Abstract:
We consider infinite-dimensional versions of the notions of the
convex hull and convex roof of a function defined on the set of quantum
states. We obtain sufficient conditions for the coincidence and
continuity of restrictions of different convex hulls of a given lower
semicontinuous function to the subset of states with bounded mean
generalized energy (an affine lower semicontinuous non-negative
function). These results are used to justify an
infinite-dimensional generalization of the convex roof
construction of entanglement monotones that is widely
used in finite dimensions. We give several examples of entanglement
monotones produced by the generalized convex roof construction.
In particular, we consider an infinite-dimensional generalization
of the notion of Entanglement of Formation and study its properties.
Keywords:
convex hull and convex roof of a function, quantum state, entanglement monotone, entanglement of formation.
Citation:
M. E. Shirokov, “On properties of the space of quantum states and their
application to the construction of entanglement monotones”, Izv. Math., 74:4 (2010), 849–882
\Bibitem{Shi10}
\by M.~E.~Shirokov
\paper On properties of the space of quantum states and their
application to the construction of entanglement monotones
\jour Izv. Math.
\yr 2010
\vol 74
\issue 4
\pages 849--882
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Linking options:
https://www.mathnet.ru/eng/im2815
https://doi.org/10.1070/IM2010v074n04ABEH002510
https://www.mathnet.ru/eng/im/v74/i4/p189
This publication is cited in the following 19 articles:
M. E. Shirokov, “On Local Continuity of Characteristics of Composite Quantum Systems”, Proc. Steklov Inst. Math., 324 (2024), 225–260
E. R. Loubenets, M. Namkung, “Conclusive Discrimination by N Sequential Receivers between r≥2 Arbitrary Quantum States”, Russ. J. Math. Phys., 30:2 (2023), 219
Maksim Shirokov, “Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki–Fannes–Winter technique”, Lett. Math. Phys., 113 (2023), 121–35
M. E. Shirokov, “Quantifying continuity of characteristics of composite quantum systems”, Phys. Scr., 98:4 (2023), 042002
A. S. Holevo, “On Optimization Problem for Positive Operator-Valued Measures”, Lobachevskii J. Math., 43:7 (2022), 1646–1650
Holevo A., “On the Classical Capacity of General Quantum Gaussian Measurement”, Entropy, 23:3 (2021), 377
Lami L., Regula B., Takagi R., Ferrari G., “Framework For Resource Quantification in Infinite-Dimensional General Probabilistic Theories”, Phys. Rev. A, 103:3 (2021), 032424
Weis S., Shirokov M., “The Face Generated By a Point, Generalized Affine Constraints, and Quantum Theory”, J. Convex Anal., 28:3 (2021), 847–870
Sakai Yu., “Generalizations of Fano'S Inequality For Conditional Information Measures Via Majorization Theory Dagger”, Entropy, 22:3 (2020), 288
Shirokov M.E., Bulinski A.V., “On Quantum Channels and Operations Preserving Finiteness of the Von Neumann Entropy”, Lobachevskii J. Math., 41:12, SI (2020), 2383–2396
Alexander S. Holevo, A. A. Kuznetsova, “The information capacity of entanglement-assisted continuous variable quantum measurement”, J. Phys. A, 53:37 (2020), 375307–17
Regula B., “Convex Geometry of Quantum Resource Quantification”, J. Phys. A-Math. Theor., 51:4 (2018), 045303
Wilde M.M., “Entanglement Cost and Quantum Channel Simulation”, Phys. Rev. A, 98:4 (2018), 042338
Sakai Yu., “Generalized Fano-Type Inequality For Countably Infinite Systems With List-Decoding”, Proceedings of 2018 International Symposium on Information Theory and Its Applications (Isita2018), IEEE, 2018, 727–731
M. E. Shirokov, “Estimates for discontinuity jumps of information characteristics of quantum systems and channels”, Problems of Information Transmission, 52:3 (2016), 239–264
Shirokov M.E., “Squashed entanglement in infinite dimensions”, J. Math. Phys., 57:3 (2016), 032203
A. S. Holevo, M. E. Shirokov, “Criterion of weak compactness for families of generalized quantum ensembles and its applications”, Theory Probab. Appl., 60:2 (2016), 320–325
Chang M., Quantum Stochastics, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge Univ Press, 2015
M. E. Shirokov, “Schmidt Number and Partially Entanglement-Breaking Channels in Infinite-Dimensional Quantum Systems”, Math. Notes, 93:5 (2013), 766–779
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