Fixed Points of Completely Positive Trace-Preserving Maps in Infinite Dimension

TL;DR

This paper proves the existence of fixed points for certain infinite-dimensional quantum channels under specific conditions, supporting Deutsch's proposal for quantum theory with closed timelike curves.

Contribution

It extends fixed point results for quantum channels to infinite dimensions under invariance conditions, using Schauder-Tychonoff theorem.

Findings

Fixed points exist for certain infinite-dimensional quantum channels.

Supports Deutsch's proposal for quantum theory with closed timelike curves.

Uses Schauder-Tychonoff fixed point theorem in the proof.

Abstract

Completely positive trace-preserving maps $S$, also known as quantum channels, arise in quantum physics as a description of how the density operator $\rho$ of a system changes in a given time interval, allowing not only for unitary evolution but arbitrary operations including measurements or other interaction with an environment. It is known that if the Hilbert space $\mathscr{H}$ that $\rho$ acts on is finite-dimensional, then every $S$ must have a fixed point, i.e., a density operator $\rho_0$ with $S(\rho_0)=\rho_0$. In infinite dimension, $S$ need not have a fixed point in general. However, we prove here the existence of a fixed point under a certain additional assumption which is, roughly speaking, that $S$ leaves invariant a certain set of density operators with bounded ``cost'' of preparation. The proof is an application of the Schauder-Tychonoff fixed point theorem. Our motivation for this question comes from a proposal of Deutsch for how to define quantum theory in a space-time with closed timelike curves; our result supports the viability of Deutsch's proposal.