Multipartite Entanglement Witnesses

Abstract

We derive a set of algebraic equations, the so-called multipartite separability eigenvalue equations. Based on their solutions, we introduce a universal method for the construction of multipartite entanglement witnesses. We witness multipartite entanglement of 10 3 coupled quantum oscillators, by solving our basic equations analytically. This clearly demonstrates the feasibility of our method for studying ultrahigh orders of multipartite entanglement in complex quantum systems.PACS numbers: 03.67. Mn, 03.65.Ud, 42.50.Dv Entanglement represents a fundamental quantum correlation between compound quantum systems. Since the early days of quantum physics this property has been used to illustrate the surprising consequences of the quantum description of nature [1,2]. Moreover, entanglement plays a fundamental role in various applications and protocols in quantum information science [3][4][5].In multipartite systems a separable state is a statistical mixture of product states [6]. A quantum state is entangled, whenever it cannot be represented in this form. Various forms of multipartite entanglement are known [7][8][9][10]. The most prominent and nonequivalent forms of entangled multipartite quantum states are the GHZ-state [11] and the W-state [12], which have been generalized to so-called cluster and graph states [13,14]. Another classification is given in terms of partial and full (or genuine) multipartite entanglement, for an introduction see e.g. [4,5]. Beyond finite dimensional systems, multipartite quantum entanglement in continuous variable systems turns out to be a cumbersome problem. Even in the case of Gaussian states, there exist multipartite entangled states, which cannot be distilled [15].High orders of multipartite entanglement are of great interest, for example, in quantum metrology. Multipartite entanglement has been shown to be essential to reach the maximal sensitivity in metrological tasks [26]. In this context, the quantum Fisher information has been used to characterize the entanglement [27][28][29].The detection of entanglement is typically done via the construction of proper entanglement witnesses [16][17][18], being equivalent to the method of positive, but not completely positive maps. A witness is an observable, which is non-negative for separable states, and it can have a negative expectation value for entangled states. For different kinds of entanglement, different types of witnesses have been considered: bipartite witnesses [17,19]; Schmidt number witnesses [20,21]; and multipartite witnesses for partial and genuine entanglement [22][23][24][25]. A systematic approach for witnessing entanglement in complex quantum systems is missing yet.Recently, we considered the construction of bipartite entanglement witnesses with the so-called separability eigenvalue equations [19]. We have shown that the same equations need to be solved to obtain entanglement quasiprobabilities, which are nonpositive distributions if and only if the corresponding quantum state is entangled [30]. Moreover, we have s...

Reliable and robust entanglement witness

“…To efficiently find α, one can make use of the technique similar to Ref. [24], in which, EW can be systematically constructed.…”

Measurement-Device-Independent Approach to Entanglement Measures

“…[19,47], in general, multipartite entanglement has a highly complex structure. However, the subset of witnesses extremal to the set of P k -separable quantum states is necessary and sufficient for detecting entanglement within P k .…”

Systematic construction of genuine-multipartite-entanglement criteria in continuous-variable systems using uncertainty relations

“…Then, we can proceed like in Eq. (29) and recognise that inequality (43) is never violated by biseparable statesρ bs with p { α| β} = 0 corresponding to the bipartitions such that { α| β} = { α} m (see Eq. (13)).…”

“…Motivated by the impossibility of PT based entanglement criteria to detect bound entanglement [27], in [28] the authors construct bipartite entanglement witnesses solving the so-called separability eigenvalue equation. The authors extend their approach in [29] deriving a set of algebraic equations, which yield the construction of arbitrary multipartite entanglement witnesses using general Hermitian operators. This approach is powerful when several measurements of different Hermitian operators are considered in order to feed the optimisation problem involved to obtain the optimal entanglement criteria with the measurements at hand.…”