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:2111.12220

Notes on quantum coherence with l_1-norm and convex-roof l_-norm

Jiayao Zhu, Jian Ma, Tinggui Zhang

Published Nov 24, 2021• 0 citations
arXivHTMLPDF

Abstract

In this work, we evaluate quantum coherence using the l_1-norm and convex-roof l_1-norm and obtain several new results. First, we provide some new general triangle-like inequalities of quantum coherence, with results better than existing ones. Second, for some special three-dimensional quantum states, a method for calculating the convex-roof l_1-norm is presented. Lastly, we offer distinct upper bounds in the l_1-norm measure of coherence based on the quantum state itself.

DOI
10.48550/arxiv.2111.12220

Related papers

Triangle-like inequalities related to coherence and entanglement negativity

Jin, Z., Li-Jost, X., Fei, S.

Quantum coherence and entanglement are two key features in quantum mechanics and play important roles in quantum information processing and quantum computation. We provide a general triangle-like inequality satisfied by the $$l_1$$-norm measure of coherence for convex combination of arbitrary n pure states of a quantum state $$\rho $$. Furthermore, we present triangle-like inequality for the convex-roof extended negativity for any states of rank 2, which gives a positive answer to a conjecture raised in Dai et al. (Phys. Rev. A 96:062308, 2017). Detailed examples are given to illustrate the relations characterized by the triangle-like inequalities.

•Jan 1, 2019•4 citations

l 1 norm of coherence is not equal to its convex roof quantifier

Jianwei Xu

Abstract Since a rigorous framework for quantifying quantum coherence was established by Baumgratz et al (2014 Phys. Rev. Lett. 113 140401), many coherence measures have been found. For a given coherence measure C , extending the values of C on pure states to mixed states by the convex roof construction, we will get a valid coherence measure C ¯ , we call C ¯ the corresponding convex roof quantifier of C . Whether C = C ¯ for a given coherence measure is an important question. In this work, we show that for the widely used coherence measure, l 1 norm of coherence C l 1 , it holds that C l 1 ≠ C l 1 ¯ .

arXiv:2110.10891•Journal of Physics A Mathematical and Theoretical•Feb 15, 2022•4 citations

Closed expressions for one-qubit states of convex roof coherence measures

Xiao-Dan Cui, C L Liu

Abstract We study the closed expressions of the convex roof coherence measures for one-qubit states in this paper. We present the analytical expressions for the convex roof coherence measures, C f ( ρ ) , of one-qubit states with C f ( φ ) := f ( c 0 2 , c 1 2 ) (where | φ ⟩ = c 0 | 0 ⟩ + c 1 | 1 ⟩ ) being convex with respect to the l 1 norm of coherence of ϕ (i.e. C l 1 ( φ ) ), such coherence measures including the coherence of formation, the geometric measure of coherence, the coherence concurrence, and the coherence rank. We further present the operational interpretations of these measures. Finally, we present the usefulness of the convex roof coherence measures C f ( φ ) being non-convex with respect to C l 1 ( φ ) by giving the necessary and sufficient conditions for the transformations p φ 1 ⊕ ( 1 − p ) φ 2 → q ϕ 1 ⊕ ( 1 − q ) ϕ 2 via incoherent operations, where ϕ i , φ j ( i , j = 1 , 2 ) are one-qubit pure states and 0 ⩽ p , q ⩽ 1 .

arXiv:2308.03116•Journal of Physics A Mathematical and Theoretical•Jul 13, 2023•0 citations

Notes on quantum coherence with $$l_1$$-norm and convex-roof $$l_1$$-norm

Jiayao Zhu, Jian Ma, Tinggui Zhang

Quantum Information Processing•Nov 23, 2021•1 citation

The Tightness of Multipartite Coherence from Spectrum Estimation

Qi-Ming Ding, Xiao‐Xu Fang, He Lu

Detecting multipartite quantum coherence usually requires quantum state reconstruction, which is quite inefficient for large-scale quantum systems. Along this line of research, several efficient procedures have been proposed to detect multipartite quantum coherence without quantum state reconstruction, among which the spectrum-estimation-based method is suitable for various coherence measures. Here, we first generalize the spectrum-estimation-based method for the geometric measure of coherence. Then, we investigate the tightness of the estimated lower bound of various coherence measures, including the geometric measure of coherence, the l1-norm of coherence, the robustness of coherence, and some convex roof quantifiers of coherence multiqubit GHZ states and linear cluster states. Finally, we demonstrate the spectrum-estimation-based method as well as the other two efficient methods. We observe that the spectrum-estimation-based method outperforms other methods in various coherence measures, which significantly enhances the accuracy of estimation.

arXiv:2104.12094•Entropy•Nov 15, 2021•1 citation

Convex geometry of quantum resource quantification

Bartosz Regula

Abstract We introduce a framework unifying the mathematical characterisation of different measures of general quantum resources and allowing for a systematic way to define a variety of faithful quantifiers for any given convex quantum resource theory. The approach allows us to describe many commonly used measures such as matrix norm-based quantifiers, robustness measures, convex roof-based measures, and witness-based quantifiers together in a common formalism based on the convex geometry of the underlying sets of resource-free states. We establish easily verifiable criteria for a measure to possess desirable properties such as faithfulness and strong monotonicity under relevant free operations, and show that many quantifiers obtained in this framework indeed satisfy them for any considered quantum resource. We derive various bounds and relations between the measures, generalising and providing significantly simplified proofs of results found in the resource theories of quantum entanglement and coherence. We also prove that the quantification of resources in this framework simplifies for pure states, allowing us to obtain more easily computable forms of the considered measures, and show that many of them are in fact equal on pure states. Further, we investigate the dual formulation of resource quantifiers, which provide a characterisation of the sets of resource witnesses. We present an explicit application of the results to the resource theories of multi-level coherence, entanglement of Schmidt number k , multipartite entanglement, as well as magic states, providing insight into the quantification of the four resources by establishing novel quantitative relations and introducing new quantifiers, such as a measure of entanglement of Schmidt number k which generalises the convex roof–extended negativity, a measure of k -coherence which generalises the ℓ 1 norm of coherence, and a hierarchy of norm-based quantifiers of k -partite entanglement generalising the greatest cross norm.

arXiv:1707.06298•Journal of Physics A Mathematical and Theoretical•Oct 4, 2017•100 citations
In this work, we evaluate quantum coherence using the l_1-norm and convex-roof l_1-norm and obtain several new results. First, we provide some new general triangle-like inequalities of quantum coherence, with results better than existing ones. Second, for some special three-dimensional quantum states, a method for calculating the convex-roof l_1-norm is presented. Lastly, we offer distinct upper bounds in the l_1-norm measure of coherence based on the quantum state itself.
Quantum coherence and entanglement are two key features in quantum mechanics and play important roles in quantum information processing and quantum computation. We provide a general triangle-like inequality satisfied by the l1l_1l1​-norm measure of coherence for convex combination of arbitrary n pure states of a quantum state ρ\rho ρ. Furthermore, we present triangle-like inequality for the convex-roof extended negativity for any states of rank 2, which gives a positive answer to a conjecture raised in Dai et al. (Phys. Rev. A 96:062308, 2017). Detailed examples are given to illustrate the relations characterized by the triangle-like inequalities.
Abstract Since a rigorous framework for quantifying quantum coherence was established by Baumgratz et al (2014 Phys. Rev. Lett. 113 140401), many coherence measures have been found. For a given coherence measure C , extending the values of C on pure states to mixed states by the convex roof construction, we will get a valid coherence measure C ¯ , we call C ¯ the corresponding convex roof quantifier of C . Whether C = C ¯ for a given coherence measure is an important question. In this work, we show that for the widely used coherence measure, l 1 norm of coherence C l 1 , it holds that C l 1 ≠ C l 1 ¯ .
Abstract We study the closed expressions of the convex roof coherence measures for one-qubit states in this paper. We present the analytical expressions for the convex roof coherence measures, C f ( ρ ) , of one-qubit states with C f ( φ ) := f ( c 0 2 , c 1 2 ) (where | φ ⟩ = c 0 | 0 ⟩ + c 1 | 1 ⟩ ) being convex with respect to the l 1 norm of coherence of ϕ (i.e. C l 1 ( φ ) ), such coherence measures including the coherence of formation, the geometric measure of coherence, the coherence concurrence, and the coherence rank. We further present the operational interpretations of these measures. Finally, we present the usefulness of the convex roof coherence measures C f ( φ ) being non-convex with respect to C l 1 ( φ ) by giving the necessary and sufficient conditions for the transformations p φ 1 ⊕ ( 1 − p ) φ 2 → q ϕ 1 ⊕ ( 1 − q ) ϕ 2 via incoherent operations, where ϕ i , φ j ( i , j = 1 , 2 ) are one-qubit pure states and 0 ⩽ p , q ⩽ 1 .
Notes on quantum coherence with l1l_1l1​-norm and convex-roof l1l_1l1​-norm
Detecting multipartite quantum coherence usually requires quantum state reconstruction, which is quite inefficient for large-scale quantum systems. Along this line of research, several efficient procedures have been proposed to detect multipartite quantum coherence without quantum state reconstruction, among which the spectrum-estimation-based method is suitable for various coherence measures. Here, we first generalize the spectrum-estimation-based method for the geometric measure of coherence. Then, we investigate the tightness of the estimated lower bound of various coherence measures, including the geometric measure of coherence, the l1-norm of coherence, the robustness of coherence, and some convex roof quantifiers of coherence multiqubit GHZ states and linear cluster states. Finally, we demonstrate the spectrum-estimation-based method as well as the other two efficient methods. We observe that the spectrum-estimation-based method outperforms other methods in various coherence measures, which significantly enhances the accuracy of estimation.
Abstract We introduce a framework unifying the mathematical characterisation of different measures of general quantum resources and allowing for a systematic way to define a variety of faithful quantifiers for any given convex quantum resource theory. The approach allows us to describe many commonly used measures such as matrix norm-based quantifiers, robustness measures, convex roof-based measures, and witness-based quantifiers together in a common formalism based on the convex geometry of the underlying sets of resource-free states. We establish easily verifiable criteria for a measure to possess desirable properties such as faithfulness and strong monotonicity under relevant free operations, and show that many quantifiers obtained in this framework indeed satisfy them for any considered quantum resource. We derive various bounds and relations between the measures, generalising and providing significantly simplified proofs of results found in the resource theories of quantum entanglement and coherence. We also prove that the quantification of resources in this framework simplifies for pure states, allowing us to obtain more easily computable forms of the considered measures, and show that many of them are in fact equal on pure states. Further, we investigate the dual formulation of resource quantifiers, which provide a characterisation of the sets of resource witnesses. We present an explicit application of the results to the resource theories of multi-level coherence, entanglement of Schmidt number k , multipartite entanglement, as well as magic states, providing insight into the quantification of the four resources by establishing novel quantitative relations and introducing new quantifiers, such as a measure of entanglement of Schmidt number k which generalises the convex roof–extended negativity, a measure of k -coherence which generalises the ℓ 1 norm of coherence, and a hierarchy of norm-based quantifiers of k -partite entanglement generalising the greatest cross norm.