[Paper Review] Entanglement transitions in the quantum Ising chain: A comparison between different unravelings of the same Lindbladian

G. Piccitto, Angelo Russomanno+1 more|arXiv (Cornell University)|Nov 22, 2021

Quantum many-body systems80 references90 citations

TL;DR

This paper investigates measurement-induced entanglement transitions in the quantum Ising chain under dephasing dissipation using two distinct unravelings of the same Lindblad master equation: quantum-state-diffusion (QSD) and quantum-jump (QJ) dynamics. While both preserve Gaussianity for large-scale simulations, they yield different phase diagrams—QSD shows a crossover from logarithmic to area-law entanglement scaling dependent on field strength and measurement rate, whereas QJ dynamics shifts this transition to different parameters, demonstrating that entanglement behavior is unraveling-dependent. The results contradict predictions from non-Hermitian Hamiltonian dynamics.

ABSTRACT

We study the dynamics of entanglement in the quantum Ising chain with dephasing dissipation in a Lindblad master equation form. We consider two unravelings which preserve the Gaussian form of the state, allowing to address large system sizes. The first unraveling gives rise to a quantum-state-diffusion dynamics, while the second one describes a specific form of quantum-jump evolution, suitably constructed to preserve Gaussianity. In the first case we find a crossover from area-law to logarithm-law entanglement scaling and draw the related phase diagram. In the second case we only find logarithm-law scaling, remarking the different entanglement behavior for different unravelings of the same Lindblad equation. Finally, we compare these outcomes with the predictions of a non-Hermitian Hamiltonian evolution, finding conflicting results.

Motivation & Objective

• To investigate how different stochastic unravelings of the same Lindblad master equation affect entanglement dynamics in open quantum systems.
• To compare entanglement scaling—specifically area-law vs. logarithmic scaling—under distinct measurement protocols in the transverse-field Ising chain.
• To assess whether non-Hermitian Hamiltonian dynamics can accurately predict entanglement transitions observed in stochastic quantum trajectories.
• To explore the role of measurement protocol in determining dynamical phases, particularly in the presence of dephasing dissipation.

Proposed method

• Uses a Lindblad master equation to model dephasing dissipation in the quantum Ising chain with a transverse field.
• Employs two unravelings: quantum-state-diffusion (QSD) and quantum-jump (QJ) dynamics, both preserving Gaussianity for efficient large-scale simulations.
• Applies the stochastic Schr"odinger equation for QSD and a jump-based formalism for QJ, ensuring both yield the same average density matrix.
• Computes entanglement entropy using the von Neumann entropy of a subsystem, tracking its scaling with system size.
• Performs numerical simulations up to system size L = 256, using efficient algorithms for Gaussian fermionic states.
• Compares results with predictions from non-Hermitian Hamiltonian evolution, which is known to approximate certain measurement-induced transitions.

Experimental results

Research questions

• RQ1 Does the choice of unraveling (QSD vs. QJ) lead to different entanglement scaling behaviors in the same Lindblad-dissipative quantum Ising chain?
• RQ2 How does the phase boundary between logarithmic and area-law entanglement scaling depend on the measurement protocol and external field strength?
• RQ3 Can the non-Hermitian Hamiltonian approximation accurately predict the entanglement transition observed in stochastic quantum trajectories?
• RQ4 How does the entanglement entropy evolve asymptotically under different unravelings, particularly near the quantum critical point?

Key findings

• The QSD unraveling reveals a crossover from logarithmic to area-law entanglement scaling as the transverse field strength increases, with the transition point shifting with system size.
• For the QJ unraveling, the same crossover occurs at significantly different parameters compared to QSD, indicating that the entanglement phase diagram is not unique for the same Lindblad equation.
• At fixed measurement rate γ = 1.5, the QJ unraveling consistently yields area-law entanglement across field strengths hf = 2, 5, 8, suggesting a different dynamical phase than predicted by QSD.
• The QSD phase diagram extends previous results to non-zero transverse fields, showing a maximum in entanglement entropy near the unitary critical point for small measurement rates.
• Numerical evidence from L ≤ 256 does not rule out the existence of a subextensive phase in the QJ case, but the observed behavior is consistent with area-law scaling.
• Neither unraveling’s results agree with predictions from non-Hermitian Hamiltonian dynamics, indicating limitations of this approximate approach.