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arXiv:1904.06562 · Quantum

Tensorization of the strong data processing inequality for quantum chi-square divergences

Yu Hua Cao, Jianfeng Lu

Published Oct 28, 2019• 5 citations
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Abstract

It is well-known that any quantum channel E satisfies the data processing inequality (DPI), with respect to various divergences, e.g., quantum χκ2 divergences and quantum relative entropy. More specifically, the data processing inequality states that the divergence between two arbitrary quantum states ρ and σ does not increase under the action of any quantum channel E . For a fixed channel E and a state σ , the divergence between output states E(ρ) and E(σ) might be strictly smaller than the divergence between input states ρ and σ , which is characterized by the strong data processing inequality (SDPI). Among various input states ρ , the largest value of the rate of contraction is known as the SDPI constant. An important and widely studied property for classical channels is that SDPI constants tensorize. In this paper, we extend the tensorization property to the quantum regime: we establish the tensorization of SDPIs for the quantum χκ1/22 divergence for arbitrary quantum channels and also for a family of χκ2 divergences (with κ≥κ1/2 ) for arbitrary quantum-classical channels.

DOI
10.22331/q-2019-10-28-199

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It is well-known that any quantum channel E satisfies the data processing inequality (DPI), with respect to various divergences, e.g., quantum χκ2 divergences and quantum relative entropy. More specifically, the data processing inequality states that the divergence between two arbitrary quantum states ρ and σ does not increase under the action of any quantum channel E . For a fixed channel E and a state σ , the divergence between output states E(ρ) and E(σ) might be strictly smaller than the divergence between input states ρ and σ , which is characterized by the strong data processing inequality (SDPI). Among various input states ρ , the largest value of the rate of contraction is known as the SDPI constant. An important and widely studied property for classical channels is that SDPI constants tensorize. In this paper, we extend the tensorization property to the quantum regime: we establish the tensorization of SDPIs for the quantum χκ1/22 divergence for arbitrary quantum channels and also for a family of χκ2 divergences (with κ≥κ1/2 ) for arbitrary quantum-classical channels.
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.
Tight Contraction Rates for Primitive Channels under Quantum fff-Divergences
Data-processing inequalities capture the phenomenon that two probability distributions can only become less distinguishable under any common post-processing. For more fine-grained inequalities, one turns to strong data-processing inequality (SDPI) constants, which give the strongest inequalities for a given channel and reference state for a fixed measure of distinguishability. These quantities have been used to quantify the rate at which time-homogeneous Markov chains contract towards a fixed point both in the classical and quantum setting. In this work, we establish that quantum fff-divergences satisfy a local reverse Pinsker inequality, which implies the asymptotic contraction rate of a primitive channel to its stationary state is upper bounded by the SDPI constant of any non-commutative χ2χ^2χ2-divergence. Using quantum-detailed balance, we establish a sufficient condition for these bounds to be tight. Finally, we apply these results to Petz, Matsumoto, and Hirche-Tomamichel fff-divergences, establishing new and strengthening previously known results.
Data-processing is a desired property of classical and quantum divergences and information measures. In information theory, the contraction coefficient measures how much the distinguishability of quantum states decreases when they are transmitted through a quantum channel, establishing linear strong data-processing inequalities (SDPI). However, these linear SDPI are not always tight and can be improved in most of the cases. In this work, we establish non-linear SDPI for quantum hockey-stick divergence for noisy channels that satisfy a certain noise criterion. We also note that our results improve upon existing linear SDPI for quantum hockey-stick divergences and also non-linear SDPI for classical hockey-stick divergence. We define FγF_γFγ​ curves generalizing Dobrushin curves for the quantum setting while characterizing SDPI for the sequential composition of heterogeneous channels. In addition, we derive reverse-Pinsker type inequalities for fff-divergences with additional constraints on hockey-stick divergences. We show that these non-linear SDPI can establish tighter finite mixing times that cannot be achieved through linear SDPI. Furthermore, we find applications of these in establishing stronger privacy guarantees for the composition of sequential private quantum channels when privacy is quantified by quantum local differential privacy.
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 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.
The data processing inequality is the most basic requirement for any meaningful measure of information. It essentially states that distinguishability measures between states decrease if we apply a quantum channel and is the centerpiece of many results in information theory. Moreover, it justifies the operational interpretation of most entropic quantities. In this work, we revisit the notion of contraction coefficients of quantum channels, which provide sharper and specialized versions of the data processing inequality. A concept closely related to data processing is partial orders on quantum channels. First, we discuss several quantum extensions of the well-known less noisy ordering and relate them to contraction coefficients. We further define approximate versions of the partial orders and show how they can give strengthened and conceptually simple proofs of several results on approximating capacities. Moreover, we investigate the relation to other partial orders in the literature and their properties, particularly with regard to tensorization. We then examine the relation between contraction coefficients with other properties of quantum channels such as hypercontractivity. Next, we extend the framework of contraction coefficients to general f-divergences and prove several structural results. Finally, we consider two important classes of quantum channels, namely Weyl-covariant and bosonic Gaussian channels. For those, we determine new contraction coefficients and relations for various partial orders.