Partial Projected Ensembles

• Partial projected ensembles are frameworks defined by conditioning on selective projections or marginalizations to reveal finer correlation structures than conventional methods.
• They enable refined diagnostics in quantum information, statistical mechanics, and machine learning by uncovering hidden phase transitions and emergent universality.
• This approach offers scalable computational advantages, precise state estimation, and enhanced robustness in high-dimensional data assimilation and clustering.

A partial projected ensemble is a general framework where an ensemble is constructed by conditioning on partial or incomplete projection operations—typically selective measurement, projection, or marginalization—on a subset of coordinates, variables, or subsystems, rather than on the full complement. This concept arises independently in quantum information, statistical mechanics, high-dimensional probability, point process theory, data assimilation, structured learning, and constraint enumeration, providing refined probes of structure, complexity, correlation, and ergodicity that are invisible to conventional (fully projected or marginal) ensembles. Formally, partial projected ensembles are ensembles of (possibly mixed) states or measures on a subsystem, generated by projective, marginal, or partial assignments on an overlapping, non-fully-complementary part of the remaining degrees of freedom.

1. Formal Definitions and General Construction Principles

In the quantum setting, a partial projected ensemble (PPE) is obtained by projectively measuring a subsystem $B_1$ and discarding (tracing out) an additional subsystem $B_2$ in a tripartite Hilbert space $A \otimes B_1 \otimes B_2$; this produces an ensemble of mixed states on $A$: $${\cal E}_{\rm PPE}^{A} = \left\{\,p(z_{B_1}),\,\rho_A(z_{B_1})\right\}_{z_{B_1}}$$ with

$$\rho_A(z_{B_1}) = \frac{ \mathrm{Tr}_{B_1 B_2}\left[ \Pi_{z_{B_1}} |\Psi\rangle\langle\Psi| \Pi_{z_{B_1}} \right] }{ p(z_{B_1}) }$$

and $$p(z_{B_1}) = \langle\Psi | \Pi_{z_{B_1}} |\Psi\rangle$$. The first moment always yields the reduced density matrix of $A$. This paradigm generalizes to classical probability, data models, and point process settings, where only a selected subset of variables or modes is projected or marginalized, generating an ensemble over the remaining variables with higher-order structure than standard marginalization.

2. Statistical Structure and Diagnostic Capability

Partial projected ensembles provide strictly finer probes of information and correlation structure than conventional ensembles. For example, in tripartite quantum systems, the Holevo information of a PPE (where only part of the complement is measured and the rest discarded) can sharply distinguish information phases—regimes where classical information about measurement outcomes is either invisible or visible to the remaining subsystem—whereas conventional bipartite entanglement measures (e.g., logarithmic negativity) cannot. In particular, there exists an MIQC (measurement-invisible quantum-correlated) phase where the Holevo information decays exponentially with system size, even as bipartite entanglement remains volume-law; above a threshold in measured subsystem size, the PPE transitions to an MVQC (measurement-visible) phase, with a nonzero volume-law Holevo information, producing a non-analytic phase boundary invisible to entanglement negativity (Sherry et al., 13 Nov 2025).

Similarly, the k-th moment (e.g., variance) of a PPE, or other statistical functionals (like trace distance of higher moments or distributions of conditional probabilities), diagnose spatiotemporal spreading, scrambling, or universality classes, exploiting the ensemble's structure beyond what is accessible from first moments (Mandal et al., 7 Aug 2025, O'Donovan et al., 19 Feb 2025).

3. Applications across Domains

Partial projected ensembles have been independently instantiated in a variety of mathematical and applied contexts:

• Quantum information and scrambling: PPEs capture the fine structure of information scrambling, tracking how information about local measurements is hidden or spread by many-body dynamics. Fluctuations in the PPE, such as the variance of conditional states or the distribution of bit-string probabilities, reveal the causal structure, lightcone dynamics, and local integrals of motion in both ergodic and many-body-localized (MBL) regimes (Mandal et al., 7 Aug 2025, Manna et al., 3 Jan 2025).
• Random clustering and classification: Ensembles are constructed by considering random projections of feature vectors to low-dimensional subspaces, fitting models (e.g., Gaussian mixture models) in each subspace, and aggregating the results via consensus. By selecting a partial set of projections and states (those providing best fit), the so-constructed "random-projection ensemble" or partial projected ensemble can outperform full-dimensional approaches, especially in ultra-high dimensions (Anderlucci et al., 2019, Silva et al., 2018, Tomita et al., 2015).
• Determinantal Point Processes (DPPs): In the theory of point processes, partial-projection DPPs (pp-DPPs) are extended L-ensembles where only a partial set of directions is forced deterministically (projection), with the remainder randomly filled according to an L-ensemble kernel. Partial-projection DPPs interpolate between projection DPPs and L-ensembles, arise as universal limits of "flattened" kernels, and provide a continuous family connecting hard and soft repulsion regimes (Barthelmé et al., 2020).
• Data assimilation and filtering: Partial projected ensembles arise in particle filtering where the likelihood or observation operator is projected onto a low-rank subspace of dynamically relevant directions (e.g., unstable/neutral modes in weather models), dramatically improving filter stability and efficiency in high dimension (Maclean et al., 2019).
• SAT/SMT and formal verification: In symbolic computation, partial projected ensembles correspond to the enumeration of minimal implicants (partial assignments) covering all satisfying assignments on a projected variable set. This minimizes redundancy and output size, compared to full enumeration, and results in an exponentially more succinct summary (Spallitta et al., 22 Oct 2024).

4. Mathematical Properties and Computational Complexity

PPEs are characterized by the following properties:

Property	PPE/pp-DPPs/Random Projection Ensembles	Conventional (full projection/marginal)
Moment structure	All moments accessible; higher moments sharply discriminate	First moment (marginal) usually sufficient
Sensitivity to erasure/projection	Exponential (quantum), polynomial (random) decay in discarded size	Aggregated effect via mixed state
Ensemble size/scalability	Compact representation via minimal or optimal coverings	Exponential in full complement size
Diagnostics of transitions/phases	Non-analytic transitions, universal lightcone structure	Often insensitive, e.g., no MIQC distinction

For instance, in quantum chaos, only higher moments of PPEs (e.g., variance of Renyi-2 entropy) provide exponentially decaying indicators for chaotic phases, as opposed to first moments which saturate in both chaotic and integrable regimes (O'Donovan et al., 19 Feb 2025). In SAT/SMT projected enumeration, aggressive implicant shrinking produces minimal disjoint coverings, reducing complexity from exponential in all variables to exponential in projected variables (Spallitta et al., 22 Oct 2024).

5. Physical and Algorithmic Significance

A central insight across applications is that partial projected ensembles interpolate between the extremes of marginalization and full projection, providing a rigorous operational interpolation between complete knowledge (pure states/full assignment) and maximal ignorance (average over unobserved/dropped degrees of freedom). By tuning the degree or location of projection, or the selection of projected subspace:

• In quantum dynamics, this allows direct experimental access to scrambling depth, lightcone velocities, and universal late-time ensemble structure (e.g., generalized Hilbert–Schmidt ensemble, Scrooge ensemble) as distinguished from GSEs in globally constrained systems (Mandal et al., 7 Aug 2025, Manna et al., 3 Jan 2025).
• In ensemble learning and clustering, this enables information aggregation from diverse low-dimensional random projections, boosting statistical efficiency, robustness to noise, and performance in high-dimensional tasks (Anderlucci et al., 2019, Silva et al., 2018, Tomita et al., 2015).
• In point process theory, the partial projection framework provides a universal boundary family for L-ensemble DPPs, capturing limits and scaling properties for flattened or singular kernels and supporting versatile repulsion structures (Barthelmé et al., 2020).

6. Representative Examples

Quantum Many-Body: Holevo Phases of PPE

A Haar-random tripartite state ($A,B_1,B_2$) shows a transition in PPE Holevo information:

• For $p<1-2\gamma,$ $\chi\rightarrow 0$ exponentially (MIQC phase).
• For $p>1-2\gamma,$ $\chi \sim (p+2\gamma-1)\ln 2\,N$ (MVQC phase). This divides a regime where measurement on $B_1$ is invisible to $A$ from one where it is visible (Sherry et al., 13 Nov 2025).

DPPs: Partial-Projection Construction

Given L-ensemble kernel $L(\epsilon) = \epsilon A + VV^T$, in the limit $\epsilon\to 0$,

• When $m>p$, the fixed-size DPP converges to a partial projection DPP based on $(A, V)$.
• This class possesses a mixture sampling structure and interpolates between soft (L-ensemble, $p=0$) and hard (full projection, $p=m$) repulsion (Barthelmé et al., 2020).

Clustering: Random Projection Ensemble

In high-dimensional clustering, one may generate $B$ independent random projections, select the $B^*$ providing the best statistical fit (BIC), and aggregate via consensus, forming a partial projected ensemble with significantly better robustness and accuracy relative to naïve high-dimensional mixture modeling (Anderlucci et al., 2019).

7. Outlook and Cross-Domain Universality

Research on partial projected ensembles establishes them as scalable, theoretically grounded tools for probing structure and diagnosing phases or transitions in high-dimensional and complex systems. Notably, their application to quantum simulation is experimentally feasible, requiring only polynomially many measurements and providing access to universal phenomena (e.g., scrambling plateau, causally sharp lightcones, etc.) (Mandal et al., 7 Aug 2025, Milekhin et al., 28 Oct 2024, Ge et al., 15 Aug 2024). In machine learning, random partial projection-based ensembles (sparse-oblique forests, projection pursuit forests) bridge performance and interpretability gaps between axis-aligned and fully oblique methods (Tomita et al., 2015, Silva et al., 2018).

A plausible implication is that PPEs may form a unifying framework for understanding emergent universality, robustness, and algorithmic efficiency across probabilistic, statistical, and physical systems, particularly in regimes where full marginalization or projection either oversimplifies or is computationally infeasible.