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Physical Review A

Gaussian channels that are eventually entanglement breaking yet asymptotically nonclassicality saving

Sagnik Garai, J. Solomon Ivan

Published Nov 29, 2018• 2 citations
Publisher

Abstract

A complete classification of single-mode bosonic Gaussian channels on the basis of being quantum-limited or entanglement breaking under $n$-fold composition is obtained. Parametric forms for all single-mode bosonic Gaussian channels that remain quantum-limited under $n$-fold composition is obtained. It is shown that the obtained parametric forms of quantum-limited amplifier, attenuator, and singular noise channels are entanglement saving, i.e., they do not break entanglement under finite $n$-fold composition. All other single-mode bosonic Gaussian channels, quantum limited or not, are shown to be eventually entanglement breaking. Nonclassicality breaking under multiple composition of a single-mode bosonic Gaussian channel is also studied. We outline a family of single-mode bosonic Gaussian channels that are eventually entanglement breaking, but asymptotically nonclassicality saving. We illustrate examples of channels that are eventually nonclassicality breaking, asymptotically nonclassicality breaking, and asymptotically nonclassicality saving.

DOI
10.1103/physreva.98.052353

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A complete classification of single-mode bosonic Gaussian channels on the basis of being quantum-limited or entanglement breaking under nnn-fold composition is obtained. Parametric forms for all single-mode bosonic Gaussian channels that remain quantum-limited under nnn-fold composition is obtained. It is shown that the obtained parametric forms of quantum-limited amplifier, attenuator, and singular noise channels are entanglement saving, i.e., they do not break entanglement under finite nnn-fold composition. All other single-mode bosonic Gaussian channels, quantum limited or not, are shown to be eventually entanglement breaking. Nonclassicality breaking under multiple composition of a single-mode bosonic Gaussian channel is also studied. We outline a family of single-mode bosonic Gaussian channels that are eventually entanglement breaking, but asymptotically nonclassicality saving. We illustrate examples of channels that are eventually nonclassicality breaking, asymptotically nonclassicality breaking, and asymptotically nonclassicality saving.
Nonclassicality and entanglement are notions fundamental to quantum-information processes involving continuous variable systems. That these two notions are intimately related has been intuitively appreciated for quite some time. An aspect of considerable interest is the behavior of these attributes of a state under the action of a noisy channel. Inspired by the notion of entanglement-breaking channels, we define the concept of nonclassicality-breaking channels in a natural manner. We show that the notion of nonclassicality breaking is essentially equivalent---in a clearly defined sense of the term ``essentially''---to the notion of entanglement breaking, as far as bosonic Gaussian channels are concerned. This is notwithstanding the fact that the very notion of entanglement breaking requires reference to a bipartite system, whereas the definition of nonclassicality breaking makes no such reference. Our analysis rests on our classification of channels into nonclassicality-based, as against entanglement-based, types of canonical forms. Our result takes ones intuitive understanding of the close relationship between nonclassicality and entanglement a step closer.
The Glauber-Sudarshan diagonal ``weight'' function provides a natural divide between the quantum-optical notion of classical and nonclassical states of continuous-variable systems. Based on this demarcation, a channel is said to be nonclassicality breaking if it outputs only classical states for any input state. We focus on multimode bosonic Gaussian channels and classify those that are nonclassicality breaking by introducing a criterion that needs to be satisfied by the matrices representing these channels. The criterion can be interpreted as a nonclassicality benchmark for these channels since it quantifies the threshold noise at which there is a complete nonclassical to classical transition of the output states, i.e., it quantifies the robustness of the nonclassicality of the outputs of the channel against Gaussian noise. We then prove a striking ``duality'' between nonclassicality-breaking and entanglement-breaking bosonic Gaussian channels.
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A complete degradability analysis of one-mode bosonic Gaussian channels is presented. We show that apart from the class of channels which are unitarily equivalent to the channels with additive classical noise, these maps can be characterized in terms of weak- and/or anti-degradability. Furthermore a new set of channels which have null quantum capacity is identified. This is done by exploiting the composition rules of one-mode Gaussian maps and the fact that anti-degradable channels cannot be used to transfer quantum information.
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We identify the quantum channels corresponding to the interaction of a Gaussian quantum state with an already formed Schwarzschild black hole. Using recent advances in the classification of one-mode bosonic Gaussian channels, we find that (with one exception) the black hole Gaussian channels lie in the nonentanglement breaking subset of the lossy channels C(loss)\mathcal{C}(\text{loss})C(loss), amplifying channels C(amp)\mathcal{C}(\mathrm{amp})C(amp) and classical-noise channels B2{\mathcal{B}}_{2}B2​. We show that the channel parameters depend on the black hole mass and the properties of the potential barrier surrounding it. This classification enables us to calculate the classical and quantum capacity of the black hole and to estimate the quantum capacity where no tractable quantum capacity expression exists today. We discuss these findings in the light of the black hole quantum information loss problem.