Direct experimental test of commutation relation via imaginary weak value

In our latest work ¹, we tested canonical commutation in an interferometric experiment. The canonical commutation relation is a tenet of quantum theory and Heisenberg’s uncertainty relation is a direct consequence of it. However, despite its fundamental role in quantum theory, surprisingly, its genuine direct experimental test has hitherto not been performed. The reason for this is the following: while in quantum mechanics observables are represented by Hermitian operators, the product of two non-commuting observables (as occurring in the commutator relation) is in general non-Hermitian and therefore cannot be measured by a usual strong (von Neumann type) measurement. In our work, we we have overcome this hurdle by measuring the imaginary part of the weak value (see here for theoretical details) of a suited path-qubit observable to verify the canonical commutation relation. The experiment is realized using a newly developed technique in our in a neutron interferometric setup (see here for details of the experiment), where the neutron’s coupled spin-energy degree of freedom serves as ancilla.

1. R. Wagner, W. Kersten, A. Danner, H. Lemmel, A. K. Pan, and S. Sponar Phys. Rev. Research 3, 023243 (2021). ↩