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arXiv:0705.1832 · Physical Review A

Optimal entanglement witnesses based on local orthogonal observables

Chengjie Zhang, Yong-Sheng Zhang, Shun Zhang, Guang‐Can Guo

Published Jul 31, 2007• 105 citations
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Abstract

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 $I$ concurrence of bipartite higher-dimensional systems.

DOI
10.1103/physreva.76.012334

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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.
We propose a family of entanglement witnesses and corresponding positive maps that are not completely positive based on local orthogonal observables. As applications the entanglement witness of a 3x3 bound entangled state [P. Horodecki, Phys. Lett. A 232, 333 (1997)] is explicitly constructed and a family of dxd bound entangled states is introduced, whose entanglement can be detected by permuting local orthogonal observables. The proposed criterion of separability can be physically realized by measuring a Hermitian correlation matrix of local orthogonal observables.
Let ${H}^{[N]}{=H}^{[{d}_{1}]}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{H}^{[{d}_{n}]}$ be a tensor product of Hilbert spaces and let ${\ensuremath{\tau}}_{0}$ be the closest separable state in the Hilbert-Schmidt norm to an entangled state ${\ensuremath{\rho}}_{0}.$ Let ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}$ denote the closest separable state to ${\ensuremath{\rho}}_{0}$ along the line segment from I/NI/NI/N to ${\ensuremath{\rho}}_{0}$ where I is the identity matrix. Following A. O. Pittenger and M. H. Rubin [Linear Algebr. Appl. 346, 75 (2002)] a witness W0{W}_{0}W0​ detecting the entanglement of ${\ensuremath{\rho}}_{0}$ can be constructed in terms of I, ${\ensuremath{\tau}}_{0},$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}.$ If representations of ${\ensuremath{\tau}}_{0}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}$ as convex combinations of separable projections are known, then the entanglement of ${\ensuremath{\rho}}_{0}$ can be detected by local measurements. G\"uhne et al. [Phys. Rev. A 66, 062305 (2002)] obtain the minimum number of measurement settings required for a class of two-qubit states. We use our geometric approach to generalize their result to the corresponding two-qudit case when d is prime and obtain the minimum number of measurement settings. In those particular bipartite cases, ${\ensuremath{\tau}}_{0}={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}.$ We illustrate our general approach with a two-parameter family of three-qubit bound entangled states for which ${\ensuremath{\tau}}_{0}\ensuremath{\ne}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}$ and we show that our approach works for n qubits. We elaborated earlier [A. O. Pittenger, Linear Algebr. App. 359, 235 (2003)] on the role of a ``far face'' of the separable states relative to a bound entangled state ${\ensuremath{\rho}}_{0}$ constructed from an orthogonal unextendible product base. In this paper the geometric approach leads to an entanglement witness expressible in terms of a constant times I and a separable density ${\ensuremath{\mu}}_{0}$ on the far face from ${\ensuremath{\rho}}_{0}.$ Up to a normalization this coincides with the witness obtained by G\"uhne et al. for the particular example analyzed there.
Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed. Using single particle feasible region, a way of constructing effective entanglement witnesses for bipartite systems is provided by exact convex optimization. Examples for some well known two qutrit quantum systems show these entanglement witnesses in most cases, provide necessary and sufficient conditions for separability of given bipartite system. Also this method is applied to a class of bipartite qudit quantum systems with details for d=3, 4 and 5.
We address the problem of optimizing entanglement witnesses when a limited fixed set of local measurements can be performed on a bipartite system, thus providing a procedure, feasible also for experiments, to detect entangled states using only the statistics of these local measurements. We completely characterize the class of entanglement witnesses of the form $W={P}^{\mathrm{\ensuremath{\Gamma}}}$, where $\mathrm{\ensuremath{\Gamma}}$ denotes partial transposition, that can be constructed from the measurements of the bipartite operators ${\ensuremath{\sigma}}_{x}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{x}, {\ensuremath{\sigma}}_{y}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{y}$, and ${\ensuremath{\sigma}}_{z}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{z}$ in the case of two-qubit systems. In particular, we consider all possible extremal decomposable witnesses within the considered class that can be defined from this set of measurements. Finally, we discuss possible extensions to higher-dimension bipartite systems when the set of available measurements is characterized by the generalized Gell-Mann matrices. We provide several examples of entanglement witnesses, both decomposable and indecomposable, that can be constructed with these limited resources.
Recently, a proper genuine multipartite entanglement measure has been found for three-qubit pure states [see Xie and Eberly, Phys. Rev. Lett. 127, 040403 (2021)PRLTAO0031-900710.1103/PhysRevLett.127.040403], but capturing useful entanglement measures for mixed states has remained an open challenge. So far, it requires not only a full tomography in experiments, but also huge calculational labor. A leading proposal was made by Gühne, Reimpell, and Werner [Phys. Rev. Lett. 98, 110502 (2007)PRLTAO0031-900710.1103/PhysRevLett.98.110502], who used expectation values of entanglement witnesses to describe a lower bound estimation of entanglement. We provide here an extension that also gives genuine upper bounds of entanglement. This advance requires only the expectation value of any Hermitian operator. Moreover, we identify a class of operators A_{1} that not only give good estimates, but also require a remarkably small number of experimental measurements. In this Letter, we define our approach and illustrate it by estimating entanglement measures for a number of pure and mixed states prepared in our recent experiments.
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.