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arXiv:1912.03258 · Physical Review Research

Direct experimental test of forward and reverse uncertainty relations

Lei Xiao, Bowen Fan, Kunkun Wang, Arun Kumar Pati, Peng Xue

Published Apr 30, 2020• 9 citations
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Abstract

This paper shows experimental tests of state dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables for photonic qutrits. These uncertainty relations are independent of any optimization and much tighter than the Robertson-Schrodinger uncertainty relation and other ones found in current literature. Also the authors test the state dependent reverse uncertainty relations for the sum of variances of two incompatible observables.

DOI
10.1103/physrevresearch.2.023106

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This paper shows experimental tests of state dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables for photonic qutrits. These uncertainty relations are independent of any optimization and much tighter than the Robertson-Schrodinger uncertainty relation and other ones found in current literature. Also the authors test the state dependent reverse uncertainty relations for the sum of variances of two incompatible observables.
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 Heisenberg-Robertson uncertainty relation presents a lower bound on the product of variance of two observables, and provides a trade-off relation of measurement errors of these two observables for any given quantum states. Recently multi-observable uncertainty relations relating on that product of variances are proposed. Here we experimentally demonstrate these multi-observable uncertainty relations are valid in a state-dependent manner and the lower bound is tight for multi-observable being incompatible on the state of the system being measured. We find the behaviour of multiple incompatible observables agrees with the predictions of quantum theory.
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.
The Heisenberg-Robertson uncertainty relation is the hallmark of quantum physics and has been widely investigated. However, it does not capture the concept of incompatible observables because it can be trivial even for incompatible observables. Recently some stronger uncertainty relations relating the sums of variances were proposed. Here we experimentally demonstrate that these stronger multiobservable uncertainty relations are valid in a state-dependent manner and that the lower bound is guaranteed to be nontrivial for multiple observables that are incompatible on the state of the system being measured. We find that the behavior of multiple high-dimensional observables agrees with the predictions of quantum theory. Our experimental results not only foster insight into a fundamental limitation of measurements with multiple observables but also contribute to the study of the precision measurement technology in quantum information processing.
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.
Uncertainty relations are old, yet potentially rewarding to explore. By introducing a quantity called the uncertainty matrix, we provide a link between purity and observable incompatibility, and derive several stronger uncertainty relations in both forward and reverse directions for arbitrary quantum states, i.e., mixed as well as pure, and arbitrary incompatible quantum observables, none of which suffer from the problem of triviality. Besides the tightness, the interpretations of terms in these uncertainty relations may be of independent inter- est. We provide the possible generalization of stronger uncertainty relations to sum of variances of more than two observables. We also demonstrate applications of techniques used here to, firstly, obtain a simple reverse quantum speed limit for quantum states undergoing Markovian dynamical evolution, and secondly, to provide a lower bound for fidelity between two quantum states.