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Journal of Modern Optics

Higher-order uncertainty relations

Alfred Wünsche

Published Feb 17, 2006• 9 citations

Abstract

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.

DOI: 10.1080/09500340500418815

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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.

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.

We present a general framework and procedure to derive uncertainty relations for observables of quantum systems in a covariant manner. All such relations are consequences of the positive semidefiniteness of the density matrix of a general quantum state. Particular emphasis is given to the action of unitary symmetry operations of the system on the chosen observables, and the covariance of the uncertainty relations under these operations. The general method is applied to the case of an

n

-mode system to recover the

Sp(2n,\,R)

-covariant multi mode generalization of the single mode Schrödinger-Robertson Uncertainty Principle; and to the set of all polynomials in canonical variables for a single mode system. In the latter situation, the case of the fourth order moments is analyzed in detail, exploiting covariance under the homogeneous Lorentz group

SO(2,\,1)

of which the symplectic group

Sp(2,\,R)

is the double cover.

The non-negativity of the density operator of a state is faithfully coded in its Wigner distribution, and this coding places on the moments of the Wigner distribution constraints arising from the non-negativity of the density operator. Working in a monomial basis for the algebra of operators on the Hilbert space of a bosonic mode, we formulate these constraints in a canonically covariant form which is both concise and explicit. Since the conventional uncertainty relation is such a constraint on the first and second moments, our result constitutes a generalization of the same to all orders. The structure constants of , in the monomial basis, are shown to be essentially the SU (2) Clebsch–Gordan coefficients. Our results have applications in quantum state reconstruction using optical homodyne tomography and, when generalized to the n -mode case, which will be done in the second part of this work, will have applications also for continuous variable quantum information systems involving non-Gaussian states.

The second-,fourth-and sixth-order fluctuations have been discussed by making use of- SU-(2)Lie algebra.On the basis of higher-order uncertainty relation,the definition of higher-order squeezing for the fluctuations of angular momentum has been put forward.In particular,the second-,fourth-and sixth-order squeezing in atomic coherent states are investigated.These methods and definition can be used for studying much higher-order squeezing.The higher-order squeezing can be generalized to the fluctuations of atomic variances thereby.

The second ,fourth and sixth order fluctuations have been discussed by making use of SU (2)Lie algebra.On the basis of higher order uncertainty relation,the definition of higher order squeezing for the fluctuations of angular momentum has been put forward.In particular,the second ,fourth and sixth order squeezing in atomic coherent states are investigated.These methods and definition can be used for studying much higher order squeezing.The higher order squeezing can be generalized to the fluctuations of atomic variances thereby.