Projected ensembles reveal information about quantum many-body dynamics beyond the reduced density matrix. Here we investigate how spatial constraints affect the emergence of this information in a kicked Ising chain with a spatially varying longitudinal field. We consider partial projected ensembles, obtained by measuring a remote region while leaving an intervening buffer unobserved. Our numerical results reveal a pronounced contrast between nearly ballistic correlation onset in the ergodic regime and strongly delayed onset in the tilt-constrained regime, corroborated by quantum mutual information. Despite these different onset scales, the conditional-state fluctuations decrease approximately exponentially with buffer length in both regimes. Full projected ensembles additionally exhibit slow relaxation and persistent measurement-basis dependence in their higher moments. We establish an information-theoretic characterization of the connected second moment and use a solvable dephasing model with effective-basis measurements to illustrate the separation of phase accumulation and coherence loss. Within this model, a factorially suppressed coupling envelope yields sublogarithmic spreading consistent with the finite-distance onset trends. Our results identify partial projected ensembles as probes of the distinction between the time required for correlations to develop and the information that remains accessible under partial observation.
Projected ensembles reveal information about quantum many-body dynamics beyond the reduced density matrix. Here we investigate how spatial constraints affect the emergence of this information in a kicked Ising chain with a spatially varying longitudinal field. We consider partial projected ensembles, obtained by measuring a remote region while leaving an intervening buffer unobserved. Our numerical results reveal a pronounced contrast between nearly ballistic correlation onset in the ergodic regime and strongly delayed onset in the tilt-constrained regime, corroborated by quantum mutual information. Despite these different onset scales, the conditional-state fluctuations decrease approximately exponentially with buffer length in both regimes. Full projected ensembles additionally exhibit slow relaxation and persistent measurement-basis dependence in their higher moments. We establish an information-theoretic characterization of the connected second moment and use a solvable dephasing model with effective-basis measurements to illustrate the separation of phase accumulation and coherence loss. Within this model, a factorially suppressed coupling envelope yields sublogarithmic spreading consistent with the finite-distance onset trends. Our results identify partial projected ensembles as probes of the distinction between the time required for correlations to develop and the information that remains accessible under partial observation.
Thermalisation and information scrambling in out-of-equilibrium quantum many-body systems are deeply intertwined: local subsystems dynamically approach thermal density matrices while their entropies track information spreading. Projected ensembles--ensembles of pure states conditioned on measurement outcomes of complementary subsystems--provide higher-order probes of thermalisation, converging at late times to universal maximum-entropy ensembles. In this work, we introduce the partial projected ensemble (PPE) as a framework to study how the spatiotemporal structure of scrambling is imprinted on projected ensembles. The PPE consists of an ensemble of mixed states induced on a subsystem by measurements on a spatially separated part of its complement, tracing out the remainder, naturally capturing scenarios involving discarded outcomes or noise-induced losses. We show that statistical fluctuations of the PPE faithfully track the causal lightcone of information spreading, revealing how scrambling dynamics are encoded in ensemble structure. In addition, we demonstrate that the probabilities of bit-string probabilities (PoPs) associated with the PPE exhibit distinct dynamical regimes and provide an experimentally accessible probe of scrambling. Both PPE fluctuations and PoPs display exponential sensitivity to the size of the discarded region, reflecting exponential degradation of quantum correlations under erasure. We substantiate these findings using the non-integrable kicked Ising chain, combining numerics in the ergodic regime with exact results at its self-dual point. We extend our analysis to a many-body localised (MBL) regime numerically, along with analytic results for the ℓ-bit model. The linear and logarithmic lightcones characteristic of ergodic and MBL regimes emerge naturally from PPE dynamics, establishing it as a powerful tool for probing scrambling and deep thermalisation.
The projected ensemble -- an ensemble of pure states on a subsystem conditioned on projective measurement outcomes on its complement -- provides a finer probe of ergodicity and information structure than the reduced density matrix of the subsystem in bipartite quantum states. This framework can be generalised to partial projected ensembles in tripartite settings, where outcomes from part of the measured subsystem are discarded, leading to ensembles of mixed states. We show that information measures defined for such ensembles, in particular the Holevo information, yield a more detailed characterisation of how quantum information is distributed between subsystems compared to conventional entanglement measures. Using exact analytical results supported by numerical results, we uncover a qualitative change in the scaling of the Holevo information with system size in partial projected ensembles generated by Haar-random states, as the relative sizes of the subsystem are varied. In one phase, the Holevo information decays exponentially with system size, while in the other it grows linearly, thereby defining distinct information phases separated by sharp transitions signalled by non-analyticities in the Holevo information. The exponentially decaying phase rigorously establishes the existence of a measurement-invisible quantum-correlated phase -- a manifestation of many-body information scrambling with no bipartite analogue. We contrast this information-phase diagram with the entanglement-phase structure of tripartite Haar-random states obtained from logarithmic negativity, and show that the Holevo information reveals additional fine structure beyond conventional entanglement measures. Finally, we show that these information phases, as characterised by the Holevo information, emerge in the dynamics of chaotic quantum circuits and discuss the associated timescales.
Disorder-free quantum many-body localization can strongly suppress transport while still enabling the dynamical buildup of computationally costly non-Clifford resources. In a tilted transverse-field Ising chain realizing disorder-free Stark many-body localization, we use the stabilizer Rényi entropy to quantify quantum magic (nonstabilizerness) and find that it remains finite and grows anomalously slowly over extended time windows before saturating to a size-dependent plateau deep in the strong-tilt regime, with pronounced initial-state selectivity. Upon increasing the Stark gradient, the long-time magic and half-chain entanglement exhibit consistent finite-size crossing behavior, indicating a crossover from ergodic dynamics to constrained localization. These results establish stabilizer-based magic as a practical complexity diagnostic of disorder-free ergodicity breaking and constrained dynamics, and provide an experimentally accessible route to benchmarking and designing near-term quantum simulators.
The dynamics of a quantum-many body system subject to measurements is naturally described by an ensemble of quantum trajectories, which can feature measurement-induced phase transitions (MIPTs). This phenomenon cannot be revealed through ensemble-averaged observables, but it requires the ability to discriminate each trajectory separately, making its experimental observation extremely challenging. We explore the fate of MIPTs under an observer's reduced ability to discriminate each measurement outcome. This introduces uncertainty in the state of the system, causing observables to probe a restricted subset of trajectories rather than a single one. By introducing an exactly-solvable Liouvillian model, we examine how long-time spatial correlations are influenced by varying degrees of trajectory averaging. We compute exactly the correlation matrix, Liouvillian gap, and entanglement negativity to demonstrate that averaging over multiple realizations introduces an effective finite lengthscale, beyond which long-range correlations are suppressed. This suggests that partial averaging over trajectories conceals the critical features of individual realizations, thereby blurring away the signatures of distinct measurement-induced phases.
We investigate the propagation of two-point density correlations in the one-dimensional Bose-Hubbard Hamiltonian in the thermodynamic limit in terms of the correlation transport distance (CTD), an experimentally measurable magnitude that characterizes the spatial spreading of correlations in time. We confirm that the integrable limits of the model exhibit CTD ballistic growth, while the onset of the chaotic phase leads to the emergence of a pronounced sub-ballistic regime, in agreement with previous results for finite systems. By a meticulous analysis of the spatio-temporal correlation profiles, we show that the correlation front nonetheless propagates ballistically for all interaction strengths, and that the chaos-induced slowdown of the CTD originates from the emergence of long-time distance-dependent correlation tails, together with an enhanced decay of the correlation front amplitude. Our results thus provide a detailed characterization of correlation transport that goes beyond a simple light-cone picture.
The investigation of ergodicity or lack thereof in isolated quantum many-body systems has conventionally focused on the description of the reduced density matrices of local subsystems in the contexts of thermalization, integrability, and localization. Recent experimental capabilities to measure the full distribution of quantum states in Hilbert space and the emergence of specific state ensembles have extended this to questions of deep thermalization, introducing the notion of the projected ensemble---ensemble of pure states of a subsystem obtained by projective measurements on its complement. While previous studies examined chaotic unitary circuits and Hamiltonian evolution in systems with or without global conserved charges, we study the projected ensemble in systems where there are an extensive number of conserved charges all of which have (quasi)local support. We employ a strongly disordered quantum spin chain that shows many-body localized dynamics over long timescales, as well as the $\ensuremath{\ell}$-bit model, a phenomenological archetype of a many-body localized system, with the charges being 1-local in the latter. In particular, we discuss the dependence of the projected ensemble on the measurement basis. Starting with random direct product states, we find that the projected ensemble constructed from time-evolved states converges to a Scrooge ensemble at late times and in the large system limit except when the measurement operator is close to the conserved charges. This is in contrast to systems with global conserved charges where the ensemble varies continuously with the measurement basis. We relate these observations to the emergence of the Porter-Thomas distribution in the probability distribution of bitstring measurement probabilities.