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

Triangle-like inequalities related to coherence and entanglement negativity

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

Published Jan 1, 2019• 4 citations
OpenAlex

Abstract

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.

Related papers

Triangle inequalities in coherence measures and entanglement concurrence

Yue Dai, Wen‐Long You, Yuli Dong +1 more

We provide detailed proofs of triangle inequalities in coherence measures and entanglement concurrence. If a rank-2 state $\ensuremath{\varrho}$ can be expressed as a convex combination of two pure states, i.e., $\ensuremath{\varrho}={p}_{1}|{\ensuremath{\psi}}_{1}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\psi}}_{1}|+{p}_{2}|{\ensuremath{\psi}}_{2}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\psi}}_{2}|$, a triangle inequality can be established as $|E(|{\mathrm{\ensuremath{\Psi}}}_{1}\ensuremath{\rangle})\ensuremath{-}E(|{\mathrm{\ensuremath{\Psi}}}_{2}\ensuremath{\rangle})|\ensuremath{\le}E(\ensuremath{\varrho})\ensuremath{\le}E(|{\mathrm{\ensuremath{\Psi}}}_{1}\ensuremath{\rangle})+E(|{\mathrm{\ensuremath{\Psi}}}_{2}\ensuremath{\rangle})$, where $|{\mathrm{\ensuremath{\Psi}}}_{1}\ensuremath{\rangle}=\sqrt{{p}_{1}}|{\ensuremath{\psi}}_{1}\ensuremath{\rangle}$ and $|{\mathrm{\ensuremath{\Psi}}}_{2}\ensuremath{\rangle}=\sqrt{{p}_{2}}|{\ensuremath{\psi}}_{2}\ensuremath{\rangle}$; $E$ can be considered either coherence measures or entanglement concurrence. This inequality displays mathematical beauty for its similarity to the triangle inequality in plane geometry. An illustrative example is given after the proof.

arXiv:1804.03840•Physical Review A•Dec 6, 2017•8 citations

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

Jiayao Zhu, Jian Ma, Tinggui Zhang

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.

arXiv:2111.12220•Nov 24, 2021•0 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

Convex-roof extended negativity as an entanglement measure for bipartite quantum systems

Soojoon Lee, Dong Pyo Chi, Sung Dahm Oh +1 more

We extend the concept of negativity, a good measure of entanglement for bipartite pure states, to mixed states by means of the convex-roof extension. We show that the measure does not increase under local quantum operations and classical communication, and derive explicit formulas for the entanglement measure of isotropic states and Werner states, applying the formalism presented by Vollbrecht and Werner [Phys. Rev. A 64, 062307 (2001)].

arXiv:quant-ph/0310027•Physical Review A•Dec 4, 2003•182 citations

Monogamy of nonconvex entanglement measures

Limin Gao, Fengli Yan, Ting Gao

One of the fundamental traits of quantum entanglement is the restricted shareability among multipartite quantum systems, namely monogamy of entanglement, while it is well known that monogamy inequalities are always satisfied by entanglement measures with convexity. Here we present a measure of entanglement, logarithmic convex-roof extended negativity (LCREN) satisfying important characteristics of an entanglement measure, and investigate the monogamy relation for logarithmic negativity and LCREN both without convexity. We show exactly that the αth power of logarithmic negativity, and a newly defined good measure of entanglement, LCREN, obey a class of general monogamy inequalities in multiqubit systems, 2⊗2⊗3 systems and 2⊗2⊗2n systems for α≥4ln2. We provide a class of general polygamy inequalities of multiqubit systems in terms of logarithmic convex-roof extended negativity of assistance (LCRENoA) for 0≤β≤2. Given that the logarithmic negativity and LCREN are not convex, these results are surprising. Using the power of the logarithmic negativity and LCREN, we further establish a class of tight monogamy inequalities of multiqubit systems, 2⊗2⊗3 systems and 2⊗2⊗2n systems in terms of the αth power of logarithmic negativity and LCREN for α≥4ln2. We also show that the βth power of LCRENoA obeys a class of tight polygamy inequalities of multiqubit systems for 0≤β≤2.

arXiv:2007.09573•Results in Physics•Nov 16, 2021•10 citations

Axiomatic and operational connections between the l1 -norm of coherence and negativity

Huangjun Zhu, Masahito Hayashi, Lin Chen

Quantum coherence plays a central role in various research areas. The ${l}_{1}$-norm of coherence is one of the most important coherence measures that are easily computable, but it is not easy to find a simple interpretation. We show that the ${l}_{1}$-norm of coherence is uniquely characterized by a few simple axioms, which demonstrates in a precise sense that it is the analog of negativity in entanglement theory and sum negativity in the resource theory of magic-state quantum computation. We also provide an operational interpretation of the ${l}_{1}$-norm of coherence as the maximum entanglement, measured by the negativity, produced by incoherent operations acting on the system and an incoherent ancilla. To achieve this goal, we clarify the relation between the ${l}_{1}$-norm of coherence and negativity for all bipartite states, which leads to an interesting generalization of maximally correlated states. Surprisingly, all entangled states thus obtained are distillable. Moreover, their entanglement cost and distillable entanglement can be computed explicitly for a qubit-qudit system.

arXiv:1704.02896•Physical Review A•Feb 28, 2018•49 citations
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.
We provide detailed proofs of triangle inequalities in coherence measures and entanglement concurrence. If a rank-2 state $\ensuremath{\varrho}$ can be expressed as a convex combination of two pure states, i.e., $\ensuremath{\varrho}={p}_{1}|{\ensuremath{\psi}}_{1}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\psi}}_{1}|+{p}_{2}|{\ensuremath{\psi}}_{2}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\psi}}_{2}|$, a triangle inequality can be established as $|E(|{\mathrm{\ensuremath{\Psi}}}_{1}\ensuremath{\rangle})\ensuremath{-}E(|{\mathrm{\ensuremath{\Psi}}}_{2}\ensuremath{\rangle})|\ensuremath{\le}E(\ensuremath{\varrho})\ensuremath{\le}E(|{\mathrm{\ensuremath{\Psi}}}_{1}\ensuremath{\rangle})+E(|{\mathrm{\ensuremath{\Psi}}}_{2}\ensuremath{\rangle})$, where $|{\mathrm{\ensuremath{\Psi}}}_{1}\ensuremath{\rangle}=\sqrt{{p}_{1}}|{\ensuremath{\psi}}_{1}\ensuremath{\rangle}$ and $|{\mathrm{\ensuremath{\Psi}}}_{2}\ensuremath{\rangle}=\sqrt{{p}_{2}}|{\ensuremath{\psi}}_{2}\ensuremath{\rangle}$; EEE can be considered either coherence measures or entanglement concurrence. This inequality displays mathematical beauty for its similarity to the triangle inequality in plane geometry. An illustrative example is given after the proof.
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 ¯ .
One of the fundamental traits of quantum entanglement is the restricted shareability among multipartite quantum systems, namely monogamy of entanglement, while it is well known that monogamy inequalities are always satisfied by entanglement measures with convexity. Here we present a measure of entanglement, logarithmic convex-roof extended negativity (LCREN) satisfying important characteristics of an entanglement measure, and investigate the monogamy relation for logarithmic negativity and LCREN both without convexity. We show exactly that the αth power of logarithmic negativity, and a newly defined good measure of entanglement, LCREN, obey a class of general monogamy inequalities in multiqubit systems, 2⊗2⊗3 systems and 2⊗2⊗2n systems for α≥4ln2. We provide a class of general polygamy inequalities of multiqubit systems in terms of logarithmic convex-roof extended negativity of assistance (LCRENoA) for 0≤β≤2. Given that the logarithmic negativity and LCREN are not convex, these results are surprising. Using the power of the logarithmic negativity and LCREN, we further establish a class of tight monogamy inequalities of multiqubit systems, 2⊗2⊗3 systems and 2⊗2⊗2n systems in terms of the αth power of logarithmic negativity and LCREN for α≥4ln2. We also show that the βth power of LCRENoA obeys a class of tight polygamy inequalities of multiqubit systems for 0≤β≤2.
Quantum coherence plays a central role in various research areas. The l1{l}_{1}l1​-norm of coherence is one of the most important coherence measures that are easily computable, but it is not easy to find a simple interpretation. We show that the l1{l}_{1}l1​-norm of coherence is uniquely characterized by a few simple axioms, which demonstrates in a precise sense that it is the analog of negativity in entanglement theory and sum negativity in the resource theory of magic-state quantum computation. We also provide an operational interpretation of the l1{l}_{1}l1​-norm of coherence as the maximum entanglement, measured by the negativity, produced by incoherent operations acting on the system and an incoherent ancilla. To achieve this goal, we clarify the relation between the l1{l}_{1}l1​-norm of coherence and negativity for all bipartite states, which leads to an interesting generalization of maximally correlated states. Surprisingly, all entangled states thus obtained are distillable. Moreover, their entanglement cost and distillable entanglement can be computed explicitly for a qubit-qudit system.