Theory of direct measurement of the quantum pseudo-distribution via its characteristic function

We propose a method for directly measuring the quantum mechanical pseudo-distribution of observable properties via its characteristic function. Vandermonde matrices of the eigenvalues play a central role in the theory. This proposal directly finds the pseudo-distribution using weak measurements of the generator of position moments (momentum translations). While the pseudo-distribution can be extracted from the data in a theory-agnostic way, it is shown that under quantum-mechanical formalism, the predicted pseudo-distribution is identified with the Kirkwood-Dirac pseudo-distribution. We discuss the construction of both the joint pseudo-distribution and a conditional pseudo-distribution, which is closely connected to weak-value physics. By permuting position and momentum measurements, we give a prescription to directly probe the canonical commutation relation and verify it for any quantum state. This work establishes the theory of a characteristic function approach to pseudo-distributions, as well as providing a constructive approach to measuring them directly.