arXiv:1102.3040

Telescopic Relative Entropy

Koenraad M. R. Audenaert

Published Feb 15, 2011• 3 citations

Abstract

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, $\traceρ^{1-p}σ^p \ge 1-(||ρ-σ||_1)/2$.

DOI: 10.48550/arxiv.1102.3040

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arXiv:1102.3040

Telescopic Relative Entropy

Koenraad M. R. Audenaert

Published Feb 15, 2011• 3 citations

arXiv HTML PDF

Abstract

\traceρ^{1-p}σ^p \ge 1-(||ρ-σ||_1)/2

DOI: 10.48550/arxiv.1102.3040

Related papers

Telescopic Relative Entropy--II Triangle inequalities

Koenraad M. R. Audenaert

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.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.

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\traceρ^{1-p}σ^p \ge 1-(||ρ-σ||_1)/2

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.

The relative entropy

S(\\rho||\\sigma)

is frequently used as a distance, or distinguishability measure between two states

\\rho

and

\\sigma

. In this paper we study the relation between this measure and a number of other measures used for that purpose, including the trace norm distance. More precisely, we derive lower and upper bounds on the relative entropy

S(\\rho||\\sigma)

in terms of various distance measures for

\\rho-\\sigma

based on unitarily invariant norms. The upper bounds can be considered as statements of continuity of the relative entropy distance in the sense of Fannes. We employ optimisation theory to obtain bounds that are as sharp as possible.