Mixed state deep thermalization

Abstract

We introduce the notion of the mixed state projected ensemble (MSPE), a collection of mixed states describing a local region of a quantum many-body system, conditioned upon measurements of the complementary region which are incomplete. This constitutes a generalization of the pure state projected ensemble in which measurements are assumed ideal and complete, and which has been shown to tend towards limiting pure state distributions depending only on symmetries of the system, thus representing a new kind of universality in quantum equilibration dubbed deep thermalization. We study the MSPE generated by solvable (1+1)d dual-unitary quantum circuit evolution, and identify the limiting mixed state distributions which emerge at late times depending on the size of the incomplete measurement, which we assume to be lossy, finding that they correspond to certain random density matrix ensembles known in the literature. We also derive the rate of the emergence of such universality. Furthermore, we investigate the quantum information properties of the states composing the ensemble, specifically their capacity to teleport quantum information between the ends of the system. The teleportation fidelity is upper bounded by the quantum conditional entropy, which we find exhibits a sharp transition from zero to maximal when the number of measurements lost matches of that the number of degrees of freedom to be teleported. Our results initiate the first investigation of deep thermalization for mixed state ensembles, which are relevant for present-day quantum simulation experiments wherein measurements are typically not perfect, and also amount to a physical and natural way of sampling from hitherto abstract random density matrix ensembles.

Figures (7)

• Figure 1: Quench dynamics leading to a mixed state projected ensemble (MSPE). (a) The conditional state $\rho_{N_{A}}^{\otimes k}(\bm{\alpha})$ consisting of $k$-pairs of forward and backward evolution ($k$-replica), which together with Born probability $P_{\bm{\alpha}}$ determines the moments of the ensemble $\rho_{N_{A}}^{(k)}$. A product state composed of maximally entangled pairs evolves under a brick-wall quantum circuit with $t$ layers. The system is bipartited into two subsystems, $A$ and $B$, where $B$ is further split into three subparts. $B$ is measured in $2$-qudit basis, with measurement outcomes from $B_2$ being lost. We consider $A$'s mixed states conditioned on the measurement outcomes in $B_1, B_3$. (b) The orange circle denotes the projective measurement on two qudits with the measurement outcome $\bm{\alpha}$ in $k$-replicas. (c) The white empty bullet corresponds to lost measurement outcomes and is proportional to the $k$-copy of the vectorized identity matrix. This effectively implements a partial trace, connecting the forward and backward time evolutions.
• Figure 2: Distance of $\rho_{N_{A}}^{(k)}$ from the generalized Hilbert-Schmidt ensemble. Left: local Haar-random circuits for system size $N$ from $6$ to $22$, where error bars indicate statistical fluctuations over different circuit realizations. Right: mixed-field Ising dynamics for $N$ from $6$ to $18$, governed by the Hamiltonian $H=\sum_{j=1}^N(h_x\sigma_x^j+h_y\sigma^j_y)+\sum_{j=1}^{N-1}J\sigma_x^j\sigma_x^{j+1}$, with $\sigma_\alpha^j$ the Pauli matrix on $j$-th qubit and $(h_x,h_y,J)=(0.8090,0.9045,1.0)$. Inset: saturated deviation $\Delta_1^{(2)}$ at late times versus the system size.
• Figure 3: Quench dynamics leading to MSPE. The symbols have the same meaning as in Fig. \ref{['fig1']}. Here, we use the unitarity to simplify the region $B_2$. We use the dashed line to indicate the states on the time-like cuts, which appear in the derivation of the results.
• Figure 4: Membrane-picture illustration of domain-wall evolution (blue lines). The figure shows the trajectory with minimal energy penalty.
• Figure S1: Quench dynamics leading to MSPE. The symbols have the same meaning as in Fig. \ref{['fig1']} in the main text. Here, we use the unitarity to simplify the $B_2$ subsystem. We also use the dashed line to represent the states on the timelike cut.

Theorems & Definitions (7)

• proof
• Theorem 1: Global Haar random states
• Lemma 1: The Levy's lemma
• Lemma 2: Coefficients in Levy's Lemma
• proof
• Lemma 3: Avege value
• proof