Details
| Original language | English |
|---|---|
| Article number | 38 |
| Pages (from-to) | 8 |
| Number of pages | 1 |
| Journal | Mathematics |
| Volume | 4 |
| Issue number | 2 |
| Publication status | Published - 1 Jun 2016 |
Abstract
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.
Keywords
- Error-disturbance tradeoff, Measurement uncertainty, Optimal transport, Semidefinite programming, Uncertainty relations
ASJC Scopus subject areas
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In: Mathematics, Vol. 4, No. 2, 38, 01.06.2016, p. 8.
Research output: Contribution to journal › Article › Research › peer review
}
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 -