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arXiv:1904.10768 · Reviews in Mathematical Physics

A strengthened data processing inequality for the Belavkin–Staszewski relative entropy

Andreas Blühm, Ángela Capel

Published Aug 22, 2019• 14 citations
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Abstract

In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). This extends previous results by Carlen and Vershynina for the relative entropy and other standard [Formula: see text]-divergences. To this end, we provide two new equivalent conditions for the equality case of the data processing inequality for the BS-entropy. Subsequently, we extend our result to a larger class of maximal [Formula: see text]-divergences. Here, we first focus on quantum channels which are conditional expectations onto subalgebras and use the Stinespring dilation to lift our results to arbitrary quantum channels.

DOI
10.1142/s0129055x20500051

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In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). This extends previous results by Carlen and Vershynina for the relative entropy and other standard [Formula: see text]-divergences. To this end, we provide two new equivalent conditions for the equality case of the data processing inequality for the BS-entropy. Subsequently, we extend our result to a larger class of maximal [Formula: see text]-divergences. Here, we first focus on quantum channels which are conditional expectations onto subalgebras and use the Stinespring dilation to lift our results to arbitrary quantum channels.
Quasi-factorization-type inequalities for the relative entropy have recently proven to be fundamental in modern proofs of modified logarithmic Sobolev inequalities for quantum spin systems. In this paper, we show some results of weak quasi-factorization for the Belavkin-Staszewski relative entropy, i.e. upper bounds for the BS-entropy between two bipartite states in terms of the sum of two conditional BS-entropies, up to some multiplicative and additive factors. A more detailed version of this paper is accessible at: https://arxiv.org/pdf/2101.10312.pdf
Fawzi and Fawzi [Quantum 5, 387 (2021)] recently defined the sharp Rényi divergence, Dα#, for α ∈ (1, ∞), as an additional quantum Rényi divergence with nice mathematical properties and applications in quantum channel discrimination and quantum communication. One of their open questions was the limit α → 1 of this divergence. By finding a new expression of the sharp divergence in terms of a minimization of the geometric Rényi divergence, we show that this limit is equal to the Belavkin–Staszewski relative entropy. Analogous minimizations of arbitrary generalized divergences lead to a new family of generalized divergences that we call kringel divergences for which we prove various properties, including the data-processing inequality.
The quantum data processing inequality asserts that quantum relative entropy is monotonically non-increasing under quantum channels. Here, we promote this fundamental inequality to an exact equality via boundary and surface Poisson integral representations across a complex strip. The relative-entropy loss is resolved into two independent, nonnegative physical mechanisms: an operational Petz recovery component governing state reconstructibility, and a boundary term measuring the mismatch of relative modular dynamics. We further express the loss as the measured relative entropy between the original state and its averaged Petz recovery state plus a nonnegative variational remainder. For the Belavkin--Staszewski (BS) loss, analogous integral representations yield a decomposition into negative log Uhlmann fidelity with an averaged reconstruction plus a nonnegative variational remainder. Furthermore, we establish recovery bounds at any prescribed modular parameter, including the canonical unrotated Petz map, with optimal square-root scaling in the relative entropy loss. The prefactors depend on distinct-eigenvalue counts or logarithmic spectral quantities, improving upon the inverse-power dependence on small eigenvalues in previous bounds.
The optimized quantum f -divergences form a family of distinguishability measures that includes the quantum relative entropy and the sandwiched R'enyi 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 howwell 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 R'enyi 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.
Strengthening the quantum data processing inequality for maximal fff-divergences via unraveling
Maximal fff-divergences, among them the Belavkin--Staszewski relative entropy, are non-increasing under completely positive trace-preserving maps, which is the quantum data processing inequality (DPI). We quantify this decrease for Lindblad evolutions with Hermitian jump operators by lifting the problem to a stochastic unraveling, in which the two states become measures on pure states evolving under a classical Fokker--Planck equation. The lift rests on the identity that the infimum of the classical fff-divergence over ensembles realizing two states is equal to their maximal fff-divergence. This infimum is attained on a common basis of pure states, which is admissible initial data for the unraveling. The result is a strengthened DPI, with the loss bounded below by the time integral of a relative fff-Fisher information. Given a two-sided bound on the unraveling's transition kernel, we derive two exponential decay estimates. For bracket-generating jump operators, the rate reaches twice the spectral gap of the symmetrized generator after an explicit delay. Without bracket generation, the kernel bound alone gives any rate below twice the spectral gap of the full generator.
In information theory, the data processing inequality is a fundamental property that describes the loss of information through noisy channels. A more refined description is given by the strong data processing inequality (SDPI) for a fixed channel and a full-rank reference state. In classical information theory, the strong data processing inequality tensorizes for a large family of fff-divergences. However, its quantum counterpart remains largely unexplored. Tensorization of SDPI has previously been shown only for some special cases, and whether it holds more generally for quantum divergences still remains open. In this work, we report two negative results: the tensorization property fails for certain quantum chi-square divergences, and it also does not hold for the quantum relative entropy.