[Paper Review] Optimized Quantum Steering and Exceptional Points

Parveen Kumar, Heinrich-Gregor Zirnstein+3 more|arXiv (Cornell University)|Jan 18, 2021

Quantum many-body systems50 references4 citations

TL;DR

This paper demonstrates that any pure or mixed quantum state can be optimally steered using weak blind measurements combined with a system Hamiltonian, where the optimization is governed by Liouvillian exceptional points. It reveals a phase transition between relaxational (second-order EP) and oscillatory (third-order EP) dynamics depending on target state purity.

ABSTRACT

The state of a quantum system may be steered towards a predesignated target state, employing a sequence of weak $ extit{blind}$ measurements (where the detector's readouts are traced out). Here we analyze the steering of a two-level system using the interplay of a system Hamiltonian and weak measurements, and show that $ extit{any}$ pure or mixed state can be targeted. We show that the optimization of such a steering protocol is underlain by the presence of Liouvillian exceptional points. More specifically, for high purity target states, optimal steering implies purely relaxational dynamics marked by a second-order exceptional point, while for low purity target states, it implies an oscillatory approach to the target state. The phase transition between these two regimes is characterized by a third-order exceptional point.

Motivation & Objective

• To investigate how weak blind measurements and system Hamiltonians can steer a two-level quantum system toward any target state.
• To identify the role of exceptional points in the Liouvillian spectrum in enabling optimal state steering.
• To characterize the dynamical regimes—relaxational vs. oscillatory—underlying optimal steering for different target state purities.
• To determine the nature of the phase transition between these dynamical regimes.

Proposed method

• Model the system using a Lindblad master equation with weak measurement and Hamiltonian evolution.
• Analyze the Liouvillian superoperator's spectrum to identify exceptional points (EPs) in the parameter space.
• Use perturbative and exact diagonalization techniques to study the dynamics near EPs.
• Classify the type of exceptional point (second- or third-order) based on the behavior of eigenvalues and eigenvectors.
• Map the transition between relaxational and oscillatory dynamics to the order of the exceptional point.
• Verify that any pure or mixed state can be targeted through optimal parameter tuning near EPs.

Experimental results

Research questions

• RQ1 Can any pure or mixed quantum state be optimally steered using weak blind measurements and a Hamiltonian?
• RQ2 How do exceptional points in the Liouvillian spectrum influence the optimality of the steering protocol?
• RQ3 What dynamical behavior emerges during optimal steering for high-purity versus low-purity target states?
• RQ4 What is the nature of the phase transition between relaxational and oscillatory dynamics in the steering process?
• RQ5 How do the orders of exceptional points (second- vs. third-order) relate to the underlying dynamical regimes?

Key findings

• Optimal steering of any pure or mixed state is achievable through the interplay of weak measurements and system Hamiltonians.
• For high-purity target states, optimal steering corresponds to purely relaxational dynamics associated with a second-order exceptional point.
• For low-purity target states, optimal steering involves an oscillatory approach to the target, linked to a third-order exceptional point.
• The transition between relaxational and oscillatory dynamics is characterized by a third-order exceptional point, marking a phase transition in the Liouvillian spectrum.
• The presence of exceptional points underlies the optimization mechanism, providing a geometric and spectral explanation for efficient state engineering.