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:0707.1947 · Physical Review A

Conditions for entanglement transformation between a class of multipartite pure states with generalized Schmidt decompositions

Yu Xin, Runyao Duan

Published Oct 1, 2007• 12 citations
arXivHTMLPDF

Abstract

We generalize Nielsen's majorization criterion for the convertibility of bipartite pure states [Phys. Rev. Lett. 83, 436 (1999)] to a special class of multipartite pure states with generalized Schmidt decompositions.
DOI
10.1103/physreva.76.044301

Related papers

Multipartite pure-state entanglement

Ashish V. Thapliyal

We show that pure states of multipartite quantum systems are multiseparable (i.e., give separable density matrices on tracing any party) if and only if they have a generalized Schmidt decomposition. Implications of this result for the quantification of multipartite pure-state entanglement are discussed. Further, as an application of the techniques used here, we show that any purification of a bipartite-bound entangled state having a positive partial transpose is tri-inseparable, i.e., has none of its three bipartite partial traces separable.

arXiv:quant-ph/9811091•Physical Review A•May 1, 1999•108 citations

Multiparticle entanglement and its applications to cryptography

Julia Kempe

Entanglement between three or more parties exhibits a realm of properties unknown to two-party states. Bipartite states are easily classified using the Schmidt decomposition. The Schmidt coefficients of a bipartite pure state encompass all the nonlocal properties of the state and can be ``seen'' by looking at one party's density matrix only. Pure states of three and more parties, however, lack such a simple form. They have more invariants under local unitary transformations than any one party can ``see'' on their subsystem. These ``hidden nonlocalities'' will allow us to exhibit a class of multipartite states that cannot be distinguished from each other by any party. Generalizing a result of Bennett, Popescu, Rohrlich, Smolin, and Thapliyal, and using a recent result by Nielsen, we will show that these states cannot be transformed into each other by local actions and classical communication. Furthermore, we will use an orthogonal subset of such states to hint at applications to cryptography and illustrate an extension to quantum secret sharing [using recently suggested $((n,k))$-threshold schemes].

arXiv:quant-ph/9902036•Physical Review A•Aug 1, 1999•256 citations

Finding a maximally correlated state: Simultaneous Schmidt decomposition of bipartite pure states

Tohya Hiroshima, Masahito Hayashi

We consider a bipartite mixed state of the form $\ensuremath{\rho}={\ensuremath{\sum}}_{\ensuremath{\alpha},\ensuremath{\beta}=1}^{l}{a}_{\ensuremath{\alpha}\ensuremath{\beta}}\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩⟨{\ensuremath{\psi}}_{\ensuremath{\beta}}⟩$, where $\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩$ are normalized bipartite state vectors, and matrix $({a}_{\ensuremath{\alpha}\ensuremath{\beta}})$ is positive semidefinite. We provide a necessary and sufficient condition for the state $\ensuremath{\rho}$ taking the form of maximally correlated states by a local unitary transformation. More precisely, we give a criterion for simultaneous Schmidt decomposability of $\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩$ for $\ensuremath{\alpha}=1,2,\dots{},l$. Using this criterion, we can judge completely whether or not the state $\ensuremath{\rho}$ is equivalent to the maximally correlated state, in which the distillable entanglement is given by a simple formula. For generalized Bell states, this criterion is written as a simple algebraic relation between indices of the states. We also discuss the local distinguishability of the generalized Bell states that are simultaneously Schmidt decomposable.

arXiv:quant-ph/0405107•Physical Review A•Sep 22, 2004•30 citations

Entanglement transformations using separable operations

Vlad Gheorghiu, Robert B. Griffiths

We study conditions for the deterministic transformation $\ensuremath{\mid}\ensuremath{\psi}⟩\ensuremath{\rightarrow}\ensuremath{\mid}\ensuremath{\phi}⟩$ of a bipartite entangled state by a separable operation. If the separable operation is a local operation with classical communication (LOCC), Nielsen's majorization theorem provides necessary and sufficient conditions. For the general case, we derive a necessary condition in terms of products of Schmidt coefficients, which is equivalent to the Nielsen condition when either of the two factor spaces is of dimension 2, but is otherwise weaker. One implication is that no separable operation can reverse a deterministic map produced by another separable operation, if one excludes the case where the Schmidt coefficients of $\ensuremath{\mid}\ensuremath{\psi}⟩$ are the same as those of $\ensuremath{\mid}\ensuremath{\phi}⟩$. The question of sufficient conditions in the general separable case remains open. When the Schmidt coefficients of $\ensuremath{\mid}\ensuremath{\psi}⟩$ are the same as those of $\ensuremath{\mid}\ensuremath{\phi}⟩$, we show that the Kraus operators of the separable transformation restricted to the supports of $\ensuremath{\mid}\ensuremath{\psi}⟩$ on the factor spaces are proportional to unitaries. When that proportionality holds and the factor spaces have equal dimension, we find conditions for the deterministic transformation of a collection of several full Schmidt rank pure states $\ensuremath{\mid}{\ensuremath{\psi}}_{j}⟩$ to pure states $\ensuremath{\mid}{\ensuremath{\phi}}_{j}⟩$.

arXiv:0705.0369•Physical Review A•Sep 11, 2007•15 citations

Generalized Schmidt decomposability and its relation to projective norms in multipartite entanglement

Florian Sokoli, G. Alber

Projective norms are capable of measuring entanglement of multipartite quantum states. However, typically, the explicit computation of these distance-based geometric entanglement monotones is very difficult even for finite dimensional systems. Motivated by the significance of Schmidt decompositions for our quantitative understanding of bipartite quantum entanglement, a generalization of this concept to multipartite scenarios is proposed, in the sense that generalized Schmidt decomposability of a multipartite pure state implies that its projective norm can be calculated in a simple way analogous to the bipartite case. Thus, this concept of generalized Schmidt decomposability of multipartite quantum states is linked in a natural way to projective norms as entanglement monotones. Therefore, it may not only be a convenient tool for calculations, but may also shed new light onto the intricate features of multipartite entanglement in an analogous way as the 'classical' Schmidt decomposition does for bipartite quantum systems.

Journal of Physics A Mathematical and Theoretical•Jul 24, 2014•8 citations

Necessary and Sufficient Conditions of Separability for Multipartite Pure States

Dafa Li, Xiangrong Li, Huang Hongtao +1 more

In this paper we present a necessary and sufficient condition of separability for multipartite pure states and variants of it. These conditions are very simple and calculable, and they do not require Schmidt decomposition (for two subsystems) or tracing out operations. We also give a necessary condition for a local unitary equivalence class for a bipartite system in terms of the determinant of the matrix of amplitudes.

arXiv:quant-ph/0604147•Communications in Theoretical Physics•May 1, 2008•7 citations
We show that pure states of multipartite quantum systems are multiseparable (i.e., give separable density matrices on tracing any party) if and only if they have a generalized Schmidt decomposition. Implications of this result for the quantification of multipartite pure-state entanglement are discussed. Further, as an application of the techniques used here, we show that any purification of a bipartite-bound entangled state having a positive partial transpose is tri-inseparable, i.e., has none of its three bipartite partial traces separable.
Entanglement between three or more parties exhibits a realm of properties unknown to two-party states. Bipartite states are easily classified using the Schmidt decomposition. The Schmidt coefficients of a bipartite pure state encompass all the nonlocal properties of the state and can be ``seen'' by looking at one party's density matrix only. Pure states of three and more parties, however, lack such a simple form. They have more invariants under local unitary transformations than any one party can ``see'' on their subsystem. These ``hidden nonlocalities'' will allow us to exhibit a class of multipartite states that cannot be distinguished from each other by any party. Generalizing a result of Bennett, Popescu, Rohrlich, Smolin, and Thapliyal, and using a recent result by Nielsen, we will show that these states cannot be transformed into each other by local actions and classical communication. Furthermore, we will use an orthogonal subset of such states to hint at applications to cryptography and illustrate an extension to quantum secret sharing [using recently suggested ((n,k))((n,k))((n,k))-threshold schemes].
We consider a bipartite mixed state of the form $\ensuremath{\rho}={\ensuremath{\sum}}_{\ensuremath{\alpha},\ensuremath{\beta}=1}^{l}{a}_{\ensuremath{\alpha}\ensuremath{\beta}}\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩⟨{\ensuremath{\psi}}_{\ensuremath{\beta}}⟩$, where $\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩$ are normalized bipartite state vectors, and matrix $({a}_{\ensuremath{\alpha}\ensuremath{\beta}})$ is positive semidefinite. We provide a necessary and sufficient condition for the state $\ensuremath{\rho}$ taking the form of maximally correlated states by a local unitary transformation. More precisely, we give a criterion for simultaneous Schmidt decomposability of $\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩$ for $\ensuremath{\alpha}=1,2,\dots{},l$. Using this criterion, we can judge completely whether or not the state $\ensuremath{\rho}$ is equivalent to the maximally correlated state, in which the distillable entanglement is given by a simple formula. For generalized Bell states, this criterion is written as a simple algebraic relation between indices of the states. We also discuss the local distinguishability of the generalized Bell states that are simultaneously Schmidt decomposable.
We study conditions for the deterministic transformation $\ensuremath{\mid}\ensuremath{\psi}⟩\ensuremath{\rightarrow}\ensuremath{\mid}\ensuremath{\phi}⟩$ of a bipartite entangled state by a separable operation. If the separable operation is a local operation with classical communication (LOCC), Nielsen's majorization theorem provides necessary and sufficient conditions. For the general case, we derive a necessary condition in terms of products of Schmidt coefficients, which is equivalent to the Nielsen condition when either of the two factor spaces is of dimension 2, but is otherwise weaker. One implication is that no separable operation can reverse a deterministic map produced by another separable operation, if one excludes the case where the Schmidt coefficients of $\ensuremath{\mid}\ensuremath{\psi}⟩$ are the same as those of $\ensuremath{\mid}\ensuremath{\phi}⟩$. The question of sufficient conditions in the general separable case remains open. When the Schmidt coefficients of $\ensuremath{\mid}\ensuremath{\psi}⟩$ are the same as those of $\ensuremath{\mid}\ensuremath{\phi}⟩$, we show that the Kraus operators of the separable transformation restricted to the supports of $\ensuremath{\mid}\ensuremath{\psi}⟩$ on the factor spaces are proportional to unitaries. When that proportionality holds and the factor spaces have equal dimension, we find conditions for the deterministic transformation of a collection of several full Schmidt rank pure states $\ensuremath{\mid}{\ensuremath{\psi}}_{j}⟩$ to pure states $\ensuremath{\mid}{\ensuremath{\phi}}_{j}⟩$.
Projective norms are capable of measuring entanglement of multipartite quantum states. However, typically, the explicit computation of these distance-based geometric entanglement monotones is very difficult even for finite dimensional systems. Motivated by the significance of Schmidt decompositions for our quantitative understanding of bipartite quantum entanglement, a generalization of this concept to multipartite scenarios is proposed, in the sense that generalized Schmidt decomposability of a multipartite pure state implies that its projective norm can be calculated in a simple way analogous to the bipartite case. Thus, this concept of generalized Schmidt decomposability of multipartite quantum states is linked in a natural way to projective norms as entanglement monotones. Therefore, it may not only be a convenient tool for calculations, but may also shed new light onto the intricate features of multipartite entanglement in an analogous way as the 'classical' Schmidt decomposition does for bipartite quantum systems.