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Journal of Optics

Entropic uncertainty minimum for angle and angular momentum

Alison M. Yao, Thomas Brougham, Electra Eleftheriadou, Miles J. Padgett, Stephen Mark Barnett

Published Sep 23, 2014• 1 citation
Publisher

Abstract

Uncertainty relations are key components in the understanding of the nature of quantum mechanics. In particular, entropic relations are preferred in the study of angular position and angular momentum states. We propose a new form of angle–angular momentum state that provides, for all practical purposes, a lower bound on the entropic uncertainty relation, , for any given angular uncertainty, thus improving upon previous bounds. We establish this by comparing this sum with the absolute minimum value determined by a global numerical search. These states are convenient to work with both analytically and experimentally, which suggests that they may be of use for quantum information purposes.

DOI
10.1088/2040-8978/16/10/105404

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Uncertainty relations are key components in the understanding of the nature of quantum mechanics. In particular, entropic relations are preferred in the study of angular position and angular momentum states. We propose a new form of angle–angular momentum state that provides, for all practical purposes, a lower bound on the entropic uncertainty relation, , for any given angular uncertainty, thus improving upon previous bounds. We establish this by comparing this sum with the absolute minimum value determined by a global numerical search. These states are convenient to work with both analytically and experimentally, which suggests that they may be of use for quantum information purposes.
The states of linear momentum that satisfy the equality in the Heisenberg uncertainty principle for position and momentum, that is the intelligent states, are also the states that minimize the uncertainty product for position and momentum. The corresponding uncertainty relation for angular momentum and angular position, however, is more complicated and the intelligent states need not be the constrained minimum uncertainty product states. In this paper, we investigate the differences between the intelligent and the constrained minimum uncertainty product states for the angular case by means of instructive approximations, a numerical iterative search and the exact solution. We find that these differences can be quite significant for particular values of angular position uncertainty and indeed may be amenable to experimental measurement with the present technology.
The equality in the uncertainty principle for linear momentum and position is obtained for states which also minimize the uncertainty product. However, in the uncertainty relation for angular momentum and angular position both sides of the inequality are state dependent and therefore the intelligent states, which satisfy the equality, do not necessarily give a minimum for the uncertainty product. In this paper, we highlight the difference between intelligent states and minimum uncertainty states by investigating a class of intelligent states which obey the equality in the angular uncertainty relation while having an arbitrarily large uncertainty product. To develop an understanding for the uncertainties of angle and angular momentum for the large-uncertainty intelligent states we compare exact solutions with analytical approximations in two limiting cases.
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 s ( s + 1 ) . 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 r ( s ) , where r ( s ) / s → 1 as s → ∞ .
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.