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arXiv:1710.08080 · IEEE Transactions on Information Theory

Recovery and the Data Processing Inequality for Quasi-Entropies

Eric A. Carlen, Anna Vershynina

Published Mar 5, 2018• 15 citations
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Abstract

We prove a number of quantitative stability bounds for the cases of equality in Petz's monotonicity theorem for quasi-relative entropies Sf(p||σ) defined in terms of an operator monotone decreasing functions f and, in particular, the Rényi relative entropies. Included in our results are bounds in terms of the Petz recovery map, but we obtain more general results. The present treatment is entirely elementary and developed in the context of finite dimensional von Neumann algebras, where the results are already non-trivial and of interest in quantum information theory.

DOI
10.1109/tit.2018.2812038

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We prove a number of quantitative stability bounds for the cases of equality in Petz's monotonicity theorem for quasi-relative entropies Sf(p||σ) defined in terms of an operator monotone decreasing functions f and, in particular, the Rényi relative entropies. Included in our results are bounds in terms of the Petz recovery map, but we obtain more general results. The present treatment is entirely elementary and developed in the context of finite dimensional von Neumann algebras, where the results are already non-trivial and of interest in quantum information theory.
Abstract Let be a finite dimensional von Neumann algebra and a von Neumann subalgebra of it. For states and on , let and be the corresponding states induced on . The data processing inequality implies that where is the relative entropy. Petz proved that there is equality if and only if , where is the Petz recovery map . We prove a quantitative version of Petz’s theorem. In it simplest form, our bound is where is the relative modular operator. Since , this yields a bound that is independent of . We also prove an analogous result with a more complicated constant in which the roles of and are interchanged on the right. Quantum information theoretic inequalities are usually much harder to prove, or differ from, their classical counterparts because classical proofs often rely on conditioning argument that do not carry over to the quantum setting. In particular, quantum conditional expectations rarely preserve expectations—something that always happens in the classical setting. We also prove a simple theorem characterizing states and subalgebras for which conditional expectations do preserve expectation with respect to , illuminating the quantum obstacle to the existence of nicely behaved conditional expectations and the origins the Petz recovery map.
We revisit and improve joint concavity/convexity and monotonicity properties of quasi-entropies due to Petz in a new fashion. Then we characterize equality cases in the monotonicity inequalities (the data-processing inequalities) of quasi-entropies in several ways as follows: Let Φ:B(H)→B(K) be a trace-preserving map such that Φ* is a Schwarz map. When f is an operator monotone or operator convex function on [0, ∞), we present several equivalent conditions for the equality SfK(Φ(ρ)‖Φ(σ))=SfΦ*(K)(ρ‖σ) to hold for given positive operators ρ, σ on H and K∈B(K). The conditions include equality cases in the monotonicity versions of Lieb’s concavity and Ando’s convexity theorems. Specializing the map Φ we have equivalent conditions for equality cases in Lieb’s concavity and Ando’s convexity. Similar equality conditions are discussed also for monotone metrics and χ2-divergences. We further consider some types of linear preserver problems for those quantum information quantities.
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.
In this paper, we discuss a refinement of quantum data processing inequality for the sandwiched quasi-relative entropy S2\mathcal{S}_2S2​ on a tracial von-Neumann algebra. The main result is a universal recoverability bound with the Petz recovery map, which was previously obtained in the finite dimensional setup.
Abstract The optimized quantum f -divergences form a family of distinguishability measures that includes the quantum relative entropy and the sandwiched Rényi relative quasi-entropy as special cases. In this paper, we establish physically meaningful refinements of the data-processing inequality for the optimized f -divergence. In particular, the refinements state that the absolute difference between the optimized f -divergence and its channel-processed version is an upper bound on how well one can recover a quantum state acted upon by a quantum channel, whenever the recovery channel is taken to be a rotated Petz recovery channel. Not only do these results lead to physically meaningful refinements of the data-processing inequality for the sandwiched Rényi relative entropy, but they also have implications for perfect reversibility (i.e. quantum sufficiency) of the optimized f -divergences. Along the way, we improve upon previous physically meaningful refinements of the data-processing inequality for the standard f -divergence, as established in recent work of Carlen and Vershynina [arXiv: 1710.02409 , arXiv: 1710.08080 ]. Finally, we extend the definition of the optimized f -divergence, its data-processing inequality, and all of our recoverability results to the general von Neumann algebraic setting, so that all of our results can be employed in physical settings beyond those confined to the most common finite-dimensional setting of interest in quantum information theory.
The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations. Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map. The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, arXiv:1410.0664] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, arXiv:quant-ph/0307170]. Our paper constitutes partial progress towards a remainder term which features just the Petz recovery map (not a rotated Petz map), a conjecture which would have many consequences in quantum information theory. A well known result states that the monotonicity of relative entropy with respect to quantum operations is equivalent to each of the following inequalities: strong subadditivity of entropy, concavity of conditional entropy, joint convexity of relative entropy, and monotonicity of relative entropy with respect to partial trace. We show that this equivalence holds true for refinements of all these inequalities in terms of the Petz recovery map. So either all of these refinements are true or all are false.