Operator-sum Representation for Bosonic Gaussian Channels

TL;DR

This paper develops Kraus representations for Bosonic Gaussian channels, revealing their structure, fixed points, and extremality, and introduces a simple set of Kraus operators for noisy channels.

Contribution

It provides a comprehensive Kraus operator framework for Bosonic Gaussian channels, including new insights into their extremality and structure of Kraus operators.

Findings

Quantum-limited channels are extremal.

Entanglement-breaking channels have rank-one Kraus operators.

Noisy Gaussian channels can be constructed from quantum-limited channels.

Abstract

Operator-sum or Kraus representations for single-mode Bosonic Gaussian channels are developed, and several of their consequences explored. Kraus operators are employed to bring out the manner in which the unphysical matrix transposition map when accompanied by injection of a threshold classical noise becomes a physical channel. The action of the quantum-limited attenuator and amplifier channels as simply scaling maps on suitable quasi-probabilities in phase space is examined in the Kraus picture. Consideration of cumulants is used to examine the issue of fixed points. In the cases of entanglement-breaking channels a description in terms of rank one Kraus operators is shown to emerge quite simply. In contradistinction, it is shown that there is not even one finite rank operator in the entire linear span of Kraus operators of the quantum-limited amplifier or attenuator families, an assertion far stronger than the statement that these are not entanglement breaking channels. A characterization of extremality in terms of Kraus operators, originally due to Choi, is employed to show that all quantum-limited Gaussian channels are extremal. The fact that every noisy Gaussian channel can be realised as product of a pair of quantum-limited channels is used to construct a discrete set of linearly independent Kraus operators for noisy Gaussian channels, including the classical noise channel, and these Kraus operators have a particularly simple structure.