Partial Projected Ensemble (PPE)

• Partial Projected Ensemble (PPE) is a framework where only part of the subsystem complement is measured, yielding mixed state ensembles with rich statistical structure.
• It generalizes traditional projected ensembles to a tripartite system, capturing higher moments and offering insights into deep thermalisation and information scrambling.
• Diagnostic measures like Holevo information and PPE fluctuations reveal distinct dynamical regimes between ergodic and many-body localized systems.

Partial projected ensemble (PPE) denotes a generalization of the projected ensemble in which only part of the complement of a subsystem is measured and part is discarded, so that the induced ensemble on the subsystem consists of mixed rather than pure states. In the tripartite formulation, a global pure state on $R\cup E\cup S$ is projectively measured on $S$, the outcomes on $S$ are retained, and the region $E$ is traced out, yielding an ensemble $$\mathcal{E}_{\text{PPE}_R}=\{p(o_S),\rho_R(o_S)\}_{o_S}$$ on $R$ (Sherry et al., 13 Nov 2025). This framework has been developed as a higher-order probe of deep thermalisation, information scrambling, and multipartite information structure, and it has also been realized in dynamical many-body settings where PPE fluctuations track linear or logarithmic lightcones depending on whether the dynamics are ergodic or many-body localized (Mandal et al., 7 Aug 2025).

1. Definition and formal structure

The starting point is the projected ensemble (PE) for a bipartite pure state $|\Psi\rangle_{AB}$. Measuring subsystem $B$ in an orthonormal basis $\{|o_B\rangle\}$ produces Born probabilities

$$p(o_B)=\langle \Psi|\Pi_{o_B}|\Psi\rangle,\qquad \Pi_{o_B}=|o_B\rangle\langle o_B|,$$

and conditional pure states $S$0 on $S$1, so that

$S$2

Its $S$3-th moment is

$S$4

with $S$5 (Sherry et al., 13 Nov 2025).

The PPE extends this to a tripartite system $S$6. One measures only $S$7 in a local product basis $S$8, retains those outcomes, and discards $S$9. The defining quantities are

$S$0

and

$S$1

The partial projected ensemble is then

$S$2

with moment hierarchy

$S$3

As in the PE, the first moment is just the reduced density matrix,

$S$4

while $S$5 moments encode nonlinear information about the ensemble of conditional states that is not reconstructible from $S$6 alone (Sherry et al., 13 Nov 2025).

A useful structural identity is that the PPE on $S$7 can be obtained by tracing out $S$8 from a PE on $S$9 built from measurements on $E$0: $E$1 This relation makes the PPE a mixed-state descendant of the PE and underlies the emergence of generalized Hilbert–Schmidt ensembles as maximum-entropy limits (Sherry et al., 13 Nov 2025).

A related construction appears in observable-projected ensembles: one measures an extensive Hermitian operator $E$2 on a region $E$3, projects onto an outcome $E$4, and traces out $E$5, obtaining a mixed-state ensemble $E$6 on $E$7 (Milekhin et al., 2024). This places PPE within a broader class of partial-measurement ensemble constructions.

2. Higher moments, maximum-entropy limits, and deep thermalisation

The PPE is motivated by the observation that reduced density matrices alone are too coarse to characterize the full distribution of conditional states generated by measurements. In PE language, this is the basis of deep thermalisation: the PE is compared not only at the level of $E$8, but through all moments up to order $E$9, or equivalently through the frame potential and proximity to a state design (Chan et al., 2024).

For PPEs, the relevant maximum-entropy object is the generalized Hilbert–Schmidt ensemble (gHSe). It is defined by taking a Haar-random pure state $$\mathcal{E}_{\text{PPE}_R}=\{p(o_S),\rho_R(o_S)\}_{o_S}$$0, tracing out $$\mathcal{E}_{\text{PPE}_R}=\{p(o_S),\rho_R(o_S)\}_{o_S}$$1, and collecting the resulting mixed states on $$\mathcal{E}_{\text{PPE}_R}=\{p(o_S),\rho_R(o_S)\}_{o_S}$$2: $$\mathcal{E}_{\text{PPE}_R}=\{p(o_S),\rho_R(o_S)\}_{o_S}$$3 with moment

$$\mathcal{E}_{\text{PPE}_R}=\{p(o_S),\rho_R(o_S)\}_{o_S}$$4

A central theorem gives a quantitative condition for a Haar-random tripartite state to generate a PPE close to gHSe: $$\mathcal{E}_{\text{PPE}_R}=\{p(o_S),\rho_R(o_S)\}_{o_S}$$5 with probability at least $$\mathcal{E}_{\text{PPE}_R}=\{p(o_S),\rho_R(o_S)\}_{o_S}$$6 if

$$\mathcal{E}_{\text{PPE}_R}=\{p(o_S),\rho_R(o_S)\}_{o_S}$$7

For extensive subsystems, assuming $$\mathcal{E}_{\text{PPE}_R}=\{p(o_S),\rho_R(o_S)\}_{o_S}$$8, this becomes asymptotically

$$\mathcal{E}_{\text{PPE}_R}=\{p(o_S),\rho_R(o_S)\}_{o_S}$$9

and the subsequent analysis requires only $R$0 (Sherry et al., 13 Nov 2025).

This gHSe limit is the mixed-state analogue of Haar design behaviour in projected ensembles. In random unitary circuits, PE design times at large local Hilbert-space dimension satisfy

$R$1

for all $R$2, while at finite $R$3,

$R$4

so higher moments require only an additional logarithmic time in $R$5 (Chan et al., 2024). A plausible implication is that partial tracing does not erase the underlying design logic, but changes the universal target from Haar to gHSe.

3. Holevo information and information phases

The principal information-theoretic diagnostic for PPEs is the Holevo information. For the ensemble

$R$6

one defines the associated classical–quantum state

$R$7

and the Holevo quantity

$R$8

It is simultaneously the mutual information between the classical register $R$9 and the quantum system $|\Psi\rangle_{AB}$0, and the average relative entropy of each $|\Psi\rangle_{AB}$1 from the mean state $|\Psi\rangle_{AB}$2 (Sherry et al., 13 Nov 2025).

For Haar-random tripartite states on $|\Psi\rangle_{AB}$3 qubits, with subsystem sizes $|\Psi\rangle_{AB}$4, $|\Psi\rangle_{AB}$5, and $|\Psi\rangle_{AB}$6, the PPE exhibits two information phases separated by the line

$|\Psi\rangle_{AB}$7

The distinction comes from spectral concentration of the conditional states $|\Psi\rangle_{AB}$8. When $|\Psi\rangle_{AB}$9, one has $B$0, the eigenvalues concentrate near $B$1, each $B$2 is close to $B$3, and

$B$4

This is the measurement-invisible quantum-correlated (MIQC) phase, in which the Holevo information decays exponentially with system size (Sherry et al., 13 Nov 2025).

When $B$5, one has $B$6, the eigenvalues concentrate near

$B$7

and the asymptotic Holevo information becomes

$B$8

This is the measurement-visible quantum-correlated (MVQC) phase, in which $B$9 scales linearly with $\{|o_B\rangle\}$0 (Sherry et al., 13 Nov 2025).

The phase line $\{|o_B\rangle\}$1 is non-analytic in the thermodynamic limit. Numerical crossing points of $\{|o_B\rangle\}$2 for different system sizes are consistent with a genuine phase transition rather than a crossover (Sherry et al., 13 Nov 2025).

The comparison with logarithmic negativity is central. The negativity phase diagram of tripartite Haar-random states contains PPT/decoupled, maximally entangled, and entanglement saturation regimes, but the Holevo-based PPE phase diagram cuts across them. In particular, there are regions in which the negativity is extensive and positive while the PPE Holevo information is exponentially small. This invalidates the common identification of entanglement visibility with measurement visibility: extensive entanglement between $\{|o_B\rangle\}$3 and $\{|o_B\rangle\}$4 does not imply that measurement outcomes on $\{|o_B\rangle\}$5 are accessible from $\{|o_B\rangle\}$6 (Sherry et al., 13 Nov 2025).

4. Dynamical PPEs and the spatiotemporal structure of scrambling

In out-of-equilibrium many-body systems, the PPE is used to probe how measurements on a source region $\{|o_B\rangle\}$7 influence a receiver region $\{|o_B\rangle\}$8 across an intervening erased region $\{|o_B\rangle\}$9. The basic fluctuation diagnostic is

$$p(o_B)=\langle \Psi|\Pi_{o_B}|\Psi\rangle,\qquad \Pi_{o_B}=|o_B\rangle\langle o_B|,$$0

If $$p(o_B)=\langle \Psi|\Pi_{o_B}|\Psi\rangle,\qquad \Pi_{o_B}=|o_B\rangle\langle o_B|,$$1, then the conditional states $$p(o_B)=\langle \Psi|\Pi_{o_B}|\Psi\rangle,\qquad \Pi_{o_B}=|o_B\rangle\langle o_B|,$$2 are effectively independent of $$p(o_B)=\langle \Psi|\Pi_{o_B}|\Psi\rangle,\qquad \Pi_{o_B}=|o_B\rangle\langle o_B|,$$3, and the PPE is trivial (Mandal et al., 7 Aug 2025).

For generic brickwork circuits, tensor-network cancellations yield the exact bound

$$p(o_B)=\langle \Psi|\Pi_{o_B}|\Psi\rangle,\qquad \Pi_{o_B}=|o_B\rangle\langle o_B|,$$4

so the PPE detects a lightcone in the $$p(o_B)=\langle \Psi|\Pi_{o_B}|\Psi\rangle,\qquad \Pi_{o_B}=|o_B\rangle\langle o_B|,$$5 plane (Mandal et al., 7 Aug 2025). In the non-integrable kicked Ising chain, numerical results sharpen this to

$$p(o_B)=\langle \Psi|\Pi_{o_B}|\Psi\rangle,\qquad \Pi_{o_B}=|o_B\rangle\langle o_B|,$$6

with $$p(o_B)=\langle \Psi|\Pi_{o_B}|\Psi\rangle,\qquad \Pi_{o_B}=|o_B\rangle\langle o_B|,$$7 below the lightcone and $$p(o_B)=\langle \Psi|\Pi_{o_B}|\Psi\rangle,\qquad \Pi_{o_B}=|o_B\rangle\langle o_B|,$$8 immediately after it (Mandal et al., 7 Aug 2025).

At late times in ergodic dynamics, the PPE approaches the gHSe. The asymptotic second moment on $$p(o_B)=\langle \Psi|\Pi_{o_B}|\Psi\rangle,\qquad \Pi_{o_B}=|o_B\rangle\langle o_B|,$$9 is

$S$00

while $S$01. Hence

$S$02

so late-time PPE fluctuations decay exponentially with the size of the discarded region (Mandal et al., 7 Aug 2025).

At the self-dual point of the kicked Ising model, exact analysis yields three dynamical regimes. For $S$03, $S$04. For

$S$05

one finds

$S$06

For

$S$07

deep thermalisation of $S$08 leads to

$S$09

The intermediate regime reflects incomplete thermalisation of $S$10, whereas the late regime is the exact gHSe limit (Mandal et al., 7 Aug 2025).

In the many-body localized regime, the same PPE fluctuation measure develops a logarithmic lightcone rather than a linear one. Numerical and $S$11-bit analyses give

$S$12

and the infinite-time fluctuations satisfy

$S$13

In the $S$14-bit description, dependence on the measured spins in $S$15 enters only through interaction terms of range $S$16, becoming dynamically relevant only for $S$17 (Mandal et al., 7 Aug 2025).

The same dynamical structure appears in the probabilities of bit-string probabilities (PoPs). For a fixed bit-string $S$18,

$S$19

Before the lightcone reaches $S$20, one has $S$21 for all outcomes and therefore

$S$22

At late times in the ergodic regime, the PoP approaches the Erlang distribution

$S$23

the expected universal form for the gHSe (Mandal et al., 7 Aug 2025).

5. Analytical frameworks and related variants

Two analytical frameworks have been especially important for PPE research. The first is the random-state and concentration-of-measure approach underlying the gHSe theorems, which uses Haar integration, Levy’s lemma, Chebyshev’s inequality, and Page-type entropy estimates to control moment convergence and eigenvalue concentration (Sherry et al., 13 Nov 2025). The second is the many-body dynamical framework, in which tensor-network contractions, dual-unitary structure, and $S$24-bit effective Hamiltonians yield exact or asymptotically exact statements about PPE fluctuations and PoPs (Mandal et al., 7 Aug 2025).

A field-theoretic partial-measurement variant is the observable-projected ensemble. Here one measures

$S$25

on a region $S$26 disjoint from $S$27, projects onto outcome $S$28, and defines

$S$29

For the free compact boson, replica calculations give

$S$30

and after summing over $S$31 the measurement-induced entanglement becomes

$S$32

Because $S$33, the correction is small, so charge measurement changes the entanglement of the compact-boson ground state only mildly (Milekhin et al., 2024). The same work shows that measuring an extensive observable produces only $S$34 outcomes, while a full projected ensemble requires $S$35 outcomes, which makes the partial-observable version much more accessible experimentally (Milekhin et al., 2024).

A distinct line of work concerns projected ensembles in systems with an extensive set of conserved charges with local or quasi-local support. In a strongly disordered Floquet spin chain and in the $S$36-bit model, the projected ensemble converges at late times and in the large-system limit to a Scrooge ensemble built from $S$37, except when the measurement operator is close to the conserved charges. For $S$38, the distances $S$39 to the Scrooge ensemble tend to zero as $S$40 and $S$41, whereas for $S$42 in the $S$43-bit model the PE does not converge to the Scrooge ensemble (Manna et al., 3 Jan 2025). A plausible implication is that a PPE formed by measuring only part of the environment and tracing the rest should inherit the same basis sensitivity: generic measurement bases lead to a single Scrooge-type limit, while bases aligned with local integrals of motion retain additional structure.

6. Conceptual significance and common misconceptions

The PPE refines several standard viewpoints on thermalisation and quantum correlations. First, it is not reducible to the ordinary reduced density matrix. The equality

$S$44

shows that $S$45 is only the first moment; the higher moments $S$46 encode the full statistics of the conditional ensemble and can distinguish states with identical $S$47 but inequivalent multipartite information structure (Sherry et al., 13 Nov 2025).

Second, the PPE is not merely a noisy version of the projected ensemble. The PE yields pure conditional states by measuring the full complement, whereas the PPE yields mixed conditional states because only part of the complement is measured and the rest is traced out. This change is not cosmetic: it replaces the Haar ensemble by the generalized Hilbert–Schmidt ensemble as the natural maximum-entropy reference and makes the Holevo information, rather than pure-state frame potentials alone, a central diagnostic (Sherry et al., 13 Nov 2025).

Third, extensive entanglement does not imply measurement visibility. The MIQC phase provides the clearest counterexample: the logarithmic negativity between $S$48 and $S$49 can be extensive while the PPE Holevo information vanishes exponentially with system size (Sherry et al., 13 Nov 2025). This shows that PPEs probe how information is organized across $S$50, $S$51, and $S$52, not merely how much bipartite entanglement is present.

Fourth, partial measurements do not necessarily produce large entanglement changes. In the charge-projected compact boson, the measurement-induced entanglement is very close to the original entanglement entropy, with only a small geometric correction (Milekhin et al., 2024). This indicates that the effect of partial projection depends strongly on which observable is measured and on the geometry.

Finally, PPEs provide an experimentally relevant route to probing scrambling. The dynamical PPE fluctuation $S$53 and the associated PoPs require only projective measurements and conditional statistics, and both show exponential sensitivity to the size of the discarded region: $S$54 This suggests that partial erasure is itself a sharp probe of scrambling structure: linear and logarithmic lightcones, as well as the degradation of quantum correlations under erasure, become visible directly in ensemble statistics rather than only in observables or entropies (Mandal et al., 7 Aug 2025).