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Analysis of Quantum Communication Protocols

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Quantum Information Theory

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Abstract

The problems of transmitting a classical message via a quantum channel (Chap. 4) and estimating a quantum state (Chaps. 3 and 6) have a classical analog. These are not intrinsically quantum-specific problems but quantum extensions of classical problems. The difficulties of these quantum extensions are mainly caused by the non-commutativity of quantum mechanics. However, quantum information processing is not merely a non-commuting version of classical information processing. There exist many quantum protocols without any classical analog. In this context, quantum information theory covers a greater field than a noncommutative analog of classical information theory. The key to these additional effects is the advantage of using entanglement treated in Chap. 8, where we examined mainly the quantification of entanglement. In this chapter, we will introduce several quantum communication protocols that are possible only by using entanglement and are therefore classically impossible. (Some of protocols introduced in this section have classical analogs.) We also examine the transmission of quantum states (quantum error correction), communication in the presence of eavesdroppers, and several other types of communication that we could not handle in Chap. 4. As seen in this chapter, the transmission of a quantum state is closely related to communication with no information leakage to eavesdroppers. The noise in the transmission of a quantum state clearly corresponds to the eavesdropper in a quantum communication.

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Notes

  1. 1.

    The superscript e of \(C_{c}^{e}\) indicates that “entangled” input is allowed.

  2. 2.

    The subscript a expresses “assistance.”

  3. 3.

    The second subscript, e, of \(C_{c,e}^{e}\) indicates the shared “entanglement.” The superscript e indicates “entangled” operations between sending systems.

  4. 4.

    The subscript r indicates “resolvability”.

  5. 5.

    Ahlswede-Winter [36] also showed that \(C_i(W)=C_i^\dagger (W)=C_c(W)\) in a different way.

  6. 6.

    By adding the states \(e^*_+\mathop {=}\limits ^{\mathrm{def}}\frac{1}{\sqrt{2}}(e_0+ i e_1)\) and \(e^*_-\mathop {=}\limits ^{\mathrm{def}}\frac{1}{\sqrt{2}}(e_0- i e_1)\) in the transmission, and by adding the measurement \(\{|e^*_+\rangle \langle e^*_+|, |e^*_-\rangle \langle e^*_-|\}\), it is possible to estimate \(\kappa \). This is called the six-state method [46–48].

  7. 7.

    Since “quantum” states are to be sent, the capacities are called quantum capacities. The subscript q indicates that “quantum” states are to be sent.

  8. 8.

    In their paper, it is mentioned that Levitin [82] showed the direct part of Theorem 4.1 in this special case.

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  1. Graduate School of Mathematics, Nagoya University, Nagoya, Aichi, Japan

    Masahito Hayashi

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Hayashi, M. (2017). Analysis of Quantum Communication Protocols. In: Quantum Information Theory. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49725-8_9

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