Details
| Original language | English |
|---|---|
| Article number | 093046 |
| Pages (from-to) | 093046 |
| Number of pages | 1 |
| Journal | New J. Phys. |
| Volume | 17 |
| Issue number | 9 |
| Publication status | Published - 25 Sept 2015 |
Abstract
In this work we study various notions of uncertainty for angular momentum in the spin-s representation of SU(2). We characterize the 'uncertainty regions' given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s. Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s, since they are saturated by states taken from the Holstein-Primakoff approximation. We show that, for all s, all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s. For large s the Maassen-Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length where as
Keywords
- angular momentum, quantum mechanics, uncertainty relations
ASJC Scopus subject areas
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In: New J. Phys., Vol. 17, No. 9, 093046, 25.09.2015, p. 093046.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Uncertainty Relations for Angular Momentum
AU - Dammeier, L.
AU - Schwonnek, R.
AU - Werner, R.F.
N1 - Publisher Copyright: © 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. Copyright: Copyright 2015 Elsevier B.V., All rights reserved.
PY - 2015/9/25
Y1 - 2015/9/25
N2 - In this work we study various notions of uncertainty for angular momentum in the spin-s representation of SU(2). We characterize the 'uncertainty regions' given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s. Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s, since they are saturated by states taken from the Holstein-Primakoff approximation. We show that, for all s, all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s. For large s the Maassen-Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length where as
AB - In this work we study various notions of uncertainty for angular momentum in the spin-s representation of SU(2). We characterize the 'uncertainty regions' given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s. Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s, since they are saturated by states taken from the Holstein-Primakoff approximation. We show that, for all s, all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s. For large s the Maassen-Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length where as
KW - angular momentum
KW - quantum mechanics
KW - uncertainty relations
UR - http://www.scopus.com/inward/record.url?scp=84943523705&partnerID=8YFLogxK
U2 - 10.1088/1367-2630/17/9/093046
DO - 10.1088/1367-2630/17/9/093046
M3 - Article
VL - 17
SP - 093046
JO - New J. Phys.
JF - New J. Phys.
SN - 1367-2630
IS - 9
M1 - 093046
ER -