[Paper Review] Control of open quantum systems dynamics

Seth Lloyd, Lorenza Viola|ArXiv.org|Aug 24, 2000

Spectroscopy and Quantum Chemical Studies3 citations

TL;DR

This paper demonstrates that arbitrary open quantum system dynamics can be universally controlled using only coherent control operations and feedback from a single 'Yes-No' measurement. By repeatedly applying two coherent operations and feeding back measurement outcomes, the system can simulate any Lindblad master equation, enabling universal control over incoherent quantum evolution.

ABSTRACT

We investigate the control resources needed to effect arbitrary quantum dynamics. We show that the ability to perform measurements on a quantum system, combined with the ability to feed back the measurement results via coherent control, allows one to control the system to follow any desired open-system evolution. Such universal control can be achieved, in principle, through the repeated application of only two coherent control operations and a simple ``Yes-No'' measurement.

Motivation & Objective

• To determine the minimal control resources required to achieve arbitrary open-system quantum dynamics.
• To investigate whether feedback control can overcome the limitations of open-loop control in incoherent quantum evolution.
• To show that a single measurement and coherent feedback can generate any Markovian open-system evolution.
• To establish a framework for constructing arbitrary Lindblad-type master equations using only two coherent operations and a simple measurement.
• To demonstrate that stroboscopic, approximate control can be made arbitrarily accurate by increasing control operation frequency.

Proposed method

• Use of a simple 'Yes-No' von Neumann measurement on an auxiliary system to couple to the system of interest.
• Implementation of coherent control via unitary operations generated by two Hamiltonians, enabling universal unitary evolution on the joint system.
• Application of average Hamiltonian techniques to simulate infinitesimal measurement interactions within a small time window.
• Feedback of classical measurement results to coherently condition subsequent control operations, effectively shaping the Lindblad operators.
• Derivation of the effective Lindblad equation in the small-time limit, showing that the resulting dynamics can emulate any single Lindblad operator.
• Composition of sequential feedback-controlled operations to realize multiple Lindblad operators, enabling universal open-system evolution.

Experimental results

Research questions

• RQ1 Can arbitrary open-system dynamics be achieved using only a finite set of coherent control operations and feedback?
• RQ2 What is the minimal measurement and control resource required to simulate any Markovian quantum evolution?
• RQ3 How does feedback enable universal control in open systems when open-loop control fails?
• RQ4 Can infinitesimal measurement interactions be used to generate arbitrary Lindblad operators through feedback?
• RQ5 What is the role of coherent control in stabilizing and shaping incoherent dynamics in open quantum systems?

Key findings

• Arbitrary open-system dynamics can be enacted using only two coherent control operations and a single 'Yes-No' measurement with feedback.
• The feedback mechanism allows the construction of any Lindblad master equation, including those with multiple Lindblad operators, by sequential application.
• The effective dynamics are stroboscopic and approximate, but the error can be made arbitrarily small by increasing the number of control operations per time step.
• The method enables the realization of any single Lindblad operator proportional to $ U\overline{X} $, where $ U $ is a unitary and $ \overline{X} $ is a positive operator.
• The derivation shows that the matrix $ \mathbf{A} $ in the Lindblad equation can be made to span any rank-one matrix, allowing universal control through composition.
• The small-time limit of the protocol yields a valid Lindblad equation with arbitrary $ L $, proving the feasibility of continuous-time Markovian evolution control.