[Paper Review] Full counting statistics as probe of measurement-induced transitions in the quantum Ising chain

Emanuele Tirrito, Alessandro Santini+2 more|arXiv (Cornell University)|Dec 19, 2022

Quantum many-body systems4 citations

TL;DR

This paper investigates measurement-induced phase transitions in the quantum Ising chain by analyzing the full counting statistics (FCS) of local magnetization under continuous projective measurements. It demonstrates that the probability distribution of the transverse magnetization undergoes a qualitative change at a critical measurement rate, revealing a transition between paramagnetic and ferromagnetic phases, with the fourth cumulant of the magnetization serving as a non-trivial probe of the transition, though numerical limitations prevent access to thermodynamic scaling.

ABSTRACT

Non-equilibrium dynamics of many-body quantum systems under the effect of measurement protocols is attracting an increasing amount of attention. It has been recently revealed that measurements may induce different non-equilibrium regimes and an abrupt change in the scaling-law of the bipartite entanglement entropy. However, our understanding of how these regimes appear, how they affect the statistics of local quantities and, finally whether they survive in the thermodynamic limit, is much less established. Here we investigate measurement-induced phase transitions in the Quantum Ising chain coupled to a monitoring environment. In particular we show that local projective measurements induce a quantitative modification of the out-of-equilibrium probability distribution function of the local magnetization. Starting from a GHZ state, the relaxation of the paramagnetic and the ferromagnetic order is analysed. In particular we describe how the probability distribution of the former shows different behaviour in the area-law and volume-law regimes.

Motivation & Objective

• To investigate how continuous projective measurements induce phase transitions in the quantum Ising chain by probing local observables.
• To examine whether measurement-induced transitions persist in the thermodynamic limit using full counting statistics of local magnetization.
• To characterize the statistical behavior of the transverse magnetization under varying measurement rates, focusing on its probability distribution and cumulants.
• To determine whether the fourth cumulant of the magnetization serves as a sensitive probe of the transition, despite numerical challenges.
• To establish a connection between the FCS of local order parameters and the emergence of distinct non-equilibrium phases in open quantum many-body systems.

Proposed method

• The study employs a quantum Ising chain coupled to a monitoring environment that continuously measures the transverse spin component via projective measurements.
• The dynamics is described using a Lindblad master equation for the averaged density matrix, with quantum trajectories used to sample stochastic realizations.
• Full counting statistics (FCS) of the local magnetization operator $ M^{xx}_ ho $ are computed via the generating function $ F_ ho( heta) $, which encodes all moments of the distribution.
• The generating function is decomposed into a trivial infinite-temperature part and a non-trivial part $ G_ ho( heta) $, whose derivatives yield the cumulants.
• The computation of higher-order moments relies on the Pfaffian representation of spin correlation functions in terms of Majorana fermion correlations.
• The fourth cumulant $ ar{ar{ heta}}_{4} $ is extracted as a non-linear response to the measurement rate, serving as a key indicator of the transition.

Experimental results

Research questions

• RQ1 How does the full counting statistics of the local transverse magnetization change across a measurement-induced phase transition in the quantum Iscing chain?
• RQ2 Can the fourth cumulant of the magnetization serve as a non-trivial probe of the transition between paramagnetic and ferromagnetic phases?
• RQ3 Does the probability distribution of the magnetization exhibit a qualitative change at a critical measurement rate, even when entanglement entropy scaling is ambiguous?
• RQ4 To what extent do the statistical properties of local observables reflect the underlying non-equilibrium phase transition, particularly in the thermodynamic limit?
• RQ5 Why does the time evolution of the fourth cumulant fail to capture the transition, despite its sensitivity in the stationary state?

Key findings

• The probability distribution of the local transverse magnetization $ M^{xx}_ ho $ exhibits a qualitative change at a critical measurement rate, signaling a transition between paramagnetic and ferromagnetic phases.
• The fourth cumulant $ ar{ar{ heta}}_{4} $ of the magnetization shows an exponential decay towards zero as the measurement rate $ \gamma $ increases, indicating a transition to the infinite-temperature state.
• The non-trivial part of the fourth cumulant $ \kappa_{t,4} $ is non-zero in the transient regime but does not reveal the phase transition clearly due to numerical limitations.
• The second cumulant is trivial due to $ \mathbb{Z}_2 $ symmetry, while the fourth cumulant captures non-linear correlations and is the first non-trivial statistical moment.
• The full counting statistics approach successfully identifies a transition in the statistics of local observables, even when entanglement entropy scaling is inconclusive.
• Numerical evaluation of FCS scales exponentially with subsystem size, limiting access to thermodynamic-size systems and preventing full characterization of the scaling behavior in the thermodynamic limit.