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:quant-ph/0509006 · Journal of Physics A Mathematical and General

Partial scaling transform of multiqubit states as a criterion of separability

Cosmo Lupo, Vladimir I. Man’ko, Giuseppe Marmo, E. C. G. Sudarshan

Published Nov 16, 2005• 14 citations
arXivHTMLPDF

Abstract

The partial scaling transform of the density matrix for multiqubit states is introduced to detect entanglement of quantum states. The transform contains partial transposition as a special case. The scaling transform corresponds to partial time scaling of subsystem (or partial Planck's constant scaling) which was used to formulate recently separability criterion for continous variables.A measure of entanglement which is a generalization of negativity measure is introduced being based on tomographic probability description of spin states.

DOI
10.1088/0305-4470/38/48/009

Related papers

The Generalized Partial Transposition Criterion for Separability of Multipartite Quantum States

Chen, K, Wu, L A

We present a generalized partial transposition separability criterion for the density matrix of a multipartite quantum system. This criterion comprises as special cases the famous Peres-Horodecki criterion and the recent realignment criterion in quant-ph/0202121 and quant-ph/0205017. It involves only straightforward matrix manipulations and is easy to apply. A quantitative measure of entanglement based on this criterion is also obtained.

•Aug 9, 2002•65 citations

The generalized partial transposition criterion for infinite-dimensional quantum systems

Siqing YanYu GuoJinchuan Hou

The positive partial transposition (PPT) criterion and the realignment criterion constitute two of the most important criteria for detecting entanglement. The general- ized partial transposition (GPT) criterion which contains the PPT criterion and the realignment criterion as special cases, provides a necessary condition for a multipartite state to be separable. Here we extend the GPT criterion to the infinite- dimensional multipartite (bipartite) case and show that it includes multipartite (bipartite) PPT criterion and multipartite (bipartite) realignment criterion as special cases as well.

•Jan 1, 2014•0 citations

Partial separability and entanglement criteria for multiqubit quantum states

Michael Seevinck, Jos Uffink

We explore the subtle relationships between partial separability and entanglement of subsystems in multiqubit quantum states and give experimentally accessible conditions that distinguish between various classes and levels of partial separability in a hierarchical order. These conditions take the form of bounds on the correlations of locally orthogonal observables. Violations of such inequalities give strong sufficient criteria for various forms of partial inseparability and multiqubit entanglement. The strength of these criteria is illustrated by showing that they are stronger than several other well-known entanglement criteria (the fidelity criterion, violation of Mermin-type separability inequalities, the Laskowski-\ifmmode \dot{Z}\else \.{Z}\fi{}ukowski criterion, and the D\"ur-Cirac criterion) and also by showing their great noise robustness for a variety of multiqubit states, including $N$-qubit Greenberger-Horne-Zeilinger states and Dicke states. Furthermore, for $N\ensuremath{\geqslant}3$ they can detect bound entangled states. For all these states, the required number of measurement settings for implementation of the entanglement criteria is shown to be only $N+1$. If one chooses the familiar Pauli matrices as single-qubit observables, the inequalities take the form of bounds on the antidiagonal matrix elements of a state in terms of its diagonal matrix elements.

arXiv:0710.5326•Physical Review A•Sep 2, 2008•110 citations

Separability Criterion for Density Matrices

Asher Peres

A quantum system consisting of two subsystems is separable if its density matrix can be written as $\ensuremath{\rho}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{A}^{}{w}_{A}{\ensuremath{\rho}}_{A}^{\ensuremath{'}}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{A}^{\ensuremath{'}\ensuremath{'}},$ where ${\ensuremath{\rho}}_{A}^{\ensuremath{'}}$ and ${\ensuremath{\rho}}_{A}^{\ensuremath{'}\ensuremath{'}}$ are density matrices for the two subsystems, and the positive weights ${w}_{A}$ satisfy $\ensuremath{\Sigma}{w}_{A}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$. In this Letter, it is proved that a necessary condition for separability is that a matrix, obtained by partial transposition of \ensuremath{\rho}, has only non-negative eigenvalues. Some examples show that this criterion is more sensitive than Bell's inequality for detecting quantum inseparability.

arXiv:quant-ph/9604005•Physical Review Letters•Aug 19, 1996•5,181 citations

The entanglement criterion of multiqubits

Hyuk‐Jae Lee, Sung Dahm Oh, Doyeol Ahn

We present an entanglement criterion for multiqubits by using the quantum correlation tensors which rely on the expectation values of the Pauli operators for a multiqubit state. Our criterion explains not only the total entanglement of the system but also the partial entanglement in subsystems. It shows that we have to consider the subsystem entanglements in order to obtain the full description for multiqubit entanglements. Furthermore, we offer an extension of the entanglement to multiqudits.

arXiv:quant-ph/0506127•Jun 16, 2005•1 citation

Entanglement in SU(2)-invariant quantum systems: The PPT criterion and others

John Schliemann

We study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix $\\rho$ is entirely determined by the diagonal elements of $\\rho$ in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for $\\rho$ having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.

•Mar 14, 2005•0 citations
The partial scaling transform of the density matrix for multiqubit states is introduced to detect entanglement of quantum states. The transform contains partial transposition as a special case. The scaling transform corresponds to partial time scaling of subsystem (or partial Planck's constant scaling) which was used to formulate recently separability criterion for continous variables.A measure of entanglement which is a generalization of negativity measure is introduced being based on tomographic probability description of spin states.
The positive partial transposition (PPT) criterion and the realignment criterion constitute two of the most important criteria for detecting entanglement. The general- ized partial transposition (GPT) criterion which contains the PPT criterion and the realignment criterion as special cases, provides a necessary condition for a multipartite state to be separable. Here we extend the GPT criterion to the infinite- dimensional multipartite (bipartite) case and show that it includes multipartite (bipartite) PPT criterion and multipartite (bipartite) realignment criterion as special cases as well.
We explore the subtle relationships between partial separability and entanglement of subsystems in multiqubit quantum states and give experimentally accessible conditions that distinguish between various classes and levels of partial separability in a hierarchical order. These conditions take the form of bounds on the correlations of locally orthogonal observables. Violations of such inequalities give strong sufficient criteria for various forms of partial inseparability and multiqubit entanglement. The strength of these criteria is illustrated by showing that they are stronger than several other well-known entanglement criteria (the fidelity criterion, violation of Mermin-type separability inequalities, the Laskowski-\ifmmode \dot{Z}\else \.{Z}\fi{}ukowski criterion, and the D\"ur-Cirac criterion) and also by showing their great noise robustness for a variety of multiqubit states, including NNN-qubit Greenberger-Horne-Zeilinger states and Dicke states. Furthermore, for $N\ensuremath{\geqslant}3$ they can detect bound entangled states. For all these states, the required number of measurement settings for implementation of the entanglement criteria is shown to be only N+1N+1N+1. If one chooses the familiar Pauli matrices as single-qubit observables, the inequalities take the form of bounds on the antidiagonal matrix elements of a state in terms of its diagonal matrix elements.
A quantum system consisting of two subsystems is separable if its density matrix can be written as $\ensuremath{\rho}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{A}^{}{w}_{A}{\ensuremath{\rho}}_{A}^{\ensuremath{'}}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{A}^{\ensuremath{'}\ensuremath{'}},$ where ${\ensuremath{\rho}}_{A}^{\ensuremath{'}}$ and ${\ensuremath{\rho}}_{A}^{\ensuremath{'}\ensuremath{'}}$ are density matrices for the two subsystems, and the positive weights wA{w}_{A}wA​ satisfy $\ensuremath{\Sigma}{w}_{A}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$. In this Letter, it is proved that a necessary condition for separability is that a matrix, obtained by partial transposition of \ensuremath{\rho}, has only non-negative eigenvalues. Some examples show that this criterion is more sensitive than Bell's inequality for detecting quantum inseparability.
We study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix rho\\rhorho is entirely determined by the diagonal elements of rho\\rhorho in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for rho\\rhorho having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.