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:1609.08016 · Physical Review A

Entanglement monotones and transformations of symmetric bipartite states

Mark Girard, Gilad Gour

Published Jan 9, 2017• 10 citations
arXivHTMLPDF

Abstract

The primary goal in the study of entanglement as a resource theory is to find conditions that determine when one quantum state can or cannot be transformed into another via local operations and classical communication operations. This is typically done through entanglement monotones or conversion witnesses. Such quantities cannot be computed for arbitrary quantum states in general, but it is useful to consider classes of symmetric states for which closed-form expressions can be found. In this paper, we show how to compute the convex roof of any entanglement monotone for all Werner states. The convex roofs of the well-known Vidal monotones are computed for all isotropic states, and we show how this method can generalize to other entanglement measures and other types of symmetries as well. We also present necessary and sufficient conditions that determine when a pure bipartite state can be deterministically converted into a Werner state or an isotropic state.

DOI
10.1103/physreva.95.012308

Related papers

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

Quantifying entanglement in terms of an operational way*

Deng-hui Yu, Chang‐shui Yu

We establish entanglement monotones in terms of an operational approach, which is closely connected with the state conversion from pure states to the objective state by the local operations and classical communications. It is shown that any good entanglement quantifier defined on pure states can induce an entanglement monotone for all density matrices. Particularly, we show that our entanglement monotone is the maximal one among all those having the same form for pure states. In some special cases, our proposed entanglement monotones turn to be equivalent to the convex roof construction, which hence gain an operational meaning. Some examples are given to demonstrate different cases.

arXiv:2109.02888•Chinese Physics B•Oct 15, 2020•3 citations

Convex Analysis in Quantum Information

Mark Girard

Convexity arises naturally in the study of quantum information. As a result, many useful tools from convex analysis can be used to give important results regarding aspects of quantum information. This thesis builds up methods using core concepts from convex analysis, including convex optimization problems, convex roof constructions, and conic programming, to study mathematical problems related to quantum entanglement. In this thesis, I develop a method for solving convex optimization problems that arise in quantum information theory by analyzing the corresponding converse problem. That is, given an element in a convex set, I determine a family of convex functions that are minimized at this point. This method is used find explicit formulas for the relative entropy of entanglement, as well as other important quantities used to quantify entanglement, and allows one to show important relationships between them. I also construct a practical algorithm that can be used to compute these quantities. This thesis also presents a method to compute convex roofs of arbitrary entanglement measures evaluated on highly symmetric bipartite states. I also establish a framework for completely characterizing quantum resource theories that are convex. For resource theories with a simple mathematical structure, this gives rise to a complete set of resource monotones that can be computed in practice using semidefinite programs. This has applications to the study of entanglement transformations.

•Jan 1, 2017•3 citations

Insufficiency of Pure-State Ensembles in Characterizing Transformations of Entangled States under LOCC

C. L. Liu, Baoqing Sun, D. L. Zhou

The conditions for transforming pure entangled states under local operations and classical communication (LOCC) are well understood. A natural question then arises: Can we determine the transformation conditions for mixed entangled states under LOCC based on the properties of their pure-state ensembles? While much effort has been devoted to this issue, in this paper, we rule out this possibility. Our findings address several open questions, including: (i) The conditions \( E_f^{cr}(ρ) \geq E_f^{cr}(σ) \) for all convex roof entanglement measures \(E_f^{cr}\) is insufficient to guarantee the existence of an LOCC transformation \(Λ^L(\cdot)\) from \(ρ\) to \(σ\); and (ii) The inequalities \(\sum_j p_j E(\varphi_j) \geq \sum_l q_l E(ψ_l)\) for all entanglement monotones \(E\) are not sufficient to ensure the existence of an LOCC transformation from \(\{p_j, \ket{\varphi_j}\}\) to \(\{q_l, \ket{ψ_l}\}\).

quant-pharXiv:2512.21665•Physical Review A•Dec 26, 2025•0 citations

Multipartite Entangled States: Transformations, Entanglement Monotones, and Applications

Wei Cui

Entanglement is one of the fundamental features of quantum information science.\nThough bipartite entanglement has been analyzed thoroughly in theory and shown to\nbe an important resource in quantum computation and communication protocols, the\ntheory of entanglement shared between more than two parties, which is called multipartite\nentanglement, is still not complete. Specifically, the classification of multipartite\nentanglement and the transformation property between different multipartite states by\nlocal operators and classical communications (LOCC) are two fundamental questions in\nthe theory of multipartite entanglement.\nIn this thesis, we present results related to the LOCC transformation between multipartite\nentangled states. Firstly, we investigate the bounds on the LOCC transformation\nprobability between multipartite states, especially the GHZ class states. By analyzing\nthe involvement of 3-tangle and other entanglement measures under weak two-outcome\nmeasurement, we derive explicit upper and lower bound on the transformation probability\nbetween GHZ class states. After that, we also analyze the transformation between\nN-party W type states, which is a special class of multipartite entangled states that\nhas an explicit unique expression and a set of analytical entanglement monotones. We\npresent a necessary and sufficient condition for a known upper bound of transformation\nprobability between two N-party W type states to be achieved.\nWe also further investigate a novel entanglement transformation protocol, the random\ndistillation, which transforms multipartite entanglement into bipartite entanglement\nii\nshared by a non-deterministic pair of parties. We find upper bounds for the random distillation\nprotocol for general N-party W type states and find the condition for the upper\nbounds to be achieved. What is surprising is that the upper bounds correspond to entanglement\nmonotones that can be increased by Separable Operators (SEP), which gives\nthe first set of analytical entanglement monotones that can be increased by SEP.\nFinally, we investigate the idea of a new class of multipartite entangled states, the\nAbsolutely Maximal Entangled (AME) states, which is characterized by the fact that\nany bipartition of the states would give a maximal entangled state between the two sets.\nThe relationship between AME states and Quantum secret sharing (QSS) protocols is\nexhibited and the application of AME states in novel quantum communication protocols\nis also explored.

•Jan 7, 2014•0 citations

Entanglement of formation of rotationally symmetric states

Kiran K. Manne, Carlton M. Caves

Computing the entanglement of formation of a bipartite state is generally difficult, but special symmetries of a state can simplify the problem. For instance, this allows one to determine the entanglement of formation of Werner states and isotropic states. We consider a slightly more general class of states, rotationally symmetric states, also known as SU(2)-invariant states. These states are invariant under global rotations of both subsystems, and one can examine entanglement in cases where the subsystems have different dimensions. We derive an analytic expression for the entanglement of formation of rotationally symmetric states of a spin-$j$ particle and a spin-${1\\over2}$ particle. We also give expressions for the I-concurrence, I-tangle, and convex-roof-extended negativity.

Quantum Information and Computation•Mar 1, 2008•6 citations
The primary goal in the study of entanglement as a resource theory is to find conditions that determine when one quantum state can or cannot be transformed into another via local operations and classical communication operations. This is typically done through entanglement monotones or conversion witnesses. Such quantities cannot be computed for arbitrary quantum states in general, but it is useful to consider classes of symmetric states for which closed-form expressions can be found. In this paper, we show how to compute the convex roof of any entanglement monotone for all Werner states. The convex roofs of the well-known Vidal monotones are computed for all isotropic states, and we show how this method can generalize to other entanglement measures and other types of symmetries as well. We also present necessary and sufficient conditions that determine when a pure bipartite state can be deterministically converted into a Werner state or an isotropic state.
We establish entanglement monotones in terms of an operational approach, which is closely connected with the state conversion from pure states to the objective state by the local operations and classical communications. It is shown that any good entanglement quantifier defined on pure states can induce an entanglement monotone for all density matrices. Particularly, we show that our entanglement monotone is the maximal one among all those having the same form for pure states. In some special cases, our proposed entanglement monotones turn to be equivalent to the convex roof construction, which hence gain an operational meaning. Some examples are given to demonstrate different cases.
Convexity arises naturally in the study of quantum information. As a result, many useful tools from convex analysis can be used to give important results regarding aspects of quantum information. This thesis builds up methods using core concepts from convex analysis, including convex optimization problems, convex roof constructions, and conic programming, to study mathematical problems related to quantum entanglement. In this thesis, I develop a method for solving convex optimization problems that arise in quantum information theory by analyzing the corresponding converse problem. That is, given an element in a convex set, I determine a family of convex functions that are minimized at this point. This method is used find explicit formulas for the relative entropy of entanglement, as well as other important quantities used to quantify entanglement, and allows one to show important relationships between them. I also construct a practical algorithm that can be used to compute these quantities. This thesis also presents a method to compute convex roofs of arbitrary entanglement measures evaluated on highly symmetric bipartite states. I also establish a framework for completely characterizing quantum resource theories that are convex. For resource theories with a simple mathematical structure, this gives rise to a complete set of resource monotones that can be computed in practice using semidefinite programs. This has applications to the study of entanglement transformations.
The conditions for transforming pure entangled states under local operations and classical communication (LOCC) are well understood. A natural question then arises: Can we determine the transformation conditions for mixed entangled states under LOCC based on the properties of their pure-state ensembles? While much effort has been devoted to this issue, in this paper, we rule out this possibility. Our findings address several open questions, including: (i) The conditions Efcr(ρ)≥Efcr(σ) E_f^{cr}(ρ) \geq E_f^{cr}(σ) Efcr​(ρ)≥Efcr​(σ) for all convex roof entanglement measures EfcrE_f^{cr}Efcr​ is insufficient to guarantee the existence of an LOCC transformation ΛL(⋅)Λ^L(\cdot)ΛL(⋅) from ρρρ to σσσ; and (ii) The inequalities ∑jpjE(φj)≥∑lqlE(ψl)\sum_j p_j E(\varphi_j) \geq \sum_l q_l E(ψ_l)∑j​pj​E(φj​)≥∑l​ql​E(ψl​) for all entanglement monotones EEE are not sufficient to ensure the existence of an LOCC transformation from {pj,∣φj⟩}\{p_j, \ket{\varphi_j}\}{pj​,∣φj​⟩} to {ql,∣ψl⟩}\{q_l, \ket{ψ_l}\}{ql​,∣ψl​⟩}.
Entanglement is one of the fundamental features of quantum information science.\nThough bipartite entanglement has been analyzed thoroughly in theory and shown to\nbe an important resource in quantum computation and communication protocols, the\ntheory of entanglement shared between more than two parties, which is called multipartite\nentanglement, is still not complete. Specifically, the classification of multipartite\nentanglement and the transformation property between different multipartite states by\nlocal operators and classical communications (LOCC) are two fundamental questions in\nthe theory of multipartite entanglement.\nIn this thesis, we present results related to the LOCC transformation between multipartite\nentangled states. Firstly, we investigate the bounds on the LOCC transformation\nprobability between multipartite states, especially the GHZ class states. By analyzing\nthe involvement of 3-tangle and other entanglement measures under weak two-outcome\nmeasurement, we derive explicit upper and lower bound on the transformation probability\nbetween GHZ class states. After that, we also analyze the transformation between\nN-party W type states, which is a special class of multipartite entangled states that\nhas an explicit unique expression and a set of analytical entanglement monotones. We\npresent a necessary and sufficient condition for a known upper bound of transformation\nprobability between two N-party W type states to be achieved.\nWe also further investigate a novel entanglement transformation protocol, the random\ndistillation, which transforms multipartite entanglement into bipartite entanglement\nii\nshared by a non-deterministic pair of parties. We find upper bounds for the random distillation\nprotocol for general N-party W type states and find the condition for the upper\nbounds to be achieved. What is surprising is that the upper bounds correspond to entanglement\nmonotones that can be increased by Separable Operators (SEP), which gives\nthe first set of analytical entanglement monotones that can be increased by SEP.\nFinally, we investigate the idea of a new class of multipartite entangled states, the\nAbsolutely Maximal Entangled (AME) states, which is characterized by the fact that\nany bipartition of the states would give a maximal entangled state between the two sets.\nThe relationship between AME states and Quantum secret sharing (QSS) protocols is\nexhibited and the application of AME states in novel quantum communication protocols\nis also explored.
Computing the entanglement of formation of a bipartite state is generally difficult, but special symmetries of a state can simplify the problem. For instance, this allows one to determine the entanglement of formation of Werner states and isotropic states. We consider a slightly more general class of states, rotationally symmetric states, also known as SU(2)-invariant states. These states are invariant under global rotations of both subsystems, and one can examine entanglement in cases where the subsystems have different dimensions. We derive an analytic expression for the entanglement of formation of rotationally symmetric states of a spin-jjj particle and a spin-1over2{1\\over2}1over2 particle. We also give expressions for the I-concurrence, I-tangle, and convex-roof-extended negativity.