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arXiv:1908.05053 · Physical Review A

Strong unitary uncertainty relations

Bing Yu, Naihuan Jing, Xianqing Li‐Jost

Published Aug 13, 2019• 20 citations
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Abstract

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.

DOI
10.1103/physreva.100.022116

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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.
Abstract We derive the lower bound of uncertainty relations of two unitary operators for a class of states based on the geometric-arithmetic inequality and Cauchy–Schwarz inequality. Furthermore, we propose a set of uncertainty relations for three unitary operators. Compared to the known bound introduced in Yu et al (2019 Phys. Rev . A 100 022116), the unitary uncertainty relations bound with our method is tighter, to a certain extent. Meanwhile, some examples are given in the paper to illustrate our conclusions.
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.
Uncertainty relations are one of the most important foundations of quantum physics. In the textbook literatures, uncertainty relations usually refer to the preparation uncertainty. Its original formulation based on variances of two observables limits on the ability to prepare an ensemble of quantum systems for which non-commuting observables will have arbitrary uncertainty. The preparation uncertainty relation has been widely investigated. On the other hand, a unitary operator is a fundamental tenet of quantum theory. Every evolution of a closed quantum system is governed by acting unitary operators on the state of the system and the evolution of an open system can be represented by acting unitary operators on an enlarged system consisting of the quantum system as a subsystem. Therefore, naturally, to understand and quantitatively capture the essence of uncertainty relations for unitary operators is important and timely. Here we report an experimental investigation of a set of uncertainty relations for two unitary operators, which are theoretically derived by using a sequence of fine-grained inequalities. We test these uncertainty relations with single photons and interferometric networks. The unitary uncertainty relation is saturated by any pure qubit state. For higher-dimensional states, it is stronger than the best known bound introduced in the previous literatures. The lower bounds of the unitary uncertainty relations can be even further strengthened by the symmetry of permutation. The experimental findings agree with the predictions of quantum theory and respect the new uncertainty relations.
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)].
We formulate uncertainty relations for arbitrary N observables. Two uncertainty inequalities are presented in terms of the sum of variances and standard deviations, respectively. The lower bounds of the corresponding sum uncertainty relations are explicitly derived. These bounds are shown to be tighter than the ones such as derived from the uncertainty inequality for two observables [Phys. Rev. Lett. 113, 260401 (2014)]. Detailed examples are presented to compare among our results with some existing ones.