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arXiv:quant-ph/9809026 · Journal of Physics A Mathematical and General

Characteristic uncertainty relations

D. A. Trifonov, Stoil Donev

Published Oct 2, 1998• 24 citations
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Abstract

New uncertainty relations for n observables are established. The relations take the invariant form of inequalities between the characteristic coefficients of order r , , of the uncertainty matrix and the matrix of mean commutators of the observables. It is shown that the second- and third-order characteristic inequalities for the three generators of and SU (2) are minimized in the corresponding group-related coherent states with maximal symmetry.

DOI
10.1088/0305-4470/31/39/016

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New uncertainty relations for n observables are established. The relations take the invariant form of inequalities between the characteristic coefficients of order r , , of the uncertainty matrix and the matrix of mean commutators of the observables. It is shown that the second- and third-order characteristic inequalities for the three generators of and SU (2) are minimized in the corresponding group-related coherent states with maximal symmetry.
Characteristic uncertainty relations and their related squeezed states are briefly reviewed and compared in accordance with the generalizations of three equivalent definitions of canonical coherent states. The standard SU(1, 1) coherent states are shown to be the unique states that minimize the Schrödinger uncertainty relation for every pair of the three generators and the Robertson relation for the three generators. The characteristic uncertainty inequalities are naturally extended to the case of several states. It is shown that these inequalities can be written in the equivalent complementary form.
Using the non-negativity of Gram determinants of arbitrary order, we derive higher-order uncertainty relations for the symmetric uncertainty matrices of corresponding order n > 2 to n Hermitean operators (n = 2 is the usual case). The special cases of third-order and fourth-order uncertainty relations are considered in detail. The obtained third-order uncertainty relations are applied to the Lie groups SU(1,1) with three Hermitean basis operators (K 1,K 2,K 0) and SU(2) with three Hermitean basis operators (J 1,J 2,J 3) where, in particular, the group-coherent states of Perelomov type and of Barut–Girardello type for SU(1,1) and the spin or atomic coherent states for SU(2) are investigated. The uncertainty relations for the determinant of the third-order uncertainty matrix are satisfied with the equality sign for coherent states and this determinant becomes vanishing for the Perelomov type of coherent states for SU(1,1) and SU(2). As an example of the application of fourth-order uncertainty relations, we consider the canonical operators (Q 1,P 1,Q 2,P 2) of two boson modes and the corresponding uncertainty matrix formed by the operators of the corresponding mean deviations, taking into account the correlations between the two modes. In two mathematical appendices, we prove the non-negativity of the determinant of correlation matrices of arbitrary order and clarify the principal structure of higher-order uncertainty relations.
We analyze the Schwarz inequality and its generalizations, as well as inequalities resulting from the Jensen inequality. They are used in quantum theory to derive the Heisenberg-Robertson (HR) and Schroedinger-Robertson (SR) uncertainty relation for two non-commuting observables and their generalizations to three or more non-commuting observables. Jensen's inequality, in turn, is helpful in deriving various the "sum uncertainty relations" for two or more observables. Using these inequalities, we derive various types of generalized uncertainty relations for more than two non--commuting observables and analyze their properties and critical points. We also study the connections between the generalizations of the HR and SR uncertainty relations for two and more observables and the correlations of these observables in the state of the quantum system under study. In this analysis, we pay special attention to the consequences of the generalized SR uncertainty relation for three non--commuting observables on their correlations in a given state of a quantum system and to the connections of this relation with the appropriate correlation matrix, whose matrix elements are the quantum versions of the Pearson coefficient. We show also that the SR uncertainty relation (including the generalized ones) can be written in an equivalent way using these Pearson coefficients.
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.
A brief review of various existing mathematical formulations of the uncertainty relations in quantum mechanics, containing variances of two or more non-commuting operators, is given. In particular, inequalities for the products of higher-order moments of a coordinate and a momentum are considered, as well as inequalities making the uncertainty relations more accurate when additional information about a quantum system is available (for example, the correlation coefficient or the degree of mixing of a quantum state characterized by the trace of the squared statistical operator). The special cases of two, three, and four operators are discussed in detail.
The close relationship between the zero-point energy, the uncertainty relation, coherent states, squeezed states, and correlated states for one mode is investigated. This group theoretic perspective of the problem enables the parametrization and identification of their multimode generalization. A simple and efficient method of determining the canonical structure of the generalized correlated states is presented. Implication of canonical commutation relations for correlations are not exhausted by the Heisenberg uncertainty relation, not even by the Schr\"odinger-Robertson uncertainty inequality, but there are relations in the multimode case that are the generalization of the Schr\"odinger-Robertson relation.