TY - JOUR

T1 - Operational distance and fidelity for quantum channels

AU - Belavkin, Viacheslav P.

AU - D'Ariano, Giacomo Mauro

AU - Raginsky, Maxim

N1 - Funding Information:
This work has been sponsored by the Multiple Universities Research Initiative (MURI) program administered by the U.S. Army Research Office under Grant No. DAAD19-00-1-0177. V.P.B. acknowledges support from EC under the program ATESIT (Contract No. IST-2000-29681). G.M.D. also acknowledges support by EC and Ministero Italiano dell’Università e della Ricerca (MIUR) through the cosponsored ATESIT project IST-2000-29681 and Cofinanziamento 2002. M.R. acknowledges the kind hospitality of the Quantum Information Theory Group at Università di Pavia, and support from MIUR under Cofinanziamento 2002 and from the European Science Foundation. The authors would like to thank the referee for several suggestions, which resulted in improved presentation.

PY - 2005/6

Y1 - 2005/6

N2 - We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is well defined for channels between finite-dimensional algebras, but it also applies to a certain class of channels between infinite-dimensional algebras (explicitly, those channels that possess an operator-valued Radon-Nikodym density with respect to the trace in the sense of Belavkin-Staszewski) and induces a metric on the set of quantum channels that is topologically equivalent to the CB-norm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantum-mechanical systems, derived from the well-known fidelity ("generalized transition probability") of Uhlmann, is topologically equivalent to the trace-norm distance.

AB - We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is well defined for channels between finite-dimensional algebras, but it also applies to a certain class of channels between infinite-dimensional algebras (explicitly, those channels that possess an operator-valued Radon-Nikodym density with respect to the trace in the sense of Belavkin-Staszewski) and induces a metric on the set of quantum channels that is topologically equivalent to the CB-norm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantum-mechanical systems, derived from the well-known fidelity ("generalized transition probability") of Uhlmann, is topologically equivalent to the trace-norm distance.

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U2 - 10.1063/1.1904510

DO - 10.1063/1.1904510

M3 - Article

AN - SCOPUS:21844465942

VL - 46

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 6

M1 - 062106

ER -