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Physical Review A

Minimum-uncertainty states for noncanonical operators

Ravinder Rupchand Puri

Published Mar 1, 1994• 35 citations
Publisher

Abstract

The usual uncertainty relation \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$\ensuremath{\ge}〈C${\mathrm{〉}}^{2}$/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}${\mathit{A}}^{2}$ and \ensuremath{\Delta}${\mathit{B}}^{2}$ when 〈C〉=0. For those operators, therefore, the general uncertainty relation \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$\ensuremath{\ge}[〈C${\mathrm{〉}}^{2}$+〈F${\mathrm{〉}}^{2}$]/4 where 〈F〉=〈AB+BA〉-2〈A〉〈B〉 is better suited to determine the lower bound on \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$. 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.

DOI
10.1103/physreva.49.2178

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Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables A and B and such vectors that the lower bound for the product of standard deviations ΔA and ΔB calculated for these vectors is zero: ΔA·ΔB≥0. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices (2×2) and (3×3) and the position–momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in PT–symmetric quantum theory and the problems associated with it are also studied.
We derive several uncertainty relations for two arbitrary unitary operators acting on physical states of a Hilbert space. We show that our bounds are tighter in various cases than the ones existing in the current literature. Using the uncertainty relation for the unitary operators, we obtain the tight state-independent lower bound for the uncertainty of two Pauli observables and anticommuting observables in higher dimensions. With regard to the minimum-uncertainty states, we derive the minimum-uncertainty state equation by the analytic method and relate this to the ground-state problem of the Harper Hamiltonian. Furthermore, the higher-dimensional limit of the uncertainty relations and minimum-uncertainty states are explored. From an operational point of view, we show that the uncertainty in the unitary operator is directly related to the visibility of quantum interference in an interferometer where one arm of the interferometer is affected by a unitary operator. This shows a principle of preparation uncertainty, i.e., for any quantum system, the amount of visibility for two general noncommuting unitary operators is nontrivially upper bounded.
The uncertainty relations (URs) of two arbitrary Hermitian and non-Hermitian incompatible operators represented by the product of variances have been confirmed theoretically and experimentally in various physical systems. However, the lower bound of the product uncertainty inequality can be null even for two noncommuting operators, i.e., a trivial case. Therefore, for two incompatible operators over the measured system state, the associated URs regarding the sum of variances are valid in a state-dependent manner, and the lower bound is guaranteed to be nontrivial. Although the sum URs formulated for Hermitian and unitary operators have been affirmed, the general forms for arbitrary non-Hermitian operators have not yet been investigated. This study presents the sum URs for non-Hermitian operators acting on system states using an appropriate Hilbert-space metric. The compatible forms of our sum inequalities with the conventional quantum mechanics are also provided via the GGG-metric formalism. Concrete examples illustrate the validity of the proposed sum URs in both PT\mathcal{PT}PT-symmetric and PT\mathcal{PT}PT-broken phases. The developed methods and results can help give an in-depth understanding of the usefulness of GGG-metric formalism in non-Hermitian quantum mechanics and the sum URs of incompatible operators within.
The Heisenberg-Robertson uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial; i.e., the lower bound can be null even for two noncompatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.
The Heisenberg–Robertson (HR) and Robertson–Schrödinger (RS) uncertainty relations are studied. Analyzing the RS uncertainty relation we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of noncommuting observables, A and B, is zero, and which differ from those described in the literature for the HR uncertainty relation. These states are not eigenstates of either the observable A or B. The correlation function for these observables in such states is equal to zero. We have also shown that the so-called “sum uncertainty relations” also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: One is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the noncommuting observables in the state under consideration.