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

The logarithmic negativity is a full entanglement monotone under measuring LOCC and PPT-operations

Martin Bodo Plenio

Published May 10, 2005• 1 citation
OpenAlex

Abstract

It is proven that the logarithmic negativity does not increase on average under positive partial transpose preserving (PPT) operation including subselection (a set of operations that incorporate local operations and classical communication (LOCC) as a subset) and, in the process, a further proof is provided that the negativity does not increase on average under the same set of operations. Given that the logarithmic negativity is obtained from the negativity applying a concave function and is itself not a convex quantity this result is surprising as convexity is generally considered as describing the local physical process of losing information. The role of convexity and in particular its relation (or lack thereof) to physical processes is discussed in this context.

Related papers

Logarithmic Negativity: A Full Entanglement Monotone That is not Convex

Martin Bodo Plenio

It is proven that logarithmic negativity does not increase on average under a general positive partial transpose preserving operation (a set of operations that incorporate local operations and classical communication as a subset) and, in the process, a further proof is provided that the negativity does not increase on average under the same set of operations. Given that the logarithmic negativity is not a convex function this result is surprising, as it is generally considered that convexity describes the local physical process of losing information. The role of convexity and, in particular, its relation (or lack thereof) to physical processes is discussed and importance of continuity in this context is stressed.

arXiv:quant-ph/0505071•Physical Review Letters•Aug 26, 2005•1,524 citations

Strict entanglement monotonicity under local operations and classical communication

Yu Jing Guo

Entanglement monotone is defined as a convex measure of entanglement that does not increase on average under local operations and classical communication (LOCC). Here we call an entanglement monotone a strict entanglement monotone (SEM) if it decreases strictly on average under LOCC. We show that, for any convex roof extended entanglement monotone that on pure states is given by a function of the reduced states, if the function is strictly concave, then it is a SEM. Moreover, we prove that the negativity and the relative entropy of entanglement, which are not defined by the convex roof structure, are also SEMs. In addition, if the squashed entanglement could be obtained by some optimal extension, then it is a SEM as well. Our results imply that entanglement is strictly decreasing on average under LOCC.

arXiv:1904.01183•Physical Review A•Feb 28, 2019•11 citations

Entanglement Cost under Positive-Partial-Transpose-Preserving Operations

K. Audenaert, Martin Bodo Plenio, Jens Eisert

We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby providing the operational interpretation for this entanglement measure. As examples we discuss general Werner states and arbitrary bipartite Gaussian states. Then we prove that for the antisymmetric Werner state PPT cost and PPT entanglement of distillation coincide. This is the first example of a truly mixed state for which entanglement manipulation is asymptotically reversible, which points towards a unique entanglement measure under PPT operations.

arXiv:quant-ph/0207146•Physical Review Letters•Jan 16, 2003•290 citations

Logarithmic negativity typically equals exact entanglement cost

Bowen Ouyang, Jonah Kudler-Flam

Quantum entanglement plays a leading role in modern understanding of physical systems, from quantum phases of matter to quantum gravity. In quantum information theory, one seeks operationally meaningful quantifiers of entanglement, which for large quantum systems are notoriously difficult to evaluate due to the lack of computationally efficient algorithms. In this Letter, we show that for large random induced mixed states the logarithmic negativity, an efficiently computable entanglement measure, generically coincides with the exact entanglement cost under positive-partial-transpose-preserving operations, thereby acquiring a precise operational interpretation. Our results establish logarithmic negativity as an exact characterization of entanglement in generic many-body states and provide a tractable route for quantifying entanglement in complex quantum systems.

quant-pharXiv:2607.01320•Jul 3, 2026•0 citations

Infinitesimal local operations and differential conditions for entanglement monotones

Ognyan Oreshkov, Todd A. Brun

Much of the theory of entanglement concerns the transformations that are possible to a state under local operations with classical communication (LOCC); however, this set of operations is complicated and difficult to describe mathematically. An idea which has proven very useful is that of the entanglement monotone: a function of the state which is invariant under local unitary transformations and always decreases (or increases) on average after any local operation. In this paper we look on LOCC as the set of operations generated by infinitesimal local operations, operations which can be performed locally and which leave the state little changed. We show that a necessary and sufficient condition for a function of the state to be an entanglement monotone under local operations that do not involve information loss is that the function be a monotone under infinitesimal local operations. We then derive necessary and sufficient differential conditions for a function of the state to be an entanglement monotone. We first derive two conditions for local operations without information loss, and then show that they can be extended to more general operations by adding the requirement of convexity. We then demonstrate that a number of known entanglement monotones satisfy these differential criteria. Finally, as an application, we use the differential conditions to construct a new polynomial entanglement monotone for three-qubit pure states. It is our hope that this approach will avoid some of the difficulties in the theory of multipartite and mixed-state entanglement.

arXiv:quant-ph/0506181•Physical Review A•Apr 13, 2006•6 citations

Irreversibility of Asymptotic Entanglement Manipulation Under Quantum Operations Completely Preserving Positivity of Partial Transpose

Xin Wang, Runyao Duan

We demonstrate the irreversibility of asymptotic entanglement manipulation under quantum operations that completely preserve the positivity of partial transpose (PPT), resolving a major open problem in quantum information theory. Our key tool is a new efficiently computable additive lower bound for the asymptotic relative entropy of entanglement with respect to PPT states, which can be used to evaluate the entanglement cost under local operations and classical communication (LOCC). We find that for any rank-two mixed state supporting on the 3⊗3 antisymmetric subspace, the amount of distillable entanglement by PPT operations is strictly smaller than one entanglement bit (ebit) while its entanglement cost under PPT operations is exactly one ebit. As a by-product, we find that for this class of states, both the Rains's bound and its regularization are strictly less than the asymptotic relative entropy of entanglement. So, in general, there is no unique entanglement measure for the manipulation of entanglement by PPT operations. We further show a computable sufficient condition for the irreversibility of entanglement distillation by LOCC (or PPT) operations.

arXiv:1606.09421•Physical Review Letters•Nov 3, 2017•32 citations
It is proven that the logarithmic negativity does not increase on average under positive partial transpose preserving (PPT) operation including subselection (a set of operations that incorporate local operations and classical communication (LOCC) as a subset) and, in the process, a further proof is provided that the negativity does not increase on average under the same set of operations. Given that the logarithmic negativity is obtained from the negativity applying a concave function and is itself not a convex quantity this result is surprising as convexity is generally considered as describing the local physical process of losing information. The role of convexity and in particular its relation (or lack thereof) to physical processes is discussed in this context.
It is proven that logarithmic negativity does not increase on average under a general positive partial transpose preserving operation (a set of operations that incorporate local operations and classical communication as a subset) and, in the process, a further proof is provided that the negativity does not increase on average under the same set of operations. Given that the logarithmic negativity is not a convex function this result is surprising, as it is generally considered that convexity describes the local physical process of losing information. The role of convexity and, in particular, its relation (or lack thereof) to physical processes is discussed and importance of continuity in this context is stressed.
Entanglement monotone is defined as a convex measure of entanglement that does not increase on average under local operations and classical communication (LOCC). Here we call an entanglement monotone a strict entanglement monotone (SEM) if it decreases strictly on average under LOCC. We show that, for any convex roof extended entanglement monotone that on pure states is given by a function of the reduced states, if the function is strictly concave, then it is a SEM. Moreover, we prove that the negativity and the relative entropy of entanglement, which are not defined by the convex roof structure, are also SEMs. In addition, if the squashed entanglement could be obtained by some optimal extension, then it is a SEM as well. Our results imply that entanglement is strictly decreasing on average under LOCC.
We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby providing the operational interpretation for this entanglement measure. As examples we discuss general Werner states and arbitrary bipartite Gaussian states. Then we prove that for the antisymmetric Werner state PPT cost and PPT entanglement of distillation coincide. This is the first example of a truly mixed state for which entanglement manipulation is asymptotically reversible, which points towards a unique entanglement measure under PPT operations.
Quantum entanglement plays a leading role in modern understanding of physical systems, from quantum phases of matter to quantum gravity. In quantum information theory, one seeks operationally meaningful quantifiers of entanglement, which for large quantum systems are notoriously difficult to evaluate due to the lack of computationally efficient algorithms. In this Letter, we show that for large random induced mixed states the logarithmic negativity, an efficiently computable entanglement measure, generically coincides with the exact entanglement cost under positive-partial-transpose-preserving operations, thereby acquiring a precise operational interpretation. Our results establish logarithmic negativity as an exact characterization of entanglement in generic many-body states and provide a tractable route for quantifying entanglement in complex quantum systems.
Much of the theory of entanglement concerns the transformations that are possible to a state under local operations with classical communication (LOCC); however, this set of operations is complicated and difficult to describe mathematically. An idea which has proven very useful is that of the entanglement monotone: a function of the state which is invariant under local unitary transformations and always decreases (or increases) on average after any local operation. In this paper we look on LOCC as the set of operations generated by infinitesimal local operations, operations which can be performed locally and which leave the state little changed. We show that a necessary and sufficient condition for a function of the state to be an entanglement monotone under local operations that do not involve information loss is that the function be a monotone under infinitesimal local operations. We then derive necessary and sufficient differential conditions for a function of the state to be an entanglement monotone. We first derive two conditions for local operations without information loss, and then show that they can be extended to more general operations by adding the requirement of convexity. We then demonstrate that a number of known entanglement monotones satisfy these differential criteria. Finally, as an application, we use the differential conditions to construct a new polynomial entanglement monotone for three-qubit pure states. It is our hope that this approach will avoid some of the difficulties in the theory of multipartite and mixed-state entanglement.
We demonstrate the irreversibility of asymptotic entanglement manipulation under quantum operations that completely preserve the positivity of partial transpose (PPT), resolving a major open problem in quantum information theory. Our key tool is a new efficiently computable additive lower bound for the asymptotic relative entropy of entanglement with respect to PPT states, which can be used to evaluate the entanglement cost under local operations and classical communication (LOCC). We find that for any rank-two mixed state supporting on the 3⊗3 antisymmetric subspace, the amount of distillable entanglement by PPT operations is strictly smaller than one entanglement bit (ebit) while its entanglement cost under PPT operations is exactly one ebit. As a by-product, we find that for this class of states, both the Rains's bound and its regularization are strictly less than the asymptotic relative entropy of entanglement. So, in general, there is no unique entanglement measure for the manipulation of entanglement by PPT operations. We further show a computable sufficient condition for the irreversibility of entanglement distillation by LOCC (or PPT) operations.