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Paper IDQ.4.1
Paper Title Entropy of a Quantum Channel: Definition, Properties, and Application
Authors Gilad Gour, University of Calgary, Canada; Mark Wilde, Louisiana State University, United States
Session Q.4: Quantum Information Theory
Presentation Lecture
Track Quantum Systems, Codes, and Information
Manuscript  Click here to download the manuscript
Virtual Presentation  Click here to watch in the Virtual Symposium
Abstract The von Neumann entropy is a central concept in physics and information theory, having a number of compelling physical interpretations. There is a certain perspective that the most fundamental notion in quantum mechanics is that of a quantum channel, as quantum states, unitary evolutions, measurements, and discarding of quantum systems can each be regarded as certain kinds of quantum channels. Thus, an important goal is to define a consistent and meaningful notion of the entropy of a quantum channel. Motivated by the fact that the entropy of a state $\rho$ can be formulated as the difference of the number of physical qubits and the "relative entropy distance" between $\rho$ and the maximally mixed state, here we define the entropy of a channel $\mathcal{N}$ as the difference of the number of physical qubits of the channel output with the "relative entropy distance" between $\mathcal{N}$ and the completely depolarizing channel. We establish that this definition satisfies all of the axioms, recently put forward in [Gour, IEEE Trans. Inf. Theory 65, 5880 (2019)], required for a channel entropy function. The task of quantum channel merging, in which the goal is for the receiver to merge his share of the channel with the environment's share, gives a compelling operational interpretation of the entropy of a channel. We define Renyi and min-entropies of a channel and establish that they satisfy the axioms required for a channel entropy function. Among other results, we also establish that a smoothed version of the min-entropy of a channel satisfies the asymptotic equipartition property.

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2021 IEEE International Symposium on Information Theory

11-16 July 2021 | Melbourne, Victoria, Australia

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