Chain Rule for the Quantum Relative Entropy

Fang, Kun; Fawzi, Omar; Renner, Renato; Sutter, David

doi:10.1103/physrevlett.124.100501

Cited by 55 publications

(60 citation statements)

References 36 publications

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“…It is unclear whether taking the limit α → 1 in this chain rule recovers the chain rule proved in [13]. The reason for this difficulty is that it remains open whether lim…”

Section: Remark 53mentioning

confidence: 90%

“…Our second application is a chain rule for the sandwiched Rényi divergence, which once again features the regularized divergence between channels. Such a chain rule was proved in [13] for the Umegaki relative entropy.…”

Section: A Chain Rule For the Sandwiched Rényi Divergencementioning

confidence: 91%

“…Next, we prove that as for states, the regularized channel divergence is equal to the regularized sandwiched Rényi divergence. Note that unlike for states, the sandwiched Rényi divergence of channels is not additive in general, see [13] for an example.…”

Section: Corollary 43 For Any α ∈ (1 ∞) and Completely Positive Mapsmentioning

confidence: 99%

“…As the regularized divergence between channels appears in the analysis of many information processing tasks, we believe this result will be useful in obtaining improved characterizations of such tasks. An example is the task of channel discrimination, for which the regularized Umegaki channel divergence governs the asymptotic error rate [13,39,43]. One could also obtain upper bounds on quantum channel capacities, such as the classical capacity, in terms of regularized divergence between channels (see e.g., [41]).…”

Section: Converging Hierarchy Of Upper Bounds On the Regularized Divergence Of Channelsmentioning

confidence: 99%

“…Multiple regimes can be considered, the most studied is the asymmetric or Stein setting where we set α n ≤ ε for some ε ∈ (0, 1) and consider the asymptotic behavior of the optimal type II error − 1 n log β n . The works [13,39,43] establish that if we take ε → 0 this is given by the regularized Umegaki relative entropy D reg (N M) 3 . Our focus here is on the strong converse regime, i.e., we require β n ≤ 2 −rn with r > D reg (N M) and we consider the behavior of α n .…”

Section: Channel Discriminationmentioning

confidence: 99%

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Defining quantum divergences via convex optimization

Fawzi

2021

Quantum

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We introduce a new quantum Rényi divergence Dα# for α∈(1,∞) defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an efficient semidefinite programming representation for states and channels, and a chain rule property. An important property of this new divergence is that its regularization is equal to the sandwiched (also known as the minimal) quantum Rényi divergence. This allows us to prove several results. First, we use it to get a converging hierarchy of upper bounds on the regularized sandwiched α-Rényi divergence between quantum channels for α>1. Second it allows us to prove a chain rule property for the sandwiched α-Rényi divergence for α>1 which we use to characterize the strong converse exponent for channel discrimination. Finally it allows us to get improved bounds on quantum channel capacities.

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“…It is unclear whether taking the limit α → 1 in this chain rule recovers the chain rule proved in [13]. The reason for this difficulty is that it remains open whether lim…”

Section: Remark 53mentioning

confidence: 90%

Section: A Chain Rule For the Sandwiched Rényi Divergencementioning

confidence: 91%

Section: Corollary 43 For Any α ∈ (1 ∞) and Completely Positive Mapsmentioning

confidence: 99%

Section: Converging Hierarchy Of Upper Bounds On the Regularized Divergence Of Channelsmentioning

confidence: 99%

Section: Channel Discriminationmentioning

confidence: 99%

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Defining quantum divergences via convex optimization

Fawzi

2021

Quantum

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Space-time generalization of mutual information

Glorioso,

Qi,

Yang

2024

J. High Energ. Phys.

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The mutual information characterizes correlations between spatially separated regions of a system. Yet, in experiments we often measure dynamical correlations, which involve probing operators that are also separated in time. Here, we introduce a space-time generalization of mutual information which, by construction, satisfies several natural properties of the mutual information and at the same time characterizes correlations across subsystems that are separated in time. In particular, this quantity, that we call the space-time mutual information, bounds all dynamical correlations. We construct this quantity based on the idea of the quantum hypothesis testing. As a by-product, our definition provides a transparent interpretation in terms of an experimentally accessible setup. We draw connections with other notions in quantum information theory, such as quantum channel discrimination. Finally, we study the behavior of the space-time mutual information in several settings and contrast its long-time behavior in many-body localizing and thermalizing systems.

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Geometric Rényi Divergence and its Applications in Quantum Channel Capacities

Fang

Fawzi

2021

Commun. Math. Phys.

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Having a distance measure between quantum states satisfying the right properties is of fundamental importance in all areas of quantum information. In this work, we present a systematic study of the geometric Rényi divergence (GRD), also known as the maximal Rényi divergence, from the point of view of quantum information theory. We show that this divergence, together with its extension to channels, has many appealing structural properties, which are not satisfied by other quantum Rényi divergences. For example we prove a chain rule inequality that immediately implies the “amortization collapse” for the geometric Rényi divergence, addressing an open question by Berta et al. [Letters in Mathematical Physics 110:2277–2336, 2020, Equation (55)] in the area of quantum channel discrimination. As applications, we explore various channel capacity problems and construct new channel information measures based on the geometric Rényi divergence, sharpening the previously best-known bounds based on the max-relative entropy while still keeping the new bounds single-letter and efficiently computable. A plethora of examples are investigated and the improvements are evident for almost all cases.

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Chain Rule for the Quantum Relative Entropy

Cited by 55 publications

References 36 publications

Defining quantum divergences via convex optimization

Defining quantum divergences via convex optimization

Space-time generalization of mutual information

Geometric Rényi Divergence and its Applications in Quantum Channel Capacities

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