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:1811.02242 · Physical Review A

Coherence measure: Logarithmic coherence number

Zhengjun Xi, Shanshan Yuwen

Published Feb 28, 2019• 25 citations
arXivHTMLPDF

Abstract

We introduce a measure of coherence which is extended from the coherence rank via the standard convex roof construction; we call it the logarithmic coherence number. This approach is parallel to the Schmidt measure in entanglement theory. We study some interesting properties of the logarithmic coherence number and show that this quantifier can be considered as a proper coherence measure. We find that the logarithmic coherence number can be calculated exactly for a large class of mixed states. We also discuss the relationships between the logarithmic coherence number and other coherence quantifiers, e.g., the relative entropy of coherence, the ${l}_{1}$-norm coherence, and average fidelity coherence. We give the relationship between coherence and entanglement in a bipartite system, and our results can be generalized to multipartite settings. Finally, we give that the creation of entanglement with bipartite incoherent operations is bounded by the logarithmic coherence number of the initial system during the process.

DOI
10.1103/physreva.99.022340

Related papers

Partial coherence versus entanglement

Sunho Kim, Chunhe Xiong, Shunlong Luo +2 more

We study partial coherence and its connection with entanglement. First, we provide a sufficient and necessary condition for bipartite pure-state transformation under partially incoherent operations: A bipartite pure state can be transformed to another one if and only if a majorization relationship holds between their partial coherence vectors. As a consequence, we introduce the concept of maximal partial coherent states in the sense that they can be used to construct any bipartite state of the same system via partial incoherent operations. Second, we provide a strategy to construct measures of partial coherence by use of symmetric concave functions. Third, we establish the relationship between partial coherence and entanglement. We show that the minimal partial coherence under local unitary transformations is also a measure of entanglement for bipartite pure states, which can be extended to all mixed states by convex-roof extension. Conversely, every entanglement measure can be constructed as a partial coherence measure, and we further show that partial coherence measures are induced through maximal entanglement under partially incoherent operations for bipartite pure states.

arXiv:2302.01760•Physical Review A•Jul 12, 2023•10 citations

Operational one-to-one mapping between coherence and entanglement measures

Huangjun Zhu, Zhihao Ma, Zhu Cao +2 more

We establish a general operational one-to-one mapping between coherence measures and entanglement measures: Any entanglement measure of bipartite pure states is the minimum of a suitable coherence measure over product bases. Any coherence measure of pure states, with extension to mixed states by convex roof, is the maximum entanglement generated by incoherent operations acting on the system and an incoherent ancilla. Remarkably, the generalized cnot gate is the universal optimal incoherent operation. In this way, all convex-roof coherence measures, including the coherence of formation, are endowed with (additional) operational interpretations. By virtue of this connection, many results on entanglement can be translated to the coherence setting, and vice versa. As applications, we provide tight observable lower bounds for generalized entanglement concurrence and coherence concurrence, which enable experimentalists to quantify entanglement and coherence of the maximal dimension in real experiments.

arXiv:1704.01935•Physical Review A•Sep 11, 2017•148 citations

Polynomial measure of coherence

You Zhou, Qi Zhao, Xiao Yuan +1 more

Coherence, the superposition of orthogonal quantum states, is indispensable in various quantum processes. Inspired by the polynomial invariant for classifying and quantifying entanglement, we first define polynomial coherence measure and systematically investigate its properties. Except for the qubit case, we show that there is no polynomial coherence measure satisfying the criterion that its value takes zero if and only if for incoherent states. Then, we release this strict criterion and obtain a necessary condition for polynomial coherence measure. Furthermore, we give a typical example of polynomial coherence measure for pure states and extend it to mixed states via a convex-roof construction. Analytical formula of our convex-roof polynomial coherence measure is obtained for symmetric states which are invariant under arbitrary basis permutation. Consequently, for general mixed states, we give a lower bound of our coherence measure.

arXiv:1704.05755•New Journal of Physics•Oct 9, 2017•8 citations

Logarithmic coherence: Operational interpretation of ℓ1 -norm coherence

Swapan Rana, Preeti Parashar, Andreas Winter +1 more

We show that the distillable coherence---which is equal to the relative entropy of coherence---is, up to a constant factor, always bounded by the ${\ensuremath{\ell}}_{1}$-norm measure of coherence (defined as the sum of absolute values of off diagonals). Thus the latter plays a similar role as logarithmic negativity plays in entanglement theory and this is the best operational interpretation from a resource-theoretic viewpoint. Consequently the two measures are intimately connected to another operational measure, the robustness of coherence. We find also relationships between these measures, which are tight for general states, and the tightest possible for pure and qubit states. For a given robustness, we construct a state having minimum distillable coherence.

arXiv:1612.09234•Physical Review A•Nov 27, 2017•46 citations

Lower bounds for genuine multipartite entanglement and coherence measures

Yue Dai, Yuli Dong, Zhenyu Xu +3 more

Experimentally quantifying entanglement and coherence are extremely important for quantum resource theory. However, because the quantum state tomography requires exponentially growing measurements with the number of qubits, it is hard to quantify entanglement and coherence based on the full information of the experimentally realized multipartite states. Fortunately, other methods have been found to directly measure the fidelity of experimental states without quantum state tomography. Here we present a fidelity-based method to derive experimentally accessible lower bounds for measures of genuine multipartite entanglement and coherence. On the one hand, the method works for genuine multipartite entanglement measures including the convex-roof extended negativity, the concurrence, the G-concurrence, and the geometric measure for genuine multipartite entanglement. On the other hand, the method also delivers observable lower bounds for the convex roof of the $l_{1}$-norm of coherence, the geometric measure of coherence, and the coherence of formation. Furthermore, all the lower bounds are based on the fidelity between the chosen pure state and the target state, and we obtain the lower bounds of several real experimental states as examples of our results.

•Oct 17, 2018•0 citations

Quantum coherence: A fundamental resource for establishing genuine multipartite correlations

Zong Wang, Zhihua Guo, Zhihua Chen +5 more

We establish the profound equivalence between the coherence measures defined via the convex roof method and two classes of genuine multipartite entanglement (GME) under certain conditions. Initially we construct two distinct classes of measures for genuine multipartite entanglement utilizing real symmetric concave functions and the convex roof technique. We then demonstrate that all coherence measures for any qudit states, defined through the convex roof approach, are identical to our two classes of GME measures of the states combined with an incoherent ancilla under an incoherent unitary operation. This relationship implies that genuine multipartite entanglement can be generated from the coherence incoherent in an initial state through the incoherent unitary operations. Furthermore, we conduct a preliminary exploration into the interplay between coherence and genuine quantum correlations such as genuine multipartite steering and genuine multipartite nonlocality. We show that the coherence exists in a qubit state if and only if genuine multipartite nonlocality is present for the special three-qubit $X$ states, obtained by performing unitary incoherent operations on the tensor product of the qubit state and the incoherent ancilla. Moreover, under the above process, we prove that the coherence of the qubit state exists if and only if the corresponding special $X$ state is genuinely multipartite steerable.

arXiv:2411.11485•Physical Review A•Mar 6, 2025•7 citations
We introduce a measure of coherence which is extended from the coherence rank via the standard convex roof construction; we call it the logarithmic coherence number. This approach is parallel to the Schmidt measure in entanglement theory. We study some interesting properties of the logarithmic coherence number and show that this quantifier can be considered as a proper coherence measure. We find that the logarithmic coherence number can be calculated exactly for a large class of mixed states. We also discuss the relationships between the logarithmic coherence number and other coherence quantifiers, e.g., the relative entropy of coherence, the l1{l}_{1}l1​-norm coherence, and average fidelity coherence. We give the relationship between coherence and entanglement in a bipartite system, and our results can be generalized to multipartite settings. Finally, we give that the creation of entanglement with bipartite incoherent operations is bounded by the logarithmic coherence number of the initial system during the process.
We study partial coherence and its connection with entanglement. First, we provide a sufficient and necessary condition for bipartite pure-state transformation under partially incoherent operations: A bipartite pure state can be transformed to another one if and only if a majorization relationship holds between their partial coherence vectors. As a consequence, we introduce the concept of maximal partial coherent states in the sense that they can be used to construct any bipartite state of the same system via partial incoherent operations. Second, we provide a strategy to construct measures of partial coherence by use of symmetric concave functions. Third, we establish the relationship between partial coherence and entanglement. We show that the minimal partial coherence under local unitary transformations is also a measure of entanglement for bipartite pure states, which can be extended to all mixed states by convex-roof extension. Conversely, every entanglement measure can be constructed as a partial coherence measure, and we further show that partial coherence measures are induced through maximal entanglement under partially incoherent operations for bipartite pure states.
We establish a general operational one-to-one mapping between coherence measures and entanglement measures: Any entanglement measure of bipartite pure states is the minimum of a suitable coherence measure over product bases. Any coherence measure of pure states, with extension to mixed states by convex roof, is the maximum entanglement generated by incoherent operations acting on the system and an incoherent ancilla. Remarkably, the generalized cnot gate is the universal optimal incoherent operation. In this way, all convex-roof coherence measures, including the coherence of formation, are endowed with (additional) operational interpretations. By virtue of this connection, many results on entanglement can be translated to the coherence setting, and vice versa. As applications, we provide tight observable lower bounds for generalized entanglement concurrence and coherence concurrence, which enable experimentalists to quantify entanglement and coherence of the maximal dimension in real experiments.
Coherence, the superposition of orthogonal quantum states, is indispensable in various quantum processes. Inspired by the polynomial invariant for classifying and quantifying entanglement, we first define polynomial coherence measure and systematically investigate its properties. Except for the qubit case, we show that there is no polynomial coherence measure satisfying the criterion that its value takes zero if and only if for incoherent states. Then, we release this strict criterion and obtain a necessary condition for polynomial coherence measure. Furthermore, we give a typical example of polynomial coherence measure for pure states and extend it to mixed states via a convex-roof construction. Analytical formula of our convex-roof polynomial coherence measure is obtained for symmetric states which are invariant under arbitrary basis permutation. Consequently, for general mixed states, we give a lower bound of our coherence measure.
We show that the distillable coherence---which is equal to the relative entropy of coherence---is, up to a constant factor, always bounded by the ${\ensuremath{\ell}}_{1}$-norm measure of coherence (defined as the sum of absolute values of off diagonals). Thus the latter plays a similar role as logarithmic negativity plays in entanglement theory and this is the best operational interpretation from a resource-theoretic viewpoint. Consequently the two measures are intimately connected to another operational measure, the robustness of coherence. We find also relationships between these measures, which are tight for general states, and the tightest possible for pure and qubit states. For a given robustness, we construct a state having minimum distillable coherence.
Experimentally quantifying entanglement and coherence are extremely important for quantum resource theory. However, because the quantum state tomography requires exponentially growing measurements with the number of qubits, it is hard to quantify entanglement and coherence based on the full information of the experimentally realized multipartite states. Fortunately, other methods have been found to directly measure the fidelity of experimental states without quantum state tomography. Here we present a fidelity-based method to derive experimentally accessible lower bounds for measures of genuine multipartite entanglement and coherence. On the one hand, the method works for genuine multipartite entanglement measures including the convex-roof extended negativity, the concurrence, the G-concurrence, and the geometric measure for genuine multipartite entanglement. On the other hand, the method also delivers observable lower bounds for the convex roof of the l1l_{1}l1​-norm of coherence, the geometric measure of coherence, and the coherence of formation. Furthermore, all the lower bounds are based on the fidelity between the chosen pure state and the target state, and we obtain the lower bounds of several real experimental states as examples of our results.
We establish the profound equivalence between the coherence measures defined via the convex roof method and two classes of genuine multipartite entanglement (GME) under certain conditions. Initially we construct two distinct classes of measures for genuine multipartite entanglement utilizing real symmetric concave functions and the convex roof technique. We then demonstrate that all coherence measures for any qudit states, defined through the convex roof approach, are identical to our two classes of GME measures of the states combined with an incoherent ancilla under an incoherent unitary operation. This relationship implies that genuine multipartite entanglement can be generated from the coherence incoherent in an initial state through the incoherent unitary operations. Furthermore, we conduct a preliminary exploration into the interplay between coherence and genuine quantum correlations such as genuine multipartite steering and genuine multipartite nonlocality. We show that the coherence exists in a qubit state if and only if genuine multipartite nonlocality is present for the special three-qubit XXX states, obtained by performing unitary incoherent operations on the tensor product of the qubit state and the incoherent ancilla. Moreover, under the above process, we prove that the coherence of the qubit state exists if and only if the corresponding special XXX state is genuinely multipartite steerable.