Mixed-State Measurement-Induced Phase Transitions in Imaginary-Time Dynamics

TL;DR

This work introduces measurement-dressed imaginary-time evolution (MDITE) to study mixed-state phase transitions arising from the competition between non-unitary imaginary-time dynamics and projective measurements. A diagrammatic representation enables efficient quantum Monte Carlo and other numerical methods to explore stationary states in large-scale, higher-dimensional systems. Numerical demonstrations in the 1D TFIM and 2D CDHM reveal mixed-state transitions not falling into known universality classes, with a robust universal ratio of critical exponents, beta over nu, along varied dynamical trajectories. The framework provides a general, experimentally relevant approach to non-unitary open-system dynamics, with potential extensions to other decoherence channels and platforms for realizing and probing mixed-state criticality in quantum matter.

Abstract

Mixed-state phase transitions have recently attracted growing attention as a new frontier in nonequilibrium quantum matter and quantum information. In this work, we introduce the measurement-dressed imaginary-time evolution (MDITE) as a novel framework to explore mixed-state quantum phases and decoherence-driven criticality. In this setup, alternating imaginary-time evolution and projective measurements generate a competition between coherence-restoring dynamics and decoherence-inducing events. While reminiscent of monitored unitary circuits, MDITE fundamentally differs in that the physics is encoded in decoherent mixed states rather than in quantum trajectories. We demonstrate that this interplay gives rise to a novel class of mixed-state phase transitions, using numerical simulations of the one-dimensional transverse-field Ising model and the two-dimensional columnar dimerized Heisenberg model. Notably, the observed transitions do not fall into any previously established universality classes. Furthermore, we provide a diagrammatic representation of the evolving state, which naturally enables efficient studies of MDITE with quantum Monte Carlo and other many-body numerical methods, thereby extending investigations of mixed-state phase transitions to large-scale and higher-dimensional systems. In addition, the representation provides a natural interpretation of the phase transitions in terms of cluster formation within the simulations. Our results highlight MDITE as a powerful paradigm for investigating non-unitary dynamics and the fundamental role of decoherence in many-body quantum systems.

Figures (13)

• Figure 1: Diagrammatic representation of the MDITE at measurement rate $p=0$, with the input (left) state given by $\rho_0 = I_{2^N} / 2^{N}$. Each node (black filled circle) represents a basis state. The label "System" indicates that the diagram applies to all qubits in the entire system. (a) For circuit depth one, after an evolution time of $\tau/2$, the initial state $\rho_0$ evolves into $\rho_1 \propto e^{-\tau H}$ on the right, where $\bra{s}$ and $\ket{s'}$ denote the bra and ket components of $\rho_1$ in Eq. \ref{['eq:rho1']}, respectively. (b) Starting from $\rho_1$ with bra and ket components $\bra{r}$ and $\ket{r}$, another evolution for time $\tau/2$ (i.e., circuit depth two) yields the output state $\rho_2 \propto e^{-2\tau H}$, where $\bra{s}$ and $\ket{s'}$ correspond to the components of $\rho_2$.
• Figure 2: Diagrammatic representations of the MDITE under deterministic projective measurements. (a) Compared with Fig. \ref{['fig:diagram_no_measure']}, the measurement on $\rho_1$ identifies the bra and ket components of $\rho_1$ in Eq. \ref{['eq:rho1']}, forming a closed loop for the measurement-averaged state $\bar{\rho}_1$. (b) The output state $\rho_2$ after the second circuit layer.
• Figure 3: Diagrammatic representation of the MDITE where only subsystem $A$ is projectively measured in a deterministic manner. The label "$A$" ("$B$") indicates that the diagram applies to all qubits in subsystem $A$ ($B$). Since the computational basis is local, the evolutions of the reduced density matrices $(\rho_0)_A = \text{Tr}_B(\rho_0)$ and $(\rho_0)_B = \text{Tr}_A(\rho_0)$ can be depicted separately in (a) and (b), respectively.
• Figure 4: For $N=2$, $k=3$, and $\{\ket{s} \equiv \ket{s_1, s_2}\}$, diagrammatic representations of two possible ensembles are shown. The subscripts $q_1$ and $q_2$ denote the reduced density matrices of qubit 1 and qubit 2, respectively. (a) In the second layer, qubit 1 is unmeasured while qubit 2 is measured; in the third layer, qubit 1 is measured while qubit 2 is unmeasured. (b) Qubit 1 is measured in both the second and third layers, whereas qubit 2 remains unmeasured throughout.
• Figure 5: When setting $(\tau,h,p)=(1,1.8,0.66)$ for the MDITE with the 1D TFIM: (a) Convergence of the second moment $\langle m^2\rangle$ with increasing circuit depth for various system sizes $L$. (b) Convergence of the Binder ratio $R_2$ with increasing circuit depth for various system sizes $L$.