Detection of entanglement with few local measurements

Abstract

We introduce a general method for the experimental detection of entanglement by performing only few local measurements, assuming some prior knowledge of the density matrix. The idea is based on the minimal decomposition of witness operators into a pseudo-mixture of local operators. We discuss an experimentally relevant case of two qubits, and show an example how bound entanglement can be detected with few local measurements. 03.67.Dd, 03.67.Hk, A central aim in the physics of quantum information is to create and detect entanglement -the resource that allows to realize various quantum protocols. Recently, much progress has been achieved experimentally in creating entangled states [1]. In every real experiment noise and imperfections are present so that the generated states, although intended to be entangled, may in fact be separable. Therefore, it is important to find efficient experimental methods to test whether a given imperfect state ρ is indeed entangled.Obviously, the ultimate goal of entanglement detection is to characterize entanglement quantitatively, and identify regions in the parameter space which allow to maximize entanglement for a particular quantum information processing task. The first step towards this ambitious goal is to detect whether a given state is entangled or not.The question of direct detection of quantum entanglement has been recently addressed in Refs. [2][3][4]. In [3,4] the authors study the case of mixed states and find efficient ways to estimate the entanglement of an unknown state. Their method is based on structural approximations of some linear maps followed by a spectrum estimation. Although experimentally viable the method is not very easy to implement and it requires further modifications in order to be performed by local measurements [5]. Here, we approach the same problem from a different perspective. We use special observables, the so-called witness operators [6,7] and their optimal decomposition into a sum of local projectors. Note that in this way we answer an open question posed recently in [8], where non-local measurements of entanglement witnesses were studied.The construction of a witness for a given arbitrary state is, in general, a formidable task. It can, however, be accomplished in typical experimental situations where one has some a priori information about the density matrix. This is always the case when the experiment is aimed at producing a certain state, rather than checking properties of an a priori unknown state. We discuss two experimentally relevant situations in this paper, namely the generation of a definite pure entangled state of two parties, and the generation of a specific bound entangled edge state. In both cases our method can be applied in arbitrary dimensions.Having constructed a witness, its measurement can be performed locally, since every observable can be decomposed in terms of a product basis in the operator space. Here we propose two ways of optimizing such local measurements. The first one consists in looking for the optimal number of l...

Nonseparable Werner states in spontaneous parametric down-conversion

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“…We find that, for any value of g, the resulting 2-photon state is a "Werner state" [24], i.e., a superposition of a maximally entangled (singlet) state with a fully mixed state. This condition is investigated thoroughly by exploiting the extensive knowledge available on the Werner states and modern techniques like the "entanglement witness" and the various forms of state entropy [25,26]. Resilience of entanglement is theoretically demonstrated for any value of g in the high loss (HL) approximation [27].…”

Robust semidefinite programming approach to the separability problem

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“…Theorem 5 A state ρ AB is entangled iff there exists a multiplier matrix (15) such that (16) and (17) hold, with X = −I. The matrices appearing in the LMI (17) are given by equations (10 − 12) and the matrix ∆ is given by (13).…”

Proposal for direct measurement of concurrence via visibility in a cavity QED system

“…The question naturally arises whether entanglement can be estimated without having to fully reconstruct the unknown state. It has been shown that the answer to this question is yes at least for the case of pure two-qubit states [7,8], although, even in this case, more than one observable need to be measured.In recent years, several methods [9,10,11,12,13] have been proposed for detecting and measuring entanglement without a full reconstruction of the state; e.g., the method [9] based on the technique of minimal and optimal tomography [14,15] performed on one of the entangled pair, the method [10] based on entanglement witness [16,17] which was realized experimentally [18], the method [11,12] based on PPT criterion [1,2], and the method [13] based on two-particle interferometry [19,20]. These methods, although much simpler than the full state reconstruction, are not completely free of experimental difficulties, as they require either controlled unitary operations or some prior knowledge about the quantum state in question, or they can detect entanglement but not measure its amount.…”