# Lie-algebraic connections between two classes of risk-sensitive performance criteria for linear quantum stochastic systems

## TL;DR

This paper explores the mathematical relationship between two types of risk-sensitive performance measures for linear quantum systems, aiming to transfer properties and develop computational tools for quantum control.

## Contribution

It establishes a Lie algebraic connection between two classes of risk-sensitive criteria for quantum systems using complex Hamiltonian kernels and symplectic factorizations.

## Key findings

- Derived a Lie algebraic link between the two risk-sensitive criteria.
- Extended properties from one criterion to the other.
- Developed state-space equations for control and filtering applications.

## Abstract

This paper is concerned with the original risk-sensitive performance criterion for quantum stochastic systems and its recent quadratic-exponential counterpart. These functionals are of different structure because of the noncommutativity of quantum variables and have their own useful features such as tractability of evolution equations and robustness properties. We discuss a Lie algebraic connection between these two classes of cost functionals for open quantum harmonic oscillators using an apparatus of complex Hamiltonian kernels and symplectic factorizations. These results are aimed to extend useful properties from one of the classes of risk-sensitive costs to the other and develop state-space equations for computation and optimization of these criteria in quantum robust control and filtering problems.