Projected Ensemble (PE) in Quantum Systems

• Projected Ensemble (PE) is a framework that conditions quantum state distributions via projective measurements, capturing higher-order correlations beyond simple reduced density matrices.
• PE methods bridge integrable and chaotic regimes by linking the framework to constructs like Generalized Gibbs Ensembles and quantum state k-designs, with measurable timescales for thermalization.
• The PE paradigm extends to data science and cosmology, enabling applications in high-dimensional classification and field inference through observable-projected ensembles.

A projected ensemble (PE) is a construction that produces an ensemble of quantum states or distributions by conditioning on projective measurements or partial projections, typically within the context of many-body physics, quantum information, cosmology, or statistical inference. The PE framework underlies several analytic and computational methodologies for capturing conditional properties, universal statistics, and deep structural features of quantum and statistical systems.

1. Formal Definition and General Framework

Given a bipartite quantum system $A \cup B$ in a pure state $\ket{\Psi}_{AB}$, a projected ensemble is defined by making a complete projective measurement on subsystem $B$. Each measurement outcome is labeled by a pattern $\boldsymbol{z}_B$ corresponding to the eigenbasis of observables (e.g., occupation numbers for fermions: $\hat n_j = c_j^\dagger c_j$). The post-measurement state on $A$ is then

$$\ket{\Psi'(\boldsymbol{z}_B)}_A = \frac{\Pi_{\boldsymbol{z}_B}\ket{\Psi}}{\sqrt{p(\boldsymbol{z}_B)}}$$

where $$p(\boldsymbol{z}_B) = \|\Pi_{\boldsymbol{z}_B}\ket{\Psi}\|^2$$. The full PE is the ensemble $$\{p(\boldsymbol{z}_B), \ket{\Psi'(\boldsymbol{z}_B)}_A\}$$.

Physical diagnostics of the PE are expressed through $k$-moment tensors,

$$\rho^{(k)}_{\rm PE} = \sum_{\boldsymbol{z}_B} p(\boldsymbol{z}_B) \left( \ket{\Psi'(\boldsymbol{z}_B)}\bra{\Psi'(\boldsymbol{z}_B)} \right)^{\otimes k}$$

which capture all higher-order correlations and distinguish the PE from the simple reduced density matrix. Analogous constructions exist for measurements of non-complete bases or of specific observables (Milekhin et al., 2024), leading to observable-projected ensembles with generally mixed states.

2. PE in Quantum Statistical Mechanics: Integrability and Deep GGE

In integrable quadratic models, such as non-interacting fermionic tight-binding chains, PEs are directly related to Generalized Gibbs Ensembles (GGE). The standard GGE,

$$\rho_{\rm GGE} = \frac{1}{Z} \exp\left(-\sum_k \beta_k \hat n(k)\right)$$

is fixed by the conserved occupation numbers $\hat n(k)$. The first moment of the PE recovers the GGE, but all higher cumulants (covariances, entanglement entropies) are encoded in the full ensemble.

A more refined description, the "deep GGE" (dGGE), constructs a random Gaussian ensemble constrained to reproduce all conservation laws. In the limit of infinite bath ($L_B \to \infty$) and long-time evolution ($t \to \infty$), the PE and dGGE are numerically and analytically indistinguishable for all moments. At infinite temperature, the dGGE reduces to a Haar-random ensemble on the manifold of Gaussian states, and the post-measurement states sample the universal Haar random Gaussian projective ensemble (2207.13628).

3. Quantum Chaos, State Designs, and Deep Thermalization

In chaotic many-body systems, convergence of the PE to a quantum state $k$-design signals "deep thermalization," i.e., the approach to Haar-random statistics not just for first moments (reduced density matrices) but for all $k$-point functions. For random unitary circuit models, the PE becomes indistinguishable from the Haar ensemble (in the sense of $k$th frame potentials and moments) at time scales governed by subsystem size $L_A$ and system parameters.

Statistical-mechanical mappings (membrane or domain-wall pictures) connect the time evolution of the frame potential $F^{(k)}$ with fluctuating geometrical objects in the circuit, determining the scaling of "design times" $t_k$. In large-$q$ random circuits, all moments thermalize simultaneously with $t_k = O(|A|)$, whereas at finite $q$ there is a logarithmic scaling $t_k = O(\ln k)$ due to rare domain-wall fluctuations (Chan et al., 2024).

In dual-unitary circuits, the PE forms an exact state-design for all $k$ at the same time, providing an explicit scenario for instantaneous deep thermalization. Dynamical purification considerations connect the failure to achieve high-order designs to monitoring (measurement-induced) phases and information-scrambling characteristics. Separation of timescales between $k=1$ and $k\gg1$ signatures probes the scrambling velocity and depth of thermalization (Ippoliti et al., 2022).

4. Projected Ensembles and Conservation Laws in Many-Body Localization

In systems with an extensive set of conserved charges with local support (e.g., disordered Floquet chains or $\ell$-bit MBL models), the structure and universality of the PE depend sensitively on the relation between the measurement basis and the integrals of motion. For generic measurement bases (not aligned with local charges), the PE in the limit of large baths and long times converges to the Scrooge ensemble constructed from the steady-state reduced density matrix,

$$\mathcal{E}_\text{Scrooge}(\rho) = \left\{ D_A \langle\psi|\rho|\psi\rangle, \frac{\sqrt{\rho}|\psi\rangle}{\sqrt{\langle\psi|\rho|\psi\rangle}} \right\}_{|\psi\rangle\sim\mathrm{Haar}}$$

which maximizes entropy subject to $\rho$ as the first moment. The higher moments and distributions, including the Porter–Thomas law for measurement outcome probabilities, are diagnostic of the universal deep-thermalized regime (Manna et al., 3 Jan 2025).

By contrast, for global conserved charges, the PE retains a mixture structure over the resolved values of the charge in $B$, reflecting the greater "rigidity" of global symmetry constraints. This leads to convex mixtures of Scrooge ensembles over charge sectors, rather than single universal ensembles.

5. Observable-Projected Ensembles and Partial Projections

PEs can be generalized to ensembles generated by projective measurements of arbitrary local observables ("observable-projected ensembles" or OPEs). Instead of projecting onto a full basis in $B$, a total operator (e.g., $Q_B = \int_B dx\,\mathcal{O}_x$) is measured, yielding an outcome $q$ and a mixed conditional state on $A$: $$\rho_{A,q} = \frac{\widetilde{\rho}_{A,q}}{p(q)}, \quad \widetilde{\rho}_{A,q} = \mathrm{Tr}_B \left[(I_A \otimes P_q)\,|\Psi\rangle\langle\Psi|\,(I_A \otimes P_q)\right]$$ where $P_q = \delta(q-Q_B)$. OPEs are analytically tractable in conformal field theory, where their moments reduce to charged correlators on replica Riemann surfaces, allowing computation of entanglement measures, Holevo information, and geometry-dependent corrections.

A significant practical advantage of OPEs is the linear scaling of the number of measurement outcomes ($O(|B|)$), as opposed to the exponential scaling for full projective PE, facilitating both analytical and experimental exploration of entanglement structures in many-body systems (Milekhin et al., 2024).

6. Projected Ensemble Methods in Data Analysis and Cosmology

Beyond quantum many-body physics, the PE framework finds application in statistical data analysis and cosmological field inference.

• In high-dimensional classification, the projected ensemble ("ensemble of randomly projected linear discriminants") averages LDA classifiers over random $k$-dimensional subspaces. In the large-sample, high-dimensional limit, this construction acts as an implicit regularizer, closely related to ridge regression. Asymptotic misclassification probabilities and estimators for optimal projection dimension can be derived via random matrix theory, with the PE ensemble outperforming traditional LDA in small-sample, high-noise settings (Niyazi et al., 2020).
• In cosmological large-scale structure, the "projected-ensemble" construction specifies the distribution of evolved projected fields (e.g., weak lensing mass maps) conditioned on fixed initial projected fields. The analytic mapping for the mean and covariance at next-to-leading order in Lagrangian perturbation theory quantifies how the initial projected information is exponentially suppressed on nonlinear scales, enabling systematic field-level likelihood evaluation and hybrid inference methods in data analysis (Hong et al., 15 Oct 2025).

7. Canonical and Symmetry-Projected Ensembles in Statistical Physics

The PE paradigm encompasses traditional approaches to symmetry projection in thermal ensembles. In the projected thermal Hartree-Fock-Bogoliubov (HFB) framework, the partition function is constructed by projecting the statistical density operator onto fixed particle number (or other good quantum numbers) via an integral over group rotations. The partition function,

$$Z_N = \frac{1}{2\pi}\int_0^{2\pi} d\varphi\, e^{-i\varphi N} \mathrm{Tr}[e^{i\varphi \hat{N}} e^{-\beta \hat{H}_{\rm HFB}}]$$

is evaluated using combinatorial symmetric functions of the eigenvalues of the rotated mean-field matrices, enabling exact evaluation of equilibrium properties, gradients, and observables in the canonical ensemble (Puddu, 2011).

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Summary Table: Key PE Variants Across Domains

System/Domain	PE Type / Projection	Core Structural Feature
Quantum chaotic chain	Full basis in $B$	Haar state design, deep thermalization
Free fermions (integrable chain)	Density operators in $B$	GGE/dGGE, Gaussian covariance structure
Many-body localized systems ($\ell$-bit)	Basis non-aligned to charges	Universal Scrooge ensemble, minimal info
Conformal field theory	Observable-projection (OPE)	Mixed-state ensembles, analytic CFT control
High-dimensional statistics	Random subspace projection	Covariance regularization, ensemble classifier
Cosmological fields	3D→2D mode projection	Conditional mean and covariance, exponential decorrelation

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Projected ensembles constitute a unifying paradigm across quantum statistical mechanics, classical field theory, data science, and cosmology. They provide rigorous characterizations of deep equilibrium and out-of-equilibrium statistics, encoding information-accessible distributions conditioned on measurements or symmetry constraints, and connect fundamental questions of universality, information loss, and state complexity to practical computational and inferential methodologies (2207.13628, Milekhin et al., 2024, Manna et al., 3 Jan 2025, Chan et al., 2024, Niyazi et al., 2020, Hong et al., 15 Oct 2025, Puddu, 2011).