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arXiv:1102.3041

Telescopic Relative Entropy--II Triangle inequalities

Koenraad M. R. Audenaert

Published Feb 15, 2011• 6 citations
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Abstract

In previous work (see arxiv:1102.3040), we have defined the telescopic relative entropy (TRE), which is a regularisation of the quantum relative entropy $S(ρ||σ)=\traceρ(\logρ-\logσ)$, by replacing the second argument $σ$ by a convex combination of the first and the second argument, $τ=aρ+(1-a)σ$ and dividing the result by $-\log a$. We also explored some basic properties of the TRE. In this follow-up paper we state and prove two upper bounds on the variation of the TRE when either the first or the second argument changes. These bounds are close in spirit to a triangle inequality. For the ordinary relative entropy no such bounds are possible due to the fact that the variation could be infinite.

DOI
10.48550/arxiv.1102.3041

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In previous work (see arxiv:1102.3040), we have defined the telescopic relative entropy (TRE), which is a regularisation of the quantum relative entropy S(ρ∣∣σ)=\traceρ(log⁡ρ−log⁡σ)S(ρ||σ)=\traceρ(\logρ-\logσ)S(ρ∣∣σ)=\traceρ(logρ−logσ), by replacing the second argument σσσ by a convex combination of the first and the second argument, τ=aρ+(1−a)στ=aρ+(1-a)στ=aρ+(1−a)σ and dividing the result by −log⁡a-\log a−loga. We also explored some basic properties of the TRE. In this follow-up paper we state and prove two upper bounds on the variation of the TRE when either the first or the second argument changes. These bounds are close in spirit to a triangle inequality. For the ordinary relative entropy no such bounds are possible due to the fact that the variation could be infinite.
We introduce the telescopic relative entropy (TRE), which is a new regularisation of the relative entropy related to smoothing, to overcome the problem that the relative entropy between pure states is either zero or infinity and therefore useless as a distance measure in this case. We study basic properties of this quantity, and find interesting relationships between the TRE and the trace norm distance. We then exploit the same techniques to obtain a new and shorter proof of a lower bound on the relative Renyi entropies in terms of the trace norm distance, $\traceρ^{1-p}σ^p \ge 1-(||ρ-σ||_1)/2$.
The quantum relative entropy is frequently used as a distance measure between two quantum states, and inequalities relating it to other distance measures are important mathematical tools in many areas of quantum information theory. We have derived many such inequalities in previous work. The present paper is a follow-up on this, and provides a sharp upper bound on the relative entropy in terms of the trace norm distance and of the smallest eigenvalues of both states concerned. The result obtained here is more general than the corresponding one from our previous work. As a corollary, we obtain a sharp upper bound on the regularised relative entropy introduced by Lendi, Farhadmotamed, and van Wonderen.
In the dynamics of open quantum systems, the backflow of information to the reduced system under study has been suggested as the actual physical mechanism inducing memory and thus leading to non-Markovian quantum dynamics. To this aim, the trace-distance revivals between distinct evolved system states have been shown to be subordinated to the establishment of system-environment correlations or changes in the environmental state. We show that this interpretation can be substantiated also for a class of entropic quantifiers. We exploit a suitably regularized version of Umegaki's quantum relative entropy, known as telescopic relative entropy, that is tightly connected to the quantum Jensen-Shannon divergence. In particular, we derive general upper bounds on the telescopic relative entropy revivals conditioned and determined by the formation of correlations and changes in the environment. We illustrate our findings by means of examples, considering in particular the Jaynes-Cummings model and a phase covariant dynamics.
Relative entropy is the standard measure of distinguishability in classical and quantum information theory. In the classical case, its loss under channels admits an exact chain rule, while in the quantum case only asymptotic, regularized chain rules are known. We establish new chain rules for quantum relative entropy that apply already in the single-copy regime. The first inequality can be naturally obtained via POVM decompositions, extending the point distributions in the classical chain rule to quantum ensemble partitions. The second gives a sufficient condition for the most natural extension of the classical result, which uses projectors as an analog for the classical point distributions. We additionally find a semiclassical chain rule where the point distributions are replaced with the projectors of the initial states, and, finally, we find a relation to previous works on strengthened data processing inequalities and recoverability. These results show that meaningful chain inequalities are possible already at the single-copy level, but they also highlight that tighter bounds remain to be found.
For quantum systems, it is shown that the relative entropy S(P,Q)=-Tr P log P+Tr P log Q of two positive semi-definite operators P and Q satisfies gamma -1 Tr(P-P 1+ gamma Q - gamma )<or=S(P,Q)<or= gamma -1 Tr(P 1- gamma Q gamma -P) for 0( gamma <or=1, and that these bounds become exact in the limit gamma to Phi . Analogous inequalities hold, for states in classical statistical mechanics or information theory, with trace replaced by integration or summation. Furthermore, the average of these bounds is, in general a better approximation to S(P,Q) than either bound alone, and the average is amenable to further improvement via repeated Richardson extrapolation. IF P and Q are Gibbs equilibrium states, then these inequalities can also be used to obtain bounds on the free energy of a perturbed system in terms of the free energy of the unperturbed system and expectations of the perturbation.