Passive states optimize the output of bosonic Gaussian quantum channels

TL;DR

This paper proves that passive input states, which are diagonal with decreasing eigenvalues, optimize the output of bosonic Gaussian channels by producing outputs that majorize those from other states with the same spectrum.

Contribution

It establishes that passive states maximize the output in bosonic Gaussian channels, extending previous results to a broader class of input states.

Findings

Passive states produce majorizing outputs compared to other states with same spectrum.

The result generalizes the vacuum input optimality to all passive states.

Majorization ordering applies to the outputs of gauge-covariant bosonic Gaussian channels.

Abstract

An ordering between the quantum states emerging from a single mode gauge-covariant bosonic Gaussian channel is proven. Specifically, we show that within the set of input density matrices with the same given spectrum, the element passive with respect to the Fock basis (i.e. diagonal with decreasing eigenvalues) produces an output which majorizes all the other outputs emerging from the same set. When applied to pure input states, our finding includes as a special case the result of A. Mari, et al., Nat. Comm. 5, 3826 (2014) which implies that the output associated to the vacuum majorizes the others.