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/9804024 · Physical Review A

Volume of the set of separable states

Karol Życzkowski, Paweł Horodecki, Anna Sanpera, Maciej Lewenstein

Published Aug 1, 1998• 1,409 citations
arXivHTMLPDF

Abstract

The question of how many entangled or, respectively, separable states there are in the set of all quantum states is considered. We propose a natural measure in the space of density matrices $\ensuremath{\varrho}$ describing $N$-dimensional quantum systems. We prove that, under this measure, the set of separable states possesses a nonzero volume. Analytical lower and upper bounds of this volume are also derived for $N=2\ifmmode\times\else\texttimes\fi{}2$ and $N=2\ifmmode\times\else\texttimes\fi{}3$ cases. Finally, numerical Monte Carlo calculations allow us to estimate the volume of separable states, providing numerical evidence that it decreases exponentially with the dimension of the composite system. We have also analyzed a conditional measure of separability under the condition of fixed purity. Our results display a clear dualism between purity and separability: entanglement is typical of pure states, while separability is connected with quantum mixtures. In particular, states of sufficiently low purity are necessarily separable.

DOI
10.1103/physreva.58.883

Related papers

Volume of the set of separable states. II

Karol Życzkowski

The problem of how many entangled or, respectively, separable states there are in the set of all quantum states is investigated. We study to what extent the choice of a measure in the space of density matrices $\ensuremath{\varrho}$ describing N-dimensional quantum systems affects the results obtained. We demonstrate that the link between the purity of the mixed states and the probability of entanglement is not sensitive to the measure chosen. Since the criterion of partial transposition is not sufficient to distinguish all separable states for $N>~8$, we develop an efficient algorithm to calculate numerically the entanglement of formation of a given mixed quantum state, which allows us to compute the volume of separable states for $N=8$ and to estimate the volume of the bound entangled states in this case.

arXiv:quant-ph/9902050•Physical Review A•Nov 1, 1999•261 citations

Volume of Separable States for Arbitrary NNN-dimensional System

Dong-Ling Deng, Jing‐Ling Chen

In a celebrated paper ([Phys. Rev. A 58, 883 (1998)]), K. Zyczkowski, P. Horodecki, A. Sanpera,and M. Lewenstein proved for the frst time a very interesting theorem that the volume of separable quantum states is nonzero. Inspired by their ideas, we obtain a general analytical lower bound of the volume of separable states (VOSS) for arbitrary N-dimensional system. Our results give quite simple and computable suffcient conditions for separability. Moreover, for bipartite system, an upper bound of the VOSS is also presented.

arXiv:0810.2020•Oct 11, 2008•0 citations

On the Bures Volume of Separable Quantum States ∗

Deping Ye

We obtain two sided estimates for the Bures volume of an arbitrary subset of the set of N × N density matrices, in terms of the Hilbert-Schmidt volume of that subset. For general subsets, our results are essentially optimal (for large N). As applications, we derive in particular nontrivial lower and upper bounds for the Bures volume of sets of separable states and for sets of states with positive partial transpose. PACS numbers: 02.40.Ft, 03.65.Db, 03.65.Ud, 03.67.Mn 1

•Nov 26, 2012•18 citations

On the Bures volume of separable quantum states

Deping Ye

We obtain two sided estimates for the Bures volume of an arbitrary subset of the set of N×N density matrices, in terms of the Hilbert–Schmidt volume of that subset. For general subsets, our results are essentially optimal (for large N). As applications, we derive, in particular, nontrivial lower and upper bounds for the Bures volume of sets of separable states and for sets of states with positive partial transpose.

arXiv:0902.1505•Journal of Mathematical Physics•Aug 1, 2009•15 citations

Relative volume of separable bipartite states

Rajeev Singh, Ravi Kunjwal, R. Simon

Every choice of an orthonormal frame in the $d$-dimensional Hilbert space of a system corresponds to one set of all mutually commuting density matrices or, equivalently, to the classical statistical state space of the system; the quantum state space itself can thus be profitably viewed as an $\text{SU}(d)$ orbit of classical state spaces, one for each orthonormal frame. We exploit this connection to study the relative volume of separable states of a bipartite quantum system. While the two-qubit case is studied in considerable analytic detail, for higher-dimensional systems we fall back on Monte Carlo. Several insights seem to emerge from our study.

arXiv:1307.1454•Physical Review A•Feb 10, 2014•10 citations

Comment on "Volume of the set of separable states"

Paul B. Slater

We demonstrate --- using the case of the two-dimensional quantum systems --- that the "natural measure on the space of density matrices describing N-dimensional quantum systems" proposed by Zyczkowski et al (quant-ph/9804024) does not belong to the class of normalized volume elements of monotone metrics on the quantum systems. Such metrics possess the statistically important property of being decreasing under stochastic mappings (coarse-grainings). We do note that the proposed measure (and certain evident variations upon it) exhibit quite specific monotonicity properties, but not of the form required for membership in the class.

•Apr 5, 1999•0 citations
The question of how many entangled or, respectively, separable states there are in the set of all quantum states is considered. We propose a natural measure in the space of density matrices \ensuremathϱ\ensuremath{\varrho}\ensuremathϱ describing NNN-dimensional quantum systems. We prove that, under this measure, the set of separable states possesses a nonzero volume. Analytical lower and upper bounds of this volume are also derived for N=2\ifmmode×\else\texttimes\fi2N=2\ifmmode\times\else\texttimes\fi{}2N=2\ifmmode×\else\texttimes\fi2 and N=2\ifmmode×\else\texttimes\fi3N=2\ifmmode\times\else\texttimes\fi{}3N=2\ifmmode×\else\texttimes\fi3 cases. Finally, numerical Monte Carlo calculations allow us to estimate the volume of separable states, providing numerical evidence that it decreases exponentially with the dimension of the composite system. We have also analyzed a conditional measure of separability under the condition of fixed purity. Our results display a clear dualism between purity and separability: entanglement is typical of pure states, while separability is connected with quantum mixtures. In particular, states of sufficiently low purity are necessarily separable.
The problem of how many entangled or, respectively, separable states there are in the set of all quantum states is investigated. We study to what extent the choice of a measure in the space of density matrices $\ensuremath{\varrho}$ describing N-dimensional quantum systems affects the results obtained. We demonstrate that the link between the purity of the mixed states and the probability of entanglement is not sensitive to the measure chosen. Since the criterion of partial transposition is not sufficient to distinguish all separable states for N> 8N>~8N> 8, we develop an efficient algorithm to calculate numerically the entanglement of formation of a given mixed quantum state, which allows us to compute the volume of separable states for N=8N=8N=8 and to estimate the volume of the bound entangled states in this case.
In a celebrated paper ([Phys. Rev. A 58, 883 (1998)]), K. Zyczkowski, P. Horodecki, A. Sanpera,and M. Lewenstein proved for the frst time a very interesting theorem that the volume of separable quantum states is nonzero. Inspired by their ideas, we obtain a general analytical lower bound of the volume of separable states (VOSS) for arbitrary N-dimensional system. Our results give quite simple and computable suffcient conditions for separability. Moreover, for bipartite system, an upper bound of the VOSS is also presented.
Every choice of an orthonormal frame in the ddd-dimensional Hilbert space of a system corresponds to one set of all mutually commuting density matrices or, equivalently, to the classical statistical state space of the system; the quantum state space itself can thus be profitably viewed as an SU(d)\text{SU}(d)SU(d) orbit of classical state spaces, one for each orthonormal frame. We exploit this connection to study the relative volume of separable states of a bipartite quantum system. While the two-qubit case is studied in considerable analytic detail, for higher-dimensional systems we fall back on Monte Carlo. Several insights seem to emerge from our study.
We demonstrate --- using the case of the two-dimensional quantum systems --- that the "natural measure on the space of density matrices describing N-dimensional quantum systems" proposed by Zyczkowski et al (quant-ph/9804024) does not belong to the class of normalized volume elements of monotone metrics on the quantum systems. Such metrics possess the statistically important property of being decreasing under stochastic mappings (coarse-grainings). We do note that the proposed measure (and certain evident variations upon it) exhibit quite specific monotonicity properties, but not of the form required for membership in the class.