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/0310061 · Physical Review A

Volume of separable states is super-doubly-exponentially small in the number of qubits

Stanisław J. Szarek

Published Sep 2, 2005• 62 citations
arXivHTMLPDF

Abstract

In this paper we give sharp two-sided estimates of the volume of the set of separable states on $N$ qubits. In particular, the magnitude of the ``effective radius'' of that set in the sense of volume is determined up to a factor which is a (small) power of $N$, and thus precisely on the scale of powers of its dimension. We also identify an ellipsoid that appears to optimally approximate the set of separable states. Additionally, one of the appendixes contains sharp estimates (by known methods) for the expected values of norms of the Gaussian unitary ensemble random matrices. We employ standard tools of classical convexity, high-dimensional probability, and geometry of Banach spaces.

DOI
10.1103/physreva.72.032304

Related papers

Tensor products of convex sets and the volume of separable states on N qudits

Guillaume Aubrun, Stanisław J. Szarek

This paper deals with estimating the volume of the set of separable mixed quantum states when the dimension of the state space grows to infinity. This has been studied recently for qubits; here we consider larger particles and conclude that, in all cases, the proportion of the states that are separable is superexponentially small in the dimension of the set. We also show that the partial transpose criterion becomes imprecise when the dimension increases, and that the lower bound ${6}^{\ensuremath{-}N∕2}$ on the (Hilbert-Schmidt) inradius of the set of separable states on $N$ qubits obtained recently by Gurvits and Barnum is essentially optimal. We employ standard tools of classical convexity, high-dimensional probability, and geometry of Banach spaces. One relatively nonstandard point is a formal introduction of the concept of projective tensor products of convex bodies, and an initial study of this concept.

arXiv:quant-ph/0503221•Physical Review A•Feb 14, 2006•74 citations

Quantum Steering Ellipsoid Volume as a Measure on the Two-Qubit States and Associated Separability and Absolute Separability Ratios

Paul B. Slater

Employing the volume of quantum steering ellipsoids (QSEs) as a measure on the fifteen-dimensional convex set of two-qubit states, we estimate the ratio of the integral of the measure over the separable states to its integral over all (separable and entangled) states to be 0.0288. This can be contrasted with the considerably larger separability ratios (probabilities) of $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$ and $\frac{25}{341}=\frac{5^2}{11 \cdot 31} \approx 0.0733138$ that various forms of evidence point to with the use of the prominent Hilbert-Schmidt and Bures measures, respectively. The questions of whether the ratio in the QSE setting can be more precisely obtained or even exactly computed, as well as whether a metric can be constructed, the volume element of which yields the measure, remain to be addressed. We also investigate related issues pertaining to absolute separability. Further, we examine the behavior of the separability probability--constant in the Hilbert-Schmidt case and decreasing in the Bures--as a function of the Bloch vector norm in the QSE instance. It appears to increase approaching the pure state boundary.

arXiv:2012.09152•Dec 16, 2020•0 citations

Volumes and hyperareas of the spaces of separable and nonseparable qubit-qutrit systems: Initial numerical analyses

Paul B. Slater

Paralleling our recent computationally-intensive work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric. At the same time, we estimate the unknown volumes and hyperareas based on a number of other monotone metrics of interest. Additionally, we estimate -- but with perhaps unavoidably diminished accuracy -- all these volume and hyperarea quantities, when restricted to the ``small'' subset of 6 x 6 density matrices that are separable (classically correlated) in nature. The ratios of separable to separable plus nonseparable volumes, then, yield corresponding estimates of the ``probabilities of separability''. We are particularly interested in exploring the possibility that a number of the various 35-dimensional volumes and 34-dimensional hyperareas, possess exact values -- which we had, in fact, conjectured to be the case for the qubit-qubit systems (N=4), with the ``silver mean'', sqrt{2}-1, appearing to play a fundamental role as regards the separable states.

arXiv:quant-ph/0405114•May 20, 2004•1 citation

Optimal ensemble length of mixed separable states

Robert Lockhart

The optimal (pure state) ensemble length of a separable state A is the minimum number of (pure) states needed in convex combination to construct A. We study the set of all separable states with optimal (pure state) ensemble length equal to k or fewer. Lower bounds on k are found below which these sets have measure 0 in the set of separable states. In the bipartite case and the multiparticle case where one of the particles has significantly more quantum numbers than the rest the lower bounds are sharp. A consequence of our results is that for all two-particle systems, except possibly those with one qubit or those with a nine-dimensional Hilbert space, and for all systems with more than two particles the optimal pure state ensemble length for a randomly picked separable state is with probability 1 greater than the state’s rank. In bipartite systems with probabilty 1 it is greater than 1/4 the rank raised to the 3/2 power and in a system of p qubits with probability 1 it is greater than 22p/(1+2p), i.e., almost the square of the rank.

arXiv:quant-ph/9908050•Journal of Mathematical Physics•Oct 1, 2000•18 citations

Optimal Ensemble Length of Mixed Separable States

Robert B. Lockhart

The optimal (pure state) ensemble length of a separable state, A, is the minimum number of (pure) product states needed in convex combination to construct A. We study the set of all separable states with optimal (pure state) ensemble length equal to k or fewer. Lower bounds on k are found below which these sets have measure 0 in the set of separable states. In the bipartite case and the multiparticle case where one of the particles has significantly more quantum numbers than the rest, the lower bounds are sharp. A consequence of our results is that for all two particle systems, except possibly those with a qubit and those with a nine dimensional Hilbert space, and for all systems with more than two particles the optimal pure state ensemble length for a randomly picked separable state is with probability 1 greater than the state’s rank. In bipartite systems with each particle having the same Hilbert space, with probablity 1 it is greater than 1/4 the rank raised to the 3/2 power and in a system with p qubits with probability 1 it is greater than 2 2p /(1 + 2p), which is almost the maximal rank squared. 1

•Aug 17, 1999•14 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
In this paper we give sharp two-sided estimates of the volume of the set of separable states on NNN qubits. In particular, the magnitude of the ``effective radius'' of that set in the sense of volume is determined up to a factor which is a (small) power of NNN, and thus precisely on the scale of powers of its dimension. We also identify an ellipsoid that appears to optimally approximate the set of separable states. Additionally, one of the appendixes contains sharp estimates (by known methods) for the expected values of norms of the Gaussian unitary ensemble random matrices. We employ standard tools of classical convexity, high-dimensional probability, and geometry of Banach spaces.
This paper deals with estimating the volume of the set of separable mixed quantum states when the dimension of the state space grows to infinity. This has been studied recently for qubits; here we consider larger particles and conclude that, in all cases, the proportion of the states that are separable is superexponentially small in the dimension of the set. We also show that the partial transpose criterion becomes imprecise when the dimension increases, and that the lower bound ${6}^{\ensuremath{-}N∕2}$ on the (Hilbert-Schmidt) inradius of the set of separable states on NNN qubits obtained recently by Gurvits and Barnum is essentially optimal. We employ standard tools of classical convexity, high-dimensional probability, and geometry of Banach spaces. One relatively nonstandard point is a formal introduction of the concept of projective tensor products of convex bodies, and an initial study of this concept.
Employing the volume of quantum steering ellipsoids (QSEs) as a measure on the fifteen-dimensional convex set of two-qubit states, we estimate the ratio of the integral of the measure over the separable states to its integral over all (separable and entangled) states to be 0.0288. This can be contrasted with the considerably larger separability ratios (probabilities) of 833=233⋅11≈0.242424\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424338​=3⋅1123​≈0.242424 and 25341=5211⋅31≈0.0733138\frac{25}{341}=\frac{5^2}{11 \cdot 31} \approx 0.073313834125​=11⋅3152​≈0.0733138 that various forms of evidence point to with the use of the prominent Hilbert-Schmidt and Bures measures, respectively. The questions of whether the ratio in the QSE setting can be more precisely obtained or even exactly computed, as well as whether a metric can be constructed, the volume element of which yields the measure, remain to be addressed. We also investigate related issues pertaining to absolute separability. Further, we examine the behavior of the separability probability--constant in the Hilbert-Schmidt case and decreasing in the Bures--as a function of the Bloch vector norm in the QSE instance. It appears to increase approaching the pure state boundary.
Paralleling our recent computationally-intensive work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric. At the same time, we estimate the unknown volumes and hyperareas based on a number of other monotone metrics of interest. Additionally, we estimate -- but with perhaps unavoidably diminished accuracy -- all these volume and hyperarea quantities, when restricted to the ``small'' subset of 6 x 6 density matrices that are separable (classically correlated) in nature. The ratios of separable to separable plus nonseparable volumes, then, yield corresponding estimates of the ``probabilities of separability''. We are particularly interested in exploring the possibility that a number of the various 35-dimensional volumes and 34-dimensional hyperareas, possess exact values -- which we had, in fact, conjectured to be the case for the qubit-qubit systems (N=4), with the ``silver mean'', sqrt{2}-1, appearing to play a fundamental role as regards the separable states.
The optimal (pure state) ensemble length of a separable state A is the minimum number of (pure) states needed in convex combination to construct A. We study the set of all separable states with optimal (pure state) ensemble length equal to k or fewer. Lower bounds on k are found below which these sets have measure 0 in the set of separable states. In the bipartite case and the multiparticle case where one of the particles has significantly more quantum numbers than the rest the lower bounds are sharp. A consequence of our results is that for all two-particle systems, except possibly those with one qubit or those with a nine-dimensional Hilbert space, and for all systems with more than two particles the optimal pure state ensemble length for a randomly picked separable state is with probability 1 greater than the state’s rank. In bipartite systems with probabilty 1 it is greater than 1/4 the rank raised to the 3/2 power and in a system of p qubits with probability 1 it is greater than 22p/(1+2p), i.e., almost the square of the rank.
The optimal (pure state) ensemble length of a separable state, A, is the minimum number of (pure) product states needed in convex combination to construct A. We study the set of all separable states with optimal (pure state) ensemble length equal to k or fewer. Lower bounds on k are found below which these sets have measure 0 in the set of separable states. In the bipartite case and the multiparticle case where one of the particles has significantly more quantum numbers than the rest, the lower bounds are sharp. A consequence of our results is that for all two particle systems, except possibly those with a qubit and those with a nine dimensional Hilbert space, and for all systems with more than two particles the optimal pure state ensemble length for a randomly picked separable state is with probability 1 greater than the state’s rank. In bipartite systems with each particle having the same Hilbert space, with probablity 1 it is greater than 1/4 the rank raised to the 3/2 power and in a system with p qubits with probability 1 it is greater than 2 2p /(1 + 2p), which is almost the maximal rank squared. 1