Many-body Projected Ensemble (MPE)

• MPE is a rigorous framework that defines pure-state output distributions from sequences of local quantum interventions in many-body systems.
• It computes higher-order statistical moments to diagnose chaos, scrambling, and deep thermalization in both unitary and measurement-monitored dynamics.
• MPE benchmarks quantum simulators by revealing emergent entanglement structures and dynamical transitions beyond traditional density-matrix approaches.

The Many-body Projected Ensemble (MPE) is a mathematically rigorous framework for probing the fine-grained statistical structure of local quantum states in many-body systems after projective measurements on complementary subsystems. Originally introduced as an operational ensemble derived from process tensors, the MPE encodes models of pure-state output distributions resulting from sequences of local interventions, offering a unified method to analyze chaos, scrambling, ergodicity, and deep thermalization in both unitary and measurement-monitored quantum dynamics (O'Donovan et al., 19 Feb 2025). Higher-order statistical moments of the MPE subsume traditional chaos quantifiers and reveal new entanglement structures not captured by reduced density matrices or low-order entropy measures, providing powerful tools for diagnosing dynamical regimes and benchmarking quantum simulators.

1. Mathematical Structure of the Many-body Projected Ensemble

Consider an $n$-step quantum process acting on system $S$ (optionally coupled to environment $E$), with initial pure state $|\rho_0\rangle$ in Hilbert space $\mathcal H_R$. The process tensor $$|\Upsilon\rangle \in \mathcal H_B \otimes \mathcal H_R$$ encodes all spacetime correlations resulting from sequences of local interventions $A_{i_k}$ at times $t_k$. An orthogonal basis of interventions $\{A_{i_1},A_{i_2},\dots,A_{i_n}\}$ is specified, satisfying

$$\sum_{i_k}A_{i_k}^\dagger A_{i_k}=\mathbb1_S,\quad \operatorname{Tr}[A_{i_k}^\dagger A_{j_k}]=\delta_{i_k j_k}.$$

For each sequence $\vec i = (i_1,\dots,i_n)$, the process generates an unnormalized output

$$|\psi_{\vec i}\rangle = A_{i_n} \circ \cdots \circ A_{i_1} [\rho_0]^{1/2} \in \mathcal H_R,$$

with norm $$p(\vec i) = \langle \psi_{\vec i}|\psi_{\vec i}\rangle$$ giving its probability. Normalized outputs are

$$|\widetilde\psi_{\vec i}\rangle = \frac{|\psi_{\vec i}\rangle}{\sqrt{p(\vec i)}}.$$

The Many-body Projected Ensemble is then

$$\mathcal E = \{p(\vec i), |\widetilde\psi_{\vec i}\rangle\}_{\vec i}.$$

This ensemble is defined for arbitrary choices of interventions and measurement protocols, allowing generalization to both deterministic (unitary) and stochastic (POVM) settings.

2. Statistical Moments and Chaos Diagnostics

The first moment operator is

$$\rho^{(1)} = \sum_{\vec i} p(\vec i) |\widetilde\psi_{\vec i}\rangle\langle\widetilde\psi_{\vec i}| = \sum_{\vec i} |\psi_{\vec i}\rangle\langle\psi_{\vec i}|,$$

which coincides with the reduced process-tensor state on $\mathcal H_R$. The second-Rényi quantum dynamical entropy (QDE) is

$$E(B:R) = S^{(2)}(\Upsilon_B) = -\log \operatorname{Tr}[(\Upsilon_B)^2] = -\log \operatorname{Tr}[(\rho^{(1)})^2],$$

recovering the Alicki–Fannes entropy in the large-$n$ limit. High QDE ($E(B:R)\approx n\log d_S$) implies maximal sensitivity: orthogonal interventions yield nearly orthogonal output states ("butterfly flutter fidelity").

A more refined probe is the spatiotemporal entanglement (STE),

$$E(BR_1:R_2) = S^{(2)}(\operatorname{Tr}_{BR_1}|\Upsilon\rangle\langle\Upsilon|) = -\log \operatorname{Tr}[(\Upsilon_{R_2})^2],$$

which quantifies simultaneous scrambling in space and time.

The $k$-th moment of the ensemble,

$$M^{(k)} = \sum_{\vec i} p(\vec i) \left(|\widetilde\psi_{\vec i}\rangle\langle\widetilde\psi_{\vec i}|\right)^{\otimes k},$$

provides access to higher-order "deep thermalization" statistics.

3. Dynamical Regimes: Chaos, Integrability, and Localization

Spectral analysis of $M^{(2)}$ discriminates chaotic from non-chaotic regimes:

• Chaos: $M^{(2)} \sim (\mathbb1 + \text{SWAP})/[d_R(d_R+1)]$; two nonzero eigenvalues $2/[d_R(d_R+1)]$, rest zero.
• Integrable/MBL: Broader spectrum, many small but nonzero eigenvalues.

Rényi-2 entropy $S^{(2)}(M^{(2)})$ saturates at $\log\binom{d_R+1}{2}$ in chaos, with exponentially vanishing variance in system size. Non-chaotic models feature suppressed entropy and persistent fluctuations.

Simulation of prototypical spin-chain models (integrable XXZ, chaotic XXZ+perturbation, Aubry–André crossover, free fermions) reveals:

• Linear growth of QDE and STE with number of interventions, saturating to near-Haar bounds exclusively in chaotic circuits.
• MPE entanglement variance decays exponentially in chaotic models, but remains constant or polynomial in integrable/MBL cases.
• Mean entanglement closely tracks the Page curve only in the chaotic regime.

4. Unified Approach to Unitary and Monitored Many-body Dynamics

The MPE framework accommodates

• Unitary evolution: Interventions are local unitaries.
• Monitored/open-system dynamics: Interventions are projectors or general POVM elements; enables study of measurement-induced randomness and entanglement transitions.

At low measurement rates, the ensemble’s statistical moments remain Haar-like in chaotic backgrounds, signaling resilience of quantum thermalization until a critical phase transition.

5. Connections to Quantum State Designs and Membrane Theory

Under repeated random unitary evolution, the MPE approaches a $k$-design: its first $k$ moments become indistinguishable from those of the Haar ensemble (Ippoliti et al., 2022, Chan et al., 2024). In dual-unitary circuits and in the large local Hilbert-space dimension ($q\to\infty$), all moments thermalize simultaneously, with design time scaling as $t_k=\mathcal O(L_A)$ for region A. In generic circuits, finite $q$ domain-wall fluctuations induce logarithmic corrections $t_k=\mathcal O(L_A)+\mathcal O(\log k)$, diagnostically mapped via frame potentials and geometric membrane models.

Table: MPE Design-Time Regimes

Regime	Scaling of $t_k$	Characteristic Physics
Dual-unitary, $q\to\infty$	$t_k \sim L_A/2\,\,\,\,\,\,\forall k$	Simultaneous thermalization
Finite $q$	$t_k \sim L_A + \log k$	Membrane fluctuations, hierarchy
Monitored	Rate-dependent	Measurement-induced transitions

The design-time hierarchy reflects the depth of information scrambling, with higher $k$ reflecting "deep thermalization" that is invisible to reduced density matrices.

6. Experimental Realization and Observables

Projected ensemble protocols have been implemented on superconducting quantum processors (Yan et al., 26 Jun 2025). Measurement circuits and state tomography reconstruct post-measurement states, enabling calculation of ensemble-averaged purity, second Rényi entropy, trace distances to Haar $k$-designs, and entanglement statistics. Observed features include:

• Approach to Haar benchmarks in ergodic sectors for both purity and entropic measures.
• PoP statistics showing Porter–Thomas behavior in chaotic cases, transitioning to Erlang or Scrooge ensemble distributions depending on conservation laws and localization.

Ensemble-averaged entropy serves as an experimentally accessible metric for information leakage, with saturation rates distinguishing between ergodic, integrable, and MBL dynamics.

7. Interpretation and Significance

The MPE paradigm supersedes conventional density-matrix approaches by capturing the full distribution of output pure states conditioned on measurement histories, exposing quantum many-body systems to deep probes of spatiotemporal complexity, chaos, and scrambling. Higher moments of the MPE reveal emergent universal ensembles (Haar in unconstrained chaos, Scrooge in constrained systems), encode the full state-design hierarchy, and underpin the operational definition of quantum thermalization beyond traditional eigenstate thermalization. The framework analytically and numerically diagnoses dynamical transitions, models membrane-driven scaling, and benchmarks quantum simulators against stringent ensemble criteria.

The MPE provides a rigorous, unified platform for characterizing deep ergodicity, quantum chaos, and information structure in complex quantum systems (O'Donovan et al., 19 Feb 2025, Ippoliti et al., 2022, Chan et al., 2024, Yan et al., 26 Jun 2025).