Hilbert-Space Return Probability

• Hilbert-space return probability is defined as the squared magnitude of the overlap between an initial pure state and its time-evolved version, encapsulating quantum recurrence dynamics.
• It offers a unifying measure by connecting the inverse participation ratio and spectral form factors, thereby distinguishing between ergodic and localized regimes.
• Applications span many-body dynamics and random matrix ensembles, providing quantitative insights into critical scaling, decoherence, and relaxation processes in quantum systems.

The Hilbert-space return probability quantifies the likelihood that a quantum system initially prepared in a pure state will be found in the same state after some temporal evolution induced by its Hamiltonian or Floquet operator. As a unifying measure across quantum statistical mechanics, many-body theory, open quantum dynamics, and critical random matrix ensembles, the return probability provides fundamental insights into ergodicity, localization, relaxation dynamics, and spectral characteristics of the underlying quantum system.

1. Formal Definition and Spectral Representation

For a closed quantum system with time-evolution operator $U(t)$ and initial pure state %%%%1%%%%, the Hilbert-space return probability at time $t$ is

$$P_{\rm ret}(t) = |\langle \psi | U(t) | \psi \rangle |^2$$

Given an energy eigenbasis $|E_\alpha\rangle$, this expands as

$$P_{\rm ret}(t) = \sum_{\alpha,\beta} p_\alpha p_\beta e^{i(E_\beta-E_\alpha)t}$$

with $p_\alpha = |\langle E_\alpha | \psi \rangle|^2$ (Cohen et al., 2015).

The long-time average for a nondegenerate spectrum is

$$\langle P_{\rm ret} \rangle = \sum_\alpha p_\alpha^2 = 1/\mathcal{P}$$

where $\mathcal{P}$ is the inverse participation ratio (IPR), directly encoding localization in the energy basis.

For ensembles, the averaged return probability links to spectral statistics: $$\langle P_{\rm ret}(t) \rangle_{\psi} = \frac{1+K(t)}{N+1}$$ where $K(t) = \frac{1}{N} |\mathrm{Tr} U(t)|^2$ is the spectral form factor (Hahn et al., 2020). This establishes a direct bridge between return probability and two-point spectral correlations.

2. Connections to Localization and Ergodicity

The return probability is a cornerstone in quantifying Hilbert-space localization, contrasting the ergodic paradigm. In localized phases, $P_{\rm ret}(t)$ remains finite at long times, with the IPR dominating its average, whereas in ergodic systems, it decays irreversibly and the participation numbers approach their maximal thermal values (Cohen et al., 2015).

A rigorous framework defines the “localization fraction”: $\mathcal{F} = \mathcal{N}_\infty / \mathcal{N}_E$ where $\mathcal{N}_\infty = [\sum_\alpha p_\alpha^2]^{-1}$ and $\mathcal{N}_E$ is the effective Hilbert-space dimension within the energy shell. $\mathcal{F} \ll 1$ signals strong localization, $\mathcal{F} \sim O(1)$ ergodicity.

Semiclassical analysis (e.g., Heller’s approach) elucidates how phase-space exploration and classical quantum correspondence determine the temporal evolution and eventual breakdown into localized or ergodic regimes. In strong localization, quantum recurrences lead to persistent $P_{\rm ret}(t)$, while in chaotic regimes, quantum spread rapidly saturates classical exploration, quenching recurrences.

3. Statistical and Ensemble Averages

The Hilbert-space-averaging method computes expectation values over random state distributions:

• For Haar-uniform $|\psi\rangle$ in $\mathbb{C}^N$,

$$\langle |\langle \chi | A | \psi \rangle |^2 \rangle_\psi = \frac{\langle \chi | AA^\dagger | \chi \rangle}{N}$$

Single and double state averages (with or without overlap constraints) admit closed analytic forms (Hahn et al., 2020).

• For transition probabilities between states with overlap $z$, the average is given by

$$\frac{1-z^2}{N-1}\langle \chi|AA^\dagger|\chi\rangle + \frac{N z^2 - 1}{N-1}|\langle\chi|A|\chi\rangle|^2$$

with further averaging possible over ensembles parametrized by statistical operators $\rho$.

Numerical studies on kicked spin chains confirm analytic predictions, including ensemble fluctuation widths and significant distinctions between integrable and chaotic regimes, manifested in spectral form factor behaviors (Hahn et al., 2020).

4. Return Probability, Autocorrelations, and Many-Body Relaxation

In interacting quantum many-body systems, the return probability reveals substantial parallels with local real-space autocorrelations. For a chain of $L$ spins-$1/2$ at infinite temperature, the averaged return probability

$$R(t) = \frac{1}{H} \sum_{I=1}^H |\langle s_I | U(t) | s_I \rangle|^2$$

approaches late-time saturation following the same functional decay laws as the local autocorrelation $C(t)$, modulo renormalized exponents (Pain et al., 2023).

Explicitly,

$$[C(t)]^\alpha \propto \frac{1}{L} \ln \frac{R(t)}{R_\infty}$$

and functional forms (power-law, stretched exponential) as observed in $C(t)$ translate to $R(t)$, with exponents/time-scales renormalized by $\alpha$.

This mapping, valid in large $L$ ergodic limits, substantiates the transferability of analytical and numerical methods between Hilbert-space return probability and real-space observable relaxation, particularly in disordered spin chains.

5. Open Quantum Dynamics and Quantized Recurrence

The concept of Hilbert-space return probability extends to iterated open quantum dynamical systems, governed by quantum channels $\Phi$ and Kraus operators. The relevant Hilbert subspace is defined as the support of all states accessible from the initial $|\Psi\rangle$: $$\Pi_{\text{rel}} = \text{supp} \left( \sum_{t=0}^\infty \rho(t) \right ), \quad \rho(t) = \Phi^t[|\Psi\rangle\langle \Psi|]$$ If $\Phi$ is unital on this subspace, the expected first-return time is strictly quantized: $\langle T \rangle = \text{dim} H_{\text{rel}}$ This result generalizes Kac’s classical lemma for Markov chains and unitary quantum walks, providing a universal measure for the accessible state-space (Sinkovicz et al., 2014). When unitality is absent, $\langle T \rangle$ varies continuously, correlating with open-system decoherence and state-space contraction.

6. Return Probability in Random Matrix Ensembles and Critical Scaling

In critical random matrix ensembles, such as the power-law random banded matrix model,

$$P_N(t) = \int_{-\infty}^\infty d\omega\, e^{-i\omega t}\, C(\omega)$$

exhibits multifractal scaling governed by the fractal dimension $d_2$ and the dynamical scaling exponent $\mu$. At large $N$,

$\lim_{t\to \infty} P_N(t) \sim N^{-d_2}$

and at large $t$,

$P_N(t) \sim t^{-\mu}$

The Chalker's ansatz yields $\mu = 1 - d_2/d$ (with $d=1$ for PLBRM), analytically confirmed up to second order in the perturbation parameter (Kravtsov et al., 2011). This framework links spatial multifractality of eigenstates with slow power-law decays of return probability, underpinning quantum criticality.

7. Return Probability in Stochastic Processes and Martingales

In the context of Hilbert-space valued martingales, the return probability is analogous to small-ball probabilities: $P(\|X_t\|_\mathcal{H} \leq R \mid X_0 = x_0)$ Under bounded increment and unit conditional variance conditions, the probability for return near the origin scales as

$$P(\|X_t\| \leq R ) \leq c \frac{R}{\sqrt{t}} e^{-\|x_0\|^2/(6 L^2 t)}$$

matching classical Gaussian random walk behaviors in infinite dimensions (Lee et al., 2014). This result extends to random walks on vertex-transitive graphs and provides sharp universal bounds for dynamical returns in high-dimensional stochastic systems.

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The Hilbert-space return probability serves as a quantitative nexus between quantum dynamics, spectral structure, localization measures, and relaxation laws, underpinning diverse phenomena ranging from thermalization in many-body systems to random matrix criticality and stochastic process theory. Its analytic calculation—whether via Haar ensemble averaging, participation ratios, spectral form factors, or martingale theory—continues to reveal core principles governing quantum recurrence, ergodicity, and localization across physical and mathematical domains.