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arXiv:quant-ph/0604044 · Physics Letters A

Does the uncertainty relation determine the quantum state?

Olga V. Man’ko, Vladimir I. Man’ko, Giuseppe Marmo, E. C. G. Sudarshan, F. Zaccaria

Published May 23, 2006• 25 citations
arXivHTMLPDF
DOI
10.1016/j.physleta.2006.04.116

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In this paper we will establish a relation between geometric uncertainty relation and the determinant of the quantum covariance matrix for mixed quantum states. We will show that determinant of the covariance matrix represents the squared metric area of a parallelogram. In this setting the geometric uncertainty relation compares a metric area to a symplectic area. Moreover, we will in details investigate relation between J-holomorphic maps and geometric uncertainty relation for mixed quantum states. We will argue that determinant of the quantum covariance matrix is equal to the harmonic energy of a holomorphic map that minimize the areas.
It is shown that the Heisenberg uncertainty relation (or soft uncertainty relation) determined by the commutation properties of operators of electromagnetic field quadratures differs significantly from the Robertson–Schrödinger uncertainty relation (or rigorous uncertainty relation) determined by the quantum correlation properties of field quadratures. In the case of field quantum states, for which mutually noncommuting field operators are quantum-statistically independent or their quantum central correlation moment is zero, the rigorous uncertainty relation makes it possible to measure simultaneously and exactly the observables corresponding to both operators or measure exactly the observable of one of the operators at a finite measurement uncertainty for the other observable. The significant difference between the rigorous and soft uncertainty relations for quantum superpositions of coherent states and the two-photon coherent state of electromagnetic field (which is a state with minimum uncertainty, according to the rigorous uncertainty relation) is analyzed.
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.