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/9908028 · Physical Review A

Bipartite-mixed-states of infinite-dimensional systems are generically nonseparable

Rob Clifton, Hans Halvorson

Published Dec 14, 1999• 40 citations
arXivHTMLPDF

Abstract

Given a bipartite quantum system represented by a Hilbert space ${\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2},$ we give an elementary argument to show that if either $\mathrm{dim}{\mathcal{H}}_{1}=\ensuremath{\infty}$ or $\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},$ then the set of nonseparable density operators on ${\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2}$ is trace-norm dense in the set of all density operators (and the separable density operators nowhere dense). This result complements recent detailed investigations of separability, which show that when $\mathrm{dim}{\mathcal{H}}_{i}<\ensuremath{\infty}$ for $i=1,2,$ there is a separable neighborhood (perhaps very small for large dimensions) of the maximally mixed state.

DOI
10.1103/physreva.61.012108

Related papers

Nonlocal correlations are generic in infinite-dimensional bipartite systems

Rob Clifton, Hans Halvorson, Adrian Kent

It was recently shown that nonseparable density operators on the Hilbert space ${\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2}$ are trace norm dense if either factor space has infinite dimension. We show here that nonlocal states, i.e., states whose correlations cannot be reproduced by any local hidden variable model, are also dense. Our constructions distinguish between the case $\mathrm{dim}{\mathcal{H}}_{1}=\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},$ where we show that states violating the Clauser-Horne-Shimony-Holt (CHSH) inequality are dense, and the case $\mathrm{dim}{\mathcal{H}}_{1}<\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},$ where we identify open neighborhoods of nonseparable states that do not violate the CHSH inequality but show that states with a subtler form of nonlocality (often called ``hidden'' nonlocality) remain dense.

arXiv:quant-ph/9909016•Physical Review A•Mar 7, 2000•17 citations

A Note on the Separability of Quantum Mixed States

Cao Huai-xin

In this note,we give a representation of density matrix for a bipartite quantum system and a neces-sary condition for separability of density matrix of mixed states in a bipartite quantum system of arbitrary dimen-tion.We show that if density matrix of mixed states in a bipartite quantum system of arbitrary dimension is separa-ble,then the trace norm of constructive matrix is less than or equal to 1.

Journal of Xianyang Normal University•Jan 1, 2008•0 citations

Nonlocal correlations are generic 1 n infinitedimensional bipartite systems

Rob Clifton

Abstract The observables of a bipartite quantum system are represented by the set of all self adjoint operators on the tensor product of two Hilbert spaces H1 ® ‘H2, whose dimensions we shall denote by d1 and d2, taking d1 :S: d2 without loss of generality. It is well known that when d1 2: 2 the states of the system can be nonseparable, and it is this possibility that much of the new technology associated with quantum information and computation theory relies upon. Prompted by concerns about whether the very noisy mixed states exploited by certain models of NMR quantum computing are truly nonseparable (Braunstein et al. 1999, Linden & Popescu 2001), de­ tailed investigations have shown that, whenever d2 < oo, there is always an open neighborhood of separable states surrounding the maximally mixed state (d1d2)­1 I® I (Zyczkowski et al. 1998, Caves

•Sep 23, 2009•0 citations

Separability Criteria for Quantum Mixed States in Terms of Trace Norm

Hui Zhao

A new necessary conditions for separability of mixed states in bipartite and multipartite quantum systems of arbitrary dimensions are presented. This method requires only a few local measurements. If a quantum state in a bipartite (or multipartite) system of arbitrary dimension is separable, we show that the trace norm of a constructive matrix is less than or equal to 1.

Chinese Physics Letters•Jun 28, 2006•3 citations

The cross states of a composite quantum system: separability and entanglement in any Hilbert space dimension

Paolo Aniello

We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.

arXiv:2602.18125•Feb 20, 2026•0 citations

Largest separable balls around the maximally mixed bipartite quantum state

Leonid Gurvits, Howard Barnum

For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral ${l}_{p}$ norms for $1<~p<~\ensuremath{\infty},$ of separable (unentangled) matrices around the identity matrix. This implies a simple and intuitively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity $\mathrm{tr}{\ensuremath{\rho}}^{2}$ not be too large. Theoretical and experimental applications of these results include algorithmic problems such as computing whether or not a state is entangled, and practical ones such as obtaining information about the existence or nature of entanglement in states reached by nuclear magnetic resonance quantum computation implementations or other experimental situations.

arXiv:quant-ph/0204159•Physical Review A•Dec 12, 2002•279 citations
Given a bipartite quantum system represented by a Hilbert space H1\ensuremath⨂H2,{\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2},H1​\ensuremath⨂H2​, we give an elementary argument to show that if either dimH1=\ensuremath∞\mathrm{dim}{\mathcal{H}}_{1}=\ensuremath{\infty}dimH1​=\ensuremath∞ or dimH2=\ensuremath∞,\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},dimH2​=\ensuremath∞, then the set of nonseparable density operators on H1\ensuremath⨂H2{\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2}H1​\ensuremath⨂H2​ is trace-norm dense in the set of all density operators (and the separable density operators nowhere dense). This result complements recent detailed investigations of separability, which show that when dimHi<\ensuremath∞\mathrm{dim}{\mathcal{H}}_{i}<\ensuremath{\infty}dimHi​<\ensuremath∞ for i=1,2,i=1,2,i=1,2, there is a separable neighborhood (perhaps very small for large dimensions) of the maximally mixed state.
It was recently shown that nonseparable density operators on the Hilbert space ${\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2}$ are trace norm dense if either factor space has infinite dimension. We show here that nonlocal states, i.e., states whose correlations cannot be reproduced by any local hidden variable model, are also dense. Our constructions distinguish between the case $\mathrm{dim}{\mathcal{H}}_{1}=\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},$ where we show that states violating the Clauser-Horne-Shimony-Holt (CHSH) inequality are dense, and the case $\mathrm{dim}{\mathcal{H}}_{1}<\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},$ where we identify open neighborhoods of nonseparable states that do not violate the CHSH inequality but show that states with a subtler form of nonlocality (often called ``hidden'' nonlocality) remain dense.
Abstract The observables of a bipartite quantum system are represented by the set of all self adjoint operators on the tensor product of two Hilbert spaces H1 ® ‘H2, whose dimensions we shall denote by d1 and d2, taking d1 :S: d2 without loss of generality. It is well known that when d1 2: 2 the states of the system can be nonseparable, and it is this possibility that much of the new technology associated with quantum information and computation theory relies upon. Prompted by concerns about whether the very noisy mixed states exploited by certain models of NMR quantum computing are truly nonseparable (Braunstein et al. 1999, Linden & Popescu 2001), de­ tailed investigations have shown that, whenever d2 < oo, there is always an open neighborhood of separable states surrounding the maximally mixed state (d1d2)­1 I® I (Zyczkowski et al. 1998, Caves
We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.
For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral lp{l}_{p}lp​ norms for $1<~p<~\ensuremath{\infty},$ of separable (unentangled) matrices around the identity matrix. This implies a simple and intuitively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity $\mathrm{tr}{\ensuremath{\rho}}^{2}$ not be too large. Theoretical and experimental applications of these results include algorithmic problems such as computing whether or not a state is entangled, and practical ones such as obtaining information about the existence or nature of entanglement in states reached by nuclear magnetic resonance quantum computation implementations or other experimental situations.