A multipartite entanglement measure based on coefficient matrices

Abstract

The quantification of quantum entanglement has been extensively studied in past years. However, many existing entanglement measures are difficult to calculate. And lots of them are introduced only for bipartite system or only for the systems constituted by qubits. In this paper, we propose an entanglement measure for multipartite system based on vector lengths and the angles between vectors of the coefficient matrices. Our entanglement measure is simple and feasible, with a remarkable geometric meaning. Furthermore, we prove that our entanglement measure satisfies the three necessary conditions which are required for any entanglement measure: (1) It vanishes if and only if the state is (fully) separable; (2) it remains invariant under local unitary transformations; and (3) it cannot increase under local operation and classical communication. Finally, we apply our entanglement measure on some computational examples. It demonstrates that our entanglement measure is capable of dealing with quantum pure states with arbitrary dimensions and parties. Meanwhile, because it only needs to compute the vector lengths and the angles between vectors of every bipartition coefficient matrix, our entanglement measure is easy to calculate.

Separability conditions based on local fine-grained uncertainty relations

“…Finally, we shortly mention applications of the separability conditions (3.8), (3.12), and (3.17) to multipartite systems. Various approaches to study multipartite entanglement are considered in [58][59][60][61][62]. In general, the problem becomes more complicated [58].…”

Entropy Fit Indices: New Fit Measures for Assessing the Structure and Dimensionality of Multiple Latent Variables

“…In the EFI the joint entropy is estimated directly using the raw data, but in the EFI.vn it's calculated using the Kronecker product of the individual factors (or the product of the individual eigenvalues per factor, as shown above). This strategy seems useful to detect superposition of states (in the present case the superposition is when two factors are merged into one) since this is a typical property of quantum information theory (Preskill, 2018;Zhao, Yang, Hung, & Li, 2015). However, if one pure state (one factor) is split into two, generating a plausible but incorrect structure, then the performance of the EFI.vn may not be very accurate.…”

Entanglement in Shor's Factoring Algorithm

“…Because the rank of it is an inherent physical property since it is invariant under SLOCC [15]. It is used to classify multiqubit states [15,16] and then a entanglement measure [11] is proposed based on it. Compare with other entanglement measures, this measure is easy to calculate and capable of dealing with quantum pure states with arbitrary dimensions and parties.…”

“…here xi(xj) represents the ith(jth) row vector of coefficient matrix, ||xi|| denotes the length of the vector xi, and |xi| is the module of xi. Then general way of characterizing the global amount of entanglement exhibited by an n-qubit state is provided by the sum of the bipartite entanglement measures associated with the possible bipartitions of the n-qubits system [10,11,17]. In this paper, the entanglement values we calculate are global entanglement.…”

“…In this paper, we investigate multipartite entanglement in Shor's factoring algorithm. The entanglement measure we used is the measure based based on coefficient matrix [11]. Differ from Ref.…”