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:2310.15476 · Journal of Physics A Mathematical and Theoretical

Trade-off relations of geometric coherence

Bingyu Hu, Ming‐Jing Zhao

Published Oct 9, 2023• 5 citations
arXivHTMLPDF

Abstract

Abstract Quantum coherence is an important quantum resource and it is intimately related to various research fields. The geometric coherence is a coherence measure both operationally and geometrically. We study the trade-off relation of geometric coherence in qubit systems. We first derive an upper bound for the geometric coherence by the purity of quantum states. Based on this, a complementarity relation between the quantum coherence and the mixedness is established. We then derive the quantum uncertainty relations of the geometric coherence on two and three general measurement bases in terms of the incompatibility respectively, which turn out to be state-independent for pure states. These trade-off relations provide the limit to the amount of quantum coherence. As a byproduct, the complementarity relation between the minimum error probability for discriminating a pure-states ensemble and the mixedness of quantum states is established.

DOI
10.1088/1751-8121/ad018a

Related papers

Trade-off relations of Hellinger-distance-based coherence

Liu Sun, Shao-Qiang Shang, Yuan-Hong Tao +1 more

Abstract Quantum coherence, a fundamental quantum resource, is critical for quantum computing, communication, and metrology. Hellinger-distance-based coherence enables robust quantum superposition quantification; however its basis-dependent behavior hinders practical use in quantum information processing. To address this, this study explores the trade-off relations of Hellinger-distance-based coherence in qubits, with core contributions: calculating its upper bound (max achievable value for any qubit state); deriving its complementary relations with geometric mixedness (higher mixedness reduces coherence); and establishing its trade-offs under mutually unbiased bases. For applications, we link the minimum error probability of pure state discrimination (via least square measurement) to geometric purity (optimizing discrimination), and connect Hellinger-distance-based coherence to nonlocal advantage (critical for quantum steering).

Physica Scripta•Mar 27, 2026•0 citations

Geometric coherence and quantum state discrimination

Chunhe Xiong, Junde Wu

Abstract The operational meaning of coherence measure lies at the very heart of the coherence theory. In this paper, we provide an operational interpretation for geometric coherence, by proving that the geometric coherence of a quantum state is equal to the minimum error probability to discriminate a set of pure states with von Neumann measurement. On the other hand, we also show that a task to ambiguously discriminate a set of linearly independent pure states can be regarded as a problem of calculating geometric coherence. That is, we reveal an equivalence relation between ambiguous quantum state discrimination and geometric coherence. Based on this equivalence, moreover, we improve the upper bound for geometric coherence and give the explicit expression of geometric coherence for a class of states. Besides, we establish a complementarity relation of geometric coherence and path distinguishability, with which the relationship between l 1 -norm of coherence and geometric coherence is obtained. Finally, with geometric coherence, we study multiple copies of quantum state discrimination and give an example to show how to discriminate two pure states.

arXiv:1801.06031•Journal of Physics A Mathematical and Theoretical•Jun 1, 2018•32 citations

Uncertainty relations for quantum coherence with respect to mutually unbiased bases

Alexey E. Rastegin

The concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corresponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases (MUBs). To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.

arXiv:1702.01409•Frontiers of Physics•Sep 1, 2017•31 citations

Complementarity relations for quantum coherence

Shuming Cheng, Michael J. W. Hall

Various measures have been suggested recently for quantifying the coherence of a quantum state with respect to a given basis. We first use two of these, the ${l}_{1}$-norm and relative entropy measures, to investigate tradeoffs between the coherences of mutually unbiased bases. Results include relations between coherence, uncertainty, and purity; tight general bounds restricting the coherences of mutually unbiased bases; and an exact complementarity relation for qubit coherences. We further define the average coherence of a quantum state. For the ${l}_{1}$-norm measure this is related to a natural ``coherence radius'' for the state and leads to a conjecture for an ${l}_{2}$-norm measure of coherence. For relative entropy the average coherence is determined by the difference between the von Neumann entropy and the quantum subentropy of the state and leads to upper bounds for the latter quantity. Finally, we point out that the relative entropy of coherence is a special case of G-asymmetry, which immediately yields several operational interpretations in contexts as diverse as frame alignment, quantum communication, and metrology, and suggests generalizing the property of quantum coherence to arbitrary groups of physical transformations.

arXiv:1508.03240•Physical Review A•Oct 2, 2015•182 citations

Estimation on Geometric Measure of Quantum Coherence*

Hai-Jun 海军 Zhang 张, Bin 斌 Chen 陈, Ming 明 Li 李 +2 more

We study the geometric measure of quantum coherence recently proposed in [Phys. Rev. Lett. 115 (2015) 020403]. Both lower and upper bounds of this measure are provided. These bounds are shown to be tight for a class of important coherent states — maximally coherent mixed states. The trade-off relation between quantum coherence and mixedness for this measure is also discussed.

arXiv:1703.04245•Communications in Theoretical Physics•Feb 1, 2017•47 citations

Geometric block-coherence and quantum state discrimination

Danting Tang, Ping Li, Mingfei Ye +1 more

Abstract Quantum coherence with respect to orthonormal bases has been studied extensively in the past few years. From the perspective of operational meaning, geometric coherence can be equal to the minimum error probability to discriminate a set of pure states (2018 J. Phys. A: Math. Theor. 51 414005). By regarding coherence as a physical resource, Baumgratz et al (2014 Phys. Rev. Lett. 113 140401) presented a comprehensive framework for coherence. Recently, geometric block-coherence as an effective block-coherence measure has been proposed. In this paper, we reveal an equivalence relationship between mixed quantum state discrimination (QSD) task and geometric block-coherence, which provides an operational interpretation for geometric block-coherence and generalizes the main result in coherence resource theory. Meanwhile, we show that partial coherence is a special case of block-coherence. By linking the relationship between geometric partial coherence and QSD tasks, we show that the value range of the two measures is the same. Finally, we reveal the relationship between geometric positive operator-valued measure-based coherence and QSD task.

Journal of Physics A Mathematical and Theoretical•Dec 7, 2021•0 citations
Abstract Quantum coherence is an important quantum resource and it is intimately related to various research fields. The geometric coherence is a coherence measure both operationally and geometrically. We study the trade-off relation of geometric coherence in qubit systems. We first derive an upper bound for the geometric coherence by the purity of quantum states. Based on this, a complementarity relation between the quantum coherence and the mixedness is established. We then derive the quantum uncertainty relations of the geometric coherence on two and three general measurement bases in terms of the incompatibility respectively, which turn out to be state-independent for pure states. These trade-off relations provide the limit to the amount of quantum coherence. As a byproduct, the complementarity relation between the minimum error probability for discriminating a pure-states ensemble and the mixedness of quantum states is established.
Abstract Quantum coherence, a fundamental quantum resource, is critical for quantum computing, communication, and metrology. Hellinger-distance-based coherence enables robust quantum superposition quantification; however its basis-dependent behavior hinders practical use in quantum information processing. To address this, this study explores the trade-off relations of Hellinger-distance-based coherence in qubits, with core contributions: calculating its upper bound (max achievable value for any qubit state); deriving its complementary relations with geometric mixedness (higher mixedness reduces coherence); and establishing its trade-offs under mutually unbiased bases. For applications, we link the minimum error probability of pure state discrimination (via least square measurement) to geometric purity (optimizing discrimination), and connect Hellinger-distance-based coherence to nonlocal advantage (critical for quantum steering).
Abstract The operational meaning of coherence measure lies at the very heart of the coherence theory. In this paper, we provide an operational interpretation for geometric coherence, by proving that the geometric coherence of a quantum state is equal to the minimum error probability to discriminate a set of pure states with von Neumann measurement. On the other hand, we also show that a task to ambiguously discriminate a set of linearly independent pure states can be regarded as a problem of calculating geometric coherence. That is, we reveal an equivalence relation between ambiguous quantum state discrimination and geometric coherence. Based on this equivalence, moreover, we improve the upper bound for geometric coherence and give the explicit expression of geometric coherence for a class of states. Besides, we establish a complementarity relation of geometric coherence and path distinguishability, with which the relationship between l 1 -norm of coherence and geometric coherence is obtained. Finally, with geometric coherence, we study multiple copies of quantum state discrimination and give an example to show how to discriminate two pure states.
The concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corresponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases (MUBs). To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.
Various measures have been suggested recently for quantifying the coherence of a quantum state with respect to a given basis. We first use two of these, the l1{l}_{1}l1​-norm and relative entropy measures, to investigate tradeoffs between the coherences of mutually unbiased bases. Results include relations between coherence, uncertainty, and purity; tight general bounds restricting the coherences of mutually unbiased bases; and an exact complementarity relation for qubit coherences. We further define the average coherence of a quantum state. For the l1{l}_{1}l1​-norm measure this is related to a natural ``coherence radius'' for the state and leads to a conjecture for an l2{l}_{2}l2​-norm measure of coherence. For relative entropy the average coherence is determined by the difference between the von Neumann entropy and the quantum subentropy of the state and leads to upper bounds for the latter quantity. Finally, we point out that the relative entropy of coherence is a special case of G-asymmetry, which immediately yields several operational interpretations in contexts as diverse as frame alignment, quantum communication, and metrology, and suggests generalizing the property of quantum coherence to arbitrary groups of physical transformations.
Abstract Quantum coherence with respect to orthonormal bases has been studied extensively in the past few years. From the perspective of operational meaning, geometric coherence can be equal to the minimum error probability to discriminate a set of pure states (2018 J. Phys. A: Math. Theor. 51 414005). By regarding coherence as a physical resource, Baumgratz et al (2014 Phys. Rev. Lett. 113 140401) presented a comprehensive framework for coherence. Recently, geometric block-coherence as an effective block-coherence measure has been proposed. In this paper, we reveal an equivalence relationship between mixed quantum state discrimination (QSD) task and geometric block-coherence, which provides an operational interpretation for geometric block-coherence and generalizes the main result in coherence resource theory. Meanwhile, we show that partial coherence is a special case of block-coherence. By linking the relationship between geometric partial coherence and QSD tasks, we show that the value range of the two measures is the same. Finally, we reveal the relationship between geometric positive operator-valued measure-based coherence and QSD task.