Coherence-induced deep thermalization transition in random permutation quantum dynamics

TL;DR

The paper identifies a coherence-driven deep thermalization transition in the projected ensemble (PE) of a local subsystem under random permutation dynamics. It analyzes two minimal models—the tilted-basis and the mixed-basis models—and shows the transition is governed by input and measurement coherence, quantified by the relative entropy of coherence $C_r$. Across the transition, the reduced density matrix remains maximally mixed at infinite temperature, while the PE switches between the Haar ensemble $\mathcal{E}_{\mathrm{Haar}}$ and the classical bit-string ensemble $\mathcal{E}_{\mathrm{Cl}}$; the tilted-basis model yields a critical point $\theta_m^* \approx 0.193\pi$ and the mixed-basis model provides an analytically solvable boundary at $\alpha_0+\alpha_m=1$ (with near-match to the observed $\theta_m^*$). Robustness is demonstrated against $r$-local RPDs (with $r\ge 3$) and across models, suggesting a universal, resource-driven deep-ergodicity-breaking mechanism with potential extensions to imaginarity and non-Gaussianity.

Abstract

We report a phase transition in the projected ensemble - the collection of post-measurement wavefunctions of a local subsystem obtained by measuring its complement. The transition emerges in systems undergoing random permutation dynamics, a type of quantum time evolution wherein computational basis states are shuffled without creating superpositions. It separates a phase exhibiting deep thermalization, where the projected ensemble is distributed over Hilbert space in a maximally entropic fashion (Haar-random), from a phase where it is minimally entropic ("classical bit-string ensemble"). Crucially, this deep thermalization transition is invisible to the subsystem's density matrix, which always exhibits thermalization to infinite-temperature across the phase diagram. Through a combination of analytical arguments and numerical simulations, we show that the transition is tuned by the total amount of coherence injected by the input state and the measurement basis, and is exhibited robustly across different microscopic models. Our findings represent a novel form of ergodicity-breaking universality in quantum many-body dynamics, characterized not by a failure of regular thermalization, but rather by a failure of deep thermalization.

Figures (3)

• Figure 1: (a) Projected ensemble (PE) formed under random permutation dynamics. A single global random permutation unitary (dotted box) models the behavior of a deep quantum circuit made of local random permutation gates (brickwork circuit). For the tilted-basis model, input states and measurement basis are uniform product states, specified by Bloch angles $(\theta_0$$,$$\phi_0)$ and $(\theta_m$$,$$\phi_m)$ respectively. (b) $k$$=$$2$ trace distances of the PE (for $N_A$$=$$2$) from the classical bit-string ensemble $\mathcal{E}_\text{Cl}$ and Haar ensemble $\mathcal{E}_\text{Haar}$, generated from the tilted-basis model with $\theta_0$$=$$\phi_0$$=$$\pi/4$, $\phi_m$$=$$0$, and variable $\theta_m$. Different intensities indicate different system sizes $N$$=$$16,18,20,22,24$ (lighter to darker). There is a common crossing at $\theta_m^*$$\approx$$0.193\pi$ for both distances across all system sizes, signaling a singular change of the limiting PE.
• Figure 2: (a) Ensemble-averaged coherence of the PE ($N_A$$=$$2$) for the mixed-basis model with $\alpha_0$$=$$0.5$, showing a transition at $\alpha_m$$=$$0.5$ (vertical dashed line) as predicted. Horizontal dashed line indicates the coherence of Haar random states. (b) Trace distances of the PE ($k$$=$$2$) confirming convergence to the classical bit-string and Haar ensembles in the deeply non-ergodic and ergodic regimes respectively.
• Figure 3: (a) Time evolution under 3-local RPDs of the ensemble-averaged coherence in the tilted-basis model, for $z$- ($x$-) basis measurements, shown in the top (bottom) panel. In both cases, the coherence saturates to that of the classical bit-string ensemble and Haar ensemble respectively. (b) Saturation value as a function of the measurement angle $\theta_m$. The critical angle $\theta_m^*$$\approx$$0.193\pi$ of the tilted-basis model from Fig. \ref{['Fig:1']}(b) is shown as a vertical dashed line.