Local Records and Global Entanglement: A Unique Multi-Partite Generalization of the Schmidt Decomposition

TL;DR

This paper introduces a unique multi-partite state decomposition based on local orthogonality, leading to a new measure of global entanglement that generalizes the Schmidt decomposition and captures GHZ-like correlations.

Contribution

It presents a novel, unique decomposition for multi-partite quantum states and defines a new entanglement measure based on local orthogonal components.

Findings

Decomposition is unique for multi-partite states with N>2.

Defines a new global entanglement measure based on branch weights.

Reduces to Schmidt decomposition and entanglement entropy for bipartite states.

Abstract

We show that there is a unique maximal decomposition of a pure multi-partite (N>2) quantum state into a sum of states which are "locally orthogonal" in the sense that the local reduced state for a term in the sum lives in its own orthogonal subspace for each subsystem. Observers can make local measurements on any subsystem and determine which "branch" they are on. The Shannon entropy of the resulting branch weights defines a new measure of global, GHZ-like entanglement, which is insensitive to local pairwise entangling operations and vanishes when there is no piece of information recorded at every subsystem. In the bi-partite (N=2) case, this decomposition reduces to the (not necessarily unique) Schmidt decomposition and the entropy reduces to the entropy of entanglement