Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Sharply Bounding the Holevo Quantity

Lin Zhang, Junde Wu, Shao-Ming Fei

Published Oct 27, 2011• 0 citations
OpenAlex

Abstract

For a quantum ensemble $\cE_3 =\set{\Pa{p_i,\rho_i}}_{i=1}^3$, M. Fannes \emph{et al.} conjectured that the entropy of the correlation matrix of $\cE_3 =\set{\Pa{p_i,\rho_i})}_{i=1}^3$ is upper bound for the Holevo quantity $\chi\Pa{\cE_3}$ of quantum ensemble $\cE_3 =\set{\Pa{p_i,\rho_i}}_{i=1}^3$. In this paper, we prove the conjecture under a strictly constraint condition.

Related papers

Matrices of fidelities for ensembles of quantum states and the Holevo quantity

M. Fannes, Fernando de Melo, Wojciech Roga +1 more

The entropy of the Gram matrix of a joint purification of an ensemble of $K$ mixed states yields an upper bound for the Holevo information $\chi$ of the ensemble. In this work we combine geometrical and probabilistic aspects of the ensemble in order to obtain meaningful bounds for $\chi$. This is done by constructing various correlation matrices involving fidelities between every pair of states from the ensemble. For $K=3$ quantum states we design a matrix of root fidelities that is positive and the entropy of which is conjectured to upper bound $\chi$. Slightly weaker bounds are established for arbitrary ensembles. Finally, we investigate correlation matrices involving multi-state correlations in relation to the Holevo quantity.

Quantum Information and Computation•May 1, 2012•18 citations

Matrices of fidelities for ensembles of quantum states and the Holevo quantity

M. Fannes, Fernando de Melo, Wojciech Roga +1 more

The entropy of the Gram matrix of a joint purification of an ensemble of K mixed states yields an upper bound for the Holevo information Chi of the ensemble. In this work we combine geometrical and probabilistic aspects of the ensemble in order to obtain useful bounds for Chi. This is done by constructing various correlation matrices involving fidelities between every pair of states from the ensemble. For K=3 quantum states we design a matrix of root fidelities that is positive and the entropy of which is conjectured to upper bound Chi. Slightly weaker bounds are established for arbitrary ensembles. Finally, we investigate correlation matrices involving multi-state fidelities in relation to the Holevo quantity.

arXiv:1104.2271•Apr 12, 2011•3 citations

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

Максим Евгеньевич Широков

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.

arXiv:2506.05335•Jun 5, 2025•0 citations

Upper Bounds for the Holevo Information Quantity and Their Use

Максим Евгеньевич Широков

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.

Problems of Information Transmission•Jul 1, 2019•3 citations

MULTI-STATE CORRELATIONS AND FIDELITIES

M. Fannes

Correlations between states in a general quantum ensemble can be encoded in a Gram matrix that depends on a prescribed joint purification. This allows to establish an upper bound for the Holevo quantity χ. Optimizing over joint purifications is a hard problem. For small ensembles the connection with matrices of fidelities is investigated. General bounds are obtained for a linearized version of χ.

International Journal of Geometric Methods in Modern Physics•Mar 1, 2012•1 citation

Upper bounds for the Holevo quantity and their use

Максим Евгеньевич Широков

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.

arXiv:1703.05671•Mar 16, 2017•0 citations
For a quantum ensemble \cE_3 =\set{\Pa{p_i,\rho_i}}_{i=1}^3, M. Fannes \emph{et al.} conjectured that the entropy of the correlation matrix of \cE_3 =\set{\Pa{p_i,\rho_i})}_{i=1}^3 is upper bound for the Holevo quantity χ\Pa\cE3\chi\Pa{\cE_3}χ\Pa\cE3​ of quantum ensemble \cE_3 =\set{\Pa{p_i,\rho_i}}_{i=1}^3. In this paper, we prove the conjecture under a strictly constraint condition.
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.
The entropy of the Gram matrix of a joint purification of an ensemble of K mixed states yields an upper bound for the Holevo information Chi of the ensemble. In this work we combine geometrical and probabilistic aspects of the ensemble in order to obtain useful bounds for Chi. This is done by constructing various correlation matrices involving fidelities between every pair of states from the ensemble. For K=3 quantum states we design a matrix of root fidelities that is positive and the entropy of which is conjectured to upper bound Chi. Slightly weaker bounds are established for arbitrary ensembles. Finally, we investigate correlation matrices involving multi-state fidelities in relation to the Holevo quantity.
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.