Abstract
The quantum error correction is a technology to transmit the quantum state via a noisy quantum channel. This chapter begins with the description of the classical error correction based on an algebraic method. Using the knowledge of the classical error correction. This chapter explains the quantum error correction. Then, based on the property of the quantum error correction, we treat the secure communication of the classical message via a quantum channel and the quantum cryptography.
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Notes
- 1.
The sum in the vector space is written as \(+\), and the sum in the finite field \(\mathbb {F}_2\) is written as \(\oplus \). Since the sum \(+\) has the same meaning as the difference \(-\) in a vector space over the finite field \(\mathbb {F}_2\), we unify them to \(+\).
- 2.
In the general noise, the probability flipping \(0\) to \(1\) dose not coincide with that the probability flipping \(1\) to \(0\). In the following, we assume that these two kinds of flipping probabilities are the same. Then, the above type description is possible.
- 3.
\(\wedge \) is the multiplication over the finite field \(\mathbb {F}_2\), and is given in Table 3.1.
- 4.
It is known that this assumption does not change the asymptotically optimal transmission rate [12].
- 5.
Renner [26] proposed the criterion \(\Vert \rho _\mathrm{AE} -\rho _{\mathop {\mathrm{mix}}\nolimits }\otimes \rho _\mathrm{E} \Vert _1\) so called universal composability, in which, the reduced density \(\rho _\mathrm{A}\) is replaced by the completely mixed state \(\rho _{\mathop {\mathrm{mix}}\nolimits }\). In our case, since \(\rho _\mathrm{A}\) is the completely mixed state, both criteria coincide with each other.
- 6.
- 7.
As another method for the security analysis, the method based on the universal2 property of the Hash function is knwon [26].
- 8.
In this book, when Alice’s basis is the same as Bob’s basis, the basis is called matched.
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Hayashi, M., Ishizaka, S., Kawachi, A., Kimura, G., Ogawa, T. (2015). Quantum Error Correction and Quantum Cryptography. In: Introduction to Quantum Information Science. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43502-1_9
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