The Quantum Esscher Transform

Abstract

The Esscher Transform is a tool of broad utility in various domains of applied probability. It provides the solution to a constrained minimum relative entropy optimization problem. In this work, we study the generalization of the Esscher Transform to the quantum setting. We examine a relative entropy minimization problem for a quantum density operator, potentially of wide relevance in quantum information theory. The resulting solution form motivates us to define the \textit{quantum} Esscher Transform, which subsumes the classical Esscher Transform as a special case. Envisioning potential applications of the quantum Esscher Transform, we also discuss its implementation on fault-tolerant quantum computers. Our algorithm is based on the modern techniques of block-encoding and quantum singular value transformation (QSVT). We show that given block-encoded inputs, our algorithm outputs a subnormalized block-encoding of the quantum Esscher transform within accuracy $\hat{I}\mu$ in $\tilde{O}(\hat{I}ºd{\log }^{2}1/\hat{I}\mu )$ queries to the inputs, where $\hat{I}º$ is the condition number of the input density operator and $d$ is the number of constraints.

26 pages