Hudson's Theorem for Wavefunctions

• Hudson's theorem for wavefunctions is a framework that characterizes non-Gaussianity by linking the presence of complex zeros in an entire analytic wavefunction to deviations from Gaussian behavior.
• The analytic continuation and Hadamard factorization allow for classification of quantum states through stellar rank, providing a clear measure of non-classical resources.
• Under Gaussian evolution, the dynamic motion of zeros offers experimental signatures, enabling practical detection of non-classicality in continuous-variable quantum systems.

Hudson’s theorem for wavefunctions generalizes the classical phase-space formulation of Hudson’s theorem, traditionally phrased in terms of the nonnegativity of the Wigner function, to a direct statement about the analytic structure and complex zeros of the wavefunction itself. This perspective reveals deep connections between non-Gaussianity, the topology of the zero-set in the complex plane, and the quantum resource structure underlying continuous variable states. Recent developments have provided rigorous theorems, analytic representations, and dynamical interpretations, positioning the zero locus of entire wavefunctions as central to the quantification and detection of non-classicality in bosonic quantum systems.

1. Formulation of Hudson’s Theorem for Wavefunctions

The classical version of Hudson’s theorem asserts that the only pure quantum states with a nonnegative Wigner function are Gaussian. In the wavefunction-centric formulation, under a suitable energy bound—specifically, if there exists $s > 1$ such that

$$\langle s^{\hat{n}} \rangle_\psi = \sum_n s^n |\psi_n|^2 < +\infty,$$

the position wavefunction %%%%2%%%% of a single-mode pure state can be analytically continued to an entire function on the complex plane of order at most 2 (Cerf et al., 31 Jul 2025).

The main result, termed the Hudson theorem for the wavefunction, states:

• For any pure bosonic state satisfying the above energy condition, the state is non-Gaussian if and only if its analytic continuation to the complex plane possesses at least one zero.
• If the entire extension of $\psi(x)$ is zero-free on $\mathbb{C}$, then, by the Hadamard–Weierstrass factorization theorem, $\psi(x)$ must necessarily be a Gaussian.

In summary, a one-to-one correspondence emerges between the existence of complex zeros in the analytically continued wavefunction and non-Gaussianity of the quantum state.

2. Structure and Classification via Complex Zeros

The analytic extension of the wavefunction—that is, continuation to an entire function—enables the use of complex analysis and, particularly, the Hadamard factorization theorem. For pure states of finite “stellar rank” $r$, the wavefunction can be expressed in the canonical form

$\psi(x) = P(x) \exp(-a x^2 + b x + c)$

where:

• $P(x)$ is a complex polynomial of degree $r$;
• $a, b, c \in \mathbb{C}$ with $\mathrm{Re}(a) > 0$;
• the number of distinct complex zeros (taking multiplicity into account) is exactly $r$.

The set of zeros thus encodes all the non-Gaussian features. In the case $r=0$, the wavefunction is a Gaussian—the unique zero-free entire example of order 2.

This structure provides an intrinsic classification: the “stellar rank” (the degree of $P$) quantifies the non-Gaussianity and has direct operational consequences for phase-space and information-theoretic characterizations.

3. Information-Theoretic and Resource Implications

The zeros of the entire wavefunction admit an information-theoretic interpretation. They act as witnesses of non-classicality: all non-Gaussian aspects of a pure bosonic state are encoded in its complex zeros (Cerf et al., 31 Jul 2025). This observation aligns with previous characterizations of non-classicality based on the “stellar rank” of the Husimi Q-function zero-set.

Within the resource-theoretic framework for continuous-variable quantum information, non-Gaussianity is a crucial resource. The presence, number, and configuration of zeros in the analytic extension of $\psi(x)$ serve as a direct, operational measure—states with higher stellar rank possess greater non-Gaussian resources.

A plausible implication is that by analyzing or measuring features related to the zero-set (for instance, “holes” in quadrature distributions), one can operationally witness and even quantify non-classical resources in laboratory experiments.

4. Dynamics of Zeros under Gaussian Evolution

The propagation of quantum states under quadratic (Gaussian) Hamiltonians preserves the Gaussian form or the stellar rank of the state. In this context, the time evolution of the zeros is governed by classical many-body dynamics.

If the initial analytic wavefunction is

$$\psi(z) = \exp[a z^2 + b z + c] \prod_k (z - \lambda_k),$$

under Gaussian evolution, the time-dependent wavefunction remains of the same form, with the zeros $\lambda_k(t)$ evolving nontrivially. The equations of motion for the zeros are given by a Calogero–Moser system:

$$\ddot{\lambda}_k = (C^2 - 4AB)\lambda_k + CE - 2BD + 8B^2 \sum_{m \ne k} \frac{1}{(\lambda_k - \lambda_m)^3},$$

where $A, B, C, D, E$ parameterize the Gaussian evolution. The motion of the zeros thus provides a classical analog for the dynamics of non-Gaussian features. Notably, under certain quadratic evolutions (such as phase shifts), complex zeros can cross onto the real axis at particular times, indicating moments when non-Gaussian features are directly observable in measured quadrature distributions.

5. Experimental Consequences and Detection Protocols

These theoretical results have immediate experimental implications. Since quadrature probability distributions in homodyne detection directly sample the position (or momentum) marginal of the quantum state, zeros in these distributions manifest as “holes.” According to the extended Hudson theorem for wavefunctions, such holes are a sufficient condition for non-Gaussianity. Thus, measuring a single quadrature and observing a vanishing probability provides an operationally simple test for non-classicality in bosonic systems.

This insight can be utilized in quantum optics protocols where non-Gaussianity is a requisite resource, such as continuous-variable quantum computation, error correction, and state engineering.

6. Extensions and Related Contexts

The structural role of wavefunction zeros and the generalized Hudson theorem have ramifications for broader settings:

• For systems described by power-law potentials, the analytic structure of wavefunctions is governed by the order $\rho$ (degree of polynomial growth) and type $\sigma$ (scale of exponential decay), and the ansatz $\psi(r) = f(r)\exp[g(r)]$ encapsulates both the nodal (polynomial zero) structure and asymptotic behavior (Yang, 2020). In Gaussian cases, $g(r)$ is quadratic and $f(r)$ is constant, reproducing Hudson’s theorem’s circumstances.
• In highly correlated many-body contexts, such as fractional quantum Hall fluids, the set of allowed wavefunctions—constructed via rigid algebraic rules (e.g., reductions in the Jastrow exponent)—forms a uniquely characterized (often highly non-Gaussian) manifold, conceptually akin to the rigidity imposed by Hudson’s theorem on positive Wigner-function states (Yang, 2013, Yang, 2013).

A plausible implication is that extensions to multimode systems, mixed states, or fermionic settings will require further refinement of the zero-set criterion, potentially involving generalized phase-space functions or polynomial invariants.

7. Summary Table of Core Results

Criterion	Gaussian State	Non-Gaussian State
Entire extension zero-free?	Yes	No (at least one zero)
Hadamard factorization	$\exp(-a x^2 + b x + c)$	$P(x)\exp(-a x^2 + b x + c)$
Wigner function sign	Nonnegative everywhere	Sign-changing
Stellar rank	0	$r \ge 1$
Calogero–Moser dynamics	Trivial (no zeros)	Classical, many-body $\lambda_k$

The presence and dynamics of complex zeros in the analytically continued wavefunction provide a foundation for quantifying non-Gaussianity, witnessing non-classical resources, and establishing an operational correspondence between phase-space and analytic-structure-based characterizations of quantum states. The Hudson theorem for wavefunctions thus offers a rigorous analytic tool foregrounding the informational content of the zero-set in the paper of bosonic quantum systems (Cerf et al., 31 Jul 2025).