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Information geometry of sandwiched Rényi \(\alpha\)-divergence. (English) Zbl 1366.81048

Summary: Information geometrical structure \((g^{(D_\alpha)},\nabla^{(D_\alpha)},\nabla^{(D_\alpha)})\) induced from the sandwiched Rényi \(\alpha\)-divergence \(D_\alpha(\rho \parallel \sigma ):=\frac{1}{\alpha (\alpha -1)}\log \mathrm{Tr} \lgroup \sigma^{\frac{1-\alpha}{2\alpha}}\rho \sigma^{\frac{1-\alpha}{2\alpha}}\rgroup^\alpha\) on a finite quantum state space \(\mathcal{S}\) is studied. It is shown that the Riemannian metric \(g^{(D_\alpha)}\) is monotone if and only if \(\alpha \in (-\infty,-1]\cup[\frac{1}{2},\infty)\), and that the quantum statistical manifold \((\mathcal{S},g^{(D_\alpha)},\nabla^{(D_\alpha)}),\nabla^{(D_\alpha)*})\) is dually flat if and only if \(\alpha =1\).

MSC:

81P16 Quantum state spaces, operational and probabilistic concepts
94A17 Measures of information, entropy
53C80 Applications of global differential geometry to the sciences
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[1] Amari S-I and Nagaoka H 2000 Methods of Information Geometry(Translations of Mathematical Monographs vol 191) (Providence, RI: American Mathematical Society) · Zbl 0960.62005
[2] Audenaert K M R and Datta N 2015 α-z-relative Rényi entropies J. Math. Phys.56 022202 · Zbl 1342.81061 · doi:10.1063/1.4906367
[3] Ay N and Amari S-I 2015 A novel approach to canonical divergences within information geometry Entropy17 8111-29 · doi:10.3390/e17127866
[4] Beigi S 2013 Sandwiched Rényi divergence satisfies data processing inequality J. Math. Phys.54 122202 · Zbl 1315.81029 · doi:10.1063/1.4838855
[5] Berta M, Fawzi O and Tomamichel M 2015 On variational expressions for quantum relative entropies (arXiv:1512.02615)
[6] Bhatia R 1997 Matrix Analysis(Graduate Texts in Mathematics vol 169) (New York: Springer) · doi:10.1007/978-1-4612-0653-8
[7] Csiszár I 1967 On topological properties of f-divergences Stud. Sci. Math. Hung.2 329-39 · Zbl 0157.25803
[8] Datta N and Leditzky F 2014 A limit of the quantum Rényi divergence J. Phys. A: Math. Theor.47 045304 · Zbl 1285.81007 · doi:10.1088/1751-8113/47/4/045304
[9] Eguchi S 1992 Geometry of minimum contrast Hiroshima Math. J.22 631-47 · Zbl 0780.53015
[10] Frank R L and Lieb E H 2013 Monotonicity of a relative Rényi entropy J. Math. Phys.54 122201 · Zbl 1350.94020 · doi:10.1063/1.4838835
[11] Furuta T 2008 Concrete examples of operator monotone functions obtained by an elementary method without appealing to Löwner integral representation Linear Algebr. Appl.429 972-80 · Zbl 1147.47015 · doi:10.1016/j.laa.2006.11.023
[12] Hasegawa H and Petz D 1997 Non-commutative extension of information geometry II Quantum Communication, Computing, and Measurement ed O Hirota et al (New York: Plenum) pp 109-18 · doi:10.1007/978-1-4615-5923-8_12
[13] Helstrom C W 1976 Quantum Detection and Estimation Theory (New York: Academic) · Zbl 1332.81011
[14] Holevo A S 1982 Probabilistic and Statistical Aspects of Quantum Theory (Amsterdam: North-Holland) · Zbl 0497.46053
[15] Mosonyi M and Ogawa T 2015 Quantum hypothesis testing and the operational interpretation of the quantum Rényi relative entropies Commun. Math. Phys.334 1617-48 · Zbl 1308.81051 · doi:10.1007/s00220-014-2248-x
[16] Mosonyi M and Ogawa T 2014 Strong converse exponent for classical-quantum channel coding (arXiv:1409.3562)
[17] Müller-Lennert M, Dupuis F, Szehr O, Fehr S and Tomamichel M 2013 On quantum Rényi entropies: a new definition and some properties J. Math. Phys.54 122203 · Zbl 1290.81016 · doi:10.1063/1.4838856
[18] Petz D 1996 Monotone metrics on matrix spaces Linear Algebr. Appl.224 81-96 · Zbl 0856.15023 · doi:10.1016/0024-3795(94)00211-8
[19] Rényi A 1961 On measures of information and entropy Proc. Symp. on Mathematical Statistics and Probability(University of California Press, Berkeley,) pp 547-61 · Zbl 0106.33001
[20] Wilde M M, Winter A and Yang D 2014 Strong converse for the classical capacity of entanglement-breaking and Hadamard channels Commun. Math. Phys.331 593-622 · Zbl 1303.81042 · doi:10.1007/s00220-014-2122-x
[21] Wigner E P and Yanase M M 1963 Information content of distributions Proc. Natl Acad. Sci. USA49 910-8 · Zbl 0128.14104 · doi:10.1073/pnas.49.6.910
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