The measurements kata covers the following topics:
- single-qubit measurements
- quantum state discrimination for both orthogonal and non-orthogonal states
Variations of quantum state discrimination tasks are covered in the paper "Quantum State Discrimination".
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Task 2.1 is an example of hypothesis testing for two pure states.
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Task 2.2 is an example of unambiguous state discrimination. See also the paper "Unambiguous quantum measurement of nonorthogonal states" for further information and hints about how to implement the unambiguous measurements required for this task.
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Task 2.3 is the so-called Wootters/Peres game. See the following three references for more information and in particular the book [3, p. 287] for a nice description of the optimal POVM.
[1] A. Holevo, "Information-theoretical aspects of quantum measurement," Problems of Information Transmission, vol. 9, no. 2, pp. 110-118 (1973)
[2] A. Peres and W. K. Wootters, "Optimal detection of quantum information," Phys. Rev. Lett., vol. 66, pp. 1119-1122, Mar. 1991.
[3] A. Peres, "Quantum Theory: Concepts and Methods," Kluwer Academic Publishers, 2002.