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arXiv:2506.05335

Upper bounds on the Holevo quantity arising from the fundamental entropic inequality

Максим Евгеньевич Широков

Published Jun 5, 2025• 0 citations
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Abstract

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 $μ$ expressed in terms of the Holevo quantities of two auxiliary ensembles $μ_+$ and $μ_-$ produced by $μ$. 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.

DOI
10.48550/arxiv.2506.05335

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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.
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).
The entropy of the Gram matrix of a joint purification of an ensemble of KKK mixed states yields an upper bound for the Holevo information χ\chiχ of the ensemble. In this work we combine geometrical and probabilistic aspects of the ensemble in order to obtain meaningful bounds for χ\chiχ. This is done by constructing various correlation matrices involving fidelities between every pair of states from the ensemble. For K=3K=3K=3 quantum states we design a matrix of root fidelities that is positive and the entropy of which is conjectured to upper bound χ\chiχ. Slightly weaker bounds are established for arbitrary ensembles. Finally, we investigate correlation matrices involving multi-state correlations in relation to the Holevo quantity.
We establish a tight upper bound for the difference in von Neumann entropies between two quantum states,ρ1\rho _{1}ρ1​andρ2\rho _{2}ρ2​. This bound is expressed in terms of the von Neumann entropies of the mutually orthogonal states derived from the Jordan-Hahn decomposition of the difference operator(ρ1−ρ2)(\rho _{1} - \rho _{2})(ρ1​−ρ2​). This yields a novel entropic inequality that implies the well-known Audenaert-Fannes (AF) inequality. In fact, it also leads to a refinement of the AF inequality. We employ this inequality to obtain a uniform continuity bound for the quantum conditional entropy of two states whose marginals on the conditioning system coincide. We additionally use it to derive a continuity bound for the quantum relative entropy in both variables. Interestingly, the fundamental entropic inequality is also valid in infinite dimensions.