Normal completely positive maps on the space of quantum operations

TL;DR

This paper extends the theory of quantum supermaps to infinite-dimensional settings, proving dilation theorems that enable their implementation via quantum circuits, thus broadening the understanding of higher-order quantum transformations.

Contribution

It generalizes the theory of quantum supermaps to separable von Neumann algebras and establishes dilation theorems analogous to classical results, facilitating their physical implementation.

Findings

Proved dilation theorems for quantum supermaps in infinite-dimensional settings.

Extended the theory to quantum superinstruments and derived a dilation theorem.

Showed all supermaps can be implemented by connecting devices in quantum circuits.

Abstract

Quantum supermaps are higher-order maps transforming quantum operations into quantum operations. Here we extend the theory of quantum supermaps, originally formulated in the finite dimensional setting, to the case of higher-order maps transforming quantum operations with input in a separable von Neumann algebra and output in the algebra of the bounded operators on a given separable Hilbert space. In this setting we prove two dilation theorems for quantum supermaps that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. Finally, we consider the case of quantum superinstruments, namely measures with values in the set of quantum supermaps, and derive a dilation theorem for them that is analogue to Ozawa's theorem for quantum instruments. The three dilation theorems presented here show that all the supermaps defined in this paper can be implemented by connecting devices in quantum circuits.