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arXiv:1004.4782 · Physical Review Letters

Universal Bounds for the Holevo Quantity, Coherent Information, and the Jensen-Shannon Divergence

Wojciech Roga, M. Fannes, Karol Życzkowski

Published Jul 23, 2010• 68 citations
arXivHTMLPDF

Abstract

The mutual information between the sender of a classical message encoded in quantum carriers and a receiver is fundamentally limited by the Holevo quantity. Using strong subadditivity of entropy, we prove that the Holevo quantity is not larger than an exchange entropy. This implies an upper bound for coherent information. Moreover, restricting our attention to classical information, we bound the transmission distance between probability distributions by their entropic distance, which is a concave function of their Hellinger distance.

DOI
10.1103/physrevlett.105.040505

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The mutual information between the sender of a classical message encoded in quantum carriers and a receiver is fundamentally limited by the Holevo quantity. Using strong subadditivity of entropy, we prove that the Holevo quantity is not larger than an exchange entropy. This implies an upper bound for coherent information. Moreover, restricting our attention to classical information, we bound the transmission distance between probability distributions by their entropic distance, which is a concave function of their Hellinger distance.
I derive a universal upper bound on the capacity of any communication channel between two distant systems. The Holevo quantity, and hence the mutual information, is at most of order EΔt/ℏ, where E is the average energy of the signal, and Δt is the amount of time for which detectors operate. The bound does not depend on the size or mass of the emitting and receiving systems, nor on the nature of the signal. No restrictions on preparing and processing the signal are imposed. As an example, I consider the encoding of information in the transverse or angular position of a signal emitted and received by systems of arbitrarily large cross section. In the limit of a large message space, quantum effects become important even if individual signals are classical, and the bound is upheld.
We investigate bounds in the transmission of classical information through quantum systems. Our focus lies in the generalized Holevo theorem, which provides a single-letter Holevo-like inequality from arbitrary quantum distance measures. Through the introduction of the [Formula: see text]-[Formula: see text]-Rényi relative entropies, which comprise known relevant quantities such as the Rényi relative entropy and the sandwiched Rényi relative entropy, we establish the Holevo–Rényi inequality. This result leads to a quantum bound for the [Formula: see text]-mutual information, suggesting new insights into communication channel performance and the fundamental limits for reliability functions in memoryless multi-letter communication channels.
We provide lower and upper bounds on the information transmission capacity of one single use of a classical-quantum channel. The lower bound is expressed in terms of the Hoeffding capacity, which we define similarly to the Holevo capacity but replacing the relative entropy with the Hoeffding distance. Similarly, our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max-relative entropy in the definition of the divergence radius of a channel.
We prove a new version of the Holevo bound employing the Hilbert-Schmidt norm instead of the Kullback-Leibler divergence. Suppose Alice is sending classical information to Bob by using a quantum channel while Bob is performing some projective measurements. We bound the classical mutual information in terms of the Hilbert-Schmidt norm by its quantum Hilbert-Schmidt counterpart. This constitutes a Holevo-type upper bound on the classical information transmission rate via a quantum channel. The resulting inequality is rather natural and intuitive relating classical and quantum expressions using the same measure.
The Holevo bound is a bound on the mutual information for a given quantum encoding. In 1996 Schumacher, Westmoreland, and Wootters [Phys. Rev. Lett. 76, 3452 (1996)] derived a bound that reduces to the Holevo bound for complete measurements, but that is tighter for incomplete measurements. The most general quantum operations may be both incomplete and inefficient. Here we show that the bound derived by SWW can be further extended to obtain one that is yet again tighter for inefficient measurements. This allows us, in addition, to obtain a generalization of a bound derived by Hall, and to show that the average reduction in the von Neumann entropy during a quantum operation is concave in the initial state, for all quantum operations. This is a quantum version of the concavity of the mutual information. We also show that both this average entropy reduction and the mutual information for pure state ensembles, are Schur concave for unitarily covariant measurements; that is, for these measurements, information gain increases with initial uncertainty.
We investigate the concept of entropy in probabilistic theories more general than quantum mechanics, with particular reference to the notion of information causality (IC) recently proposed by Pawlowski et al (2009 arXiv:0905.2292). We consider two entropic quantities, which we term measurement and mixing entropy. In the context of classical and quantum theory, these coincide, being given by the Shannon and von Neumann entropies, respectively; in general, however, they are very different. In particular, while measurement entropy is easily seen to be concave, mixing entropy need not be. In fact, as we show, mixing entropy is not concave whenever the state space is a non-simplicial polytope. Thus, the condition that measurement and mixing entropies coincide is a strong constraint on possible theories. We call theories with this property monoentropic . Measurement entropy is subadditive, but not in general strongly subadditive. Equivalently, if we define the mutual information between two systems A and B by the usual formula I ( A : B )= H ( A )+ H ( B )- H ( AB ), where H denotes the measurement entropy and AB is a non-signaling composite of A and B , then it can happen that I ( A : BC )< I ( A : B ). This is relevant to IC in the sense of Pawlowski et al : we show that any monoentropic non-signaling theory in which measurement entropy is strongly subadditive, and also satisfies a version of the Holevo bound, is informationally causal, and on the other hand we observe that Popescu–Rohrlich boxes, which violate IC, also violate strong subadditivity. We also explore the interplay between measurement and mixing entropy and various natural conditions on theories that arise in quantum axiomatics.