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

arXiv:1803.09385 · Frontiers of Physics

Quantifying the quantumness of ensembles via unitary similarity invariant norms

Xianfei Qi, Ting Gao, Fengli Yan

Published Apr 10, 2018• 12 citations
arXivHTMLPDF

Abstract

The quantification of the quantumness of a quantum ensemble has theoretical and practical significance in quantum information theory. We propose herein a class of measures of the quantumness of quantum ensembles using the unitary similarity invariant norms of the commutators of the constituent density operators of an ensemble. Rigorous proof shows that they share desirable properties for a measure of quantumness, such as positivity, unitary invariance, concavity under probabilistic union, convexity under state decomposition, decreasing under coarse graining, and increasing under fine graining. Several specific examples illustrate the applications of these measures of quantumness in studying quantum information.

DOI
10.1007/s11467-018-0773-3

Related papers

Quantifying the quantumness of ensembles

Nan Li, Shunlong Luo, Yuanyuan Mao

We quantify the quantumness of quantum ensembles in terms of noncommutativity between pairs of the square root of the constituent states. We prove that this measure satisfies desirable and intuitive properties required naturally for a measure of quantumness, such as positivity, unitary invariance, subadditivity, concavity under probabilistic union, convexity under state decomposition, decreasing under coarse graining, and increasing under fine graining. Some applications and implications are indicated.

Physical Review A•Aug 25, 2017•17 citations

Moments of Quantum Channel Ensembles

Matthew Duschenes, Diego García-Martín, Zoë Holmes +1 more

Moments of ensembles of unitaries play a central role in quantum information theory as they capture the statistical properties of dynamics of systems with some form of randomness. Indeed, concepts such as approximate $t$-designs arise when comparing how close an associated moment operator of a given unitary ensemble is to that of another, reference ensemble. Despite the importance of moment operators, their properties have not been as explored for quantum channels. In this work we develop a theoretical framework to compute moment operators for ensembles of quantum channels, for all moment orders $t$, with a special focus on determining ensembles that can be used as points of reference. By deriving hierarchies between ensembles, via inequalities of their moment operator norms, we give them operational meaning, and define useful concepts such as that of channel $t$-designs. Finally, we perform theoretical and numerical studies which show that different types of noise can decrease the norm of the moment operators (e.g., depolarizing noise), as well as increase it (e.g., amplitude damping noise), and generalize noise-induced concentration phenomena to channel-design-induced phenomena. Along the way, we find a block-orthogonal basis for permutations, which greatly simplifies our analyses, and may be of independent interest in moment calculations.

quant-pharXiv:2511.12700•PRX Quantum•Nov 18, 2025•1 citation

Post-measurement quantumness

Yuan Sun, Shunlong Luo

Abstract A quantum measurement (e.g., positive-operator valued measure, or more generally, instrument) performed on a system state yields a post-measurement ensemble (a family of outcome states together with the corresponding outcome probabilities), which may have quantumness due to the impossibility to perfectly discriminate the constituent states. In order to characterize information contents and structure of a measurement, it is desirable to investigate the post-measurement ensemble. In this work, we study post-measurement quantumness (i.e., quantumness of the post-measurement ensemble) by introducing a Gram matrix summarizing the interaction between a measurement and a state, and explore its implications and consequences. Gram matrix of an ensemble can be regarded as a quantum state (density operator) in an auxiliary system with a computational basis labelling the measurement outcomes. Our basic idea is to exploit coherence of the Gram matrix (with respect to the computational basis of the auxiliary system) and employ it to quantify post-measurement quantumness . We propose several quantifiers of quantumness and reveal their basic properties. We further evaluate explicitly quantumness of measurements arising from several important quantum channels, and illustrate a comparison between them. This provides a method for investigating measurements from the perspective of coherence of Gram matrix.

Physica Scripta•Oct 29, 2021•2 citations

Quantumness of ensemble via coherence of Gram matrix

Yuan Sun, Shunlong Luo, Xiangyun Lei

The Gram matrix of a set of vectors, which encapsulates the relations between the constituent vectors, plays an important role in the exploration of both geometric and information-theoretic aspects of quantum state space. In view of its usefulness and importance, we study the Gram matrix of an ensemble of pure states (a set of pure states with a prior probability distribution) and reveal its fundamental properties. We highlight and exploit the fact that the Gram matrix of an ensemble can be formally regarded as a bona fide quantum state. The key idea is to employ coherence (with respect to the computational basis) of the Gram matrix (regarded as a quantum state) to quantify quantumness of the corresponding ensemble. In particular, we propose to use the l 1 -norm of coherence and the relative entropy of coherence, of the Gram matrix, as two significant quantifiers of quantumness. We illustrate the effectiveness and power of these quantifiers of quantumness by evaluating them for several important ensembles arising from quantum measurement and quantum cryptography. We further compare the quantumness based on the Gram matrix with various other quantifiers of quantumness in the literature, and show that although they are closely related in general, they have subtle differences and capture different aspects of ensembles, which shed light on the complexity of quantum ensembles.

Europhysics Letters (EPL)•May 1, 2021•7 citations

Frobenius-norm-based measures of quantum coherence and asymmetry

Yao Yao, Guohui Dong, Xing Mao Xiao +1 more

We formulate the Frobenius-norm-based measures for quantum coherence and asymmetry respectively. In contrast to the resource theory of coherence and asymmetry, we construct a natural measure of quantum coherence inspired from optical coherence theory while the group theoretical approach is employed to quantify the asymmetry of quantum states. Besides their simple structures and explicit physical meanings, we observe that these quantities are intimately related to the purity (or linear entropy) of the corresponding quantum states. Remarkably, we demonstrate that the proposed coherence quantifier is not only a measure of mixedness, but also an intrinsic (basis-independent) quantification of quantum coherence contained in quantum states, which can also be viewed as a normalized version of Brukner-Zeilinger invariant information. In our context, the asymmetry of N-qubit quantum systems is considered under local independent and collective transformations. In- triguingly, it is illustrated that the collective effect has a significant impact on the asymmetry measure, and quantum correlation between subsystems plays a non-negligible role in this circumstance.

arXiv:1605.00789•Scientific Reports•Aug 25, 2016•52 citations

Coherence measures induced by norm functions

Yangping Jing, Chi-Kwong Li, Edward Poon +1 more

Which matrix norms induce proper measures for quantifying quantum coherence? We study this problem for two important classes of norms and show that (i) coherence measures cannot be induced by any unitary similarity invariant norm and (ii) the ℓq,p-norm induces a coherence measure if and only if q = 1 and 1 ≤ p ≤ 2, thus giving a new class of coherence measures with simple closed forms that are easy to compute. These results extend and unify previously known facts about norm-induced coherence measures and lead to a broader framework for understanding what functionals can be coherence measures.

arXiv:2008.04362•Journal of Mathematical Physics•Apr 1, 2021•7 citations
The quantification of the quantumness of a quantum ensemble has theoretical and practical significance in quantum information theory. We propose herein a class of measures of the quantumness of quantum ensembles using the unitary similarity invariant norms of the commutators of the constituent density operators of an ensemble. Rigorous proof shows that they share desirable properties for a measure of quantumness, such as positivity, unitary invariance, concavity under probabilistic union, convexity under state decomposition, decreasing under coarse graining, and increasing under fine graining. Several specific examples illustrate the applications of these measures of quantumness in studying quantum information.
We quantify the quantumness of quantum ensembles in terms of noncommutativity between pairs of the square root of the constituent states. We prove that this measure satisfies desirable and intuitive properties required naturally for a measure of quantumness, such as positivity, unitary invariance, subadditivity, concavity under probabilistic union, convexity under state decomposition, decreasing under coarse graining, and increasing under fine graining. Some applications and implications are indicated.
Moments of ensembles of unitaries play a central role in quantum information theory as they capture the statistical properties of dynamics of systems with some form of randomness. Indeed, concepts such as approximate ttt-designs arise when comparing how close an associated moment operator of a given unitary ensemble is to that of another, reference ensemble. Despite the importance of moment operators, their properties have not been as explored for quantum channels. In this work we develop a theoretical framework to compute moment operators for ensembles of quantum channels, for all moment orders ttt, with a special focus on determining ensembles that can be used as points of reference. By deriving hierarchies between ensembles, via inequalities of their moment operator norms, we give them operational meaning, and define useful concepts such as that of channel ttt-designs. Finally, we perform theoretical and numerical studies which show that different types of noise can decrease the norm of the moment operators (e.g., depolarizing noise), as well as increase it (e.g., amplitude damping noise), and generalize noise-induced concentration phenomena to channel-design-induced phenomena. Along the way, we find a block-orthogonal basis for permutations, which greatly simplifies our analyses, and may be of independent interest in moment calculations.
Abstract A quantum measurement (e.g., positive-operator valued measure, or more generally, instrument) performed on a system state yields a post-measurement ensemble (a family of outcome states together with the corresponding outcome probabilities), which may have quantumness due to the impossibility to perfectly discriminate the constituent states. In order to characterize information contents and structure of a measurement, it is desirable to investigate the post-measurement ensemble. In this work, we study post-measurement quantumness (i.e., quantumness of the post-measurement ensemble) by introducing a Gram matrix summarizing the interaction between a measurement and a state, and explore its implications and consequences. Gram matrix of an ensemble can be regarded as a quantum state (density operator) in an auxiliary system with a computational basis labelling the measurement outcomes. Our basic idea is to exploit coherence of the Gram matrix (with respect to the computational basis of the auxiliary system) and employ it to quantify post-measurement quantumness . We propose several quantifiers of quantumness and reveal their basic properties. We further evaluate explicitly quantumness of measurements arising from several important quantum channels, and illustrate a comparison between them. This provides a method for investigating measurements from the perspective of coherence of Gram matrix.
The Gram matrix of a set of vectors, which encapsulates the relations between the constituent vectors, plays an important role in the exploration of both geometric and information-theoretic aspects of quantum state space. In view of its usefulness and importance, we study the Gram matrix of an ensemble of pure states (a set of pure states with a prior probability distribution) and reveal its fundamental properties. We highlight and exploit the fact that the Gram matrix of an ensemble can be formally regarded as a bona fide quantum state. The key idea is to employ coherence (with respect to the computational basis) of the Gram matrix (regarded as a quantum state) to quantify quantumness of the corresponding ensemble. In particular, we propose to use the l 1 -norm of coherence and the relative entropy of coherence, of the Gram matrix, as two significant quantifiers of quantumness. We illustrate the effectiveness and power of these quantifiers of quantumness by evaluating them for several important ensembles arising from quantum measurement and quantum cryptography. We further compare the quantumness based on the Gram matrix with various other quantifiers of quantumness in the literature, and show that although they are closely related in general, they have subtle differences and capture different aspects of ensembles, which shed light on the complexity of quantum ensembles.
We formulate the Frobenius-norm-based measures for quantum coherence and asymmetry respectively. In contrast to the resource theory of coherence and asymmetry, we construct a natural measure of quantum coherence inspired from optical coherence theory while the group theoretical approach is employed to quantify the asymmetry of quantum states. Besides their simple structures and explicit physical meanings, we observe that these quantities are intimately related to the purity (or linear entropy) of the corresponding quantum states. Remarkably, we demonstrate that the proposed coherence quantifier is not only a measure of mixedness, but also an intrinsic (basis-independent) quantification of quantum coherence contained in quantum states, which can also be viewed as a normalized version of Brukner-Zeilinger invariant information. In our context, the asymmetry of N-qubit quantum systems is considered under local independent and collective transformations. In- triguingly, it is illustrated that the collective effect has a significant impact on the asymmetry measure, and quantum correlation between subsystems plays a non-negligible role in this circumstance.
Which matrix norms induce proper measures for quantifying quantum coherence? We study this problem for two important classes of norms and show that (i) coherence measures cannot be induced by any unitary similarity invariant norm and (ii) the ℓq,p-norm induces a coherence measure if and only if q = 1 and 1 ≤ p ≤ 2, thus giving a new class of coherence measures with simple closed forms that are easy to compute. These results extend and unify previously known facts about norm-induced coherence measures and lead to a broader framework for understanding what functionals can be coherence measures.