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arXiv:quant-ph/0204159 · Physical Review A

Largest separable balls around the maximally mixed bipartite quantum state

Leonid Gurvits, Howard Barnum

Published Dec 12, 2002• 279 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.66.062311

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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∞,1<~p<~\ensuremath{\infty},1< p< \ensuremath∞, 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 tr\ensuremathρ2\mathrm{tr}{\ensuremath{\rho}}^{2}tr\ensuremathρ2 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.
For finite-dimensional bipartite quantum systems, we find the size of the largest balls, in spectral lpl_plp​ norms for 1lepleinfty1 \\le p \\le \\infty1lepleinfty, of separable (unentangled) matrices around the identity matrix. We discuss corollaries and applications to density matrices and bulk quantum information processing such as high-temperature nuclear magnetic resonance.
We show that for an m-partite quantum system, there is a ball of radius ${2}^{\ensuremath{-}(m/2\ensuremath{-}1)}$ in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices. This can be used to derive an \ensuremath{\epsilon} below which mixtures of \ensuremath{\epsilon} of any density matrix with $1\ensuremath{-}\ensuremath{\epsilon}$ of the maximally mixed state will be separable. The \ensuremath{\epsilon} thus obtained is exponentially better (in the number of systems) than existing results. This gives a number of qubits below which nuclear magnetic resonance with standard pseudopure-state preparation techniques can access only unentangled states; with parameters realistic for current experiments, this is 23 qubits (compared to 13 qubits via earlier results). A ball of radius 1 is obtained for multipartite states separable over the reals.
Abstract The largest separable ball is a meaningful geometric sufficient condition for separability of bipartite quantum states which tells that all the states in the unit ball in Frobenius norm centered at the identity matrix are separable. In this paper, we propose an algorithm that can improve the capability of this criterion to detect the separability outside the unit ball by solving an optimization problem based on the invertible local operators, that is, to optimize ( A ⊗ B ) ρ ( A ⊗ B ) † − 1 F with respect to the invertible square matrices A and B . The numerical examples have shown that our algorithm is more powerful than the original separable ball criterion.
Abstract Based on the generalized Bloch representation, we study the separability and entanglement of arbitrary dimensional multipartite quantum states. Some sufficient and some necessary criteria are presented. For certain states, these criteria together are both sufficient and necessary. Detailed examples show that our criteria are better than some existing ones in identifying separability. Based on these criteria, the largest separable ball around the maximally mixed state for arbitrary multi-qubit systems is found, and it is proved that its radius defined via the l 1-norm of the Bloch vector is the constant 1. Furthermore, the criteria in this paper can be implemented experimentally.
In this study, we investigate the problem of determining the maximum purity for absolutely separable and absolutely PPT quantum states. From the geometric viewpoint, this problem is equivalent to asking for the exact Euclidean radius of the smallest ball around the maximally mixed state that encompasses the set of all absolutely separable or absolutely PPT states. Our results provide an analytic solution for two qubit states. Based on numerical computation, we propose a conjectured maximum purity for absolutely separable qubit-qudit states and absolutely PPT qutrit-qudit states.