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

Quantum Information and Computation

Majorization and the interconversion of bipartite states

Michael A. Nielsen, Guifré Vidal

Published Jul 1, 2001• 177 citations
Publisher

Abstract

This review both introduces some elements of majorization theory and describes recent results on bipartite entanglement transformations, with special emphasis being placed on explaining the connections between these two topics.
DOI
10.26421/qic1.1-5

Related papers

Majorization and the interconversion of bipartite states

A NielsenMichael, VidalGuifré

Majorization is a powerful, easy-to-use and flexible tool which arises frequently in quantum mechanics as a consequence of fundamental connections between unitarity and the majorization relation. E...

Quantum Information and Computation•Jan 1, 2001•3 citations

Multipartite Entanglement Theory With Entanglement-Nonincreasing Operations

Alexander Streltsov

A key problem in quantum information science is to determine optimal protocols for the interconversion of entangled states shared between remote parties. While for two parties a large number of results in this direction is available, the multipartite setting still remains a major challenge. In this article, this problem is addressed by extending the resource theory of entanglement for multipartite systems beyond the standard framework of local operations and classical communication. Specifically, we consider transformations capable of introducing a small, controllable increase of entanglement of a state, with the requirement that the increase can be made arbitrarily small. We demonstrate that in this adjusted framework, the transformation rates between multipartite states are fundamentally dictated by the bipartite entanglement entropies of the respective quantum states. Remarkably, this approach allows the reduction of tripartite entanglement to its bipartite analog, indicating that every pure tripartite state can be reversibly synthesized from a suitable number of singlets distributed between pairs of parties.

arXiv:2305.18999•IEEE Transactions on Information Theory•Jan 9, 2025•2 citations

A Brief Overview of Bipartite and Multipartite Entanglement Measures

Saeed Haddadi, Mohammad Bohloul

arXiv:1805.03471•International Journal of Theoretical Physics•Sep 29, 2018•50 citations

Interconversion of pure Gaussian states requiring non-Gaussian operations

Michael G. Jabbour, Raúl García−Patrón, Nicolas J. Cerf

We analyze the conditions under which local operations and classical communication enable entanglement transformations within the set of bipartite pure Gaussian states. A set of necessary and sufficient conditions had been found in [Quant. Inf. Comp. 3, 211 (2003)] for the interconversion between such states that is restricted to Gaussian local operations and classical communication. Here, we exploit majorization theory in order to derive more general (sufficient) conditions for the interconversion between bipartite pure Gaussian states that goes beyond Gaussian local operations. While our technique is applicable to an arbitrary number of modes for each party, it allows us to exhibit surprisingly simple examples of 2 x 2 Gaussian states that necessarily require non-Gaussian local operations to be transformed into each other.

arXiv:1409.8217•Physical Review A•Jan 12, 2015•9 citations

Entanglement and majorization in (1+1)-dimensional quantum systems (7 pages)

Román Orús

Physical Review A•Jan 1, 2005•0 citations

Discrete And Differentiable Entanglement Transformations

David Sauerwein, Katharina Schwaiger, Barbara Kraus

The study of transformations among pure states via Local Operations assisted by Classical Communication (LOCC) plays a central role in entanglement theory. The main emphasis of these investigations is on the deterministic, or probabilistic transformations between two states and mainly tools from linear algebra are employed. Here, we go one step beyond that and analyze all optimal protocols. We show that for all bipartite and almost all multipartite (of arbitrarily many d-level systems) pairs of states, there exist infinitely many optimal intermediate states to which the initial state can first be transformed (locally) before it is transformed to the final state. The success probability of this transformation is nevertheless optimal. We provide a simple characterization of all intermediate states. We generalize this concept to differentiable paths in Hilbert space that connect two states of interest. With the help of survival analysis we determine the success probability for the continuous transformation along such a path and derive necessary and sufficient conditions on the optimality. We show, in strong contrast to previous results on state transformations, that optimal paths are characterized as solutions of a differential equation. Whereas, for almost all pairs of states, there exist infinitely many optimal paths, we present examples of pairs of states for which not even a single optimal intermediate state exists. Furthermore, we introduce a physically motivated distance measure, the interconversion metric, and show that (generically) any minimal geodesic with respect to the interconversion metric is an optimal path. Moreover, we identify infinitely many easily computable entanglement monotones for generic multipartite pure states. We show that a given finite set of these entanglement monotones can be used to completely characterize the entanglement contained in a generic state.

arXiv:1808.02819•Aug 8, 2018•3 citations
A key problem in quantum information science is to determine optimal protocols for the interconversion of entangled states shared between remote parties. While for two parties a large number of results in this direction is available, the multipartite setting still remains a major challenge. In this article, this problem is addressed by extending the resource theory of entanglement for multipartite systems beyond the standard framework of local operations and classical communication. Specifically, we consider transformations capable of introducing a small, controllable increase of entanglement of a state, with the requirement that the increase can be made arbitrarily small. We demonstrate that in this adjusted framework, the transformation rates between multipartite states are fundamentally dictated by the bipartite entanglement entropies of the respective quantum states. Remarkably, this approach allows the reduction of tripartite entanglement to its bipartite analog, indicating that every pure tripartite state can be reversibly synthesized from a suitable number of singlets distributed between pairs of parties.
We analyze the conditions under which local operations and classical communication enable entanglement transformations within the set of bipartite pure Gaussian states. A set of necessary and sufficient conditions had been found in [Quant. Inf. Comp. 3, 211 (2003)] for the interconversion between such states that is restricted to Gaussian local operations and classical communication. Here, we exploit majorization theory in order to derive more general (sufficient) conditions for the interconversion between bipartite pure Gaussian states that goes beyond Gaussian local operations. While our technique is applicable to an arbitrary number of modes for each party, it allows us to exhibit surprisingly simple examples of 2 x 2 Gaussian states that necessarily require non-Gaussian local operations to be transformed into each other.
The study of transformations among pure states via Local Operations assisted by Classical Communication (LOCC) plays a central role in entanglement theory. The main emphasis of these investigations is on the deterministic, or probabilistic transformations between two states and mainly tools from linear algebra are employed. Here, we go one step beyond that and analyze all optimal protocols. We show that for all bipartite and almost all multipartite (of arbitrarily many d-level systems) pairs of states, there exist infinitely many optimal intermediate states to which the initial state can first be transformed (locally) before it is transformed to the final state. The success probability of this transformation is nevertheless optimal. We provide a simple characterization of all intermediate states. We generalize this concept to differentiable paths in Hilbert space that connect two states of interest. With the help of survival analysis we determine the success probability for the continuous transformation along such a path and derive necessary and sufficient conditions on the optimality. We show, in strong contrast to previous results on state transformations, that optimal paths are characterized as solutions of a differential equation. Whereas, for almost all pairs of states, there exist infinitely many optimal paths, we present examples of pairs of states for which not even a single optimal intermediate state exists. Furthermore, we introduce a physically motivated distance measure, the interconversion metric, and show that (generically) any minimal geodesic with respect to the interconversion metric is an optimal path. Moreover, we identify infinitely many easily computable entanglement monotones for generic multipartite pure states. We show that a given finite set of these entanglement monotones can be used to completely characterize the entanglement contained in a generic state.