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Problems of Information Transmission

Upper Bounds for the Holevo Information Quantity and Their Use

Максим Евгеньевич Широков

Published Jul 1, 2019• 3 citations
Publisher

Abstract

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.

DOI
10.1134/s0032946019030013

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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.
We present a family of easily computable upper bounds for the Holevo quantity of 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 appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many cases improve existing estimates in the literature. We also present upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on metric divergence of the ensemble. The specification of this upper bound for the multi-mode quantum oscillator is tight for large energy. The above results are used to obtain tight upper bounds for the Holevo capacity of finite-dimensional and infinite-dimensional energy-constrained quantum channels depending on metric characteristics of the channel output.
We start with Fannes’ type and Winter’s type tight (uniform) continuity bounds for the quantum conditional mutual information and their specifications for states of special types. Then we analyse continuity of the Holevo quantity with respect to nonequivalent metrics on the set of discrete ensembles of quantum states. We show that the Holevo quantity is continuous on the set of all ensembles of m states with respect to all the metrics if either m or the dimension of underlying Hilbert space is finite and obtain Fannes’ type tight continuity bounds for the Holevo quantity in this case. In the general case, conditions for local continuity of the Holevo quantity for discrete and continuous ensembles are found. Winter’s type tight continuity bound for the Holevo quantity under constraint on the average energy of ensembles is obtained and applied to the system of quantum oscillators. The above results are used to obtain tight and close-to-tight continuity bounds for basic capacities of finite-dimensional channels (significantly refining the Leung-Smith continuity bounds).
We show how the fundamental entropic inequality proved recently in [arXiv:2408.15306] can be applied to obtain a useful relation for the Holevo quantity of discrete and continuous ensembles of quantum states. This relation gives a tight upper bound on the Holevo quantity of a given ensemble μμμ expressed in terms of the Holevo quantities of two auxiliary ensembles μ+μ_+μ+​ and μ−μ_-μ−​ produced by μμμ. Among others, this implies quite accurate upper bounds on the Holevo quantity of a discrete ensemble of quantum states expressed via the probabilities and the metric characteristics of an ensemble.
Abstract We obtain continuity bounds for basic information characteristics of quantum channels depending on their input dimension (if it is finite) and on the input energy bound (if the input dimension is infinite). We pay special attention to the case in which a multimode quantum oscillator is an input system. First, we prove continuity bounds for the output conditional mutual information for a single channel and for copies of a channel. Then we obtain estimates for the variation of the output Holevo quantity with respect to simultaneous variations of a channel and of an input ensemble. As a result, tight and close-to-tight continuity bounds for basic capacities of quantum channels, which depend on the input dimension, are obtained. They complement the Leung–Smith continuity bounds, which depend on the output dimension. Finally, we obtain tight and close-to-tight continuity bounds for basic capacities of infinite-dimensional energy-constrained channels with respect to the energy-constrained Bures distance generating the strong convergence of quantum channels.
Finite-dimensional restrictions of the channel describing elementary quantum random walk are discussed in this paper. Capacity of such channels can be accounted via Holevo upper bound, which is evaluated here. Upper estimation is the difference between maximal and minimal output entropy, and lower estimation is given by an ensemble of states with minimal output entropy, such that the output entropy for medium state is as big as possible. For these computations, we use complementary channels because they are much easier to deal with. Both estimations coincide in case of dimension 2. Moreover, in this case the complementary channel is unital cubit, so that the Holevo upper bound equals to the capacity. In large dimensions, the asymptotical estimation is an interval of length log⁡(e2)\log(\frac{e}{2})log(2e​) .
We describe analytical properties of the average output entropy of a quantum channel as a function of a pair (channel, input ensemble). In particular, tight semicontinuity bounds for this function with the rank/energy constraints are obtained by using the modified semicontinuity bounds for the quantum conditional entropy of quantum-classical states and a special approximation technique. Several applications are considered. New semicontinuity and continuity bounds for the output Holevo information of a channel as a function of a pair (channel, input ensemble) are obtained. The semicontinuity bound for the entanglement of formation with the rank constraint obtained in [1] is improved. In the preliminary part, some results concerning ensembles of quantum states are presented. In particular, a new useful metric on the set of generalized ensembles is proposed and explored. The concept of passive energy of an ensemble introduced here plays an important role in the article.