Abstract
This chapter gives a concise description of the fundamental concepts of quantum information and quantum communication, which is pertinent to the discussions in the subsequent chapters. Beginning with the basic set of rules that dictate quantum mechanics, the chapter explains the most general ways to describe quantum states, measurements, and state transformations. Convenient mathematical tools are also presented to provide an intuitive picture of a qubit, which is the simplest unit of quantum information. The chapter then elaborates on the distinction between quantum communication and classical communication, with emphasis on the role of quantum entanglement as a communication resource. Quantum teleportation and dense coding are then explained in the context of optimal resource conversions among quantum channels, classical channels, and entanglement.
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Notes
- 1.
The operational meaning of two states being different is that a measurement exists on the physical system that can show the difference statistically.
- 2.
Rule 1 also implies that we exclude any cases where a physical law such as the superselection rule imposes an additional restriction on the preparable states.
- 3.
Given s′, we may choose the dimensions of \(\mathcal{H}_{E}\) and \(\mathcal{H}_{E'}\) such that they satisfy \(\dim \mathcal{H}_{E'} \geq s'\) and \(\dim \mathcal{H}_{A}\dim \mathcal{H}_{E} =\dim \mathcal{H}_{A'}\dim \mathcal{H}_{E'}\).
- 4.
Regard (j, k) as a single index with the values 1, …, s′, where \(s' =\sum _{j}t^{(j)}\).
- 5.
There is an equivalent method to define the fidelity as \(F(\hat{\rho }_{1},\hat{\rho }_{2}) = (\mathrm{Tr}\sqrt{\hat{\rho }_{1 }^{1/2 }\hat{\rho }_{2 } \hat{\rho }_{1 }^{1/2}})^{2}\) [6, 7]. In some of the literature, the quantity \(\sqrt{F(\hat{\rho }_{1 },\hat{\rho }_{2 } )}\) is referred to as the fidelity.
- 6.
An explicit form of Alice’s transformation is \(\vert \varPhi _{l,m}\rangle _{AB} = (\hat{X}_{A}^{l}\hat{Z}_{A}^{m} \otimes \hat{ 1}_{B})\vert \varPhi _{0,0}\rangle _{AB}\), which is obtained from Eq. (1.46).
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Koashi, M. (2016). Quantum Information Theory for Quantum Communication. In: Yamamoto, Y., Semba, K. (eds) Principles and Methods of Quantum Information Technologies. Lecture Notes in Physics, vol 911. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55756-2_1
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