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IEEE International Conference on Acoustics Speech and Signal Processing

On operators providing shift-invariant uncertainty limit

Milos I. Doroslovacki

Published Dec 24, 2002• 1 citation
Publisher

Abstract

The uncertainty relation is introduced by using operators which are not necessarily self-adjoint. In general, the uncertainty (lower) limit is different for different functions. Operators are found that define the second moments and the uncertainty limit for their product such that the limit is the same for functions obtained from each other by applying unitary support-size preserving operators, e.g. by shifting in time and frequency. In this case, for all functions of finite energy the limit is related to the minimum support size in the time-frequency domain. Moreover, it is possible to find the discrete-time counterpart of the classical uncertainty relation for continuous-time functions. The operator calculus presents results from signal analysis in a more standard form extensively used in quantum mechanics.

DOI
10.1109/icassp.1996.550141

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The uncertainty relation is introduced by using operators which are not necessarily self-adjoint. In general, the uncertainty (lower) limit is different for different functions. Operators are found that define the second moments and the uncertainty limit for their product such that the limit is the same for functions obtained from each other by applying unitary support-size preserving operators, e.g. by shifting in time and frequency. In this case, for all functions of finite energy the limit is related to the minimum support size in the time-frequency domain. Moreover, it is possible to find the discrete-time counterpart of the classical uncertainty relation for continuous-time functions. The operator calculus presents results from signal analysis in a more standard form extensively used in quantum mechanics.
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Abstract Uncertainty relations quantify the joint localisation in time respectively space domain and Fourier domain. For that reason they are an important tool in the design of wavelets. Uncertainty relations in higher dimension are most often realized as the tensor product of one dimensional uncertainty relations, whence these uncertainty relations are not invariant under rotations. The property of invariance with respect to rotation is however very desirable for image processing. We will give a new uncertainty relation for sets of self‐adjoint operators that yields the desired invariance in many examples. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We derive an uncertainty relation for two unitary operators which obey a commutation relation of the form UV=e(i phi) VU. Its most important application is to constrain how much a quantum state can be localized simultaneously in two mutually unbiased bases related by a discrete fourier transform. It provides an uncertainty relation which smoothly interpolates between the well-known cases of the Pauli operators in two dimensions and the continuous variables position and momentum. This work also provides an uncertainty relation for modular variables, and could find applications in signal processing. In the finite dimensional case the minimum uncertainty states, discrete analogues of coherent and squeezed states, are minimum energy solutions of Harper's equation, a discrete version of the harmonic oscillator equation.
The usual uncertainty relation \ensuremath{\Delta}A2{\mathit{A}}^{2}A2\ensuremath{\Delta}B2{\mathit{B}}^{2}B2\ensuremath{\ge}〈C〉2{\mathrm{〉}}^{2}〉2/4 between two Hermitian operators A and B satisfying the noncanonical commutation relation [A,B]=iC, where C is not a constant multiple of the unit operator, fails to give a nontrivial lower bound on the product of the variances \ensuremath{\Delta}A2{\mathit{A}}^{2}A2 and \ensuremath{\Delta}B2{\mathit{B}}^{2}B2 when 〈C〉=0. For those operators, therefore, the general uncertainty relation \ensuremath{\Delta}A2{\mathit{A}}^{2}A2\ensuremath{\Delta}B2{\mathit{B}}^{2}B2\ensuremath{\ge}[〈C〉2{\mathrm{〉}}^{2}〉2+〈F〉2{\mathrm{〉}}^{2}〉2]/4 where 〈F〉=〈AB+BA〉-2〈A〉〈B〉 is better suited to determine the lower bound on \ensuremath{\Delta}A2{\mathit{A}}^{2}A2\ensuremath{\Delta}B2{\mathit{B}}^{2}B2. The implications of the general uncertainty relation and the properties of the minimum-uncertainty states, i.e., the states for which the general uncertainty relation is satisfied with equality, are discussed. The minimum-uncertainty states are found to fall in two different classes. One is the usually studied class for which 〈F〉=0, that is, the case when the usual uncertainty relation holds. The other is the hitherto unnoticed class for 〈C〉=0, that is, when the usual uncertainty relation is redundant. The squeezing properties of the present class of minimum-uncertainty states is discussed by defining squeezing in the light of the general uncertainty relation.
Abstract In this paper we discuss some aspects of the Heisenberg uncertainty relation, mostly from the point of view of non self-adjoint operators. Some equivalence results, and some refinements of the inequality, are deduced, and some relevant examples are discussed. We also begin a sort of dynamical analysis of the relation, in connection with what has been recently called γ-dynamics and γ-symmetries, and we discuss in some details the role of different scalar products in our analysis. The case of self-adjoint operators is recovered as a special case of our general settings.