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Source Coding in Quantum Systems

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Quantum Information Theory

Part of the book series: Graduate Texts in Physics ((GTP))

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Abstract

Nowadays, data compression software has become an indispensable tool for current network system. Why is such a compression possible? Information commonly possesses redundancies. In other words, information possesses some regularity. If one randomly types letters of the alphabet, it is highly unlikely letters that form a meaningful sentence or program. Imagine that we are assigned a task of communicating a sequence of 1000 binary digits via telephone. Assume that the binary digits satisfies the following rule: The 2nth and \((2n+1)\)th digits of this sequence are the same. Naturally, we would not read out all 1000 digits of the sequence; we would first explain that the 2nth and \((2n+1)\)th digits are the same, and then read out the even-numbered (or odd-numbered) digits. We may even check whether there is any further structure in the sequence. In this way, compression software works by changing the input sequence of letters (or numbers) into another sequence of letters that can reproduce the original sequence, thereby reducing the necessary storage. The compression process may therefore be regarded as an encoding. This procedure is called source coding in order to distinguish it from the channel coding examined in Chap. 4. Applying this idea to the quantum scenario, the presence of any redundant information in a quantum system may be similarly compressed to a smaller quantum memory for storage or communication. However, in contrast to the classical case, we have at least two distinct scenarios. The task of the first scenario is saving memory in a quantum computer. This will be relevant when quantum computers are used in practice. In this case, a given quantum state is converted into a state on a system of lower size (dimension). The original state must then be recoverable from the compressed state. Note that the encoder does not know what state is to be compressed. The task in the second scenario is to save the quantum system to be sent for quantum cryptography. In this case, the sender knows what state to be sent. This provides the encoder with more options for compression. In the decompression stage, there is no difference between the first and second scenarios, since their tasks are conversions from one quantum system to another. In this chapter, the two scenarios of compression outlined above are discussed in detail.

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Notes

  1. 1.

    The subscript q indicates the “quantum” memory.

  2. 2.

    Even though the error of universal quantum variable-length source code converges to 0, its convergence rate is not exponential [9].

  3. 3.

    The subscript c denotes classical memory.

  4. 4.

    The last subscript r denotes shared “randomness.”

  5. 5.

    The last subscript e denotes the shared “entanglement” while the superscript e denotes entangled input.

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

    Masahito Hayashi

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  1. Masahito Hayashi
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Hayashi, M. (2017). Source Coding in Quantum Systems. In: Quantum Information Theory. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49725-8_10

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  • DOI: https://doi.org/10.1007/978-3-662-49725-8_10

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