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arXiv:1604.02831 · Journal of Physics A Mathematical and Theoretical

Preservation of a quantum Rényi relative entropy implies existence of a recovery map

Anna Jenčová

Published Jan 3, 2017• 31 citations
arXivHTMLPDF

Abstract

Abstract It is known that a necessary and sufficient condition for equality in the data processing inequality (DPI) for the quantum relative entropy is the existence of a recovery map. We show that equality in DPI for a sandwiched Rényi relative α -entropy with α > 1 is also equivalent to this property. For the proof, we use an interpolating family of L p -norms with respect to a state.

DOI
10.1088/1751-8121/aa5661

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Abstract It is known that a necessary and sufficient condition for equality in the data processing inequality (DPI) for the quantum relative entropy is the existence of a recovery map. We show that equality in DPI for a sandwiched Rényi relative α -entropy with α > 1 is also equivalent to this property. For the proof, we use an interpolating family of L p -norms with respect to a state.
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We provide a transparent, simple, and unified treatment of recent results on the equality conditions for the data-processing inequality of the sandwiched quantum Rényi divergence, including the statement that the equality in the data-processing implies recoverability via the Petz recovery map for the full range of the Rényi parameter α recently proven by Jenčová [J. Phys. A: Math. Theor. 50, 085303 (2017)]. We also obtain a new set of equality conditions, generalizing a previous result by Leditzky et al. [Lett. Math. Phys. 107, 61 (2017)].
The data processing inequality states that the quantum relative entropy between two states ρ and σ can never increase by applying the same quantum channel N to both states. This inequality can be strengthened with a remainder term in the form of a distance between ρ and the closest recovered state (R∘N)(ρ), where R is a recovery map with the property that σ=(R∘N)(σ). We show the existence of an explicit recovery map that is universal in the sense that it depends only on σ and the quantum channel N to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.
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Abstract Given two density matrices ρ and σ , there are a number of different expressions that reduce to the α -Rényi relative entropy of ρ with respect to σ in the classical case; i.e. when ρ and σ commute. Only those expressions for which the data processing inequality (DPI) is valid are of potential interest as quantum divergences in quantum information theory. Audenaert and Datta have made a conjecture on the validity of the DPI for an interesting family of quantum generalizations of the α -Rényi relative entropies, the -Rényi relative entropies. They and others have contributed to the partial solution of this conjecture. We review the problem, its context, and the methods that have been used to obtain the results that are known at present, presenting a unified treatment of developments that have unfolded in a number of different papers.
In this paper, we discuss the quantum data processing inequality and its refinements that are physically meaningful in the context of approximate recoverability. An important conjecture regarding this due to Seshadreesan et al. in [Rényi squashed entanglement, discord and relative entropy differences, J. Phys. A: Math. Theor. 48(39) (2015) 395303] is disproved. We prove some inequalities capturing universal approximate recoverability with the Petz recovery map for the sandwiched quasi and Rényi relative entropies for the parameter [Formula: see text]. We also obtain convexity theorems on some parametrized versions of the relative entropy and fidelity, which can be of independent interest.