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arXiv:1803.11065 · Physical Review A

Optimization of ultrafine entanglement witnesses

Shu‐Qian Shen, Ti-Run Xu, Shao-Ming Fei, Xianqing Li‐Jost, Ming Li

Published Mar 28, 2018• 9 citations
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Abstract

The ultrafine entanglement witness introduced by Shahandeh et al. [Phys. Rev. Lett. 118, 110502 (2017)] can seamlessly and easily improve any standard entanglement witness. In this paper, by combining the constraint and the test operators, we rotate the hyperplane determined by the test operator and further improve the original ultrafine entanglement witness. In particular, we present a series of ultrafine entanglement witnesses that not only can detect entangled states that the original ultrafine entanglement witnesses cannot detect, but also have the merits that the original ultrafine entanglement witnesses have.

DOI
10.1103/physreva.97.032343

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The ultrafine entanglement witness introduced by Shahandeh et al. [Phys. Rev. Lett. 118, 110502 (2017)] can seamlessly and easily improve any standard entanglement witness. In this paper, by combining the constraint and the test operators, we rotate the hyperplane determined by the test operator and further improve the original ultrafine entanglement witness. In particular, we present a series of ultrafine entanglement witnesses that not only can detect entangled states that the original ultrafine entanglement witnesses cannot detect, but also have the merits that the original ultrafine entanglement witnesses have.
Abstract An entanglement witness is an observable with the property that a negative expectation value signals the presence of entanglement. The question arises how a witness can be improved if the expectation value of a second observable is known, and methods for doing this have recently been discussed as so-called ultrafine entanglement witnesses. We present several results on the characterization of entanglement given the expectation values of two observables. First, we explain that this problem can naturally be tackled with the method of the Legendre transformation, leading even to a quantification of entanglement. Second, we present necessary and sufficient conditions that two product observables are able to detect entanglement. Finally, we explain some fallacies in the original construction of ultrafine entanglement witnesses (Shahandeh et al 2017 Phys. Rev. Lett . 118 110502).
Entanglement witnesses are invaluable for efficient quantum entanglement certification without the need for expensive quantum state tomography. Yet, standard entanglement witnessing requires multiple measurements and its bounds can be elusive as a result of experimental imperfections. Here, we introduce and demonstrate a novel procedure for entanglement detection which simply and seamlessly improves any standard witnessing procedure by using additional available information to tighten the witnessing bounds. Moreover, by relaxing the requirements on the witness operators, our method removes the general need for the difficult task of witness decomposition into local observables. We experimentally demonstrate entanglement detection with our approach using a separable test operator and a simple fixed measurement device for each agent. Finally, we show that the method can be generalized to higher-dimensional and multipartite cases with a complexity that scales linearly with the number of parties.
An entanglement witness (EW) is an operator that allows the detection of entangled states. We give necessary and sufficient conditions for such operators to be optimal, i.e., to detect entangled states in an optimal way. We show how to optimize general EW, and then we particularize our results to the nondecomposable ones; the latter are those that can detect positive partial transpose entangled states (PPTES's). We also present a method to systematically construct and optimize this last class of operators based on the existence of ``edge'' PPTES's, i.e., states that violate the range separability criterion [Phys. Lett. A 232, 333 (1997)] in an extreme manner. This method also permits a systematic construction of nondecomposable positive maps (PM's). Our results lead to a sufficient condition for entanglement in terms of nondecomposable EW's and PM's. Finally, we illustrate our results by constructing optimal EW acting on $H={C}^{2}\ensuremath{\bigotimes}{C}^{4}.$ The corresponding PM's constitute examples of PM's with minimal ``qubit'' domains, or---equivalently---minimal Hermitian conjugate codomains.
This paper is concerned with all tests for continuous-variable entanglement that arise from linear combinations of second moments or variances of canonical coordinates, as they are commonly used in experiments to detect entanglement. All such tests for bi-partite and multi-partite entanglement correspond to hyperplanes in the set of second moments. It is shown that all optimal tests, those that are most robust against imperfections with respect to some figure of merit for a given state, can be constructed from solutions to semi-definite optimization problems. Moreover, we show that for each such test, referred to as an entanglement witness based on second moments, there is a one-to-one correspondence between the witness and a stronger product criterion, which amounts to a nonlinear witness, based on the same measurements. This generalizes the known product criteria. The presented tests are all applicable also to non-Gaussian states. To provide a service to the community, we present the documentation of two numerical routines, FullyWit and MultiWit , which have been made publicly available.
Entanglement witnesses are of special interest for distinguishing between separable and entangled states. For three-qubit states, a family of optimal entanglement witnesses has been introduced by Jafarizadeh et al. [Eur. Phys. J. D 50 (2008) 107]. We give an alternative demonstration for some elements of this family and show that they are able to detect bound entangled states. To illustrate this fact, we consider two families of bound entangled three-qubit states, introduced by Kay [Phys. Rev. A 83 (2011) 020303] and Kye [J. Phys. A, Math. Theor. 48 (2015) 235303], and show that these witnesses detect the entanglement of all states of these two families.
Quantum Mechanics Quantum entanglement produces correlations between particles that cannot be described classically, and it is a resource sought out in many applications to leverage the quantum advantage over classical operation. However, determining whether particles are entangled is a complex process that can require a set of measurements (the witness) to establish the extent of the correlations between them. Shanhandeh et al. describe a protocol that should simplify the job of the witness somewhat. Using already known information about the systems, they demonstrate that entangled and unentangled states can be distinguished using fewer measurements. Extendable to larger systems, the simplified but enhanced witness should prove useful in quantum devices for which verifying entanglement is required. Phys. Rev. Lett. 118 , 110502 (2017).