# A quantum Karhunen-Loeve expansion and quadratic-exponential functionals for linear quantum stochastic systems

## TL;DR

This paper develops a quantum version of the Karhunen-Loeve expansion for quantum Wiener processes and applies it to evaluate quadratic-exponential functionals in risk-sensitive control of open quantum systems.

## Contribution

It introduces a quantum Karhunen-Loeve expansion for quantum Wiener processes and uses it to analyze performance criteria in quantum control systems.

## Key findings

- Quantum Wiener process expansion as sinusoidal series with independent operators
- Representation of system variables in linear quantum stochastic differential equations
- Application to risk-sensitive control performance evaluation

## Abstract

This paper extends the Karhunen-Loeve representation from classical Gaussian random processes to quantum Wiener processes which model external bosonic fields for open quantum systems. The resulting expansion of the quantum Wiener process in the vacuum state is organised as a series of sinusoidal functions on a bounded time interval with statistically independent coefficients consisting of noncommuting position and momentum operators in a Gaussian quantum state. A similar representation is obtained for the solution of a linear quantum stochastic differential equation which governs the system variables of an open quantum harmonic oscillator. This expansion is applied to computing a quadratic-exponential functional arising as a performance criterion in the framework of risk-sensitive control for this class of open quantum systems.