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arXiv:0712.3867

A Separation between Divergence and Holevo Information for Ensembles

Rahul Jain, Ashwin Nayak, Yi Su

Published Dec 22, 2007• 0 citations
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Abstract

The notion of divergence information of an ensemble of probability distributions was introduced by Jain, Radhakrishnan, and Sen in the context of the ``substate theorem''. Since then, divergence has been recognized as a more natural measure of information in several situations in quantum and classical communication. We construct ensembles of probability distributions for which divergence information may be significantly smaller than the more standard Holevo information. As a result, we establish that lower bounds previously shown for Holevo information are weaker than similar ones shown for divergence information.

DOI
10.48550/arxiv.0712.3867

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The notion of divergence information of an ensemble of probability distributions was introduced by Jain, Radhakrishnan, and Sen in the context of the ``substate theorem''. Since then, divergence has been recognized as a more natural measure of information in several situations in quantum and classical communication. We construct ensembles of probability distributions for which divergence information may be significantly smaller than the more standard Holevo information. As a result, we establish that lower bounds previously shown for Holevo information are weaker than similar ones shown for divergence information.
The notion of divergence information of an ensemble of probability distributions was introduced by Jain, Radhakrishnan and Sen in Jain et al. (2002; 2009) in the context of the ‘substate theorem’. Since then, divergence has been recognised as a more natural measure of information in several situations in both quantum and classical communication. We construct ensembles of probability distributions for which divergence information may be significantly smaller than the more standard Holevo information. As a result, we establish that bounds previously shown for Holevo information are weaker than similar ones shown for divergence information.
The Quantum Substate Theorem due to Jain, Radhakrishnan, and Sen (2002) gives us a powerful operational interpretation of relative entropy, in fact, of the observational divergence of two quantum states, a quantity that is related to their relative entropy. Informally, the theorem states that if the observational divergence between two quantum states rho, sigma is small, then there is a quantum state rho' close to rho in trace distance, such that rho' when scaled down by a small factor becomes a substate of sigma. We present new proofs of this theorem. The resulting statement is optimal up to a constant factor in its dependence on observational divergence. In addition, the proofs are both conceptually simpler and significantly shorter than the earlier proof.
This paper is primarily concerned with the development and application of quantum bounds on mutual information, although some of the methods developed can be applied to any figure of merit indicating degree of correlation, such as coincidence rate. Three basic techniques for obtaining bounds are described: mappings between joint-measurement and communication correlation contexts; a duality relation for quantum ensembles and quantum measurements; and an information exclusion principle [M. J. W. Hall, Phys. Rev. Lett. 74, 3307 (1995)]. Results include a proof of Holevo's communication bound from a joint-measurement inequality; a measurement-dependent dual to Holevo's bound; lower bounds for mutual information under ensemble and measurement constraints; information exclusion relations for measurements described by probability-operator measures; a proof that Glauber coherent states are optimal signal states for quantum communication based on (noisy) optical heterodyne detection; and an information inequality for quantum eavesdropping. Relations between the three techniques are used to further obtain upper bounds for quantum information, and to extend the information exclusion principle to a joint-measurement context.
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.
We present a family of easily computable upper bounds for the Holevo (information) quantity of an ensemble of quantum states depending on a reference state (as a free parameter). These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that an appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many cases improve the estimates existing in the literature. We also present an upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on the metric divergence of an ensemble. In the case of a multi-mode quantum oscillator, this upper bound is tight for large energy. Upper bounds for the Holevo capacity of finite-dimensional quantum channels depending on metric characteristics of the channel output are obtained.
Many problems of statistical and quantum mechanics can be established in terms of a distance between probability distributions. The present work is devoted to review some of the most relevant applications of the notion of distance in the context of statistical mechanics, quantum mechanics and information theory. Although we make a general overview of the most frequently used distances between probability distributions, we will center our presentation in a distance known as the Jensen-Shannon divergence both in their classical and quantum versions. For the classical one we present its main properties and we discuss its relevance as a segmentation tool for symbolic sequences. In the quantum case we show that the quantum Jensen-Shannon divergence is an adequate measure of entanglement.