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Acta Physica Sinica

The uncertainty relations and minimum uncertainty states

Wenji Deng, Xu Yun-Hua, Liu Ping, (1)华南理工大学材料科学研究所，广州 510640; (2)华南理工大学物理系，广州 510640

Published Jan 1, 2003• 1 citation
Publisher

Abstract

Based upon some basic properties of Hermitian operators and Hilbert state vector s, the Heisenberg's uncertainty relations are analyzed in details. The necessary and sufficient condition for the minimum uncertainty states is re-obtained dire ctly from our proof. We propose also the problem of squeezed states for any coup le of dynamical variables, and solve it by using the bosonic creation and annihi lation operators.

DOI
10.7498/aps.52.2961

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Based upon some basic properties of Hermitian operators and Hilbert state vector s, the Heisenberg's uncertainty relations are analyzed in details. The necessary and sufficient condition for the minimum uncertainty states is re-obtained dire ctly from our proof. We propose also the problem of squeezed states for any coup le of dynamical variables, and solve it by using the bosonic creation and annihi lation operators.
Based upon the Heisenberger's uncertainty relations for mixed states, the minimu muncertainty and squeezing effects are studies. Although all minimum uncertainty states must be pure states, the nucertainty of one dynamical variable can be squeezed in some pure states or mixed states, even if they are not minimum uncert ainty states for the pair of dynamical variables. In addition, we present the ge neral uncertainty relation of statistics, which has the similar mathmatical stru cture of Heisenberg's indeterminacy principle, but has no relation with quantum interfernce.
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.
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.
Minimum uncertainty states of the conventional Heisenberg uncertainty relation have been extensively studied and are often regarded as the most classical quantum states from the perspective of uncertainty, providing valuable insight into the nature of quantumness and its potential applications. In this work, we investigate the minimum uncertainty states associated with an information-theoretic refinement of the Heisenberg uncertainty relation in general spin systems. Using two different approaches, the matrix formulation and the Wick symbol representation, we derive explicit expressions for the states that saturate the uncertainty bound. We show that spin coherent states indeed achieve minimum uncertainty, consistent with their conventional identification as the classical states of spin systems. Moreover, we also identify additional classes of minimum uncertainty states beyond the coherent family. Finally, we compare the spin-system results with the previously studied bosonic case and elucidate the origin of the differences between the two settings.
An introductory survey on the Schroedinger uncertainty relation and its minimization states is presented with minimal number of formulas and some historical points. The case of the two canonical observables, position and momentum, is discussed in greater detail: basic properties of the two subsets of minimization states (canonical squeezed and coherent states) are reviewed and compared. The case of two non-canonical observables is breafly outlined. Stanfard SU(1,1) and SU(2) group-related coherent states can be defined as states that minimize Schroedinger inequality for the three pairs of generators simultaneously. The symmetry of the Heisenberg and Schroedinger relations is also discussed, and two natural generalizations to the cases of several observables and several states are noted.