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Chinese Physics Letters

Separability of Bipartite Superoperator Based on Witness

Zhang Shun, Zheng-Wei Zhou, Guo Guang-Can

Published Feb 1, 2009• 3 citations
Publisher

Abstract

Based on the isomorphic relation between operator space L(H) and Hilbert space H ⊗2 , Cirac et al. mapped the global superoperator to a mixed state E which has the same separability of the initial superoperator. Zhang et al. [Phys. Rev. A 76 (2007) 012334] provided a calculable lower bound for both the linear and nonlinear witness. We use this bound to detect the entanglement of E to judge the separability of the initial superoperator. With the help of local orthogonal observables, we directly describe the separable condition of superoperator by its each operator. Lastly, using the lower bound of the nonlinear witness, we provide a calculable entanglement factor of bipartite superoperator.

DOI
10.1088/0256-307x/26/2/020304

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Based on the isomorphic relation between operator space L(H) and Hilbert space H ⊗2 , Cirac et al. mapped the global superoperator to a mixed state E which has the same separability of the initial superoperator. Zhang et al. [Phys. Rev. A 76 (2007) 012334] provided a calculable lower bound for both the linear and nonlinear witness. We use this bound to detect the entanglement of E to judge the separability of the initial superoperator. With the help of local orthogonal observables, we directly describe the separable condition of superoperator by its each operator. Lastly, using the lower bound of the nonlinear witness, we provide a calculable entanglement factor of bipartite superoperator.
The Hilbert space for [Formula: see text], [Formula: see text]-dimensional, quantum systems, [Formula: see text], contains the subspace of states that are invariant under any permutation, the symmetric sector, [Formula: see text]. Separable states must have a nonzero component on the symmetric sector. More in detail, if [Formula: see text] denotes the orthogonal projection onto [Formula: see text], [Formula: see text], for a bound [Formula: see text] and for any separable state [Formula: see text]. Therefore, if one knows a (lower bound for) [Formula: see text], the operator [Formula: see text] is an entanglement witness for the system. In this paper we calculate sharp bounds [Formula: see text] analytically for the case [Formula: see text] and general [Formula: see text], and numerically for the case of [Formula: see text] qubits ([Formula: see text]). We show with an example that the resulting entanglement witnesses are able to detect entanglement of PPT states.
We focus on determining the separability of an unknown bipartite quantum state $\ensuremath{\rho}$ by invoking a sufficiently large subset of all possible entanglement witnesses given the expected value of each element of a set of mutually orthogonal observables. We review the concept of an entanglement witness from the geometrical point of view and use this geometry to show that the set of separable states is not a polytope and to characterize the class of entanglement witnesses (observables) that detect entangled states on opposite sides of the set of separable states. All this serves to motivate a classical algorithm which, given the expected values of a subset of an orthogonal basis of observables of an otherwise unknown quantum state, searches for an entanglement witness in the span of the subset of observables. The idea of such an algorithm, which is an efficient reduction of the quantum separability problem to a global optimization problem, was introduced by [Ioannou et al., Phys. Rev. A 70, 060303(R)], where it was shown to be an improvement on the naive approach for the quantum separability problem (exhaustive search for a decomposition of the given state into a convex combination of separable states). The last section of the paper discusses in more generality such algorithms, which, in our case, assume a subroutine that computes the global maximum of a real function of several variables. Despite this, we anticipate that such algorithms will perform sufficiently well on small instances that they will render a feasible test for separability in some cases of interest (e.g., in $3\ifmmode\times\else\texttimes\fi{}3$ dimensional systems).
We show that the entanglement witnesses based on local orthogonal observables, which were introduced by Yu and Liu [Phys. Rev. Lett. 95, 150504 (2005)] and G\"uhne et al. [Phys. Rev. A 74, 010301(R) (2006)] in linear and nonlinear forms, respectively, can be optimized. As applications, we use our method to calculate the optimal nonlinear witnesses of pure bipartite states and to show a lower bound on the III concurrence of bipartite higher-dimensional systems.
Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.
Quantum correlations and entanglement in identical-particle systems have been a puzzling question which has attracted vast interest and widely different approaches. A novel approach is introduced by Kraus \emph{et al.}, [Phys. Rev. A \textbf{79}, 012306 (2009)] based on pairing correlations in fermionic systems and the use of witness formalism to detect pairing. In this contribution, this approach has been extended to bosonic systems and separability bounds based on pairing correlations for fermions and bosons have been obtained. A two-particle annihilation operator is used for constructing a two-particle observable as a candidate witness. Two different types of separability definition is introduced for bosonic systems and the separability bounds associated with each type are discussed.
Employing a recently proposed separability criterion we develop analytical lower bounds for the concurrence and for the entanglement of formation of bipartite quantum systems. The separability criterion is based on a nondecomposable positive map which operates on state spaces with even dimension N >= 4, and leads to a class of nondecomposable optimal entanglement witnesses. It is shown that the bounds derived here complement and improve the existing bounds obtained from the criterion of positive partial transposition and from the realignment criterion.