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.

💡 Research Summary

The paper presents a comprehensive theory and experimental protocol for directly measuring quantum pseudo‑distributions—specifically the Kirkwood‑Dirac (KD) distribution—through the use of characteristic functions and weak measurements. The authors begin by recalling that non‑commuting observables cannot be described by a genuine joint probability distribution; instead one must resort to quasi‑probability (pseudo‑distribution) representations such as the Wigner, Husimi, Glauber‑Sudarshan, and KD distributions. While many such representations exist, it has been unclear which one is naturally selected by experimental data without imposing a particular theoretical model.

To resolve this, the authors adopt a “classical‑style” statistical assumption: the conditional expectation value of an observable’s power, given a preparation and a post‑selected final measurement, should be expressible as a weighted sum over a conditional pseudo‑distribution exactly as in classical probability theory. They then design a three‑step experiment repeated many times: (i) prepare a quantum system in a known state |ψ⟩, (ii) perform a weak (infinitesimally disturbing) measurement of the n‑th power of an observable  (for n = 0,…,d‑1 in a d‑dimensional Hilbert space), and (iii) perform a strong projective measurement of a second, non‑commuting observable ̂B, post‑selecting on a particular eigenstate |b_j⟩. The weak‑measurement outcomes give the conditioned moments ⟨Âⁿ⟩_{ψ,b_j}=⟨b_j|Âⁿ|ψ⟩⟨b_j|ψ⟩.

These moments are assembled into a column vector A, while the unknown conditional pseudo‑distribution Q_i|j forms another column vector Q. The relationship between them is linear: A = V·Q, where V is a real Vandermonde matrix built from the eigenvalues a_i of  (each row contains powers of a_i up to a_i^{d‑1}). Because the eigenvalues are assumed non‑degenerate, V is invertible with determinant ∏_{i<j}(a_j−a_i) ≠ 0. The inverse V⁻¹ can be expressed analytically via Lagrange interpolation polynomials, enabling efficient O(d²) numerical inversion.

Solving Q = V⁻¹A yields the conditional pseudo‑distribution directly from experimental data, without any a‑priori model. The authors then demonstrate that this Q must coincide with the conditional KD distribution K_i|j = ⟨b_j|a_i⟩⟨a_i|ψ⟩⟨b_j|ψ⟩. The proof rests on the quantum weak‑value formula ⟨Âⁿ⟩_{ψ,b_j}=∑_i a_iⁿ K_i|j and the uniqueness of the solution to the linear system. Hence, the requirement that classical‑style conditional expectations be reproduced uniquely selects the KD distribution as the appropriate pseudo‑distribution.

The paper extends the method to joint distributions of two non‑commuting observables. By measuring mixed moments C_{n,m}=⟨ψ|Âⁿ ̂B^m|ψ⟩, one defines a two‑parameter characteristic function Z(λ,χ)=⟨ψ|e^{iλÂ}e^{iχ̂B}|ψ⟩. The moments are obtained as derivatives of Z at λ=χ=0. Two Vandermonde matrices V_A and V_B (constructed from the eigenvalues of  and ̂B) relate the moment matrix C to the joint pseudo‑distribution Q_{i,j} via C = V_A·Q·V_B^T. Inverting both Vandermonde matrices yields Q_{i,j}=V_A⁻¹·C·(V_B⁻¹)^T, which again matches the standard KD joint distribution K_{i,j}=⟨b_j|a_i⟩⟨a_i|ψ⟩⟨b_j|ψ⟩.

For continuous‑variable systems, the authors replace the discrete Vandermonde construction with Fourier analysis. The characteristic function Z(λ,χ)=⟨ψ|e^{iλx̂}e^{iχp̂}|ψ⟩ is measured experimentally; its inverse Fourier transform gives the phase‑space pseudo‑distribution (the KD distribution in the x‑p representation). This formalism naturally accommodates infinite‑dimensional Hilbert spaces.

A concrete experimental proposal is outlined. The generator of position moments, the momentum translation operator e^{i p̂ Δx}, is weakly coupled to an ancilla (e.g., a probe beam) to obtain the characteristic function for position. Afterward, a strong momentum measurement is performed, providing the post‑selection. By scanning Δx and recording the real and imaginary parts of the weak‑measurement pointer shift, the full Z(λ,χ) can be reconstructed. Applying the inversion procedure yields the KD distribution K(x,p). Moreover, by evaluating ⟨