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

Epsilon-measure of coherence

Zhengjun Xi, Shanshan Yuwen

Published Oct 9, 2018• 0 citations
OpenAlex

Abstract

To estimate the difference between the expected state and the actually prepared state and quantify quantum coherence contained in actually prepared state, we discuss the smooth version of coherence quantifiers which depend on a precision parameter $\epsilon$. In this paper, we call it the $\epsilon$-measure of coherence, it can be interpreted as the minimal coherence guaranteed to present in an $\epsilon$ ball around given quantum state. We find that the $\epsilon$-measure of any coherence monotone is still a coherence monotone, but it does not satisfy monotonicity on average under incoherent operations. We show the $\epsilon$-measure of coherence is continuous even if the original coherence quantifier is not. In particular, we study some properties of the $\epsilon$-measure of distance-based coherence quantifier.

Related papers

Epsilon-smooth measure of coherence

Zhengjun Xi, Shanshan Yuwen

In this paper, by minimizing the coherence quantifiers over all states in an $\ensuremath{\epsilon}$ ball around a given state, we define a generalized smooth quantifier, called the $\ensuremath{\epsilon}$-smooth measure of coherence. We use it to estimate the difference between the expected state and the actually prepared state and quantify quantum coherence contained in an actually prepared state, and it can been interpreted as the minimal coherence guaranteed to present in an $\ensuremath{\epsilon}$ ball around given quantum state. We find that the $\ensuremath{\epsilon}$-smooth measure of any coherence monotone is still a coherence monotone, but it does not satisfy monotonicity on average under incoherent operations. We show the $\ensuremath{\epsilon}$-smooth measure of coherence is continuous even if the original coherence quantifier is not. We also study the $\ensuremath{\epsilon}$-smooth measure of distance-based coherence quantifiers, and some interesting properties are given. Moreover, we discuss the dual form of the $\ensuremath{\epsilon}$-smooth measure of coherence by maximizing over all states in an $\ensuremath{\epsilon}$ ball around the given state and show that the dual $\ensuremath{\epsilon}$-smooth measure of coherence provides an upper bound of one-shot coherence distillation.

arXiv:1810.03882•Physical Review A•Jan 7, 2019•16 citations

Epsilon measures of state-based quantum resource theory

Yu Luo, Fanxu Meng, Youle Wang

The quantification of state-based quantum ``resources'' such as entanglement, coherence, and nonstabilizer states lies at the heart of quantum science and technology, providing potential advantages over classical methods. In a realistic scenario, due to the imperfections and uncertainties in physical devices, we are unable to perfectly prepare or detect the true quantum states. Consequently, it is necessary to study the quantification of quantum resources under such circumstances. In this work, by focusing on the state-based quantum resource theory, we introduce a family of resource measures called $\ensuremath{\epsilon}$ measure that relies on a precision parameter to address this issue. This family of resource measures inherits the fundamental properties of the original resource measure, such as weak monotonicity, convexity, monogamy, and so forth. Furthermore, the $\ensuremath{\epsilon}$ measure remains continuous irrespective of whether the original measure is continuous or not. We also investigate the $\ensuremath{\epsilon}$ measure of distance-based resource quantifiers, and some interesting properties are presented. As part of the applications, we derived several formulas for the $\ensuremath{\epsilon}$ measure of coherence, nonstabilizerness, asymmetry, nonuniformity, and imaginarity. Additionally, we offered an upper bound for the $\ensuremath{\epsilon}$ measure of the resource rank. Finally, we outline how this work can be extended to channel-based resources among others.

Physical Review A•May 8, 2024•9 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

Fidelity and trace-norm distances for quantifying coherence

Lian-He Shao, Zhengjun Xi, Heng Fan +1 more

We investigate the coherence measures induced by fidelity and trace norm, based on the coherence quantification recently proposed by Baumgratz et al. [T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. We show that the fidelity of coherence does not in general satisfy the monotonicity requirement as a measure of coherence under the subselection of the measurement condition. We find that the trace norm of coherence can act as a measure of coherence for qubits and some special class of qutrits with some restrictions on the incoherent operators, while the general case needs to be explored further.

arXiv:1410.8327•Physical Review A•Apr 15, 2015•272 citations

Robustness of Coherence: An Operational and Observable Measure of Quantum Coherence

Carmine Napoli, Thomas R. Bromley, Marco Cianciaruso +3 more

Quantifying coherence is an essential endeavor for both quantum foundations and quantum technologies. Here, the robustness of coherence is defined and proven to be a full monotone in the context of the recently introduced resource theories of quantum coherence. The measure is shown to be observable, as it can be recast as the expectation value of a coherence witness operator for any quantum state. The robustness of coherence is evaluated analytically on relevant classes of states, and an efficient semidefinite program that computes it on general states is given. An operational interpretation is finally provided: the robustness of coherence quantifies the advantage enabled by a quantum state in a phase discrimination task.

arXiv:1601.03781•Physical Review Letters•Apr 12, 2016•581 citations

Coherence as uncertainty

Yuan Sun, Shunlong Luo

Coherence of a quantum state (with respect to a fixed incoherent basis) is usually quantified by distancelike quantities, among which a particularly convenient and intuitive quantifier of coherence is based on the Hilbert-Schmidt distance between the quantum state and its dephased counterpart. This quantifier has a simple structure and many nice properties, although it is not a coherence monotone in the conventional resource-theoretic framework of coherence. Here we reveal its information-theoretic significance by showing that it coincides with uncertainty, as quantified by the variance of the state in the incoherent basis. The key point here is to regard the state as an observable, and to regard the incoherent basis as an ensemble of states rather than as measurement operators. Our result provides a natural decomposition of coherence into components along the incoherent basis. Furthermore, in terms of the Tsallis 2-entropy, which is also a measure of uncertainty, we provide two alternative interpretations of coherence: as increase of uncertainty caused by decoherence and as the conditional Tsallis 2-entropy in the context of purification. An intrinsic relation between the maximal coherence and the Brukner-Zeilinger invariant information is also established. These identifications of coherence with increase of uncertainty lead us to interpret coherence as a manifestation of quantum uncertainty, which may have implications for both quantum foundations and applications.

Physical Review A•Apr 23, 2021•28 citations
To estimate the difference between the expected state and the actually prepared state and quantify quantum coherence contained in actually prepared state, we discuss the smooth version of coherence quantifiers which depend on a precision parameter ϵ\epsilonϵ. In this paper, we call it the ϵ\epsilonϵ-measure of coherence, it can be interpreted as the minimal coherence guaranteed to present in an ϵ\epsilonϵ ball around given quantum state. We find that the ϵ\epsilonϵ-measure of any coherence monotone is still a coherence monotone, but it does not satisfy monotonicity on average under incoherent operations. We show the ϵ\epsilonϵ-measure of coherence is continuous even if the original coherence quantifier is not. In particular, we study some properties of the ϵ\epsilonϵ-measure of distance-based coherence quantifier.
In this paper, by minimizing the coherence quantifiers over all states in an $\ensuremath{\epsilon}$ ball around a given state, we define a generalized smooth quantifier, called the $\ensuremath{\epsilon}$-smooth measure of coherence. We use it to estimate the difference between the expected state and the actually prepared state and quantify quantum coherence contained in an actually prepared state, and it can been interpreted as the minimal coherence guaranteed to present in an $\ensuremath{\epsilon}$ ball around given quantum state. We find that the $\ensuremath{\epsilon}$-smooth measure of any coherence monotone is still a coherence monotone, but it does not satisfy monotonicity on average under incoherent operations. We show the $\ensuremath{\epsilon}$-smooth measure of coherence is continuous even if the original coherence quantifier is not. We also study the $\ensuremath{\epsilon}$-smooth measure of distance-based coherence quantifiers, and some interesting properties are given. Moreover, we discuss the dual form of the $\ensuremath{\epsilon}$-smooth measure of coherence by maximizing over all states in an $\ensuremath{\epsilon}$ ball around the given state and show that the dual $\ensuremath{\epsilon}$-smooth measure of coherence provides an upper bound of one-shot coherence distillation.
The quantification of state-based quantum ``resources'' such as entanglement, coherence, and nonstabilizer states lies at the heart of quantum science and technology, providing potential advantages over classical methods. In a realistic scenario, due to the imperfections and uncertainties in physical devices, we are unable to perfectly prepare or detect the true quantum states. Consequently, it is necessary to study the quantification of quantum resources under such circumstances. In this work, by focusing on the state-based quantum resource theory, we introduce a family of resource measures called $\ensuremath{\epsilon}$ measure that relies on a precision parameter to address this issue. This family of resource measures inherits the fundamental properties of the original resource measure, such as weak monotonicity, convexity, monogamy, and so forth. Furthermore, the $\ensuremath{\epsilon}$ measure remains continuous irrespective of whether the original measure is continuous or not. We also investigate the $\ensuremath{\epsilon}$ measure of distance-based resource quantifiers, and some interesting properties are presented. As part of the applications, we derived several formulas for the $\ensuremath{\epsilon}$ measure of coherence, nonstabilizerness, asymmetry, nonuniformity, and imaginarity. Additionally, we offered an upper bound for the $\ensuremath{\epsilon}$ measure of the resource rank. Finally, we outline how this work can be extended to channel-based resources among others.
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 the coherence measures induced by fidelity and trace norm, based on the coherence quantification recently proposed by Baumgratz et al. [T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. We show that the fidelity of coherence does not in general satisfy the monotonicity requirement as a measure of coherence under the subselection of the measurement condition. We find that the trace norm of coherence can act as a measure of coherence for qubits and some special class of qutrits with some restrictions on the incoherent operators, while the general case needs to be explored further.
Quantifying coherence is an essential endeavor for both quantum foundations and quantum technologies. Here, the robustness of coherence is defined and proven to be a full monotone in the context of the recently introduced resource theories of quantum coherence. The measure is shown to be observable, as it can be recast as the expectation value of a coherence witness operator for any quantum state. The robustness of coherence is evaluated analytically on relevant classes of states, and an efficient semidefinite program that computes it on general states is given. An operational interpretation is finally provided: the robustness of coherence quantifies the advantage enabled by a quantum state in a phase discrimination task.
Coherence of a quantum state (with respect to a fixed incoherent basis) is usually quantified by distancelike quantities, among which a particularly convenient and intuitive quantifier of coherence is based on the Hilbert-Schmidt distance between the quantum state and its dephased counterpart. This quantifier has a simple structure and many nice properties, although it is not a coherence monotone in the conventional resource-theoretic framework of coherence. Here we reveal its information-theoretic significance by showing that it coincides with uncertainty, as quantified by the variance of the state in the incoherent basis. The key point here is to regard the state as an observable, and to regard the incoherent basis as an ensemble of states rather than as measurement operators. Our result provides a natural decomposition of coherence into components along the incoherent basis. Furthermore, in terms of the Tsallis 2-entropy, which is also a measure of uncertainty, we provide two alternative interpretations of coherence: as increase of uncertainty caused by decoherence and as the conditional Tsallis 2-entropy in the context of purification. An intrinsic relation between the maximal coherence and the Brukner-Zeilinger invariant information is also established. These identifications of coherence with increase of uncertainty lead us to interpret coherence as a manifestation of quantum uncertainty, which may have implications for both quantum foundations and applications.