Entropy generation in Gaussian quantum transformations: applying the replica method to continuous-variable quantum information theory

Christos N Gagatsos,Alexandros I Karanikas,Georgios Kordas,Nicolas J Cerf

doi:10.1038/npjqi.2015.8

Abstract

AbstractIn spite of their simple description in terms of rotations or symplectic transformations in phase space, quadratic Hamiltonians such as those modelling the most common Gaussian operations on bosonic modes remain poorly understood in terms of entropy production. For instance, determining the quantum entropy generated by a Bogoliubov transformation is notably a hard problem, with generally no known analytical solution, while it is vital to the characterisation of quantum communication via bosonic channels. Here we overcome this difficulty by adapting the replica method, a tool borrowed from statistical physics and quantum field theory. We exhibit a first application of this method to continuous-variable quantum information theory, where it enables accessing entropies in an optical parametric amplifier. As an illustration, we determine the entropy generated by amplifying a binary superposition of the vacuum and a Fock state, which yields a surprisingly simple, yet unknown analytical expression.

Highlights

Gaussian transformations are ubiquitous in quantum physics, playing a major role in quantum optics, quantum field theory, solid-state physics or black-hole physics.[1]
The Bogoliubov transformations resulting from Hamiltonians that are quadratic in mode operators are among the most significant Gaussian transformations, well known to model superconductivity[2] and describing a much wider range of physical situations, from squeezing or amplification in the context of quantum optics[3,4,5,6] to Unruh radiation in an accelerating frame[7,8,9] or even Hawking radiation as emitted by a black hole.[10,11,12]
We focus on Gaussian bosonic transformations, which are at the heart of so-called Gaussian quantum information theory.[13]

Summary

Introduction

Gaussian transformations are ubiquitous in quantum physics, playing a major role in quantum optics, quantum field theory, solid-state physics or black-hole physics.[1]. We demonstrate that the replica method can be successfully exploited to overcome this central problem and find an exact analytical expression for the entropy generated by Gaussian processes acting on non-trivial bosonic states. Calculating the von Neumann entropy Sðρ^Þ 1⁄4 - trðρ^ lnρ^Þ of a where τ 1⁄4 tanhξ, from which we find bosonic mode that is found in state ρ^ at the output of a Gaussian transformation is often an intractable task because it requires turning to state space and finding the infinite-size vector of trÀρ^nmÁ

Results

Conclusion