Information phases of partial projected ensembles generated from random quantum states and scrambling dynamics

TL;DR

This work develops a information-theoretic framework for partial projected ensembles (PPEs) in tripartite quantum systems, using the Holevo information $\chi({\cal E}_{{\rm PPE}_R})$ to probe how quantum information is distributed between a subsystem $R$ and a measured complement $S$ when part of the bath is discarded. By connecting PPEs to universal ensemble descriptions (Haar and generalized Hilbert–Schmidt, gHSe) and exploiting spectral degeneracy in the large-$N$ limit, the authors derive an information-phase diagram with two distinct regimes: a measurement-invisible quantum-correlated (MIQC) phase where $\chi$ decays exponentially with system size, and a measurement-visible quantum-correlated (MVQC) phase where $\chi$ grows linearly with system size; the phase boundary is $p_c=1-2\gamma$. They analytically establish the MIQC/MVQC separation in part of the parameter space and corroborate the full phase diagram numerically; they further show the same phases arise dynamically in chaotic circuits, with all-to-all circuits displaying $t_*\sim\ln N$ scaling in MIQC and 1+1D brickwork circuits showing $t_*\sim N$ due to spatial locality. Overall, the results provide a fine-grained, operationally meaningful diagnostic of information scrambling beyond entanglement measures and have implications for deep thermalisation, lossy measurements, and the organization of information in many-body quantum systems.

Abstract

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.

Figures (7)

• Figure 1: (a) Schematic phase diagram for the Holevo information $\chi({\cal E}_{{\rm PPE}_R})$ contrasted with the phase diagram of the entanglement between $R$ and $S$ (from shapourian2021negativity). The entanglement phases are labelled, while the line of separation between the measurement-visible and measurement-invisible is denoted by the red line, with the solid part representing what was obtained from the analytical calculations and the dashed part denoting the extrapolation using the numerical data. (b) Heatmap plot for $\chi({\cal E}_{{\rm PPE}_R})$, from which the data was used to extrapolate the line that separates the measurement-visible and measurement-invisible phases. The data was generated for tripartite Haar-random states over $24$ qubits at various partitions of $R$, $S$ and $E$ corresponding to various points $(p,\gamma)$. The black horizontal and vertical dashed lines indicate two representative slices for which finite-size scaling data is presented in Fig. \ref{['fig:Holevo_crossing']}, with the red stars indicating the corresponding critical points.
• Figure 2: Plots of $\chi(\mathcal{E}_{\mathrm{PPE_R}})$ across two different slices of the $p$-$\gamma$ parameter space for different system sizes $N$. The data shows a clear crossing for different $N$ indicating a transition. Top panel shows the data for fixed $\gamma=\frac{1}{3}$ with varying $p$. Bottom panel shows the transition for fixed $p=\frac{1}{4}$ with varying $\gamma$. These slices are marked in Fig. \ref{['fig:phase-dia']}(b) by the black dashed horizontal and vertical lines respectively, and the critical points are marked by red stars as also indicated there. Insets show the same data but on logarithmic scales to highlight the exponential decay of $\chi(\mathcal{E}_{\mathrm{PPE_R}})$ with $N$ in the MIQC phase.
• Figure 3: Dynamics of the Holevo information, $\chi(t)$, for the 2-local all-to-all circuit. The data is for $\gamma=1/4$ and $\tau=0.5$. The three rows correspond to three different values of $p$, representative of the MIQC and MVQC phases and the critical point, as mentioned in the panels. The left column shows the dynamics of $\chi(t)$ as a function of $t$ for different $N=8,12,16,20,24$ (lighter to darker colours). The right columns show the infinite-time saturation value as a function of $N$. Note that the data for $p=1/3$ (MIQC phase) in the top row is on logarithmic scales.
• Figure 4: Approach of $\chi(t)$ to its infinite-time value $\chi_{\rm sat}$ (see Eq. \ref{['eq:approach']}) for the 2-local all-to-all circuit. The collapse of the data for different $N$ (different colour intensities, same as in Fig. \ref{['fig:HI-alltoall']}) when plotted against $t/\ln N$ in the MIQC phase and against $t$ in the MVQC phase and the critical point suggests the scaling of $t_\ast$, the timescale associated to emergence of the information phases, to be of the form in Eq. \ref{['eq:tast-alltoall']}.
• Figure 5: Dynamics of the Holevo information, $\chi(t)$, for the 1+1D brickwork circuit. The data is for $\gamma=1/4$ and $\tau=0.8$. The three rows correspond to three different values of $p$ (same as in Fig. \ref{['fig:HI-alltoall']}), representative of the MIQC and MVQC phases and the critical point. The left column shows the dynamics of $\chi(t)$ as a function of $t$ for different $N=8,12,16,20,24$ (lighter to darker colours), whereas the the right column shows the infinite-time saturation value as a function of $N$. The exponential decay and linear growth with $N$ of $\chi_{\rm sat}(N)$ is evident from the fits denoted by the black dashed lines.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• proof
• proof