[Paper Review] Measurement phase transitions in the no-click limit as quantum phase transitions of a non-hermitean vacuum

Caterina Zerba, Alessandro Silva|arXiv (Cornell University)|Jan 18, 2023

Quantum Mechanics and Applications6 citations

TL;DR

This paper establishes a direct link between measurement-induced entanglement phase transitions in non-Hermitian quantum systems and the quantum phase transitions of the non-Hermitian vacuum. By analyzing the no-click limit of stochastic dynamics in the Transverse Field Ising Chain and Long-Range Kitaev Chain, it shows that entanglement entropy scaling—bounded or logarithmic—mirrors the spectral gap structure of the imaginary part of the quasi-particle spectrum, generalizing the area-law theorem to non-Hermitian systems.

ABSTRACT

We study dynamical phase transitions occurring in the stationary state of the dynamics of integrable many-body non-hermitian Hamiltonians, which can be either realized as a no-click limit of a stochastic Schrödinger equation or using spacetime duality of quantum circuits. In two specific models, the Transverse Field Ising Chain and the Long Range Kitaev Chain, we observe that the entanglement phase transitions occurring in the stationary state have the same nature as that occurring in the vacuum of the non-hermitian Hamiltonian: bounded entanglement entropy when the imaginary part of the quasi-particle spectrum is gapped and a logarithmic growth for gapless imaginary spectrum. This observation suggests the possibility to generalize the area-law theorem to non-Hermitian Hamiltonians.

Motivation & Objective

• To understand the origin of entanglement phase transitions in the no-click limit of continuous measurement dynamics.
• To establish a correspondence between the entanglement properties of the stationary state in stochastic dynamics and the vacuum of a non-Hermitian Hamiltonian.
• To generalize the area-law theorem to non-Hermitian systems by relating entanglement scaling to the spectral gap of the imaginary part of the quasi-particle spectrum.
• To analyze integrable models—Transverse Field Ising Chain and Long-Range Kitaev Chain—under local measurements to test the universality of this correspondence.
• To explore how long-range interactions affect the nature of the entanglement transition in the no-click limit.

Proposed method

• Formalizing the no-click limit as a non-Hermitian Hamiltonian dynamics via the stochastic Schrödinger equation, where measurement outcomes are suppressed.
• Mapping the non-Hermitian Hamiltonian to a quadratic form in fermionic operators, enabling diagonalization via Bogoliubov transformation.
• Analyzing the spectrum of the non-Hermitian Hamiltonian to identify the imaginary part of quasi-particle energies, which determines the entanglement scaling.
• Computing the bipartite entanglement entropy using the Majorana fermion correlation matrix derived from the ground state of the non-Hermitian Hamiltonian.
• Using the generalized area-law conjecture: bounded entanglement when the imaginary spectrum has a gap, logarithmic scaling when the gap closes.
• Applying spacetime duality to connect the no-click dynamics to a non-Hermitian evolution, enabling analytical treatment in integrable models.

Experimental results

Research questions

• RQ1 Does the entanglement phase transition in the no-click limit of measurement-driven dynamics correspond to a quantum phase transition in the vacuum of a non-Hermitian Hamiltonian?
• RQ2 Can the area-law theorem be generalized to non-Hermitian systems based on the spectral gap of the imaginary part of the quasi-particle spectrum?
• RQ3 How does long-range pairing in the Kitaev chain affect the entanglement transition in the no-click limit?
• RQ4 What is the role of integrability in establishing the correspondence between the stationary state of stochastic dynamics and the non-Hermitian vacuum?
• RQ5 Why do exponentially rare no-click trajectories exhibit the same entanglement phase transition as generic trajectories with quantum jumps?

Key findings

• The entanglement entropy in the no-click limit of the Transverse Field Ising Chain matches the entanglement of the non-Hermitian vacuum: bounded when the imaginary part of the spectrum has a gap, logarithmic when the gap closes.
• For the Long-Range Kitaev Chain with power-law decay exponent $ d < 1 $, the imaginary spectrum remains gapless at all measurement rates $ ho $, leading to logarithmic entanglement scaling regardless of $ ho $, consistent with the generalized area-law conjecture.
• For $ d > 1 $, a phase transition occurs at a critical measurement rate $ ho_c $: logarithmic scaling for $ ho < ho_c $ (gapless imaginary spectrum), and bounded entanglement for $ ho > ho_c $ (gapped spectrum), confirming the generalized area-law theorem.
• The non-Hermitian vacuum of the Bogoliubov quasiparticles captures the full entanglement structure of the no-click trajectory, establishing a one-to-one correspondence between the two.
• The entanglement scaling is fully determined by the spectral gap of $ ext{Im}[ ext{eigenvalues of the non-Hermitian Hamiltonian}] $, not by the real part or other spectral features.
• The correspondence holds despite the non-Hermitian Hamiltonian being non-diagonalizable in the standard sense, with the vacuum state defined via a non-orthogonal basis of left and right eigenstates.