Characterizing Multipartite Entanglement with Moments of Random Correlations

Abstract

The experimental detection of multipartite entanglement usually requires a number of appropriately chosen local quantum measurements which are aligned with respect to a previously shared common reference frame. The latter, however, can be a challenging prerequisite e.g. for satellitebased photonic quantum communication, making the development of alternative detection strategies desirable. One possibility for avoiding the distribution of classical reference frames is to perform a number of local measurements with settings distributed uniformly at random. In this work we follow such a treatment and show that an improved detection and characterization of multipartite entanglement is possible by combining statistical moments of different order. To do so, we make use of designs which are pseudo-random processes allowing to link the present entanglement criteria to ordinary reference frame independent ones. The strengths of our methods are illustrated in various cases starting with two qubits and followed by more involved multipartite scenarios.

Entanglement characterization using quantum designs

“…In order to achieve this we have to perform several rounds of random measurements and seek a statistical treatment in terms of the moments of the randomly measured correlation functions. As we have assumed that each correlation function is characterized by a set of Haar random unitaries {U n } n=1,...,N , we can define these moments as follows [21]:…”

“…In several recent works it has been shown how to go beyond such procedures by relaxing also the assumption of being able to measure a fixed set of local observables and instead allow only for local measurements with settings drawn uniformly at random [19][20][21][22][23][24][25][26][27]. The common idea of these approaches is to measure a certain correlation function, and average the result over random local unitaries applied to the state.…”

“…Moreover, it has recently been demonstrated that a combination of statistical moments beyond the second order can lead to further improvements in terms of entanglement detection [21]. In particular, it was shown that novel reference-frame-independent criteria for the detection and also characterization of multipartite entanglement can be derived by expressing the respective moments in terms of spherical designs, i.e., pseudo-random processes allowing to mimic uniform averages over the sphere.…”

“…After recalling for completeness some of the results from Ref. [21], we first discuss the characterization of multipartite entanglement classes based on the first two non-vanishing moments in small multipartite systems consisting of three and four qubits. Finally, we prove two novel criteria enabling the discrimination of W -type entangled mixed states for an arbitrary number of parties.…”

“…2 we introduce the necessary theoretical tools from Ref. [21], i.e., the moments of random correlation functions and the concept of unitary and spherical designs (see Sec. 2.1 and 2.2, respectively).…”