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Optics Express

Experimental demonstration of strong unitary uncertainty relations

Dengke Qu, Kunkun Wang, Lei Xiao, Xiang Zhan, Peng Xue

Published Aug 23, 2021• 11 citations
Publisher

Abstract

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.

DOI
10.1364/oe.438774

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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 are the hallmarks of quantum physics and have been widely investigated since its original formulation. To understand and quantitatively capture the essence of preparation uncertainty in quantum interference, the uncertainty relations for unitary operators need to be investigated. Here, we report the first experimental investigation of the uncertainty relations for general unitary operators. In particular, we experimentally demonstrate the uncertainty relation for general unitary operators proved by Bagchi and Pati [ Phys. Rev. A94, 042104 (2016)], which places a non-trivial lower bound on the sum of uncertainties and removes the triviality problem faced by the product of the uncertainties. The experimental findings agree with the predictions of quantum theory and respect the new uncertainty relation.
We derive and experimentally investigate a strong uncertainty relation valid for any n unitary operators, which implies the standard uncertainty relation and others as special cases, and which can be written in terms of geometric phases. It is saturated by every pure state of any n-dimensional quantum system, generates a tight overlap uncertainty relation for the transition probabilities of any n+1 pure states, and gives an upper bound for the out-of-time-order correlation function. We test these uncertainty relations experimentally for photonic polarization qubits, including the minimum uncertainty states of the overlap uncertainty relation, via interferometric measurements of generalized geometric phases.
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.
The uncertainty relation, as one of the fundamental principles of quantum physics, captures the incompatibility of noncommuting observables in the preparation of quantum states. In this work, we derive two strong and universal uncertainty relations for N(N ≥ 2) observables with discrete and bounded spectra, one in multiplicative form and the other in additive form. To verify their validity, for illustration, we implement in the spin-1/2 system an experiment with single-photon measurement. The experimental results exhibit the validity and robustness of these uncertainty relations, and indicate the existence of stringent lower bounds.
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.
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.