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Journal of Applied and Theoretical Physics Research

A Note on the Strong Unitary Uncertainty Relations

Ting-Gui Zhang

Published Feb 20, 2019• 0 citations
Publisher

Abstract

Uncertainty relations satisfied by the product of variances of arbitrary n oservables have attracted much attention.In a recent article [Phys.Rev.Lett.120,230402(2018)], the authors provided so called strong unitary uncertainty relations for a set of unitary matrices by using the positivity property of the Gram matrix, from which uncertainty relations satisfied by two quantum mechanical observables are obtained.We derive the explicit uncertainty relations satisfied by n quantum mechanical observables from such Gram matrix approach.By some algebraic transformations, we show that these uncertainty relations are just the same as the ones derived from a positive semi-definite Hermitian matrix generated by the mean values of n observables in [Scientific Reports 6,31192 (2016)].

DOI
10.24218/jatpr.2019.18

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We investigate the product form uncertainty relations of variances for n (n ≥ 3) quantum observables. In particular, tight uncertainty relations satisfied by three observables has been derived, which is shown to be better than the ones derived from the strengthened Heisenberg and the generalized Schrödinger uncertainty relations, and some existing uncertainty relation for three spin-half operators. Uncertainty relation of arbitrary number of observables is also derived. As an example, the uncertainty relation satisfied by the eight Gell-Mann matrices is presented.
In this paper we provide a set of uncertainty principles for unitary operators using a sequence of inequalities with the help of the geometric-arithmetic mean inequality. As these inequalities are ``fine-grained'' compared with the well-known Cauchy-Schwarz inequality, our framework naturally improves the results based on the latter. As such, the unitary uncertainty relations based on our method outperform the best known bound introduced in Phys. Rev. Lett. 120, 230402 (2018). to some extent. Explicit examples of unitary uncertainty relations are provided to back our claims.
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.
Uncertainty relations occupy a fundamental position in quantum mechanics. We propose stronger variance-based uncertainty relations for the product and sum of variances of two incompatible observables in a finite dimensional Hilbert space. It is shown that the new uncertainty relations provide near-optimal state-dependent bounds, which can be useful for quantum metrology, entanglement detection etc. in quantum information theory. It is further shown that the uncertainty relations are related to the spreads of the distribution of measurement outcomes caused by incompatible observables. Intuitively, this means that the ability of learning the distribution has both the upper and lower bounds. Combination of these bounds provides naturally an uncertainty interval which captures the essence of uncertainty in quantum theory. Finally, we explain how to employ entropic uncertainty relations to derive lower bounds for the product of variances of incompatible observables.
Abstract Unitary uncertainty relations provide a theoretical framework that enhances our understanding of the principles underlying quantum mechanics and its applications in quantum information science. In this study, we derive both the unitary uncertainty relation and the weighted unitary uncertainty relation based on the sum variance for arbitrary pairs of unitary operators. By applying the arithmetic geometric mean inequality, we obtain a lower bound that is tighter than the one provided by Bagchi and Pati. [Physical Review A 94,042104] for two unitary operators. To illustrate our results, we include examples of both the unitary uncertainty relation and the weighted uncertainty relation based on sum variance.