arXiv:1202.3329

Measurements and majorization

Daniel J. Lehmann

Published Feb 15, 2012• 0 citations

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.

DOI: 10.48550/arxiv.1202.3329

Related papers

Some properties of n-party entanglement under LOCC operations

Daniel J. Lehmann

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.

arXiv:1202.3322Feb 15, 20120 citations

The relation between majorization theory and quantum information from entanglement monotones perspective

Volkan Erol

Entanglement has been studied extensively for understanding the mysteries of non-classical correlations between quantum systems. In the bipartite case, there are well known monotones for quantifying entanglement such as concurrence, relative entropy of entanglement (REE) and negativity, which cannot be increased via local operations. The study on these monotones has been a hot topic in quantum information [1-7] in order to understand the role of entanglement in this discipline. It can be observed that from any arbitrary quantum pure state a mixed state can obtained. A natural generalization of this observation would be to consider local operations classical communication (LOCC) transformations between general pure states of two parties. Although this question is a little more difficult, a complete solution has been developed using the mathematical framework of the majorization theory [8]. In this work, we analyze the relation between entanglement monotones concurrence and negativity with respect to majorization for general two-level quantum systems of two particles.

AIP conference proceedingsJan 1, 20169 citations

Quantification and Conversion of Entanglement

Reinhold A. Bertlmann, Nicolai Friis

Abstract In this chapter the quantification of entanglement is discussed. Beginning with the pure-state entanglement measure called the entropy of entanglement, we discuss the paradigm of local operations and classical communication (LOCC) and its relation to majorization via Nielsen’s majorization theorem. We then turn to the asymptotic setting and discuss the formation and distillation of entanglement and the related entanglement measures: entanglement cost and distillable entanglement, and we examine the notions of distillability and bound entanglement. This brings us to a more general discussion of entanglement measures and monotones, and their desired properties, during which we present the entanglement of formation and concurrence, squashed entanglement, as well as the tangle and its relation to monogamy of entanglement, but also quantities like the relative entropy of entanglement, the Hilbert-Schmidt measure, and the (logarithmic) negativity. Finally we turn to the construction of entanglement witnesses and their geometric interpretation via the Bertlmann-Narnhofer-Thirring theorem

Sep 7, 20230 citations

Classical Correlations and Entanglement in Quantum Measurements

Vlatko Vedral

We analyze a quantum measurement where the apparatus is initially in a mixed state. We show that the amount of information gained in a measurement is not equal to the amount of entanglement between the system and the apparatus, but is instead equal to the degree of classical correlations between the two. As a consequence, we derive an uncertainty-like expression relating the information gain in the measurement and the initial mixedness of the apparatus. Final entanglement between the environment and the apparatus is also shown to be relevant for the efficiency of the measurement.

arXiv:quant-ph/0207116Physical Review LettersFeb 3, 2003289 citations

Complementary relationships between entanglement and measurement

Michael Steiner, R. W. Rendell

Complementary relationships exist among interference properties of particles such as pattern visibility, predictability, and distinguishability. Additionally relationships between average information gain Ḡ and measurement disturbance F for entangled spin pairs are well established. This article examines whether a similar complementary relationship exists between entanglement and measurement. For qubit systems, both measurements on a single system and measurements on a bipartite system are considered in regard to entanglement. It is proven that Ē + D ≤ 1 holds, where Ē is the average entanglement after a measurement is made and D is a measure of the measurement disturbance of a single measurement. Assuming measurements on a bipartite system shared by Alice and Bob, it is shown that Ē + Ḡ ≤ 1, where Ḡ is the maximum average information gain that Bob can obtain regarding Alice’s result. These results are generalized to arbitrary initial mixed states and non-Hermitian operators. In the case of maximally entangled initial states, it is found that D ≤ EL and Ḡ ≤ EL, where EL is the loss of entanglement due to measurement by Alice. We conclude that the amount of disturbance and average information gain one can achieve is strictly limited by entanglement.

arXiv:2401.17537Academia quantum.Aug 29, 20242 citations

Entanglement measures: state ordering vs local operations

Mário Ziman, Vladimír Bužek

A set of all states of a bi-partite quantum system can be divided into subsets each of which contains states with the same degree of entanglement. In this paper we address a question whether local operations (without classical communication) affect the entanglement-induced state ordering. We show that arbitrary unilocal channel (i.e., a channel that acts on one sub-system of a bi-partite system only) might change the ordering for an arbitrary nontrivial measure of entanglement. A slightly weaker result holds for the maximally entangled states. In particular, the maximally entangled states might not remain the most entangled ones at the output of a unilocal noise channel.

arXiv:0707.4401Jul 30, 20072 citations