Subsystem Complexity Dynamics

• Subsystem Complexity Dynamics is defined by the evolution of a quantum subsystem’s reduced density matrix under chaotic Hamiltonians, capturing coherence loss and entanglement generation.
• Analytic methods, including noncentral correlated Wishart ensembles and the Marčenko–Pastur law, reveal explicit spectral properties and transition behaviors.
• The framework quantitatively connects unitary evolution with decoherence and dynamical phase transitions, informing protocols for random quantum state generation.

Subsystem complexity dynamics encompasses the time-dependent and structural behavior of a subsystem’s reduced density matrix when the total (closed) quantum system evolves unitarily under a sufficiently complex or quantum chaotic Hamiltonian. This domain addresses not only the decay of coherences, purity, and entanglement within the subsystem, but also explores the emergence of statistical properties such as spectral distributions, convergence rates to “randomness,” and the occurrence of dynamical phase transitions in eigenvalue statistics. The statistical modeling is grounded in noncentral correlated Wishart ensembles and reveals signatures such as transitions in spectral properties and sharp phase changes—features that are central to understanding decoherence, entanglement generation, and complexity buildup in many-body quantum systems.

1. Unitary Evolution and Random Matrix Modeling

The evolution of a closed quantum system is governed by a time-independent Hamiltonian $H$, with the full system’s state evolving via $|\psi(t)\rangle = U^t|\psi(0)\rangle$ where $U^t = \exp(-Ht)$. Focusing on a subsystem of Hilbert space dimension $N$ coupled to an environment of dimension $M$, the subsystem density matrix is $$\rho(t) = \operatorname{Tr}_{\text{env}} |\psi(t)\rangle\langle\psi(t)|$$. The problem considers $H$ as a random matrix with eigenvectors distributed according to the Haar measure (the Random Density Matrix Model, RDMM), facilitating exact analytic averaging of subsystem properties over all initial states.

The time evolution of subsystem observables $A$ obeys

$$\langle A(t)\rangle = f_t A(0), \quad f_t = \frac{1}{NM}\sum_j e^{-E_j t}$$

where $E_j$ are the eigenvalues of $H$. The reduced density matrix averages as

$$\langle \rho(t)\rangle = \frac{(NM)^2|f_t|^2-1}{(NM)^2-1} \rho^{(0)} + \frac{NM^2(1-|f_t|^2)}{(NM)^2-1} \mathbb{1}_N$$

with $\rho^{(0)}$ the initial reduced density matrix and $\mathbb{1}_N$ the $N\times N$ identity.

2. Decoherence, Purity, and Statistical Fluctuations

The decoherence of the subsystem is precisely encoded in the off-diagonal elements of $\rho(t)$, with their decay directly linked to the decay of $|f_t|^2$. The subsystem’s purity $I(t) = \operatorname{Tr}[\rho(t)^2]$ is analytically tractable:

$$I(t) = I_r + B [\mathcal{N}^2|f_t|^4+\mathcal{N} v_t+|f_{2t}|^2-4|f_t|^2]$$

where $I_r = \frac{N+M}{1+NM}$ (the typical random state purity), $\mathcal{N}=NM$, $v_t = [f_t^2]^*f_{2t} + f_t^2[f_{2t}]^*$, and $$B = \frac{(N-1)(M-1)}{(\mathcal{N}+3)(\mathcal{N}+1)(\mathcal{N}-1)}$$. For a Gaussian Unitary Ensemble (GUE) Hamiltonian in the large $NM$ limit, $f_t$ is approximated by $g_t = J_1(2t)/t$ (with $J_1$ the Bessel function), yielding asymptotic purity

$$I(t) \approx g_t^4 + \frac{(1-g_t^4)(N+M)}{NM} + \frac{2(g_t^2 g_{2t} - g_t^4)}{NM} + \mathcal{O}(1/\mathcal{N}^2)$$

The variance of the largest eigenvalue $\lambda_1$ is

$$\sigma^2(\lambda_1) \approx \frac{2g_t^2 + 2g_t^2 g_{2t} - 4g_t^4}{NM} + \mathcal{O}\left(\frac{1}{(NM)^2}\right)$$

demonstrating fluctuations in the eigenvalue spectrum driven by the chaotic evolution.

3. Noncentral Correlated Wishart Ensemble and Spectral Statistics

To describe the full statistics of $\rho(t)$, the reduced density matrix is mapped onto a noncentral correlated Wishart ensemble (n-CWE). The model

$W = ZZ^\dagger, \quad Z = Y + \sqrt{\xi} X$

with $X$ an $N\times M$ matrix of i.i.d. Gaussian variables ($\langle|X_{jk}|^2\rangle=1/(NM)$), $Y$ a shift matrix encoding the nonzero mean, and $\xi$ a positive-definite matrix of correlations. The shift and correlation structure are chosen as

$$Y = g_t A(0), \quad \sigma^2\xi = \langle\rho(t)\rangle - |g_t|^2 \rho^{(0)}$$

with $\sigma^2=1/N$. For large $N$, $M$, one can set $\xi \simeq (1-h_t^2)\mathbb{1}$, $h_t \simeq g_t^2$. The bulk eigenvalue distribution is given by a rescaled Marčenko-Pastur (MP) law:

$$p_{MP}(\lambda) = \frac{\kappa}{2\pi\lambda\sigma_{\text{eff}}^2}\sqrt{(\lambda_+ - \lambda)(\lambda - \lambda_-)}$$

with $\kappa = M/N$, $\sigma_{\text{eff}}^2 = (1-r)\sigma^2$, and spectral edges $\lambda_\pm$. The largest eigenvalue remains isolated from the bulk for $r > 1/(N\sqrt{\kappa})$, with mean value

$$\langle\lambda_1\rangle = \frac{\sigma^2[(Nr + 1 - r)(Nr\kappa + 1 - r)]}{Nr\kappa}$$

4. Convergence to Random State and Timescales

The transition from an arbitrary initial state to a random (maximally mixed) state is governed by the decay of $g_t$, which is the Fourier transform of the spectral density of $H$. The time to convergence, i.e., when the purity $I(t)$ becomes indistinguishable from the random state purity $I_r$, is reached when $g_t \sim 1/N^{1/4}$, with $I_r \sim 1/N$. If $H$ is GUE, $g_t$ vanishes at order $\tau \sim 1$ (set by the spectral width), while for Hamiltonians with exponential spectra, $\tau \sim \log N$. When implementing such dynamics by quantum circuits (as a sequence of two-qubit gates), the convergence time scales as $\sim n^2$, with $n = \log_2 N$, aligning with protocols for generating random quantum states.

5. Eigenvalue Collisions and Dynamical Phase Transitions

A prominent feature of subsystem complexity is the occurrence of eigenvalue “collisions”: The largest eigenvalue $\lambda_1$ of the reduced density matrix remains separated from the MP bulk for most times, colliding with the upper bulk edge whenever $g_t$ (and thus $r$) oscillates through certain values. At the collision point, the gap $\langle\lambda_1\rangle-\langle\lambda_2\rangle$ closes as $\sim 1/N^2$, and the fluctuation statistics of $\lambda_1$ shift from Gaussian to Tracy–Widom form. This sequence of collisions signals a cascade of dynamical phase transitions in the subsystem’s spectral properties, reflecting the complex interplay between memory of initial conditions and randomization induced by chaotic evolution.

6. Broader Implications for Subsystem Complexity

The analytic expressions for purity, eigenvalue statistics, and convergence time reveal that random matrix descriptions—via n-CWE mapping—provide accurate predictions for the entire evolution of subsystem complexity. Key implications include:

• The loss of memory of the initial state, quantified by explicit time dependence of $f_t$ and $g_t$, and rapid approach to a Haar-random (or maximally mixed) regime.
• The connection between decay times and system parameters (spectral bandwidth, system size, and gate implementation).
• The identification of phase transition points in the spectral evolution—a direct manifestation of dynamical complexity tied to entanglement generation and decoherence.

These results inform protocols for generating random quantum states, benchmarking quantum chaos, and understanding universal features of subsystem dynamics in complex quantum systems. The analysis—using explicit formulas and statistical modeling—integrates chaotic quantum evolution, non-equilibrium statistical mechanics, and random matrix theory to provide a quantitative and predictive framework for subsystem complexity dynamics.