Sharp uncertainty relations for number and angle

Paul Busch; Jukka Kiukas; Reinhard F Werner

doi:10.1063/1.5030101

Sharp uncertainty relations for number and angle

Paul Busch, Jukka Kiukas, Reinhard F Werner

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study uncertainty relations for pairs of conjugate variables like number and angle, of which one takes integer values and the other takes values on the unit circle. The translation symmetry of the problem in either variable implies that measurement uncertainty and preparation uncertainty coincide quantitatively, and the bounds depend only on the choice of two metrics used to quantify the difference of number and angle outputs, respectively. For each type of observable, we discuss two natural choices of metric and discuss the resulting optimal bounds with both numerical and analytical methods. We also develop some simple and explicit (albeit not sharp) lower bounds, using an apparently new method for obtaining certified lower bounds to ground state problems.

Original language	English
Article number	042102
Number of pages	16
Journal	Journal of Mathematical Physics
Volume	59
Issue number	4
Early online date	5 Apr 2018
DOIs	https://doi.org/10.1063/1.5030101
Publication status	Published - Apr 2018

Bibliographical note

Published by AIP Publishing. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

Keywords

quantum mechanics
uncertainty relations

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10.1063/1.5030101

1604.00566v2
Published by AIP Publishing. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details
Accepted author manuscript, 784 KBLicence: Unspecified

https://arxiv.org/abs/1604.00566

Cite this

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title = "Sharp uncertainty relations for number and angle",

abstract = "We study uncertainty relations for pairs of conjugate variables like number and angle, of which one takes integer values and the other takes values on the unit circle. The translation symmetry of the problem in either variable implies that measurement uncertainty and preparation uncertainty coincide quantitatively, and the bounds depend only on the choice of two metrics used to quantify the difference of number and angle outputs, respectively. For each type of observable, we discuss two natural choices of metric and discuss the resulting optimal bounds with both numerical and analytical methods. We also develop some simple and explicit (albeit not sharp) lower bounds, using an apparently new method for obtaining certified lower bounds to ground state problems.",

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AU - Werner, Reinhard F

N1 - Published by AIP Publishing. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

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N2 - We study uncertainty relations for pairs of conjugate variables like number and angle, of which one takes integer values and the other takes values on the unit circle. The translation symmetry of the problem in either variable implies that measurement uncertainty and preparation uncertainty coincide quantitatively, and the bounds depend only on the choice of two metrics used to quantify the difference of number and angle outputs, respectively. For each type of observable, we discuss two natural choices of metric and discuss the resulting optimal bounds with both numerical and analytical methods. We also develop some simple and explicit (albeit not sharp) lower bounds, using an apparently new method for obtaining certified lower bounds to ground state problems.

AB - We study uncertainty relations for pairs of conjugate variables like number and angle, of which one takes integer values and the other takes values on the unit circle. The translation symmetry of the problem in either variable implies that measurement uncertainty and preparation uncertainty coincide quantitatively, and the bounds depend only on the choice of two metrics used to quantify the difference of number and angle outputs, respectively. For each type of observable, we discuss two natural choices of metric and discuss the resulting optimal bounds with both numerical and analytical methods. We also develop some simple and explicit (albeit not sharp) lower bounds, using an apparently new method for obtaining certified lower bounds to ground state problems.

KW - quantum mechanics

KW - uncertainty relations

UR - https://arxiv.org/abs/1604.00566

U2 - 10.1063/1.5030101

DO - 10.1063/1.5030101

M3 - Article

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JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 4

M1 - 042102

ER -

Sharp uncertainty relations for number and angle

Abstract

Bibliographical note

Keywords

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