Some properties of n-party entanglement under LOCC operations

TL;DR

This paper extends Nielsen's characterization of 2-party quantum protocols to n-party systems, showing local operations cannot decrease the expected spectra of local mixed states, but the full characterization does not generalize.

Contribution

It generalizes the 'only if' part of Nielsen's theorem to n-party protocols with mixed states, and demonstrates limitations of spectral properties in characterizing entanglement.

Findings

Local spectra can only become purer in expectation under LOCC.

The 'if' part of Nielsen's theorem does not extend to n-party systems.

Spectral properties of local mixed states do not fully characterize entanglement classes.

Abstract

Nielsen characterized in full those 2-party quantum protocols of local operations and classical communication that transform, with probability one, a pure global initial state into a pure global final state. The present work considers the generalization of Nielsen's characterization to n-party protocols. It presents a sweeping generalization of the only if part of Nielsen's result. The result presented here pertains also to protocols that do not generate a final state for sure, it considers arbitrary mixed initial states instead of pure states and n-party protocols for an arbitrary n. In this very general setting, local operations and classical communication can never decrease the expected spectra of the local mixed states in the majorization ordering. In other terms, the local states can only become purer (weakly) in expectation. The proof also provides an improvement on Nielsen's. The if part of Nielsen's characterization does not generalize. This is shown by studying the entanglement of three qubits. It is shown that one can find pure states of a system of three qubits that are not equivalent under unitary local operations but define local mixed states on all subparts of the system that have the same spectra. Neither equivalence of pure states under local unitary operations nor accessibility under LOCC operations among a system of three qubits can be characterized by properties of the spectra of the local mixed states.