An Uncertainty Relation for the Orbital Angular Momentum Operator

Fakhri, H.; Sayyah-Fard, M.

doi:10.1007/s10701-016-9988-8

Cited by 5 publications

(1 citation statement)

References 24 publications

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“…In conclusion, we have presented a class of stateindependent tight SURs based on algebraic properties, and our scheme shows how to generalize to other algebras. Inequalities (3) and ( 6) were known previously [19,[44][45][46][53][54][55] as is (implicitly) inequality (4) [48], but bounds were not explicitly stated nor was their common algebraic origin from a similar derivation. Furthermore the state-independent nature of the tight lower bound was not investigated.…”

Section: −24mentioning

confidence: 99%

State-independent uncertainty relations

et al. 2018

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The standard state-dependent Heisenberg-Robertson uncertainly-relation lower bound fails to capture the quintessential incompatibility of observables as the bound can be zero for some states. To remedy this problem, we establish a class of tight (i.e., inequalities are saturated) variance-based sum-uncertainty relations derived from the Lie algebraic properties of observables and show that our lower bounds depend only on the irreducible representation assumed carried by the Hilbert space of state of the system. We illustrate our result for the cases of the Weyl-Heisenberg algebra, special unitary algebras up to rank 4, and any semisimple compact algebra. We also prove the usefulness of our results by extending a known variance-based entanglement detection criterion.For ∆w 2 signifying the variance of measurement outcomes for the observable w, Heisenberg's uncertainty relation for position x and momentum p iswhere [x, p] = i1, and 1 is the identity operator. Eq. (1) fortuitously has a constant lower bound due to the appealing algebraic properties of the commutator of x and p. Robertson's generalization to ∆A 2 ∆B 2 ≥ | [A, B] | 2 /4 for arbitrary observables A and B more generally has a state-dependent lower bound [1], and so fails to capture the intrinsic incompatibility of noncommuting observables [2,3]. This cannot be amended as the underlying product of uncertainties is null whenever one of the uncertainties is null, an observation that provided impetus for the emergence of uncertainty relations [4-9] that eschew variance in favor of entropy. Properly assessing uncertainty is important for foundational quantum mechanics [10][11][12] and for quantum information and communication [13][14][15][16]; variance is closer than entropy for practical quantum mechanics, a driving motivation behind research into sum-uncertainty relations (SURs), which deliver state-independent lower bounds [17][18][19][20][21][22][23][24][25]. Here we discuss SURs by showing connections with the algebras of observables, with examples of the Weyl-Heisenberg wh, special unitary su(n) and su(1, 1) and generally semi-simple compact algebras thereby extending the range to applications of SURs in areas such as [26,27] and quantum information [28] where u(n) or su(n) symmetries are prevalent.Indeed, single-photon multi-path quantum optical interferometry provides a convenient way to realize SU (n) symmetry [29][30][31][32] with the experimental signature obtained via sampling photodetection of the photon emerging from each of the n output ports, both by direct detection and by adding special post-processing interferometers at the output followed by photodetection. Photodetection sampling statistics obtained in these ways yield uncertainties from estimates second-order cumulants for the distributions and, through this process, our uncer-(a) (b) FIG. 1: Sum of variances is a measure of total uncertainty. Given a (green) box with the uncertainties as edges, the sum of variances is the squared length of the (red) diagonal. [Here 30000 (blue) points ...

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Section: −24mentioning

confidence: 99%

State-independent uncertainty relations

et al. 2018

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q-coherent states associated with the noncommutative complex plane Cq2
Fakhri
¹
,
Sayyah-Fard
²

2017
Annals of Physics
10
0
0
0
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sl(2)-modules by sl(2)-coherent states

Fakhri

Sayyah-Fard

2016

Journal of Mathematical Physics

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Irreducible sp(4)-module with highest weight, labeled by the azimuthal and magnetic quantum numbers l and m, is split into the direct sums of the irreducible su(2)- and su(1, 1)-submodules in four different ways: finite integer unitary irreducible subspaces corresponding to the orbital angular momentum algebra su(2), infinite positive discrete series of su(1, 1) with an arbitrary half-integer Bargmann index, and the positive and negative discrete series of su(1, 1) with both the Bargmann indices 1/4 and 3/4. Even and odd coherent states for the positive su(1, 1)-submodules with the Bargmann indices 1/4 and 3/4 are constructed and it is shown that they enjoy the property of completeness by two appropriate positive definite measures. We show that the even and odd coherent states themselves form the positive discrete series of su(1, 1) with the Bargmann indices 1/4 and 3/4, respectively. For these even and odd coherent states, we consider the uncertainty relations for the x- and y-components of the angular momentum as well as the generators of the negative discrete series of su(1, 1) with the Bargmann indices 1/4 and 3/4.

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An Uncertainty Relation for the Orbital Angular Momentum Operator

Cited by 5 publications

References 24 publications

State-independent uncertainty relations

State-independent uncertainty relations

q-coherent states associated with the noncommutative complex plane Cq2
Fakhri
¹
,
Sayyah-Fard
²

2017
Annals of Physics
10
0
0
0
View full text Add to dashboard Cite

sl(2)-modules by sl(2)-coherent states

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An Uncertainty Relation for the Orbital Angular Momentum Operator

Cited by 5 publications

References 24 publications

State-independent uncertainty relations

State-independent uncertainty relations

q-coherent states associated with the noncommutative complex plane Cq2Fakhri1, Sayyah-Fard2 2017Annals of Physics10000View full textAdd to dashboardCite

sl(2)-modules by sl(2)-coherent states

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Resources

About

q-coherent states associated with the noncommutative complex plane Cq2
Fakhri
¹
,
Sayyah-Fard
²

2017
Annals of Physics
10
0
0
0
View full text Add to dashboard Cite