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:2301.12295 · APL Quantum

Quantifying coherence with principal diagonal elements of density matrix

Manis Hazra, Debabrata Goswami

Published Sep 1, 2024• 2 citations
arXivHTMLPDF

Abstract

Being the key resource in quantum physics, the proper quantification of coherence is of utmost importance. Amid complex-looking functionals in quantifying coherence, we set forth a simple and easy-to-evaluate approach: principal diagonal difference of coherence (CPDD), which we prove to be non-negative, self-normalized, and monotonic (under any incoherent operation). In an attempt to extend CPDD in the domain of mixed states, we propose a simple, generic expression (applicable to other pure state coherence measures as well). To validate this theory, we consider a fictitious two-qubit system (both interacting and non-interacting) and, through the laser pulse–system interaction (semi-classical approach), compare the coherence evolution of CPDD with the relative entropy of coherence (Cr⋅e) and the l1-norm of coherence (Cl1), in both pure- and mixed-state regimes. The results from both the numerical and analytical (for a two-level system) studies clearly demonstrate a better sensitivity of CPDD in the close proximity of the maximally coherent states (MCS) than Cr⋅e or Cl1. To the best of our knowledge, CPDD is the only quantifier (among all existing coherence quantifiers) that is a linear function of density-matrix elements.

DOI
10.1063/5.0209604

Related papers

Two novel pure-state coherence measures in quantifying coherence

Manis Hazra, Debabrata Goswami

Quantification is one of the primary goals of quantum coherence resource theory. Here, we put forward two coherence measures, analytically prove their validity in the pure-state regime following the axiomatic definition of a legitimate pure-state coherence measure (PSCM), and provide their general normalized forms. We further discuss their extensions in the domain of mixed states with the assistance of the convex roof of coherence, and top coherence (quantification in terms of pure state coherence) and define their classes (coherence \emph{monotone} or \emph{measure}); in this regard, we thoroughly investigate the top coherence class. For the quantitative demonstration, we pick up different laser pulse-two-qubit interaction scenarios and compare the evolutions of these two coherence measures along with \(l_{1}\)-norm of coherence and relative entropy of coherence; this study also signifies the importance of coherence in probing an atomic or molecular system.

Quantum Information and Computation•Oct 1, 2024•0 citations

Investigating quantum coherence from discrete Wigner function

Lin Yin, Ming‐Da Huang, Yu Ya-Fei +1 more

Quantum coherence is an essential ingredient in quantum information processing and plays an important role in quantum computation. Therefore, it is a hot issue about how to quantify the coherence of quantum states in theoretical framework. The coherence effect of a state is usually described by the off-diagonal elements of its density matrix with respect to a particular reference basis. Recently, based on the established notions from quantitative theory of entanglement, a resource theory of coherence quantification has been proposed[1,2]. In the theory framework, a proper measure of coherence should satisfy three criteria: the coherence should be zero for all incoherent state; the coherence should not increase under mixing quantum states; the coherence should not increase under incoherent operations. Then, a number of coherence measures have been suggested, such as l1 norm of coherence and the relative entropy of coherence[2]. Wigner function is known as an important tool to study the non-classical property of quantum states for continuous-variable quantum systems. It has been generalized to finite-dimensional Hilbert spaces, and named as discrete Wigner function[9-16]. The magic property of quantum states, which promotes stabilizer computation to universal quantum computation, can be generally measured by the absolute sum of the negative items (negativity sum) in the discrete Wigner function of the observed quantum states. In this paper we investigate quantum coherence from the view of discrete Wigner function. From the definition of the discrete Wigner function of the quantum systems with odd prime dimensions, for a given density matrix we analyze in phase space the performance of its diagonal and off-diagonal items. We find that, the discrete Wigner function of a quantum state contains two aspects: the true quantum coherence and the classical mixture, where the part of classical mixture can be excluded by only considering the discrete Wigner function of the diagonal items of the density matrix. Thus, we propose a possible measure method for quantum coherence from the discrete Wigner function of the off-diagonal items of the density matrix. We show that the proposed measure method satisfies the criteria (C1) and (C2) of coherence measure perfectly. For the criteria (C3), we give a numerical proof in three-dimensional quantum system. Meanwhile, we compare the proposed coherence measure with l1 norm coherence, and get an inequality relationship between them. Finally, an inequality is obtained to discuss the relation between quantum coherence and the negativity sum of discrete Wigner function, which shows that the quantum coherence is only necessary but not sufficient for quantum computation speed-up.

Acta Physica Sinica•Jan 1, 2017•2 citations

Quantifying coherence of quantum measurements

Kyunghyun Baek, Adel Sohbi, Jaehak Lee +2 more

Abstract In this work we investigate how to quantify the coherence of quantum measurements. First, we establish a resource theoretical framework to address the coherence of measurement and show that any statistical distance can be adopted to define a coherence monotone of measurement. For instance, the relative entropy fulfills all the required properties as a proper monotone. We specifically introduce a coherence monotone of measurement in terms of off-diagonal elements of positive-operator-valued measure components. This quantification provides a lower bound on the robustness of measurement-coherence that has an operational meaning as the maximal advantage over all incoherent measurements in state discrimination tasks. Finally, we propose an experimental scheme to assess our quantification of measurement-coherence and demonstrate it by performing an experiment using a single qubit on IBM Q processor.

arXiv:2008.03999•New Journal of Physics•Aug 7, 2020•24 citations

Quantum Coherence Dispersion

Fernando Parisio

We investigate how quantum coherence can be distributed among the several off-diagonal elements of an arbitrary density matrix. An easily computable quantity that captures this variability notion is proposed and it is argued that it presents marked features of complexity quantifiers. It turns out that this coherence dispersion ($Δ_{\rm c}$) is maximized for intermediate values of an appropriate entropy (the relative entropy of coherence), a prevalent signature of complexity quantifiers across different fields, from evolutionary biology to linguistics and information science. The final part of the manuscript is dedicated to the connection between the proposed framework and non-equilibrium systems in the quantum regime.

quant-pharXiv:2512.12342•Dec 16, 2025•0 citations

Two novel pure-state coherence measures in quantifying coherence

Manis Hazra, Debabrata Goswami

In the resource theory of coherence, the quantification of quantum-state coherence is an important task. In this regard, the key ingredients are the various coherence monotones (or measures). There are few coherence-monotone classes that solely depend on other coherence measures defined for all the pure states; in other words, they rely on the pure state coherence measures (PSCM). Here, we set forth two such novel PSCMs, and validate each of them through the fulfillment of all four necessary conditions. In addition, we delve into the most recent (as per our knowledge) coherence-monotone class based on the innovative idea of quantifying coherence in terms of pure-state coherence, further redefine it, and, through the study of convexity under mixing, justify why this coherence monotone class cannot be treated as a coherence-measure class in general.

arXiv:2403.07146•Mar 11, 2024•0 citations

Quantifying quantum coherence with quantum Fisher information

X. N. Feng, L. F. Wei

Quantum coherence is one of the old but always important concepts in quantum mechanics, and now it has been regarded as a necessary resource for quantum information processing and quantum metrology. However, the question of how to quantify the quantum coherence has just been paid the attention recently (see, e.g., Baumgratz et al. PRL, 113. 140401 (2014)). In this paper we verify that the well-known quantum Fisher information (QFI) can be utilized to quantify the quantum coherence, as it satisfies the monotonicity under the typical incoherent operations and the convexity under the mixing of the quantum states. Differing from most of the pure axiomatic methods, quantifying quantum coherence by QFI could be experimentally testable, as the bound of the QFI is practically measurable. The validity of our proposal is specifically demonstrated with the typical phase-damping and depolarizing evolution processes of a generic single-qubit state, and also by comparing it with the other quantifying methods proposed previously.

Scientific Reports•Nov 8, 2017•39 citations
Being the key resource in quantum physics, the proper quantification of coherence is of utmost importance. Amid complex-looking functionals in quantifying coherence, we set forth a simple and easy-to-evaluate approach: principal diagonal difference of coherence (CPDD), which we prove to be non-negative, self-normalized, and monotonic (under any incoherent operation). In an attempt to extend CPDD in the domain of mixed states, we propose a simple, generic expression (applicable to other pure state coherence measures as well). To validate this theory, we consider a fictitious two-qubit system (both interacting and non-interacting) and, through the laser pulse–system interaction (semi-classical approach), compare the coherence evolution of CPDD with the relative entropy of coherence (Cr⋅e) and the l1-norm of coherence (Cl1), in both pure- and mixed-state regimes. The results from both the numerical and analytical (for a two-level system) studies clearly demonstrate a better sensitivity of CPDD in the close proximity of the maximally coherent states (MCS) than Cr⋅e or Cl1. To the best of our knowledge, CPDD is the only quantifier (among all existing coherence quantifiers) that is a linear function of density-matrix elements.
Quantification is one of the primary goals of quantum coherence resource theory. Here, we put forward two coherence measures, analytically prove their validity in the pure-state regime following the axiomatic definition of a legitimate pure-state coherence measure (PSCM), and provide their general normalized forms. We further discuss their extensions in the domain of mixed states with the assistance of the convex roof of coherence, and top coherence (quantification in terms of pure state coherence) and define their classes (coherence \emph{monotone} or \emph{measure}); in this regard, we thoroughly investigate the top coherence class. For the quantitative demonstration, we pick up different laser pulse-two-qubit interaction scenarios and compare the evolutions of these two coherence measures along with l1l_{1}l1​-norm of coherence and relative entropy of coherence; this study also signifies the importance of coherence in probing an atomic or molecular system.
Quantum coherence is an essential ingredient in quantum information processing and plays an important role in quantum computation. Therefore, it is a hot issue about how to quantify the coherence of quantum states in theoretical framework. The coherence effect of a state is usually described by the off-diagonal elements of its density matrix with respect to a particular reference basis. Recently, based on the established notions from quantitative theory of entanglement, a resource theory of coherence quantification has been proposed[1,2]. In the theory framework, a proper measure of coherence should satisfy three criteria: the coherence should be zero for all incoherent state; the coherence should not increase under mixing quantum states; the coherence should not increase under incoherent operations. Then, a number of coherence measures have been suggested, such as l1 norm of coherence and the relative entropy of coherence[2]. Wigner function is known as an important tool to study the non-classical property of quantum states for continuous-variable quantum systems. It has been generalized to finite-dimensional Hilbert spaces, and named as discrete Wigner function[9-16]. The magic property of quantum states, which promotes stabilizer computation to universal quantum computation, can be generally measured by the absolute sum of the negative items (negativity sum) in the discrete Wigner function of the observed quantum states. In this paper we investigate quantum coherence from the view of discrete Wigner function. From the definition of the discrete Wigner function of the quantum systems with odd prime dimensions, for a given density matrix we analyze in phase space the performance of its diagonal and off-diagonal items. We find that, the discrete Wigner function of a quantum state contains two aspects: the true quantum coherence and the classical mixture, where the part of classical mixture can be excluded by only considering the discrete Wigner function of the diagonal items of the density matrix. Thus, we propose a possible measure method for quantum coherence from the discrete Wigner function of the off-diagonal items of the density matrix. We show that the proposed measure method satisfies the criteria (C1) and (C2) of coherence measure perfectly. For the criteria (C3), we give a numerical proof in three-dimensional quantum system. Meanwhile, we compare the proposed coherence measure with l1 norm coherence, and get an inequality relationship between them. Finally, an inequality is obtained to discuss the relation between quantum coherence and the negativity sum of discrete Wigner function, which shows that the quantum coherence is only necessary but not sufficient for quantum computation speed-up.
Abstract In this work we investigate how to quantify the coherence of quantum measurements. First, we establish a resource theoretical framework to address the coherence of measurement and show that any statistical distance can be adopted to define a coherence monotone of measurement. For instance, the relative entropy fulfills all the required properties as a proper monotone. We specifically introduce a coherence monotone of measurement in terms of off-diagonal elements of positive-operator-valued measure components. This quantification provides a lower bound on the robustness of measurement-coherence that has an operational meaning as the maximal advantage over all incoherent measurements in state discrimination tasks. Finally, we propose an experimental scheme to assess our quantification of measurement-coherence and demonstrate it by performing an experiment using a single qubit on IBM Q processor.
We investigate how quantum coherence can be distributed among the several off-diagonal elements of an arbitrary density matrix. An easily computable quantity that captures this variability notion is proposed and it is argued that it presents marked features of complexity quantifiers. It turns out that this coherence dispersion (ΔcΔ_{\rm c}Δc​) is maximized for intermediate values of an appropriate entropy (the relative entropy of coherence), a prevalent signature of complexity quantifiers across different fields, from evolutionary biology to linguistics and information science. The final part of the manuscript is dedicated to the connection between the proposed framework and non-equilibrium systems in the quantum regime.
In the resource theory of coherence, the quantification of quantum-state coherence is an important task. In this regard, the key ingredients are the various coherence monotones (or measures). There are few coherence-monotone classes that solely depend on other coherence measures defined for all the pure states; in other words, they rely on the pure state coherence measures (PSCM). Here, we set forth two such novel PSCMs, and validate each of them through the fulfillment of all four necessary conditions. In addition, we delve into the most recent (as per our knowledge) coherence-monotone class based on the innovative idea of quantifying coherence in terms of pure-state coherence, further redefine it, and, through the study of convexity under mixing, justify why this coherence monotone class cannot be treated as a coherence-measure class in general.
Quantum coherence is one of the old but always important concepts in quantum mechanics, and now it has been regarded as a necessary resource for quantum information processing and quantum metrology. However, the question of how to quantify the quantum coherence has just been paid the attention recently (see, e.g., Baumgratz et al. PRL, 113. 140401 (2014)). In this paper we verify that the well-known quantum Fisher information (QFI) can be utilized to quantify the quantum coherence, as it satisfies the monotonicity under the typical incoherent operations and the convexity under the mixing of the quantum states. Differing from most of the pure axiomatic methods, quantifying quantum coherence by QFI could be experimentally testable, as the bound of the QFI is practically measurable. The validity of our proposal is specifically demonstrated with the typical phase-damping and depolarizing evolution processes of a generic single-qubit state, and also by comparing it with the other quantifying methods proposed previously.