Bounds on Information Combining With Quantum Side Information

Hirche, Christoph; Reeb, David M.

doi:10.1109/tit.2018.2842180

Cited by 8 publications

(6 citation statements)

References 82 publications

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“…A similar result has been proven for the sum of binary random variables [15]. Entropic inequalities with classical conditioning easily follow from the corresponding unconditioned inequalities and from the definition (3) of conditional entropy via Jensen's inequality (see e.g.…”

Section: Our Contributionsupporting

confidence: 59%

The entropy power inequality with quantum conditioning

Palma¹

2019

J. Phys. A: Math. Theor.

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The conditional Entropy Power Inequality is a fundamental inequality in information theory, stating that the conditional entropy of the sum of two conditionally independent vector-valued random variables each with an assigned conditional entropy is minimum when the random variables are Gaussian. We prove the conditional Entropy Power Inequality in the scenario where the conditioning system is quantum. The proof is based on the heat semigroup and on a generalization of the Stam inequality in the presence of quantum conditioning. The Entropy Power Inequality with quantum conditioning will be a key tool of quantum information, with applications in distributed source coding protocols with the assistance of quantum entanglement.

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Section: Our Contributionsupporting

confidence: 59%

The entropy power inequality with quantum conditioning

Palma¹

2019

J. Phys. A: Math. Theor.

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“…Furthermore, considering the variety of applications of the classical IB function it would be interesting to see which of them translate to the quantum setting. Finally, the classical IB function is closely related to entropic bounds on information combining, and our results might help to better understand their quantum generalization [23].…”

Section: Discussionmentioning

confidence: 79%

“…( 3), h(x) is the binary entropy and ⋆ denotes the binary convolution. This is an important example as it plays a crucial role in the theory of classical and quantum information combining [22], [23]. Now, using the reasoning after Eq.…”

Section: Numerics and Examplesmentioning

confidence: 99%

Convexity and Operational Interpretation of the Quantum Information Bottleneck Function

Datta

Hirche

Winter

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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In classical information theory, the information bottleneck method (IBM) can be regarded as a method of lossy data compression which focusses on preserving meaningful (or relevant) information. As such it has of late gained a lot of attention, primarily for its applications in machine learning and neural networks. A quantum analogue of the IBM has recently been defined, and an attempt at providing an operational interpretation of the so-called quantum IB function as an optimal rate of an information-theoretic task, has recently been made by Salek et al. The interpretation given by these authors is however incomplete, as its proof is based on the conjecture that the quantum IB function is convex. Our first contribution is the proof of this conjecture.Secondly, the expression for the rate function involves certain entropic quantities which occur explicitly in the very definition of the underlying information-theoretic task, thus making the latter somewhat contrived. We overcome this drawback by pointing out an alternative operational interpretation of it as the optimal rate of a bona fide information-theoretic task, namely that of quantum source coding with quantum side information at the decoder, which has recently been solved by Hsieh and Watanabe. We show that the quantum IB function characterizes the rate region of this task,We similarly show that the related privacy funnel function is concave (both in the classical and quantum case). However, we comment that it is unlikely that the quantum privacy funnel function can characterize the optimal asymptotic rate of an information theoretic task, since even its classical version lacks a certain essential additivity property.

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“…As quantum conditioning is significantly more complex, the same argument does not hold for quantum reference systems [5,39].…”

Section: Channel Preorders In Quantum Information Theory and Related ...mentioning

confidence: 99%

On contraction coefficients, partial orders and approximation of capacities for quantum channels

Hirche¹,

Rouzé²,

França³

2022

Quantum

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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.

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Bounds on Information Combining With Quantum Side Information

Cited by 8 publications

References 82 publications

The entropy power inequality with quantum conditioning

The entropy power inequality with quantum conditioning

Convexity and Operational Interpretation of the Quantum Information Bottleneck Function

On contraction coefficients, partial orders and approximation of capacities for quantum channels

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