Measurements and majorization

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.