Arrow of Time as an indicator of Measurement-Induced Phase Transitions

TL;DR

This paper introduces the arrow of time as a new thermodynamic measure to diagnose measurement-induced phase transitions in monitored quantum systems, providing an analytical demonstration of its critical behavior.

Contribution

It develops a thermodynamic perspective based on the arrow of time, linking it to phase transitions in quantum systems and analytically characterizing its critical behavior.

Findings

The arrow of time (AoT) is a nonlinear functional of the density matrix.

AoT exhibits nonanalytic behavior at the phase transition.

Analytical solution of a quantum circuit model shows AoT's critical exponent.

Abstract

Measurement-induced phase transitions (MIPTs) in monitored quantum systems are typically diagnosed using entanglement-based measures. Here, we develop a complementary thermodynamic perspective based on the arrow of time (AoT), which arises from the intrinsic irreversibility of the quantum measurements driving these transitions. We study the AoT - defined as the logarithmic ratio of forward and backward trajectory probabilities - across a family of models exhibiting MIPTs. We find that, like entanglement entropy, the AoT is a nonlinear functional of the averaged density matrix; however, in contrast to entanglement, it is associated with a local operator. To determine whether the AoT exhibits critical behavior, we formulate and exactly solve a model of a random quantum circuit with non-projective measurements. This allows us to analytically demonstrate that the AoT displays nonanalytic behavior and identify its critical exponent. Our results establish the AoT as a novel diagnostic for phase transitions in monitored quantum systems.