×

Criterion of weak compactness for families of generalized ensembles and its corollaries. (English. Russian original) Zbl 1339.81014

Theory Probab. Appl. 60, No. 2, 320-325 (2016); translation from Teor. Veroyatn. Primen. 60, No. 2, 402-408 (2015).
Summary: One of the central results of the theory of probability measures on metric spaces is Prokhorov’s theorem on weak compactness of a subset of probability measures. In the present paper this theorem is used to obtain a criterion of weak compactness for families of generalized quantum ensembles, i.e., Borel probability measures on the set of quantum states, and its applications in quantum information theory are considered.

MSC:

81P16 Quantum state spaces, operational and probabilistic concepts
81P45 Quantum information, communication, networks (quantum-theoretic aspects)
94A40 Channel models (including quantum) in information and communication theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Probabilistic and Statistical Aspects of Quantum Theory, A. S. Holevo, 2nd ed., Edizioni della Normale, Pisa, 2011
[2] Introduction to Quantum Probability Theory, T. A. Sarymsakov, FAN, Tashkent, 1985 (in Russian)
[3] Quantum Systems, Channels, Information. A Mathematical Introduction, A. S. Holevo, DeGruyter, Berlin, 2012 · Zbl 1332.81003
[4] Optimal signal ensembles, B. Schumacher; M. D. Westmoreland, Phys. Rev. A, 63 (2001), pp. 2308–2312
[5] Convergence of Probability Measures, P. Billingsley, John Willey & Sons, New York, 1968 · Zbl 0172.21201
[6] Probability Measures on Metric Spaces, K. R. Parthasarathy, Academic Press, New York, London, 1967 · Zbl 0153.19101
[7] Convergence of random processes and limit theorems in probability theory, Yu. V. Prokhorov, Theory Probab. Appl., 1 (1956), pp. 157–214
[8] Entanglement-assisted capacities of constrained quantum channels, A. S. Holevo, Theory Probab. Appl, 48 (2004), pp. 243–255 · Zbl 1056.94006
[9] Elements of Convex and Strongly Convex Analysis, E. S. Polovinkin; M. V. Balashov, Fizmatlit, Moscow, 2004 (in Russian) · Zbl 1181.26028
[10] Convex Analysis, R. Rockafellar, Princeton University Press, Princeton, NJ, 1970 · Zbl 0193.18401
[11] On the notion of entanglement in Hilbert space, A. S. Holevo\c{b} M. E. Shirokov\c{b}; R. F. Werner, Russian Math. Surveys, 60 (2005), pp. 153–154
[12] Continuous ensembles and the capacity of infinite-dimensional quantum channels, A. S. Holevo; M. E. Shirokov, Theory Probab. Appl., 50 (2006), pp. 86–98 · Zbl 1093.81013
[13] Convex hulls of varieties and entanglement measures based on the roof construction, T. J. Osborne, Quantum Inform. Comput., 7 (2007), pp. 209–227 · Zbl 1152.81792
[14] Properties of the space of quantum states and monotone characteristics of entanglement, M. E. Shirokov, Izv. Math., 74 (2010), pp. 849–882 · Zbl 1210.46056
[15] Generalized compactness in linear spaces and its application, V. U. Protasov; M. E. Shirokov, Sb. Math., 200 (2009), pp. 697–722 · Zbl 1183.46003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.