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Journal of Optical Communications and Networking

Information Geometrical Analysis of Additivity of Optical Quantum Channels

László Gyöngyösi, Sándor Imre

Published Dec 16, 2010• 9 citations
Publisher

Abstract

The additivity property of quantum channels is still an open question and an exciting subject of current research. There are some nonunital channels for which strict additivity is known, although the general rule for nonunital quantum channels is still not proven. We analyze the additivity of the amplitude damping channel, which is an important channel in physical implementations and optical communications. The effect of amplitude damping has great importance in optical communications, since this channel model describes energy dissipation. We show an efficient information computational geometric method to analyze the additivity property of the amplitude damping quantum channel, using quantum Delaunay tessellation on the Bloch ball and quantum relative entropy as a distance measure.

DOI
10.1364/jocn.3.000048

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The additivity property of quantum channels is still an open question and an exciting subject of current research. There are some nonunital channels for which strict additivity is known, although the general rule for nonunital quantum channels is still not proven. We analyze the additivity of the amplitude damping channel, which is an important channel in physical implementations and optical communications. The effect of amplitude damping has great importance in optical communications, since this channel model describes energy dissipation. We show an efficient information computational geometric method to analyze the additivity property of the amplitude damping quantum channel, using quantum Delaunay tessellation on the Bloch ball and quantum relative entropy as a distance measure.
The amplitude damping quantum channel has great importance in practical quantum communications, since this channel model describes energy dissipation. The additivity of the amplitude damping quantum channel model is still an open question and an exciting subject of current research. We show an efficient information computational geometric method to analyze the additivity property of the amplitude damping quantum channel, using quantum Delaunay tessellation on the Bloch ball and quantum relative entropy as a distance measure.
Quantum channel additivity is currently an active area of research in Quantum Information Theory. The additivity conjectures have emerged from an attempt to find a closed form expression for the capacity of a noisy quantum channel. To this day, strict additivity for quantum channel capacity has been conjectured, but not proven. The open questions related to additivity of quantum channel capacities can be discussed by our novel approach. We show an efficient computational geometric method to analyze the additivity property, using quantum Delaunay tessellation on the Bloch ball and quantum relative entropy as distance measure.
Traditionally, Quantum Information, and Quantum Communication specifically, have been focused on qubit-based architectures. Recent results, however, highlighted that higher dimensional architectures (qudit-based) may present advantages both in terms of communication and computation; a family of channels called Multi-level Amplitude Damping (MAD) channels, which are a possible qudit generalization of the well known Amplitude Damping Channels, is able to model energy decay processes that may happen during signal transmission. In this work, the Quantum Capacity of 4-dimensional MAD's is studied, relying on a technique for computing it even outside of degradable and antidegradable conditions. We also characterized the complete region of antidegradability and degradability in the parameter space for a generic d-dimensional MAD using both analytical and semi-numerical methods.
The superactivation of zero-capacity quantum channels makes it possible to use two zero-capacity quantum channels with a positive joint capacity at the output. Currently, we have no theoretical results for describing all possible combinations of superactive zero-capacity channels, hence there should be many possible - still undiscovered - combinations. Our method gives an algorithmic solution to the superactivation problem, and provides an efficient algorithmic solution for discovering “superactive” zero-capacity optical quantum channels. We analyze the superactivation property of the amplitude damping quantum channel model. This model has great importance in practical communications, since this channel model describes energy dissipation due to the noise of the optical channel. To discover these superactive zero-capacity channels, an extremely large set of possible quantum states has to be analyzed, however an efficient algorithmic framework is still missing for this purpose. We show a fundamentally new method of finding the conditions for “superactive” zero-capacity optical quantum channels.
The primary objective of quantum Shannon theory is to evaluate the capacity of quantum channels. In spite of the existence of rigorous coding theorems that quantify the transmission of information through quantum channels, superadditivity effects limit our understanding of the channel capacities. In this paper, we mainly focus on a family of channels known as multilevel amplitude damping channels. We investigate some of the information capacities of the simplest member of the multilevel amplitude damping channel, a qutrit channel, in the presence of correlations between successive applications of the channel. We find the upper bounds of the single-shot classical capacities and calculate the quantum capacities associated with a specific class of maps after investigating the degradability property of the channels. Additionally, the quantum and classical capacities of the channels have been computed in entanglement-assisted scenarios.
We study quantum channels that are close to another channel with weakly additive Holevo information, and we derive upper bounds on their classical capacity. Examples of channels with weakly additive Holevo information are entanglement-breaking channels, unital qubit channels, and Hadamard channels. Related to the method of approximate degradability, we define approximation parameters for each class above, which measure how close an arbitrary channel is to satisfying the respective property. This gives us upper bounds on the classical capacity in terms of functions of the approximation parameters, as well as an outer bound on the dynamic capacity region of a quantum channel. Since these parameters are defined in terms of the diamond distance, the upper bounds can be computed efficiently using semidefinite programming (SDP). We exhibit the usefulness of our method with two example channels: a convex mixture of amplitude damping and depolarizing noise and a composition of amplitude damping and dephasing noise. For both channels, our bounds perform well in certain regimes of the noise parameters in comparison to a recently derived SDP upper bound on the classical capacity. Along the way, we define the notion of a generalized channel divergence (which includes the diamond distance as an example), and we prove that for jointly covariant channels these quantities are maximized by purifications of a state invariant under the covariance group. This latter result may be of independent interest.