Exact Hilbert-space ergodicity from continuous monitoring

Technical Summary: Exact Hilbert-space Ergodicity from Continuous Monitoring

Problem Statement
Quantum many-body systems are generally expected to evolve toward equilibrium, a process often justified by the concept of Hilbert-space ergodicity. This principle posits that pure-state evolution explores the Hilbert space uniformly, up to physical constraints. In the absence of constraints, this is rigorously defined by the generation of a Haar-random ensemble. However, many physical systems cannot achieve full Haar-randomness due to constraints (e.g., conserved charges or symmetries) yet still exhibit chaotic and scrambling behaviors. Recent theoretical progress suggests that the "Scrooge ensemble"—a Haar-random ensemble deformed by a target density matrix $\sigma$—provides the correct universal description for such constrained systems. The Scrooge ensemble minimizes accessible information among all ensembles with a fixed average density matrix $\sigma$. While the ergodicity of time-independent Hamiltonian dynamics and projective ensembles has been linked to Scrooge ensembles, a rigorous mechanism to engineer exact Scrooge ensembles as the unique late-time equilibrium of quantum trajectories has been lacking.

Methodology
The authors propose a scheme to generate the Scrooge ensemble through continuous monitoring of a quantum many-body system. The protocol involves:

1. Continuous Weak Measurements: The system is subjected to a sequence of weak measurements implemented via coupling to ancillas initialized in $|0\rangle$ and measured in the $x$-basis. This process is described by a stochastic Schrödinger equation (SSE) driven by Wiener processes.
2. Operator Design: The jump operators $c_\alpha$ governing the monitoring are constructed as $c_\alpha = \sqrt{\kappa D p_\alpha} \sqrt{\sigma} U_\alpha$, where $\sigma$ is the target density matrix and $\{p_\alpha, U_\alpha\}$ is a unitary ensemble.
3. Conditions on Unitaries: The unitary ensemble $\{U_\alpha\}$ is required to satisfy two conditions:
   • Unitary 1-design: $\sum_\alpha p_\alpha U_\alpha Q U_\alpha^\dagger = \frac{\text{Tr}[Q]}{D} I$ for any operator $Q$.
   • Phase Neutrality: $\sum_\alpha p_\alpha U_\alpha \otimes U_\alpha = 0$.
     The authors note that these conditions are significantly weaker than requiring the jump operators to form a full unitary $t$-design or be Haar-random.

Key Contributions and Theoretical Results
The paper provides a rigorous proof that under the proposed monitoring scheme, the ensemble of quantum trajectories converges exactly to the Scrooge ensemble associated with $\sigma$. The proof proceeds in two main stages:

1. Mapping to Classical Stochastic Dynamics:
   The authors parameterize the pure state $|\psi\rangle$ in the eigenbasis of $\sigma$ using amplitude variables $y_i = |\psi_i|^2$ and relative phase variables $\phi_i$. They derive the Fokker-Planck equation governing the evolution of the probability distribution $p(y, \phi, t)$ of these classical variables.

   • The drift vector for amplitudes is shown to be $v_i^{(a)} = \kappa(\lambda_i - y_i)$, driving the system toward the eigenvalues of $\sigma$.
   • The diffusion matrices for amplitudes and phases are derived explicitly, relying on the unitary 1-design and phase-neutral conditions to simplify the noise correlations.
   • Crucially, the mixed diffusion terms between amplitudes and phases vanish due to the symmetry constraints.

2. Stationarity and Uniqueness:

   • Stationarity: The authors verify that the Scrooge probability distribution, $p_{sc}(y, \phi) \propto H(y)^{-(D+1)}$ (where $H(y) = \sum y_k/\lambda_k$), is a stationary solution of the derived Fokker-Planck equation. This is done by showing the probability current vanishes pointwise.
   • Uniqueness: Using Doob's theorem, the authors prove the uniqueness of the steady state. They establish that the diffusion matrix is uniformly positive definite (uniformly elliptic) on the projective Hilbert space ($\mathbb{C}P^{D-1}$). This ensures that the diffusion process connects all states, preventing the system from getting trapped in non-ergodic subspaces, thereby guaranteeing convergence to the unique Scrooge ensemble from any initial state.

Numerical Results
The authors numerically validate their theoretical predictions using qubit systems ($N=2, 4$) with target density matrices corresponding to thermal states of the quantum Ising model (both full-rank and symmetry-projected).

• Convergence: Simulations show that the distance between the trajectory ensemble moments and the target Scrooge moments decays rapidly to zero.
• Relaxation Rate: The decay of distances for moments $n=1, 2, 3$ follows an exponential decay with rate $\kappa$, consistent with the relaxation rate of the first moment derived from the Lindblad master equation. This suggests that the relaxation of all moments is governed by the same leading eigenmode of the Fokker-Planck operator.
• Experimental Benchmarking: To address the experimental challenge of full state tomography (which scales exponentially), the authors propose using quantum-classical cross-correlators ($C^{QC}_2$). Numerical results demonstrate that the deviation of this correlator from the Scrooge prediction decays exponentially, providing an experimentally accessible diagnostic for the emergence of the Scrooge ensemble.

Significance and Claims
The paper claims to establish a rigorous mechanism for the emergence of exact Hilbert-space ergodicity under an arbitrary prescribed density matrix via continuous measurements.

• Weak Conditions: A primary claim is that generating the maximally random ensemble (Scrooge) does not require maximally random jump operators. Instead, the much weaker condition of a deformed unitary 1-design suffices.
• Engineering Randomness: The results demonstrate that higher-order statistical properties of quantum states can be systematically engineered via measurement design.
• Practical Utility: The work provides a practical route for investigating constrained random-state ensembles on quantum devices. By preparing exact Scrooge ensembles, the authors suggest this could serve as a resource for studying information hiding, state indistinguishability, and the universal statistical properties of complex quantum states.
• Generality: The mechanism is broadly applicable across different quantum platforms as it does not rely on specific microscopic details of the system Hamiltonian, only on the design of the monitoring operators.

The authors remain modest regarding future directions, noting that while the observed relaxation rate $\kappa$ for higher moments is consistent across orders, an analytical proof of whether this reflects a deeper spectral property of the Fokker-Planck operator remains an open theoretical question. They also suggest extending the approach to ensembles with additional conserved quantities as a potential future direction.