Connections of geometric measure of entanglement of pure symmetric states to quantum state estimation

TL;DR

This paper explores the geometric measure of entanglement for pure symmetric states, linking it to quantum state estimation, and provides methods for calculating and characterizing this entanglement in various symmetric states, including experimental considerations.

Contribution

It establishes a connection between geometric entanglement and quantum state estimation, introduces a method for computation, and derives analytical formulas for specific symmetric states.

Findings

Proved additivity of GM for certain symmetric states.

Derived analytical formulas for GM of states from MUBs and SIC-POVMs.

Proposed experimental schemes for state creation and measurement.

Abstract

We study the geometric measure of entanglement (GM) of pure symmetric states related to rank-one positive-operator-valued measures (POVMs) and establish a general connection with quantum state estimation theory, especially the maximum likelihood principle. Based on this connection, we provide a method for computing the GM of these states and demonstrate its additivity property under certain conditions. In particular, we prove the additivity of the GM of pure symmetric multiqubit states whose Majorana points under Majorana representation are distributed within a half sphere, including all pure symmetric three-qubit states. We then introduce a family of symmetric states that are generated from mutually unbiased bases (MUBs), and derive an analytical formula for their GM. These states include Dicke states as special cases, which have already been realized in experiments. We also derive the GM of symmetric states generated from symmetric informationally complete POVMs (SIC~POVMs) and use it to characterize all inequivalent SIC~POVMs in three-dimensional Hilbert space that are covariant with respect to the Heisenberg--Weyl group. Finally, we describe an experimental scheme for creating the symmetric multiqubit states studied in this article and a possible scheme for measuring the permanent of the related Gram matrix.