Abstract
Quantum entanglement is a striking feature of quantum mechanics, and clarifying its properties is crucially important for the development of quantum information technology. In recent years, the theory of entanglement has been rapidly developed along with quantum information theory. In particular, introducing the concept of local operations and classical communication enables us to quantify the amount of entanglement, and leads to our much better understanding of quantum entanglement. In this chapter, the various properties of quantum entanglement are explained.
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Notes
- 1.
- 2.
- 3.
Precisely, the exact cycle of (7.20) is impossible even in the limit \(n \rightarrow \infty \). This is because the two conversion rates coincide only in the leading order of \(n\), and moreover the final state of (7.18) is close to but not equal to the initial state of (7.14). To complete the above cycle exactly, we need to consume the excess entanglement of the amount \(o(n)\) [14].
- 4.
However, even in a probabilistic way, it is impossible to create any entangled state from unentangled states by LOCC.
- 5.
In the case of bipartite pure states, the Schmidt decomposition is applicable and they are classified by the number of terms in the Schmidt decomposition, because it is equal to the rank of the reduced density operators.
- 6.
They are indeed equivalent in \({\mathbb C}^2\otimes {\mathbb C}^n\) for \(n\ge 2\).
- 7.
Due to the fact that any positive map \(\varGamma \) in \({\mathbb C}^2\otimes {\mathbb C}^2\) and \({\mathbb C}^2\otimes {\mathbb C}^3\) is written as \(\varGamma =\varTheta _1+\varTheta _2\circ T\) with \(\varTheta _1\) and \(\varTheta _2\) being completely positive maps (\(T\) denotes transposition).
- 8.
Note that, even though \(E\) satisfies (1) and (2), \(E^{\infty }\) does not necessarily satisfy them. If \(E\) also satisfies subadditivity, however, \(E^{\infty }\) automatically satisfies the weak monotonicity (\(1^{\prime }\)) and (2) [34].
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Hayashi, M., Ishizaka, S., Kawachi, A., Kimura, G., Ogawa, T. (2015). Quantum Entanglement. In: Introduction to Quantum Information Science. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43502-1_7
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