key distribution and coin tossing
,” in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing (IEEE, New York, 1984
), pp. 175
–179
.3.
Bennett
, C.
, Crépeau
, C.
, Jozsa
, R.
, and Langlois
, D.
, “A quantum bit commitment scheme provably unbreakable by both parties
,” Proceedings of the 34th Annual IEEE Symposium on the Foundations of Computer Science, 1993
, pp. 362
–371
.4.
Brassard
, G.
, Crépeau
, C.
, Mayers
, D.
, and Salvail
, L.
, “A brief review on the impossibility of quantum bit commitment
,” quant-ph/9712023.5.
Bratteli
, O.
, and Robinson
, D.
, Operator Algebras and Quantum Statistical Mechanics (Springer, New York, 1987
), Vol. 1
.6.
Bub
, J.
, “The quantum bit commitment theorem
,” Found. Phys.
31
, 735
–756
(2001
).7.
Cassinelli
, G.
, De Vito
, E.
, and Levrero
, A.
, “On the decompositions of a quantum state
,” J. Math. Anal. Appl.
210
, 472
–483
(1997
).8.
Clifton
, R.
, Feldman
, D.
, Halvorson
, H.
, Redhead
, M.
, and Wilce
, A.
, “Superentangled states
,” Phys. Rev. A
58
, 135
(1998
).9.
Davies
, E.
, Quantum Theory of Open Systems (Academic, New York, 1976
).10.
Hughston
, L.
, Jozsa
, R.
, and Wootters
, W.
, “A complete classification of quantum ensembles having a given density matrix
,” Phys. Lett. A
183
, 14
–18
(1993
).11.
Kadison
, R.
and Ringrose
, J.
, Fundamentals of the Theory of Operator Algebras (American Mathematical Society, Providence, RI, 1997
).12.
Kent
, A.
, “Unconditionally secure bit commitment
,” Phys. Rev. Lett.
83
, 1447
(1999
).13.
Keyl
, M.
, Schlingemann
, D.
, and Werner
, R.
, “Infinitely entangled states
,” Quantum Inf. Comput.
3
, 281
–306
(2003
).14.
Kitaev
, A.
, Mayers
, D.
, and Preskill
, J.
, “Superselection rules and quantum protocols
,” Phys. Rev. A
69
, 052326
(2004
).15.
Lo
, H.-K.
, and Chau
, H. F.
, “Is quantum bit commitment really possible?
” Phys. Rev. Lett.
78
, 3410
(1997
).16.
Mayers
, D.
, “Unconditionally secure quantum bit commitment is impossible
,” in Proceedings of the Fourth Workshop on Physics and Computation, Boston, 1996
, pp. 224
–228
.17.
Mayers
, D.
, “Unconditionally secure quantum bit commitment is impossible
,” Phys. Rev. Lett.
78
, 3414
(1997
).18.
19.
Ozawa
, M.
, “Quantum measuring processes of continuous observables
,” J. Math. Phys.
25
, 79
–87
(1984
).20.
Pedersen
, G.
, -algebras and Their Automorphism Groups (Academic, New York, 1979
).21.
22.
Schwartz
, J.
, “Two finite, non-hyperfinite, non-isomorphic factors
,” Commun. Pure Appl. Math.
16
, 19
–26
(1963
).23.
Srinivas
, M.
, “Collapse postulate for observables with continuous spectra
,” Commun. Math. Phys.
71
, 131
–158
(1980
).24.
Takesaki
, M.
, Theory of Operator Algebras (Springer, New York, 1979
).25.
Tomita
, M.
, “Harmonic analysis on locally compact groups
,” Math. J. Okayama Univ.
5
, 133
–193
(1956
).
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