Entanglement characterization using quantum designs

Abstract

We present in detail a statistical approach for the reference-frame-independent detection and characterization of multipartite entanglement based on moments of randomly measured correlation functions. We start by discussing how the corresponding moments can be evaluated with designs, linking methods from group and entanglement theory. Then, we illustrate the strengths of the presented framework with a focus on the multipartite scenario. We discuss a condition for characterizing genuine multipartite entanglement for three qubits, and we prove criteria that allow for a discrimination of W-type entanglement for an arbitrary number of qubits.

A Moment for Random Measurements

“…Furthermore, in [19] the authors discuss moments of distributions of correlations for witnessing SLOCC classes, which allows to decide if a state is reversibly convertible into, say, a W state. Figure 6 shows sampled four-qubit states in the R (2) -R (4) plane.…”

“…The solid black line encloses the SLOCC class of W states, whereas the dashed line surrounds the convex hull of the W states. Figure from [19].…”

“…In contrast, additional consideration of R (4) does not improve the ability for detection significantly. More generally, the authors of [19] derive a witness for Conv(W (n) ) for an arbitrary number of n qubits. If the second moment R (2) of a distribution of correlations is larger than…”

“…the n-qubit state is not a member of the mixed W class Conv(W (n) ) [19]. For 4 qubits, the threshold is χ (4) = 4/81 ≈ 0.049.…”

“…In this perspective article, we will discuss a work of Ketterer et al, which recently appeared in Quantum [19]. Before we will review their means of entanglement detection and classification, we will introduce the general concept of random measurements, provide an intuitive understanding for them and give context by discussing other methods for detection and classification of entanglement in this scenario.…”