Eötvös University, Institute of Mathematics, Pázmány Péter sétány 1/C, Budapest, 1117 Hungary
Rényi Institute, Budapest, Reáltanoda u. 13-15, 1053 Hungary
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Abstract
Integral representations of quantum relative entropy, and of the directional second and higher order derivatives of von Neumann entropy, are established, and used to give simple proofs of fundamental, known data processing inequalities: the Holevo bound on the quantity of information transmitted by a quantum communication channel, and, much more generally, the monotonicity of quantum relative entropy under trace-preserving positive linear maps – complete positivity of the map need not be assumed. The latter result was first proved by Müller-Hermes and Reeb, based on work of Beigi. For a simple application of such monotonicities, we consider any `divergence’ that is non-increasing under quantum measurements, such as the concavity of von Neumann entropy, or various known quantum divergences. An elegant argument due to Hiai, Ohya, and Tsukada is used to show that the infimum of such a `divergence’ on pairs of quantum states with prescribed trace distance is the same as the corresponding infimum on pairs of binary classical states. Applications of the new integral formulae to the general probabilistic model of information theory, and a related integral formula for the classical Rényi divergence, are also discussed.
Popular summary
A binary reduction principle for generalized divergences is also presented, leading, in particular, to an improved Pinsker-style lower bound for the Holevo quantity of two quantum states in terms of their trace distance.
The paper is already cited by two preprints that apply the main result in essential ways:
[Anna Jencová, Recoverability of quantum channels via hypothesis testing, arXiv:2303.11707] and [Christoph Hirche, Marco Tomamichel, Quantum Rényi and $f$-divergences from integral representations, arXiv:2306.12343].
► BibTeX data
► References
[1] S. Beigi: Sandwiched Rényi divergence satisfies data processing inequality, Journal of Mathematical Physics 54.12 (2013): 122202.
https://doi.org/10.1063/1.4838855
[2] R. Blume-Kohout, H. K. Ng, D. Poulin, L. Viola: Information-preserving structures: A general framework for quantum zero-error information. Physical Review A 82 (6), 062306.
https://doi.org/10.1103/PhysRevA.82.062306
[3] F. Hiai, M. Ohya, and M. Tsukada: Sufficiency, KMS Condition and Relative Entropy in von Neumann Algebras, Pacific J. Math. 96, 99–109 (1981).
https://doi.org/10.2140/pjm.1981.96.99
[4] F. Hiai, M. Mosonyi: Different quantum $f$-divergences and the reversibility of quantum operations. Reviews in Mathematical Physics 29 (7), 1750023.
https://doi.org/10.1142/S0129055X17500234
[5] C. Hirche, M. Tomamichel, Quantum Rényi and $f$-divergences from integral representations, arXiv:2306.12343.
https://doi.org/10.48550/arXiv.2306.12343
arXiv:2306.12343
[6] A. S. Holevo: Bounds for the Quantity of Information Transmitted by a Quantum Communication Channel, Probl. Peredachi Inf., 9:3 (1973), 3–11; Problems Inform. Transmission, 9:3 (1973), 177–183.
[7] A. Jenčová: Recoverability of quantum channels via hypothesis testing, e-print arXiv:2303.11707.
https://doi.org/10.48550/arXiv.2303.11707
arXiv:2303.11707
[8] I. H. Kim: Modulus of convexity for operator convex functions, J. Math. Phys. 55, 082201 (2014).
https://doi.org/10.1063/1.4890292
[9] I. H. Kim, M. B. Ruskai: Bounds on the concavity of quantum entropy. J. Math. Phys. 55 (2014), no. 9, 092201, 5 pp.
https://doi.org/10.1063/1.4895757
[10] H. Li, Monotonicity of optimized quantum $f$-divergence, arXiv:2104.12890.
https://doi.org/10.48550/arXiv.2104.12890
arXiv:2104.12890
[11] E. H. Lieb, M. B. Ruskai: Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. 14, 1938–1941 (1973).
https://doi.org/10.1063/1.1666274
[12] G. Lindblad: Completely positive maps and entropy inequalities. Commun. Math. Phys. 40 (1975), 147–151.
https://doi.org/10.1007/BF01609396
[13] A. Müller-Hermes, D. Reeb: Monotonicity of the quantum relative entropy under positive maps. Ann. Henri Poincaré 18 (2017), no. 5, 1777–1788.
https://doi.org/10.1007/s00023-017-0550-9
[14] Dénes Petz: Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. Communications in Mathematical Physics, 105(1):123–131, March 1986.
https://doi.org/10.1007/bf01212345
[15] Dénes Petz: Sufficiency of channels over von Neumann algebras. Quarterly Journal of Mathematics, 39(1):97–-108, 1988.
https://doi.org/10.1093/qmath/39.1.97
[16] Martin Plávala: General probabilistic theories: An introduction. arXiv:2103.07469.
https://doi.org/10.48550/arXiv.2103.07469
arXiv:2103.07469
[17] F. Ticozzi, L. Viola: Quantum Information Encoding, Protection, and Correction from Trace-Norm Isometries, Physical Review A 81 (3), 032313.
https://doi.org/10.1103/PhysRevA.81.032313
[18] I. Sason, S. Verdú, $f$-divergence inequalities, IEEE Transactions on Information Theory 62 (2016), no. 11, 5973–6006.
https://doi.org/10.1109/TIT.2016.2603151
[19] H. Umegaki, Conditional expectation in an operator algebra, III, Kōdai Math. Sem. Rep. 11 (1959), 51–64.
https://doi.org/10.2996/kmj/1138844157
[20] D. Virosztek: The metric property of the quantum Jensen-Shannon divergence. Advances in Mathematics 380:107595.
https://doi.org/10.1016/j.aim.2021.107595
[21] M. M. Wilde, Optimized quantum $f$-divergences and data processing, J. Phys. A: Math. Theor. 51 (2018) 374002.
https://doi.org/10.1088/1751-8121/aad5a1
Cited by
[1] Anna Jenčová, “Recoverability of quantum channels via hypothesis testing”, arXiv:2303.11707, (2023).
[2] Christoph Hirche and Marco Tomamichel, “Quantum Rényi and $f$-divergences from integral representations”, arXiv:2306.12343, (2023).
The above citations are from SAO/NASA ADS (last updated successfully 2023-09-08 02:23:21). The list may be incomplete as not all publishers provide suitable and complete citation data.
On Crossref’s cited-by service no data on citing works was found (last attempt 2023-09-08 02:23:19).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
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- Source: https://quantum-journal.org/papers/q-2023-09-07-1102/
