Study of Gaussian Channels: Classification of one mode Bosonic Channels, Kraus Representation and a result from ESD [HBNI MSc 6]

Abstract

This review starts with the basic Hilbert space structure for continuous variable systems, explaining the pre-requisites to be able to define a Gaussian state. After that a preliminary study of Gaussian channels is undertaken. The criteria for a quantum channel being Gaussian is laid out. In accordance with these criteria a classification of one-mode bosonic Gaussian channels with a single-mode environment is given. The Kraus representation for the various channels is then obtained and used to verify some of the pre-existing properties of the various channels (non-classicality breaking, entanglement breaking, extremality of the channel, etc). After this the entanglement sudden death (ESD) of a two-mode Gaussian state under the action of a Gaussian channel is studied. This channel comprises of two mutually exclusive channels, each of which is acting on one of the two-modes of the Gaussian states. The channel action comprises of interaction of the mode with a thermal bath. Both the channels, interacting with the two modes separately are at the same temperature. In the study for ESD, it is discovered that for all non-zero temperatures, all entangled-two-mode Gaussian states undergo ESD at some time or the other. For the zero temperature case S. Goyal and S. Ghosh have proved that ESD won’t occur for a set kind of states. We have tried to generalize this result i.e. found another set of two mode Gaussian states which also won’t undergo ESD under the channel action. It is desired to know if there are some two-mode Gaussian states which will undergo ESD at zero temperature. We find that there are some states which undergo ESD while interacting with a zero-temperature thermal bath.