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arXiv:1607.07331

Uncertainty relations for incompatible observables: Newer results versus modified strategies

Kamal Bhattacharyya

Published Jul 25, 2016• 0 citations
arXivHTMLPDF

Abstract

Two special situations where the standard uncertainty product inequality appears to be useless are modified. One such case is noted to also trivialize the recently-introduced alternatives [Phys. Rev. Lett. 113, 260401 (2014); Sci. Rep. 6, 23201 (2016)] involving sums of variances. A careful discussion is presented on the experimental justifications of some of the relations [Phys. Rev. A 93, 052108 (2016)] using qutrit and qubit states. Alternative bypass routes are put forward to tackle this situation, with and without involving any auxiliary state. This latter strategy is noted to be vital in an entirely different context concerned with the quality of approximate stationary states. The other case is more frustrating, but an effective method is advanced. En route, the recent alternatives are also simplified to easily accommodate even the cases of more than two observables. In favorable circumstances, an easy option in function space is obtained by virtue of symmetry that does not involve any auxiliary state. Pilot calculations reveal the advantages of our endeavor.

DOI
10.48550/arxiv.1607.07331

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We prove a few state-dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Robertson-Schr\"odinger uncertainty relation and other ones existing in the current literature. Also, we derive a state-dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only can we not prepare quantum states for which two incompatible observables can have sharp values, but also we have both the lower and the upper limits on the variances of quantum mechanical observables at a fundamental level.
The uncertainty relations (URs) of two arbitrary Hermitian and non-Hermitian incompatible operators represented by the product of variances have been confirmed theoretically and experimentally in various physical systems. However, the lower bound of the product uncertainty inequality can be null even for two noncommuting operators, i.e., a trivial case. Therefore, for two incompatible operators over the measured system state, the associated URs regarding the sum of variances are valid in a state-dependent manner, and the lower bound is guaranteed to be nontrivial. Although the sum URs formulated for Hermitian and unitary operators have been affirmed, the general forms for arbitrary non-Hermitian operators have not yet been investigated. This study presents the sum URs for non-Hermitian operators acting on system states using an appropriate Hilbert-space metric. The compatible forms of our sum inequalities with the conventional quantum mechanics are also provided via the GGG-metric formalism. Concrete examples illustrate the validity of the proposed sum URs in both PT\mathcal{PT}PT-symmetric and PT\mathcal{PT}PT-broken phases. The developed methods and results can help give an in-depth understanding of the usefulness of GGG-metric formalism in non-Hermitian quantum mechanics and the sum URs of incompatible operators within.
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.
In quantum mechanics, the variance-based Heisenberg-type uncertainty relations are a series of mathematical inequalities posing the fundamental limits on the achievable accuracy of the state preparations. In contrast, we construct and formulate two quantum uncertainty equalities, which hold for all pairs of incompatible observables and indicate the new uncertainty relations recently introduced by L. Maccone and A. K. Pati [Phys. Rev. Lett. 113, 260401 (2014)]. In fact, we obtain a series of inequalities with hierarchical structure, including the Maccone-Pati's inequalities as a special (weakest) case. Furthermore, we present an explicit interpretation lying behind the derivations and relate these relations to the so-called intelligent states. As an illustration, we investigate the properties of these uncertainty inequalities in the qubit system and a state-independent bound is obtained for the sum of variances. Finally, we apply these inequalities to the spin squeezing scenario and its implication in interferometric sensitivity is also discussed.
The Heisenberg-Robertson uncertainty relation quantitatively expresses the impossibility of jointly sharp preparation of incompatible observables. However, it does not capture the concept of incompatible observables because it can be trivial even for two incompatible observables. We experimentally demonstrate that the new stronger uncertainty relations proposed by Maccone and Pati [Phys. Rev. Lett. 113, 260401 (2014)] relating to the sum of variances are valid in a state-dependent manner and that the lower bound is guaranteed to be nontrivial when two observables are incompatible on the state of the system being measured. The behavior we find agrees with the predictions of quantum theory and obeys the new uncertainty relations even for the special states which trivialize the Heisenberg-Robertson relation. We realize a direct measurement model and perform an experimental investigation of the strengthened relations.