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AIP conference proceedings

Information Geometric Solution to Additivity of Amplitude-Damping Quantum Channel

László Gyöngyösi, Sándor Imre, Timothy C. Ralph, Ping Koy Lam

Published Jan 1, 2011• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.3630141

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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.
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.
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.
Abstract: We derive an improved upper bound on the two-way-assisted capacities of the qubit amplitude-damping channel with decay probability γ\gammaγ, given by log⁡2[1+(1−γ)γγ/(1−γ)]\log_2[1+(1-\gamma)\gamma^{\gamma/(1-\gamma)}]log2​[1+(1−γ)γγ/(1−γ)]. It applies to quantum communication, entanglement distribution, private communication, and secret-key agreement, and strictly improves the PLOB relative-entropy bound, the analytic max-relative-entropy bound, and the optimized balanced squashed-entanglement bound. At high damping, a known triple-rail protocol attains more than 99.54%99.54\%99.54% of its leading coefficient. Our method extends sector teleportation simulation to qubits: phase rotations and permutations give exact simulations within fixed-excitation subspaces while preserving the two-level input structure. We determine the exact, additive relative entropy of entanglement of the amplitude-damping resources and the exact regularized entanglement gain at fixed input population. We then derive explicit bounds for generalized amplitude damping, which includes thermal excitation. Combining the amplitude-damping gain with unital and entanglement-breaking channels gives a closed expression that recovers the zero-temperature result and vanishes throughout the entanglement-breaking region. It improves established benchmarks over broad parameter ranges, while the exact entanglement of the generalized resources remains open. Both results admit refinements under an average-excitation constraint. The bounds provide quantitative limits for adaptive communication through relaxing qubits and show how a sector simulation can remain useful beyond channels whose resource entanglement is known exactly.
In this work, we investigate the convergence rate of the quantum Blahut-Arimoto (BA) algorithm proposed for computing the coherent information of less noisy quantum channels. Previous work established exponential convergence under certain restrictive assumptions on the channel. Using a Taylor expansion around a positive definite optimizer, we show that, for an injective degradable channel whose degrading channel has a relative-entropy contraction coefficient strictly less than one, the algorithm converges at an exponential rate after sufficiently many iterations. We further verify this condition for qubit amplitude-damping (AD) channels in the relevant parameter regime. In addition, by analyzing the Bogoliubov-Kubo-Mori (BKM) metric on the Bloch ball, we derive an explicit characterization of the relative-entropy contraction coefficient of qubit AD channels. The proposed approach is further extended to calculate the relative contraction coefficient and the relative expansion coefficient of two qubit AD 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.