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Entanglement and Locality Restrictions

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Quantum Information Theory

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Abstract

Quantum mechanics violates daily intuition not only because the measured outcome can only be predicted probabilistically but also because of a quantum-specific correlation called entanglement. It is believed that this type of correlation does not exist in macroscopic objects. Entanglement can be used to produce nonlocal phenomena. States possessing such correlations are called entangled states (or states that possess entanglement). A state on a bipartite system is called called a maximally entangled state or an EPR state when it has the highest degree of entanglement among these states. Historically, the idea of a nonlocal effect due to entanglement was pointed out by Einstein, Podolsky, and Rosen; hence, the name EPR state. In order to transport a quantum state over a long distance, we have to retain its coherence during its transmission. However, it is often very difficult because the transmitted system can be easily correlated with the environment system. If the sender and receiver share an entangled state, the sender can transport his/her quantum state to the receiver without transmitting it, as explained in Chap. 9. This protocol is called quantum teleportation and clearly explains the effect of entanglement in quantum systems. Many other effects of entanglement have also been examined, some of which are given in Chap. 9. However, it is difficult to take advantage of entanglement if the shared state is insufficiently entangled. Therefore, we investigate how much of a maximally entangled state can be extracted from a state with a partially entangled state. Of course, if we allow quantum operations between two systems, we can always produce maximally entangled states. Therefore, we examine cases where locality conditions are imposed to our possible operations.

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Notes

  1. 1.

    When the measurement is employed in the class \(\emptyset \), it is required that Alice and Bob obtain the same outcome.

  2. 2.

    A large part of the discussion relating to entanglement fidelity and information quantities relating to entanglement (to be discussed in later sections) was first done by Schumacher [5].

  3. 3.

    Since the form of this inequality is similar to the Fano inequality, it is called quantum Fano inequality. However, it cannot be regarded as a quantum extension of the Fano inequality (2.35). The relationship between the two formulas is still unclear.

  4. 4.

    If the operation \(\kappa _n\) in C has a larger output system than \(\mathbb {C}^{d_n} \otimes \mathbb {C}^{d_n}\), there exists an operation \(\kappa _n'\) in C with the output system \(\mathbb {C}^{d_n} \otimes \mathbb {C}^{d_n}\) such that \(\Vert ~|\Phi _{d_n} \rangle \langle \Phi _{d_n}|- \kappa _n (\rho ^{\otimes n})\Vert _1 \ge \Vert ~|\Phi _{d_n} \rangle \langle \Phi _{d_n}|- \kappa _n' (\rho ^{\otimes n})\Vert _1\).

  5. 5.

    The subscript c denotes “common randomness.”

  6. 6.

    If we employ the original definition, the inequality (8.232) cannot shown by the concavity of \(\log \).

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

    Masahito Hayashi

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Hayashi, M. (2017). Entanglement and Locality Restrictions. In: Quantum Information Theory. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49725-8_8

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