Large deviations and conditioned monitored quantum systems: a tensor network approach

Abstract

Coexistence of different dynamical phases is a hallmark of glassy dynamics. This is well-studied in classical systems where the underlying theoretical framework is that of large deviation theory. The presence of a similar phase coexistence has been suggested in monitored quantum many-body systems, but the lack of suitable methods has yet prevented a systematic large deviation analysis. Here we present a tensor network framework that allows the application of large deviation theory to large quantum systems. Building on this, we locate a series of first-order dynamical phase transitions in a monitored discrete-time many-body quantum dynamics, at the level of the trajectory space. Crucially, our approach provides access not only to large-deviation statistics but also to conditioned quantum many-body states, enabling a microscopic characterization of the dynamical phases and their coexistence.

Figures (7)

• Figure 1: Collision model and dynamical phase diagram. (a) A chain of two-level systems ($\mathrm{S}$, green) interacting according to Eq. \ref{['eq:Rydberg_Hamiltonian']} is coupled site-wise to ancilla qubits ($\mathrm{A}$, purple). At the end of each discrete time interval $\Delta t$, the ancillas are measured and reset. The measurement outcomes define space–time quantum trajectories. (b) Example of a single trajectory for a chain of length $L=60$ at interaction strength $V/\Omega=5.875$. Each pixel represents the measurement outcome for one site at a particular time. (c) Dynamical phase diagram in the interaction-bias $(V/\Omega,s)$ plane obtained from the activity $a(s)$ (see main text) for $L=60$. The dashed line at $s=0$ highlights the physical dynamics. Crosses mark the estimated location of the first-order transition line, with red crosses indicating points where the transition occurs exactly at $s=0$.
• Figure 2: Tensor network representation of trajectory ensembles. Tensors are represented as boxes or circles with green (purple) legs corresponding to indices in the system (ancilla) basis. (a) Initial states of the system and ancilla. (b) $k-$th Kraus operator. (c) Quantum channel obtained by contracting the Kraus operators over the ancilla index $k$, interpreted as an MPS transfer matrix. (d) Local bias and projector onto the ancilla state $\ket{k}$ acting on the ancilla legs. (e) Tilted channel obtained by inserting the bias operator on the ancilla legs. (f) MPS contraction representing the trajectory probability $\pi(k_1,\ldots k_T)$. (g) Extension to a many-body chain by approximating the Kraus map as an MPO.
• Figure 3: Signatures of dynamical phase coexistence. (a,b,c): For $V/\Omega =5.875$ (red circles) the SCGF $\theta(s)$ (a) develops a non-analyticity near $s=0$ as system size increases, as the activity $a(s)$ (b) exhibits a rapidly sharpening crossover. The rate function $-\phi(a)$ (c) broadens and exhibits a Maxwell construction, corresponding to the convex hull of an underlying bimodal distribution. In contrast, for $V/\Omega =2$ (blue diamonds), $\theta(s)$ and $a(s)$ are smooth, and $-\phi(a)$ remains narrow with a peak at the stationary value $a\approx0.5$, with no noticeable system size dependence. (d,e): Finite-size analysis of the crossing point of $\theta(s)$ from system sizes $L=20,$ 40, 60. At $s\approx 0$, perturbation theory predicts a linear behavior for the active branch (solid grey line). The crossing is identified by extrapolating the inactive branch until it intercepts this linear behavior. For $V/\Omega =2$ (d) the crossing is far from $s=0$. For $V/\Omega =5.875$ (e) we linearly fit the smallest computed $s>0$ values (dotted black lines) to find the interception point (black crosses). Extrapolating these values with $1/L$ (inset) yields the value $s^* = (2 \pm 3) \times 10^{-5}$, consistent with a transition at $s=0$ in the thermodynamic limit. Results were computed with Trotter step $\Delta t/10$, and bond dimension $D_{\max}=96$, and error bars denote the difference with respect to the results with $D_\mathrm{max}=64$ (see SM for details SM).
• Figure 4: String correlator in the conditioned ensemble. (a) String correlator [Eq. \ref{['eq:string_correl']}] as a function of the string length $\ell$ for $L=20$, averaged over the two central sites of the chain. Results are shown for $V/\Omega=2$ (blue diamonds) and $V/\Omega=5.875$ (red circles). (b) Probability distribution of the activity $p(a)$ obtained from 1000 sampled trajectories, showing enhanced weight at low activity in the coexistence regime.
• Figure S1: Trotterization. (a) MPO representation of $e^{-i H_{\mathrm{S}} \Delta t / N}$ obtained from the even–odd decomposition (bond dimension 2). (b) MPO representation of $e^{-i H_{\mathrm{I}} \Delta t / N}$ with product structure (bond dimension 1). (c) MPO for a single Trotter step obtained by contracting (a) and (b). (d) Kraus operator constructed as the contraction of $N$ Trotter steps.