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Quantum Measurements and State Reduction

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Quantum Information Theory

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

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Abstract

In quantum mechanics, the state reduction due to a measurement is called the collapse of a wavefunction. Its study is often perceived as a somewhat mystical phenomenon because of the lack of proper understanding. As a result, the formalism for the state reduction is often somewhat inadequately presented. However, as will be explained in Sect. 7.1, the state reduction due to a measurement follows automatically from the formulation of quantum mechanics, as described in Sect. 1.2. Starting with the formulation of quantum mechanics given in Sects. 1.2 and 1.4, we give a detailed formulation of the state reduction due to a measurement. In Sect. 7.2, we discuss the relation with the uncertainty relation using these concepts. Finally, in Sect. 7.4, we propose a measurement with negligible state reduction.

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Notes

  1. 1.

    Normalization here implies the division of the matrix by its trace such that its trace is equal to 1.

  2. 2.

    It can also be understood as follows: the state reduction due to any measurement by PVM can be characterized as the state reduction satisfying the projection hypothesis, followed by the state evolution \(\kappa _\omega \). Indeed, many texts state that the state reduction due to any measurement is given by the projection hypothesis. Theorem 7.2 guarantees their correctness in a sense.

  3. 3.

    As discussed in Sect. 6.2, \(\varDelta _4({\varvec{\kappa }},Y,\rho )\) also has the meaning of the amount of the loss of the SLD Fisher information. Therefore, this inequality is interesting from the point of view of estimation theory. It indicates the naturalness of the SLD inner product. This is in contrast to the naturalness of the Bogoljubov inner product from a geometrical viewpoint.

  4. 4.

    The equality holds when an appropriate POVM \({\varvec{M}}\) is performed in a quantum two-level system [11]. For its more general equality condition, see Exercise 7.17.

  5. 5.

    This definition of c is the generalization of that in Theorem 7.6. See Exercise 7.22.

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Authors and Affiliations

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

    Masahito Hayashi

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Hayashi, M. (2017). Quantum Measurements and State Reduction. In: Quantum Information Theory. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49725-8_7

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  • DOI: https://doi.org/10.1007/978-3-662-49725-8_7

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