Detecting $k$-nonseparability and $k$-partite Entanglement with Generalized Skew Information and Mutually Unbiased Measurements

TL;DR

The paper addresses detecting $k$-nonseparability and $k$-partite entanglement in multipartite quantum systems by combining generalized Wigner-Yanase skew information $I^s(\rho,A)$ with Mutually Unbiased Measurements (MUMs). It derives two main, sufficient criteria (Theorems 3 and 4) that bound the total $I^s$ over a fixed MUM set for $k$-separable and $k$-producible states, enabling detection of $k$-nonseparability and $(k+1)$-partite entanglement when violated; the supremum over $s\in[-\infty,0]$ is shown to be optimally achieved at $s=-\infty$ in practice. The authors illustrate the effectiveness with examples on $6$- and $11$-partite states and demonstrate a network-depth application where the depth corresponds to the level of $k$-producibility. These results provide a practical, measurement-based approach to certify multipartite entanglement and entanglement depth, with bounds largely independent of the number of parties and adaptable to network scenarios.

Abstract

Multipartite quantum entanglement, as a core quantum resource, is fundamental to the advancement of quantum science and technology. In multipartite quantum systems, there are two kinds of quantum entanglement: $k$-nonseparability and $k$-partite entanglement. In this paper, we propose sufficient criteria for detecting $k$-nonseparability and $k$-partite entanglement by using the generalized Wigner-Yanase skew information and mutually unbiased measurements. Examples are given to demonstrate the detection capability and advantages of these criteria. As an application, an example of recognizing the networks by detecting the depth of quantum networks is given.

Figures (1)

• Figure 1: Three different quantum networks.

Theorems & Definitions (7)

• Proposition 1
• Proposition 2
• Theorem 3
• Theorem 4
• Example 5
• Example 6
• Example 7