Quantifying entanglement of arbitrary-dimensional multipartite pure states in terms of the singular values of coefficient matrices

TL;DR

This paper introduces a new entanglement measure for multipartite pure states based on the singular values of coefficient matrices, linking entanglement quantification to matrix analysis.

Contribution

It establishes the Manhattan distance of averaged partial entropies (MAPE) as an entanglement measure and connects it with the singular values of coefficient matrices for arbitrary-dimensional states.

Findings

MAPE is a valid entanglement measure for pure states.

The relation between coefficient matrix rank and entanglement degree is demonstrated.

Entanglement properties of Dicke, GHZ, and D3^n states are analyzed.

Abstract

The entanglement quantification and classification of multipartite quantum states are two important research fields in quantum information. In this work, we study the entanglement of arbitrary-dimensional multipartite pure states by looking at the averaged partial entropies of various bipartite partitions of the system, namely, the so-called Manhattan distance ($l_1$ norm) of averaged partial entropies (MAPE), and it is proved to be an entanglement measure for pure states. We connected the MAPE with the coefficient matrices, which are important tools in entanglement classification and reexpressed the MAPE for arbitrary-dimensional multipartite pure states by the nonzero singular values of the coefficient matrices. The entanglement properties of the $n$-qubit Dicke states, arbitrary-dimensional Greenberger-Horne-Zeilinger states, and $D_3^n$ states are investigated in terms of the MAPE, and the relation between the rank of the coefficient matrix and the degree of entanglement is demonstrated for symmetric states by two examples.