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ON THE EXISTENCE OF PHYSICAL TRANSFORMATIONS BETWEEN SETS OF QUANTUM STATES

ANTHONY CHEFLES

Department of Physical Sciences, University of Hertfordshire, Hatfield AL10 9AB, Herts, UK

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RICHARD JOZSA

Department of Computer Science, University of Bristol, Merchant Venturers Building, Bristol BS8 1UB, UK

Corresponding author.

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, and

ANDREAS WINTER

Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK

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https://doi.org/10.1142/S0219749904000031Cited by:66 (Source: Crossref)

Abstract

Let A={ρ₁,…,ρ_n} be a given set of quantum states. We consider the problem of finding necessary and sufficient conditions on another set B={σ₁,…,σ_n} that guarantee the existence of a physical transformation taking ρ_i to σ_i for all i. Uhlmann has given an elegant such condition when both sets comprise pure states. We give a simple proof of this condition and develop some consequences. Then we consider multi-probabilistic transformations between sets of pure states which leads to conditions for the problem of transformability between A and B when one set is pure and the other is arbitrary.

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References

A. Uhlmann, Rep. Math. Phys. 9, 273 (1976), DOI: 10.1016/0034-4877(76)90060-4. Crossref, Google Scholar
R. Jozsa, J. Mod. Opt. 41, 2314 (1994). Web of Science, Google Scholar
R. Jozsa and J. Schlienz, Phys. Rev. A 62, 012301-1 (1999). Crossref, Web of Science, Google Scholar
P. Alberti and A. Uhlmann, Existence and Density Theorems for Stochastic Maps on Commutative C*-Algebras, Math. Nachr. 97, 279 (1980); P. Alberti and A. Uhlmann, Dissipative Motion in State Spaces, Vol. 33 of Teubner-Texte zur Mathematik (Teubner Verlag Leipzig, Leipzig, 1981); P. Alberti and A. Uhlmann, Stochasticity and Partial Order—Doubly Stochastic Maps and Unitary Mixing, Vol. 9 of Mathematics and Its Applications (D.Reidel Publ. Company, Dordrecht, Boston, London, 1982) . Google Scholar
P. Alberti and A. Uhlmann, Rep. Math. Phys. 18, 163 (1980), DOI: 10.1016/0034-4877(80)90083-X. Crossref, Web of Science, Google Scholar
A. Uhlmann, Annalen der Physik 42, 524 (1985). Web of Science, Google Scholar
P. Alberti, private communication . Google Scholar
A. Carlini and M. Sasaki, Geometrical conditions for CPTP maps and their applica tion to a quantum repeater and a state-dependent quantum cloning machine, preprint available at , quant-ph/0304011 . Google Scholar
C. W. Helstrom , Quantum detection and estimation theory ( Academic Press , New York , 1976 ) . Google Scholar
P. Alberti, Rep. Math. Phys. 51, 87 (2003), DOI: 10.1016/S0034-4877(03)80005-8. Crossref, Web of Science, Google Scholar
A. Uhlmann, Wiss. Z. KMU Leipzig, Math.-Naturwiss. R. 34(6), 580 (1985). Google Scholar
A. Chefles, Phys. Lett. A 270, 14 (2000), DOI: 10.1016/S0375-9601(00)00291-7. Crossref, Web of Science, Google Scholar
A. Chefles, Phys. Rev. A 65, 052314 (2002), DOI: 10.1103/PhysRevA.65.052314. Crossref, Web of Science, Google Scholar
M. Nielsen and I. Chuang , Quantum Computation and Quantum Information Theory ( Cambridge University Press , 2000 ) . Google Scholar
E. B. Davies and J. T. Lewis, Comm. Math. Phys. 17, 239 (1970), DOI: 10.1007/BF01647093. Crossref, Web of Science, Google Scholar
P. Alberti, (2003) Transforming vector states: a toolkit, unpublished manuscript, 23pp . Google Scholar