Quantum State Complexity Quantifier

• The topic introduces a quantifier that uses Rényi entropies to measure the structural richness of quantum states beyond traditional metrics.
• It combines global information content with spatial localization properties to compare and distinguish quantum states in systems with central potentials.
• The approach ensures stability through near continuity and normalization, linking the new measure with classical complexity quantifiers like LMC.

A complexity quantifier of quantum states defines a precise, operational means of assigning a quantitative value to the "structural richness"—the informational and spatial organization—of quantum states. Unlike entanglement entropy or simple measures of uncertainty, such quantifiers are constructed to characterize, compare, and distinguish quantum states according to properties such as their global information content, spatial localization, and the transformation resources required for their preparation. The Rényi complexity ratio (RCR) is a theoretically robust and practically effective example, specifically developed for three- and higher-dimensional quantum systems with central potentials, capable of capturing both global and local features of quantum probability densities.

1. Definition and Mathematical Foundations

The Rényi complexity ratio (RCR) is built upon Rényi entropies of two quantum probability densities (or more generally, two non-negative, normalized densities) $f(\mathbf{r})$ and $g(\mathbf{r})$, both defined over an effective domain $\Omega \subseteq \mathbb{R}^D$. For any order $\alpha > 0$, $\alpha \neq 1$, the Rényi entropy of density $f$ is

$$\mathcal{R}^{(\alpha)}_{f} = \frac{1}{1-\alpha}\ln\left[\int_{\Omega} f(\mathbf{r})^\alpha d\mathbf{r}\right].$$

The Rényi complexity ratio of orders $(\alpha, \beta)$ is then

$$C^{(\alpha, \beta)}_{(f,g)} = \exp\left(\mathcal{R}_f^{(\alpha)} - \mathcal{R}_g^{(\beta)}\right).$$

In the special case $f = g$, $C^{(\alpha, \beta)}_{(f,f)} = 1$, providing normalization and indicating no relative complexity. This construction generalizes classical statistical complexity measures and links directly to broader classes of measures such as generalized or shape Rényi complexities.

2. Fundamental Properties and Theoretical Characterization

The RCR obeys a suite of rigorous mathematical properties:

• Symmetry and Inversion: $$C^{(\alpha, \beta)}_{(f,g)} \cdot C^{(\beta, \alpha)}_{(g,f)} = 1$$, so complexity inverses under order and exchange.
• Normalization: $C^{(\alpha,\beta)}_{(f,f)} = 1$ for any $\alpha, \beta$.
• Monotonicity: The RCR is a nonincreasing (resp. increasing) function of $\alpha$ (resp. $\beta$) if the other parameter is fixed, reflecting the monotonicity of Rényi entropy.
• Majorization: If $f$ "majorizes" $g$, i.e., is more spread out in the formal sense, then for $0 < \alpha < 1$, $$\mathcal{R}^{(\alpha)}_f \leq \mathcal{R}^{(\alpha)}_g$$, yielding bounds on the RCR.
• Scaling and Composition: Under scaling or composition of densities (e.g., sums of sub-densities), the RCR transforms in predictable, tractable ways.
• Near Continuity: Five theorems establish that if $f_1$ and $f_2$ are $\delta$-neighbors (their difference exceeds $\delta$ only on a set of measure zero under the Lebesgue measure), then $$|C^{(\alpha, \beta)}_{(f_1, g)} - C^{(\alpha, \beta)}_{(f_2, g)}| < \epsilon$$ for any preassigned $\epsilon > 0$ and sufficiently small $\delta>0$. The same property holds with respect to the $g$ argument. This guarantees stability under small functional perturbations.

3. Localization, Effective Domains, and Practical Interpretation

A central theme of the RCR analysis is localization. Quantum densities in three or more dimensions are naturally factorized in spherical coordinates,

$\rho(\mathbf{r}) = f(r)\,g(\theta)\,h(\phi),$

where each factor may be defined on an "effective domain" (i.e., regions giving non-negligible integral contribution). The rigorous notion of effective domain under the Lebesgue measure ensures consistency for densities arising from quantum states with varying localization characteristics. The RCR, by combining Rényi entropies of different densities or different projections of the same density, inherits sensitivity to both global spread and spatial localization properties.

4. Applications to Quantum Systems with Central Potentials

The RCR is particularly powerful for systems with central potentials, where it can compare and discriminate between quantum states even when they share identical spectra:

• Pseudoharmonic Oscillator: $$V^{3D}(r) = D_e\left(\frac{r}{r_e} - \frac{r_e}{r}\right)^2$$. The eigenstates yield densities whose Rényi entropies can be calculated exactly for integral orders.
• Isospectral Potentials: Generated by supersymmetric transformations, such potentials share energy spectra with the pseudoharmonic oscillator, but have distinct eigenfunctions (and thus different spatial densities).
• Diatomic Molecules: The RCR was evaluated for experimentally relevant diatomics such as CO, NO, N$_2$, CH, H$_2$, and ScH, using ab initio density functions.

Empirical findings include:

• The Rényi entropy and RCR of two different densities reflect differences in spatial localization, with large differences in $C^{(\alpha,\beta)}_{(f,g)}$ signaling pronounced structural variations.
• In high-lying states, irrational effective quantum numbers can render the Rényi entropy negative, and the RCR approaches zero, indicating highly distinct localization properties.

5. Relation to Other Statistical Complexity Measures

The RCR functional unifies and extends earlier statistical complexity quantifiers:

• The case $g = f$, $C^{(\alpha, \beta)}_{(f,f)}$ reduces the RCR to a "generalized Rényi complexity" (GRC).
• Limit cases, such as $C^{(1,2)}_{(f,f)}$, connect to the structural entropy of the density.
• By appropriate choices of order and density, the RCR framework includes the López–Ruiz–Mancini–Calbet (LMC) complexity and its continuous versions.

This comparison is concrete in the following table:

Measure	Definition	Special Case of RCR
Generalized Rényi	$C^{(\alpha,\beta)}_{(f,f)}$	$C^{(\alpha,\beta)}_{(f,g)}$ with $f=g$
LMC Complexity (cont.)	$C_f = \exp(S_f)\cdot\exp(-R_f)$	$C^{(1,2)}_{(f,f)}$
Structural entropy	$\ln C^{(1,2)}_{(f,f)}$	Logarithm of LMC

6. Mathematical Rigour: Theorems on Continuity and Stability

The near continuity theorems proved in the paper establish mathematical robustness:

• δ-Neighboring Functions: If $f_1, f_2$ are δ-neighbors (differ only on a set of measure zero), $C^{(\alpha,\beta)}_{(f_1,g)}$ differs from $C^{(\alpha,\beta)}_{(f_2,g)}$ by arbitrarily little.
• Multidimensional Stability: These results generalize to higher-dimensional domains and to densities separated into radial, polar, and azimuthal contributions.

Thus, the RCR provides a well-defined, stable quantifier for comparing the complexity of quantum probability densities, with rigorous bounds ensuring that numerical or experimental errors in the density do not spuriously affect the result.

7. Significance and Interpretative Insight

The RCR captures both the global and local features of quantum probability densities:

• Structural Sensitivity: By leveraging Rényi entropy, it discriminates between quantum states not only by spectrum but by the finer structure of their densities.
• Robustness: Near continuity and Lebesgue measure permanence make the RCR suitable for practical applications typified by numerical noise or experimental uncertainties.
• Comparison of Degenerate Structures: In cases such as isospectral potentials, the RCR quantifies differences in informational content even when traditional energetic measures fail to distinguish state families.

This quantifier therefore enables nuanced distinctions between quantum states relevant in molecular physics, quantum information, and spectral theory, extending the landscape of complexity measures beyond standard entropy-based approaches.

• (Nath, 2020) Properties of Rényi complexity ratio of quantum states for central potential