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

Technical Summary: Theory of Direct Measurement of the Quantum Pseudo-Distribution via its Characteristic Function

Problem Statement
In quantum mechanics, the non-commutation of observables prevents the unambiguous joint assignment of their properties, rendering a standard joint probability distribution impossible. While various pseudo-distributions (e.g., Wigner, Husimi, Glauber-Sudarshan) exist to represent quantum states, the field faces a lack of uniqueness: it is unclear which pseudo-distribution is the "correct" one for a given experiment, and many existing methods for measuring them rely on scanning weak measurements over all eigenprojectors, which is resource-intensive. Furthermore, connecting these distributions to non-classical phenomena (such as contextuality) is complicated by the infinite number of possible representations. The authors address the need for a theory-agnostic method to reconstruct a unique pseudo-distribution directly from experimental data without a priori assumptions about its form.

Methodology
The paper proposes a framework based on three consecutive temporal events: preparation of an initial state $|\psi\rangle$, a non-invasive (weak) measurement of the moments of an observable $\hat{A}$, and a final projective measurement in a basis $\{|b_j\rangle\}$.

1. Discrete Variable Approach: For a Hilbert space of dimension $d$, the authors analyze the conditional expectation values of the powers of an observable, $\langle \hat{A}^n \rangle_{\psi}^{b_j}$, for $n = 0, \dots, d-1$. They formulate a matrix equation $\vec{A} = V \cdot \vec{Q}$, where $\vec{A}$ contains the measured conditioned moments, $\vec{Q}$ represents the unknown conditional pseudo-distribution, and $V$ is a Vandermonde matrix constructed from the eigenvalues of $\hat{A}$. By inverting $V$, the pseudo-distribution is uniquely determined.
2. Characteristic Function Approach: The authors extend this to continuous variables and joint distributions by utilizing the characteristic function. They define a weak characteristic function $Z(\chi) = \langle e^{i\chi \hat{O}} \rangle_{\psi}^{\phi}$, which can be measured directly via a single experimental setup with a tunable parameter, rather than measuring an infinite number of moments. The pseudo-distribution is then recovered via an inverse Fourier transform of this characteristic function.
3. Joint and Reversed Measurements: The method is generalized to two non-commuting observables ($\hat{A}$ and $\hat{B}$) by constructing a joint characteristic function $Z(\lambda, \chi) = \langle e^{i\lambda \hat{A}} e^{i\chi \hat{B}} \rangle$. The authors propose two experimental configurations: one measuring position followed by momentum, and another reversing the order (momentum followed by position).

Key Results

• Uniqueness of the Kirkwood-Dirac Distribution: The paper demonstrates that if one requires the quantum conditional expectation values to satisfy the same statistical structure as classical conditional expectations (i.e., $\langle \hat{A}^n \rangle = \sum a_i^n Q_{i|j}$), the resulting pseudo-distribution is uniquely identified as the Kirkwood-Dirac (KD) distribution.
• Direct Measurement Protocol: The authors provide a constructive method to measure the KD distribution directly. In the discrete case, this involves inverting a Vandermonde matrix derived from moment measurements. In the continuous case, it involves measuring the characteristic function (the expectation value of an exponential operator) and performing an inverse Fourier transform.
• Verification of Canonical Commutation: By comparing the joint pseudo-distributions obtained from the $x \to p$ and $p \to x$ measurement sequences, the authors show that the difference in their correlation functions yields the canonical commutation relation $[\hat{x}, \hat{p}] = i\hbar$. This provides a direct experimental probe of the commutation relation for any quantum state.
• Connection to Weak Values: The conditional pseudo-distribution is shown to be equivalent to the weak value of the projector onto the eigenstates of the measured observable.

Significance and Claims
The paper claims to establish a theory for a characteristic function approach to pseudo-distributions that offers a "theory-agnostic" reconstruction of the quantum state's representation.

• Theoretical Unification: It resolves the ambiguity of which pseudo-distribution to use by showing that the KD distribution is the unique solution that preserves the classical statistical form of conditional expectations in the quantum regime.
• Experimental Efficiency: The proposed method is presented as more efficient than previous techniques that required scanning weak measurements over all eigenprojectors. The "all at once" nature of the characteristic function measurement encodes global properties of the state.
• Foundational Verification: The work provides a direct, operational prescription to verify the canonical commutation relation, linking the complex nature of the KD distribution directly to the fundamental non-commutativity of quantum mechanics.
• Potential for Tomography: The authors suggest that this approach may offer a more resource-efficient method for quantum state tomography in continuous variable systems compared to inverse Radon transformations or maximum-likelihood techniques, particularly for non-Gaussian states.

The paper concludes by noting that independent work on related topics has been conducted, and it positions the KD distribution as the natural pseudo-distribution for experiments where conditional expectations coincide with classical predictors.