Fock majorization in bosonic quantum channels with a passive environment

Abstract

We introduce a class of quantum channels called passive-environment bosonic channels. These channels are relevant from a quantum thermodynamical viewpoint because they correspond to the energy-preserving linear coupling of a bosonic system with a bosonic environment that is in a passive state (no energy can be extracted from it by using a unitary transformation) followed by discarding the environment. The Fock-majorization relation defined in [New J. Phys. 18, 073047 (2016)] happens to be especially useful in this context as, unlike regular majorization, it connects the disorder of a state together with its energy. Our main result here is the preservation of Fock majorization across all passive-environment bosonic channels. This implies a similar preservation property for regular majorization over the set of passive states, and it also extends to passive-environment channels whose Stinespring dilation involves an active Gaussian unitary. Beyond bosonic systems, the introduced class of passiveenvironment operations naturally generalizes thermal operations and is expected to provide new insights into the thermodynamics of quantum systems.

Continuous majorization in quantum phase space

“…3. We will prove conjecture (21) for the set of Wigner-positive states of the form (22), denoted as W restr + , which has previously been studied in [26,28]. It is obvious to see that W restr + forms a convex set as well since any convex mixture of Wigner-positive states is Wigner-positive, but the boundary of this set is nonetheless non trivial, see Fig.…”

“…This expresses that, in the sense of majorization theory, the most fundamental (Wigner-positive) state is the vacuum state, i.e., the ground state of the Hamiltonian of the harmonic oscillator. Note that conjecture (21) goes beyond the scope of quantum optical states and applies to the phase space associated with any canonical pair (x, p). Furthermore, it is unrelated to the Hamiltonian of the system : the (positive) Wigner function of any state of the system must always be majorized by function (17).…”

“…3. At the same time, this set is simple enough to enable a fully analytical proof of the conjecture (21).…”

“…As such, it finds application in the study of entanglement transformations [9,10], in the discrimination of (distillable) entangled states [11,12], in the derivation of quantum uncertainty relations [13][14][15], or via so-called thermo-majorization in the context of quantum thermodynamics [16], to cite a few. In the last years, majorization theory has also proven especially useful in the framework of bosonic quantum systems, which are our interest here, serving as an instrumental tool for the investigation of entropic inequalities that are paramount in the computation of the optimal communication rates of quantum communication systems [17][18][19][20][21].…”

All phase-space linear bosonic channels are approximately Gaussian dilatable

“…Another reason to study Gaussian dilatable channels besides the fact that they constitute one of the few analytically treatable classes of non-Gaussian operations, is that they provide a systematic way of investigating classes of operations with certain physically meaningful properties, e.g. those that can be implemented by means of passive optics and arbitrary states [4] or passive optics with passive ancillary states [38], the latter being motivated from a thermodynamic context, and also for obtaining the operator-sum representations of the corresponding channels [5,39].…”