Rigorous definition of the effective phase-space dimension of a quantum state and the existence of a universal relation $g^{(2)}(0)=f(D)$ for singular $P$-distributions

The equal-time normalized second-order intensity correlation function $g^{(2)}(0)$ of a single light mode takes the familiar values 1, 2, and 3 for a coherent state, a thermal state, and a strongly squeezed vacuum. Working in the Glauber-Sudarshan $P$-representation, Leuchs, Glauber, and Schleich (Physica Scripta 90, 108007 (2015)) showed that these values reflect the number of phase-space dimensions the $P$-distribution occupies (zero, two, and one, respectively), and observed that photon number states, whose $P$-distributions are 'narrower' than that of the zero-dimensional coherent state, suggest the notion of 'negative' dimension. The paper explicitly leaves open the rigorous, operational definition of such a dimension $D$ - valid for singular $P$-distributions built from delta functions and their derivatives - and hence the existence and form of a universal relation $g^{(2)}(0)=f(D)$; it also leaves open whether an invariant, physically motivated dimensional measure exists (the heuristic rests on one moment, $\langle|\alpha|^4\rangle$) and how the interpretation extends beyond the paradigmatic examples to general nonclassical single-mode states, and it contrasts 'negative dimension' inferred from normally ordered $P$-based moments (a consequence of differentiations of delta functions) with the negative quasi-probability regions of the Wigner function. This problem asks for exactly that: a rigorous definition of the effective phase-space dimension covering singular quasi-probability distributions, a proof or refutation of the universal relation $g^{(2)}(0)=f(D)$, and the resulting value (or proved impossibility) of negative and fractional dimensions for number states; the source-discussed contrast to Wigner-function negativity calls for a parallel clarification. A 2024 follow-up by the same group derived and experimentally tested the analogous Wigner-representation formula, but only for Gaussian states; the definitional core remains open.

Background

For a single mode of the radiation field, with annihilation and creation operators $\hat a$ and $\hat a^\dagger$ and photon number $\hat n=\hat a^\dagger\hat a$, the statistics most directly accessible to photodetection are expressed by normally ordered products, in which creation operators stand to the left of annihilation operators (an ideal detector cannot absorb a virtual photon borrowed from the vacuum). The workhorse observable is the normalized second-order correlation function at zero delay, $g^{(2)}(0)\equiv\langle(\hat a^\dagger)^2\hat a^2\rangle/\langle\hat a^\dagger\hat a\rangle^2$. It equals 1 for a coherent state (the most classical state of light, with Poissonian statistics), 2 for a thermal (chaotic) state, $3+1/\langle\hat n\rangle$ for a squeezed vacuum (a state with reduced noise in one field quadrature, i.e. in one real component of the field amplitude), approaching 3 for strong squeezing, and $1-1/n$ for a number (Fock) state with exactly $n$ photons. Values below 1 ('photon antibunching', sub-Poissonian statistics) are a standard witness of nonclassicality.

Normally ordered moments are naturally evaluated in the Glauber-Sudarshan $P$-representation, introduced by Glauber and Sudarshan in 1963: the density operator is expanded diagonally in coherent states, $\hat\rho=\int d^2\alpha\,P(\alpha)|\alpha\rangle\langle\alpha|$, where $|\alpha\rangle$ is an eigenstate of $\hat a$ and the complex amplitude $\alpha=\alpha_r+i\alpha_i$ labels a point of the two-dimensional phase space with real coordinates $\alpha_r$ and $\alpha_i$. Operator moments then become phase-space averages over $P$, in particular $g^{(2)}(0)=\langle|\alpha|^4\rangle/\langle|\alpha|^2\rangle^2$. Classical states admit positive, regular $P$-densities, whereas nonclassical states force $P$ to become a singular generalized function: a two-dimensional Dirac delta for a coherent state, infinitely many derivatives of delta functions for a squeezed vacuum, and a finite-order Laguerre polynomial of the two-dimensional Laplacian ($\partial^2/\partial\alpha_r^2+\partial^2/\partial\alpha_i^2$) acting on a delta for a number state. The Wigner function is the alternative phase-space representation based on symmetric operator ordering; it is always regular but can take negative values for nonclassical states.

In two companion papers, Leuchs, Glauber, and Schleich advanced a geometric reading of the familiar numbers above: the values $g^{(2)}(0)=1,2,3$ arise from the number of dimensions the state occupies in phase space. In the $P$-picture, the thermal state fills both phase-space coordinates isotropically (a two-dimensional Gaussian), the strongly squeezed vacuum factorizes so that only one coordinate carries weight (a one-dimensional object), and the coherent state is a point (a zero-dimensional object). In the Wigner-function companion paper, an exact moment formula links $g^{(2)}(0)$ to the state's shape in phase space, and 'negative dimensions' for number states were associated with the negative quasi-probability values of their Wigner functions; in the $P$-picture the same states are 'narrower' than the zero-dimensional coherent state because a finite number of differentiations act on the delta function, again suggesting negative dimension.

This dimension counting, however, rests on examples and on one specific diagnostic: the rotationally symmetric moment $\langle|\alpha|^4\rangle$. The authors themselves emphasize that the moments $\langle|\alpha|^2\rangle$ and $\langle|\alpha|^4\rangle$ 'probe aspects of dimensionality ... [but] are no substitute for a proper definition', that the negative-dimension notion for number states 'depends crucially on our choice of measure of dimensionality', and that only after an appropriate definition of $D$ is found 'will we be able to establish a formula expressing $g^{(2)}(0)$ in terms of $D$'. The rigorous definition was explicitly deferred to future work, and the question of what, if anything, is universal in the relation between photon statistics and phase-space dimension was left open.

Problem Statement

For a single mode of the radiation field with annihilation operator $\hat a$, the normalized second-order intensity correlation at zero delay is $g^{(2)}(0)\equiv\langle(\hat a^\dagger)^2\hat a^2\rangle/\langle\hat a^\dagger\hat a\rangle^2$; in the Glauber-Sudarshan $P$-representation, in which the density operator is written $\hat\rho=\int d^2\alpha\,P(\alpha)|\alpha\rangle\langle\alpha|$ in terms of coherent states $|\alpha\rangle$ (eigenstates of $\hat a$) labeled by the complex phase-space amplitude $\alpha$, it equals the ratio of phase-space moments $\langle|\alpha|^4\rangle/\langle|\alpha|^2\rangle^2$. Leuchs, Glauber, and Schleich showed heuristically that $g^{(2)}(0)$ 'is a measure of the dimensionality of the quantum state in phase space', but left the notion of dimension itself undefined. The problem is to make this rigorous, or to prove rigorously that it cannot be made rigorous in the intended sense:

1. Definition for singular distributions. Give a rigorous, operational definition of the effective phase-space dimension $D$ of the quantum state of a single optical mode as represented by its $P$-distribution, which is well-defined and computable not only for regular densities but for the singular generalized functions that arise for nonclassical states (delta functions and their derivatives, including the infinite-derivative squeezed-vacuum $P$ and the finite-derivative number-state $P$).

2. Invariance of the measure. The heuristic assignment uses one diagnostic, the moment $\langle|\alpha|^4\rangle$ relative to $\langle|\alpha|^2\rangle$. Determine whether an invariant, physically motivated dimensional measure - possibly fractal-like, and admitting unambiguous positive, zero, fractional, or negative values - can be defined for singular quasi-probability distributions, reproducing the assignments $D=0$ for a coherent state, $D=1$ for a strongly squeezed vacuum, and $D=2$ for a thermal state; or prove that the assignment is necessarily measure-dependent and characterize that dependence.

3. The universal relation. With $D$ so defined, either establish a formula $g^{(2)}(0)=f(D)$ valid for the states of the domain, identifying $f$; or prove that no such universal relation exists, exhibiting the obstruction (for example, states of equal $D$ with different $g^{(2)}(0)$).

4. Negative dimensions for number states. Determine what value of $D$ the definition assigns to number states, whose $g^{(2)}(0)=1-1/n<1$ lies below the coherent-state value and whose $P$-distributions are 'narrower' than that of the zero-dimensional coherent state - in particular whether negative (or fractional) values arise consistently; or prove that no consistent negative-dimension extension exists.

5. Scope beyond the examples. Establish general conditions under which $g^{(2)}(0)$ reflects an effective phase-space dimensionality for single-mode states with regular as well as singular $P$-distributions, rather than only the paradigmatic examples (thermal Gaussian, coherent delta, factorized squeezed, finite-derivative number-state $P$); the definition should hold generally within the single-mode framework and not be reverse-engineered from these examples.

6. Two senses of negativity. Clarify whether the 'negative dimension' inferred from normally ordered $P$-based moments (driven by differentiations of delta functions) and the negative quasi-probability regions of the Wigner function (from which negative dimensions were inferred in the companion paper) express one deeper geometric property of the state, or are representation-dependent diagnostics.

These items are facets of one objective: to convert the heuristic interpretation 'the value of $g^{(2)}(0)$ is a consequence of the dimensionality of the quantum state in phase space' into a rigorous theory (definition plus universal relation) or a rigorous no-go statement. The objective is to give a rigorous definition of $D$ valid for the singular $P$-distributions of single-mode states - or to prove that the intended assignment is necessarily measure-dependent - and, on that basis, to settle whether a formula $g^{(2)}(0)=f(D)$ can be established, including the value or proved impossibility of negative $D$ for number states. Item 6 restates a contrast the source itself draws (differentiation-induced negative dimensions in the $P$-picture versus negativity-induced ones in the Wigner picture) and calls for its clarification; it is source-discussed context.

References

1. Gerd Leuchs, Roy J. Glauber, and Wolfgang P. Schleich, 'Intensity-intensity correlations determined by dimension of quantum state in phase space: P-distribution', Physica Scripta 90, 108007 (2015). DOI: 10.1088/0031-8949/90/10/108007, https://doi.org/10.1088/0031-8949/90/10/108007
2. Gerd Leuchs, Roy J. Glauber, and Wolfgang P. Schleich, 'Dimension of quantum phase space measured by photon correlations', Physica Scripta 90, 074066 (2015). DOI: 10.1088/0031-8949/90/7/074066, https://doi.org/10.1088/0031-8949/90/7/074066
3. Mojdeh Shikhali Najafabadi, Luis L. Sanchez-Soto, Kun Huang, Julien Laurat, Hanna Le Jeannic, and Gerd Leuchs, 'Intensity correlations in the Wigner representation', Philosophical Transactions of the Royal Society A 382, 20230337 (2024). DOI: 10.1098/rsta.2023.0337, arXiv:2407.12901, https://doi.org/10.1098/rsta.2023.0337
4. Sreenath K. Manikandan and Frank Wilczek, 'Complementary probes of gravitational radiation states', Physical Review A 112, 043716 (2025). DOI: 10.1103/83tt-tt57, arXiv:2505.11422, https://doi.org/10.1103/83tt-tt57
5. G. Mouloudakis and P. Lambropoulos, 'Revisiting photon-statistics effects on multiphoton ionization', Physical Review A 97, 053413 (2018). DOI: 10.1103/PhysRevA.97.053413, https://doi.org/10.1103/PhysRevA.97.053413
6. Roy J. Glauber, 'Photon Correlations', Physical Review Letters 10, 84 (1963). DOI: 10.1103/PhysRevLett.10.84, https://doi.org/10.1103/PhysRevLett.10.84
7. E. C. G. Sudarshan, 'Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams', Physical Review Letters 10, 277 (1963). DOI: 10.1103/PhysRevLett.10.277, https://doi.org/10.1103/PhysRevLett.10.277