--- title: Partial Projected Ensemble (PPE) url: https://www.emergentmind.com/topics/partial-projected-ensemble-ppe type: topic --- # Partial Projected Ensemble (PPE) 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 \(R\) [2511.10595]. 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 [2508.05632]. ## 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 \(|\psi_A(o_B)\rangle\) on \(A\), so that \[ \mathcal{E}_{\text{PE}_A}=\{p(o_B),|\psi_A(o_B)\rangle\}_{o_B}. \] Its \(k\)-th moment is \[ \rho^{(k)}_{\text{PE}_A}=\sum_{o_B}p(o_B)\big(|\psi_A(o_B)\rangle\langle\psi_A(o_B)|\big)^{\otimes k}, \] with \(\rho^{(1)}_{\text{PE}_A}=\rho_A\) [2511.10595]. The PPE extends this to a tripartite system \(R,E,S\). One measures only \(S\) in a local product basis \(\{|o_S\rangle\}\), retains those outcomes, and discards \(E\). The defining quantities are \[ p(o_S)=\langle \Psi|\Pi_{o_S}|\Psi\rangle,\qquad \Pi_{o_S}=|o_S\rangle\langle o_S|, \] and \[ \rho_R(o_S)=\frac{\mathrm{Tr}_{ES}\!\big[\Pi_{o_S}|\Psi\rangle\langle\Psi|\Pi_{o_S}\big]}{p(o_S)}. \] The partial projected ensemble is then \[ \mathcal{E}_{\text{PPE}_R}=\{p(o_S),\rho_R(o_S)\}_{o_S}, \] with moment hierarchy \[ \rho^{(k)}_{\text{PPE}_R}=\sum_{o_S}p(o_S)\,[\rho_R(o_S)]^{\otimes k}. \] As in the PE, the first moment is just the reduced density matrix, \[ \rho^{(1)}_{\text{PPE}_R}=\rho_R, \] while \(k\ge 2\) moments encode nonlinear information about the ensemble of conditional states that is not reconstructible from \(\rho_R\) alone [2511.10595]. A useful structural identity is that the PPE on \(R\) can be obtained by tracing out \(E\) from a PE on \(R\cup E\) built from measurements on \(S\): \[ \rho^{(k)}_{\text{PPE}_R} =\mathrm{Tr}_{\mathcal H_E}^{\otimes k}\!\left[\rho^{(k)}_{\text{PE}_{R\cup E}}\right]. \] 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 [2511.10595]. A related construction appears in observable-projected ensembles: one measures an extensive Hermitian operator \(Q_B=\int_B dx\,\mathcal O(x)\) on a region \(B\subset \bar A\), projects onto an outcome \(q\), and traces out \(\overline{A\cup B}\), obtaining a mixed-state ensemble \(\{p_q,\rho_{A,q}\}\) on \(A\) [2410.21397]. 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 \(\rho_A\), but through all moments up to order \(k\), or equivalently through the frame potential and proximity to a state design [2402.16939]. For PPEs, the relevant maximum-entropy object is the generalized Hilbert–Schmidt ensemble (gHSe). It is defined by taking a Haar-random pure state \(|\psi\rangle_{R\cup E}\), tracing out \(E\), and collecting the resulting mixed states on \(R\): \[ \mathcal{E}_{\text{gHSe}_R} =\{\mathrm{Tr}_E(|\psi\rangle\langle\psi|):\psi\sim \mathrm{Haar}_{R\cup E}\}, \] with moment \[ \rho^{(k)}_{\text{gHSe}_R} =\mathrm{Tr}_{\mathcal H_E}^{\otimes k}\!\left[\rho^{(k)}_{\text{Haar}_{R\cup E}}\right]. \] A central theorem gives a quantitative condition for a Haar-random tripartite state to generate a PPE close to gHSe: \[ \left\Vert \rho^{(k)}_{\text{PPE}_R} -\rho^{(k)}_{\text{gHSe}_R} \right\Vert_1 \le \epsilon \] with probability at least \(1-\delta\) if \[ D_S D_E \ge \frac{18\pi^3 (2k-1)^2 D_R^{4k-1}}{\epsilon^2} \left(2k\ln D_R + \ln \frac{2}{\delta}\right). \] For extensive subsystems, assuming \(\epsilon\sim 2^{-\alpha N}\), this becomes asymptotically \[ \gamma \le \frac{1-2\alpha}{4k}, \] and the subsequent analysis requires only \(k=2\) [2511.10595]. 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 \[ t_k=L_A/2\qquad (\text{large }q), \] for all \(k\), while at finite \(q\), \[ t_k\sim t_1+\frac{\ln k}{v_2 s_{\rm eq}}, \] so higher moments require only an additional logarithmic time in \(k\) [2402.16939]. 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 \[ \mathcal{E}_{\text{PPE}_R}=\{p(o_S),\rho_R(o_S)\}, \] one defines the associated classical–quantum state \[ \rho_{\rm CQ}=\sum_{o_S}p(o_S)\,\rho_R(o_S)\otimes |o_S\rangle\langle o_S|, \] and the Holevo quantity \[ \chi(\mathcal{E}_{\text{PPE}_R}) = S(\rho_R)-\sum_{o_S}p(o_S)\,S(\rho_R(o_S)). \] It is simultaneously the mutual information between the classical register \(o_S\) and the quantum system \(R\), and the average relative entropy of each \(\rho_R(o_S)\) from the mean state \(\rho_R\) [2511.10595]. For Haar-random tripartite states on \(N\) qubits, with subsystem sizes \(|R|=\gamma N\), \(|S|=pN\), and \(|E|=(1-p-\gamma)N\), the PPE exhibits two information phases separated by the line \[ p_c(\gamma)=1-2\gamma. \] The distinction comes from spectral concentration of the conditional states \(\rho_R(o_S)\). When \(p\le 1-2\gamma\), one has \(D_R1-2\gamma\), one has \(D_R>D_E\), the eigenvalues concentrate near \[ (\underbrace{1,\dots,1}_{D_E},\underbrace{0,\dots,0}_{D_R-D_E})/D_E, \] and the asymptotic Holevo information becomes \[ \chi(\mathcal{E}_{\text{PPE}_R}) \to N\ln 2\,(p+2\gamma-1). \] This is the measurement-visible quantum-correlated (MVQC) phase, in which \(\chi\) scales linearly with \(N\) [2511.10595]. The phase line \(p_c=1-2\gamma\) is non-analytic in the thermodynamic limit. Numerical crossing points of \(\chi\) for different system sizes are consistent with a genuine phase transition rather than a crossover [2511.10595]. 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 \(R\) and \(S\) does not imply that measurement outcomes on \(S\) are accessible from \(R\) [2511.10595]. ## 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 \(S\) influence a receiver region \(R\) across an intervening erased region \(E\). The basic fluctuation diagnostic is \[ \Delta(t,L_E) =\frac{1}{2}\,\big\|\rho_R^{(2)}(t,L_E)-\rho_R(t,L_E)^{\otimes 2}\big\|_1. \] If \(\Delta=0\), then the conditional states \(\rho_R(o_S)\) are effectively independent of \(o_S\), and the PPE is trivial [2508.05632]. For generic brickwork circuits, tensor-network cancellations yield the exact bound \[ \Delta(t,L_E)=0\qquad \text{for}\qquad t<\left\lfloor\frac{L_E+3}{4}\right\rfloor, \] so the PPE detects a lightcone in the \((t,L_E)\) plane [2508.05632]. In the non-integrable kicked Ising chain, numerical results sharpen this to \[ t_\ast(L_E)=\left\lfloor \frac{L_E}{2}\right\rfloor, \] with \(\Delta(t,L_E)=0\) below the lightcone and \(\Delta>0\) immediately after it [2508.05632]. At late times in ergodic dynamics, the PPE approaches the gHSe. The asymptotic second moment on \(R\) is \[ \rho_R^{(2)}(\infty) =\frac{D_E^2\,\mathbb I_R^{\otimes 2}+D_E\,\mathbb S_R^{\otimes 2}} {D_R D_E(D_R D_E+1)}, \] while \(\rho_R(\infty)=\mathbb I_R/D_R\). Hence \[ \Delta_\infty(L_E)\sim \frac{1}{D_E}\sim 2^{-L_E}, \] so late-time PPE fluctuations decay exponentially with the size of the discarded region [2508.05632]. At the self-dual point of the kicked Ising model, exact analysis yields three dynamical regimes. For \(t