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Fundamentals of Information Theory

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Quantum Zero-Error Information Theory

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Abstract

This chapter introduces some elementary concepts regarding information theory. First, we present entropy and other measures of information. Then, we discuss a very important quantity in classical information theory, the capacity of a discrete noisy channel. In the second part of this chapter, we give a brief introduction to the quantum information theory. We start with the von Neumann entropy and other measures of information. Then, we introduce the mathematical formulation of quantum channels, including the Choi-Jamiołkowski isomorphism. Accessible information, Holevo quantity and quantum channel capacities are discussed; the classical capacity of quantum channels is presented, as well as a brief introduction to the other kinds of quantum channel capacities.

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Notes

  1. 1.

    For the sake of simplicity, we consider \(\mathcal{H}_{1} = \mathcal{H}_{2} = \mathcal{H}\).

  2. 2.

    The Schmidt decomposition of a bipartite quantum state \(\vert \psi \rangle _{AB} \in \mathcal{H}_{A} \otimes \mathcal{H}_{B}\) is given by | ψ〉 AB  = ∑ i λ i  | i〉 A  | i〉 B , where λ i  ≥ 0, ∑ i λ i 2 = 1, and | i〉 A , | i〉 B are orthonormal basis for \(\mathcal{H}_{A}\), \(\mathcal{H}_{B}\), respectively. The cardinality of {λ i }, including multiplicity, is known as Schmidt rank or Schmidt number of | ψ〉 AB .

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Authors and Affiliations

  1. Superior School of Technology, Amazonas State University, Manaus, AM, Brazil

    Elloá B. Guedes

  2. Center of Electric Engineering and Informatics, Campina Grande Federal University, Campina Grande, PB, Brazil

    Francisco Marcos de Assis

  3. School of Science and Technology, Rio Grande do Norte Federal University, Natal, RN, Brazil

    Rex A. C Medeiros

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  1. Elloá B. Guedes
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  2. Francisco Marcos de Assis
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  3. Rex A. C Medeiros
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Guedes, E.B., de Assis, F.M., Medeiros, R.A.C. (2016). Fundamentals of Information Theory. In: Quantum Zero-Error Information Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-42794-2_3

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  • DOI: https://doi.org/10.1007/978-3-319-42794-2_3

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  • Publisher Name: Springer, Cham

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