How to measure the canonical commutation relation $\mathbf{[\hat{x},\hat{p}]=i\hbar}$ in quantum mechanics with weak measurement?

Abstract

The quantum weak value draws many attentions recently from theoretical curiosity to experimental applications. Now we design an unusual weak measuring procedure as the pre-selection, mid-selection and post-selection to study the correlation function of two weak values, which we called the weak correlation function. In this paper, we proposed an weak measurement experiment to measure the canonical commutator $[\hat{x},\hat{p}]=i\hbar$ in quantum mechanics. Furthurmore, we found the intriguing equivalence between the canonical commutation relation and Riemann hypothesis, and then obtained the weak value of nontrivial Riemann zeros. Finally, as an nontrivial example of weak correlations, we also passed successfully a testing on the (anti-)commutators of Pauli operators, which followed the experimental setup of the landmark paper of Aharonov, et al. in 1988. Our proposed experiments could hopefully test the fundamental canonical relationship in quantum worlds and trigger more testing experiments on weak correlations.