Measurements and majorization

TL;DR

This paper explores how measurements affect the majorization relation among quantum states, providing new proofs and generalizations for the behavior of entangled states under local and global measurements.

Contribution

It offers a new proof of Nielsen and Vidal's result on spectrum majorization after POVM measurements and extends the characterization of LOCC transformations to n-party entanglement.

Findings

POVM measurements cannot decrease the expected spectrum, preserving information.

Unrecorded bi-stochastic measurements do not reduce the initial state's majorization.

Local measurements by one agent do not alter other agents' local states in n-party systems.

Abstract

Majorization is an outstanding tool to compare the purity of mixed states or the amount of information they contain and also the degrees of entanglement presented by such states in tensor products. States are compared by their spectra and majorization defines a partial order on those. This paper studies the effect of measurements on the majorization relation among states. It, then, proceeds to study the effect of local measurements on the agents sharing an entangled global state. If the result of the measurement is recorded, Nielsen and Vidal showed that the expected spectrum after any P.O.V.M. measurement majorizes the initial spectrum, i.e., a P.O.V.M. measurement cannot, in expectation, reduce the information of the observer. A new proof of this result is presented and, as a consequence, the only if part of Nielsen's characterization of LOCC transformations is generalized to n-party entanglement. If the result of a bi-stochastic measurement is not recorded, the initial state majorizes the final state, i.e., no information may be gained by such a measurement. This strengthens a result of A. Peres. In the n-party setting, no local trace preserving measurement by Alice can change the local state of another agent.