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

Andrew N. Jordan, David R. M. Arvidsson-Shukur, Aephraim M. Steinberg

TL;DR

This work addresses how to reconstruct a quantum pseudo-distribution for non-commuting observables in a theory-independent way by linking conditional moments to a Kirkwood-Dirac (KD) distribution. It develops a discrete method using Vandermonde-matrix inversion to infer the conditional KD distribution from weak-measured moments, and extends to multi-observable and continuous-variable settings via characteristic functions and inverse Fourier transforms. A concrete experimental protocol is proposed to directly measure the characteristic function, enabling direct KD-distribution reconstruction and a direct test of the canonical commutation relation $[\hat x,\hat p]=i\hbar$ through reversed measurement order. The results offer a practical route to quantum-state tomography for continuous variables and deepen the foundational connection between weak values, KD quasi-probabilities, and fundamental commutation relations.

Abstract

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.

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

TL;DR

This work addresses how to reconstruct a quantum pseudo-distribution for non-commuting observables in a theory-independent way by linking conditional moments to a Kirkwood-Dirac (KD) distribution. It develops a discrete method using Vandermonde-matrix inversion to infer the conditional KD distribution from weak-measured moments, and extends to multi-observable and continuous-variable settings via characteristic functions and inverse Fourier transforms. A concrete experimental protocol is proposed to directly measure the characteristic function, enabling direct KD-distribution reconstruction and a direct test of the canonical commutation relation through reversed measurement order. The results offer a practical route to quantum-state tomography for continuous variables and deepen the foundational connection between weak values, KD quasi-probabilities, and fundamental commutation relations.

Abstract

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.
Paper Structure (13 sections, 50 equations, 1 figure)

This paper contains 13 sections, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Pseudo-Distribution Construction method
  4. Qubit example
  5. Two-observable measurements
  6. Multiple-observable measurement
  7. Continuous variables
  8. Proposed experiment for direct measurement of the characteristic function
  9. First x, Then p
  10. First p, Then x
  11. Direct Measurement of the Canonical Commutation Relation
  12. Conclusions
  13. Acknowledgments

Figures (1)

  • Figure 1: Schematic of the proposed experiment to directly measure the pseudo-distribution characteristic function. The optical configuration consists of these elements: (a) single-photon source and (b) beam expanding lens, (c) mask to imprint the quantum state of the single-photon $\Psi(x)$, (d) optional focusing lens with focal length $f$, (e) weak periodically modulated birefringence, (f) focusing lens with focal length $f$, (g) polarizer, (h) Single photon camera. The camera is placed in the Fourier plane of the lens.