On the Choi-Jamiolkowski Correspondence in Infinite Dimensions

TL;DR

This paper rigorously formulates the Choi-Jamiolkowski correspondence in infinite-dimensional quantum systems, clarifying the properties of CJ forms and operators, especially for entanglement-breaking and Bosonic Gaussian channels.

Contribution

It provides a rigorous mathematical framework for the CJ correspondence in infinite dimensions, avoiding limiting procedures and characterizing bounded CJ operators for specific channels.

Findings

No limiting procedure needed for defining CJ forms in infinite dimensions

Characterization of bounded CJ operators for entanglement-breaking channels

Explicit expressions for CJ forms of Bosonic Gaussian channels

Abstract

We give a mathematical formulation for the Choi-Jamiolkowski (CJ) correspondence in the infinite-dimensional case in the form close to one used in quantum information theory. We show that there is no need to use a limiting procedure to define "unnormalized maximally entangled state" and the corresponding analog of the Choi matrix since they can be defined rigorously as, in general, nonclosable forms on an appropriate dense subspace. The properties of these forms are discussed in Sec. 2. An important question is: when the CJ form is given by a bounded operator. This is the case for entanglement-breaking channels: we prove this in Sec. 3 along with a version of a result of Wolf et al. characterizing CJ operators which correspond to such channels by giving precise definitions of a separable operator and a relevant integral. In Sec. 4 we obtain explicit expressions for CJ forms and operators defining a general Bosonic Gaussian channel. In Sec. 5 we give a decomposition of the Gaussian CJ form into product of the four principal types and a necessary and sufficient condition for existence of the bounded CJ operator.