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arXiv:1509.00086 · Journal of Physics A Mathematical and Theoretical

A bipartite separable ball and its applications

Shu‐Qian Shen, Ming Li, Lei Li

Published Feb 4, 2015• 5 citations
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Abstract

In this paper, based on a matrix norm, we first present a ball of separable un-normalized states around the identity matrix for the bipartite quantum system, which is larger than the separable ball in Frobenius norm. Then the proposed ball is used to get not only simple sufficient conditions for the separability of pseudopure states and the states with strong positive partial transposes, but also a separable ball centered at the identity matrix for the multipartite quantum system.

DOI
10.1088/1751-8113/48/9/095302

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In this paper, based on a matrix norm, we first present a ball of separable un-normalized states around the identity matrix for the bipartite quantum system, which is larger than the separable ball in Frobenius norm. Then the proposed ball is used to get not only simple sufficient conditions for the separability of pseudopure states and the states with strong positive partial transposes, but also a separable ball centered at the identity matrix for the multipartite quantum system.
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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.
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 We show that around any m-partite product state ρ p r o d = ρ 1 ⊗ … ⊗ ρ m of full rank (that is d e t ( ρ p r o d ) ≠ 0 ) , there exists a finite-sized closed ball of separable states centered around ρ p r o d whose radius is β := 2 1 − m / 2 λ m i n ( ρ p r o d ) . Here, λ m i n ( ρ p r o d ) is the smallest eigenvalue of ρ p r o d . We are assuming that the total Hilbert space is finite dimensional and we use the notion of distance induced by the Frobenius norm. Applying a scaling relation, we also give a new and simple sufficient criterion for multipartite separability based on trace: T r [ ρ ρ p r o d ] 2 / T r
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 generalize the definition of strong positive partial transpose (SPPT) to the multipartite system. The tripartite case was first considered by Yu and Zhao [Int. J. Theor. Phys. 54, 292 (2014)]. In this extension, unfortunately, desired properties such as the PPT of SPPT states and the separability of super and pure SPPT states are not preserved. In contrast, this paper provides an alternative generalization to multipartite cases with these properties preserved. We also provide sufficient conditions for the separability of SPPT states.