Direct experimental test of commutation relation via imaginary weak value

Abstract

The canonical commutation relation is the hallmark of quantum theory, and Heisenberg's uncertainty relation is a direct consequence of it. Although various formulations of uncertainty relations have been tested in experiments a direct test of the commutation relation has hitherto not been performed. The reason for this is that whereas in quantum mechanics, observables are usually represented by Hermitian operators, the product of two noncommuting observables as occurring in the commutator relation is, in general, non-Hermitian. Here, we present a theoretical scheme for a direct test of the commutation relation and report its experimental realization. More precisely, we provide an experimental verification of the canonical commutation relation between two qubit observables based on a measurement scheme widely known as “weak measurement.” In our neutron interferometric setup, we perform the measurement of a single anomalous weak value of a relevant path-qubit observable where the coupled spin-energy degree of freedom serves as an ancilla.