TY - JOUR

T1 - Measurement Uncertainty for Finite Quantum Observables

AU - Schwonnek, R.

AU - Reeb, D.

AU - Werner, R. F.

N1 - Publisher Copyright: © 2016 by the authors. Copyright: Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - Measurement uncertainty relations are lower bounds on the errors of any approximate joint measurement of two or more quantum observables. The aim of this paper is to provide methods to compute optimal bounds of this type. The basic method is semidefinite programming, which we apply to arbitrary finite collections of projective observables on a finite dimensional Hilbert space. The quantification of errors is based on an arbitrary cost function, which assigns a penalty to getting result x rather than y, for any pair (x, y). This induces a notion of optimal transport cost for a pair of probability distributions, and we include an Appendix with a short summary of optimal transport theory as needed in our context. There are then different ways to form an overall figure of merit from the comparison of distributions. We consider three, which are related to different physical testing scenarios. The most thorough test compares the transport distances between the marginals of a joint measurement and the reference observables for every input state. Less demanding is a test just on the states for which a "true value" is known in the sense that the reference observable yields a definite outcome. Finally, we can measure a deviation as a single expectation value by comparing the two observables on the two parts of a maximally-entangled state. All three error quantities have the property that they vanish if and only if the tested observable is equal to the reference. The theory is illustrated with some characteristic examples.

AB - Measurement uncertainty relations are lower bounds on the errors of any approximate joint measurement of two or more quantum observables. The aim of this paper is to provide methods to compute optimal bounds of this type. The basic method is semidefinite programming, which we apply to arbitrary finite collections of projective observables on a finite dimensional Hilbert space. The quantification of errors is based on an arbitrary cost function, which assigns a penalty to getting result x rather than y, for any pair (x, y). This induces a notion of optimal transport cost for a pair of probability distributions, and we include an Appendix with a short summary of optimal transport theory as needed in our context. There are then different ways to form an overall figure of merit from the comparison of distributions. We consider three, which are related to different physical testing scenarios. The most thorough test compares the transport distances between the marginals of a joint measurement and the reference observables for every input state. Less demanding is a test just on the states for which a "true value" is known in the sense that the reference observable yields a definite outcome. Finally, we can measure a deviation as a single expectation value by comparing the two observables on the two parts of a maximally-entangled state. All three error quantities have the property that they vanish if and only if the tested observable is equal to the reference. The theory is illustrated with some characteristic examples.

KW - Error-disturbance tradeoff

KW - Measurement uncertainty

KW - Optimal transport

KW - Semidefinite programming

KW - Uncertainty relations

UR - http://www.scopus.com/inward/record.url?scp=85006127032&partnerID=8YFLogxK

U2 - 10.3390/math4020038

DO - 10.3390/math4020038

M3 - Article

VL - 4

SP - 8

JO - Mathematics

JF - Mathematics

IS - 2

M1 - 38

ER -