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I've been looking into different aspects of distinguishing two pure quantum states. I've ended up reading a lot of books/papers covering things like "accessible information", but there haven't been too many explanations on how to find optimal measurements.

The book by (Kaye, Laflamme, Mosca) outlines a simple procedure in their appendix, which requires sending the two states [itex] | \Psi_Y \rangle, |\Psi_x \rangle [/itex] to the states

[tex] \cos(\theta) | 0 \rangle + \sin(\theta) |1 \rangle [/tex]

[tex] \sin(\theta) |0 \rangle + \cos(\theta) |1 \rangle [/tex]

where [itex] 0 \leq \theta \leq \frac \pi4 [/itex] and [itex] \langle \Psi_y | \Psi_x \rangle = \sin(2\theta) [/itex]. However, I'm uncertain as to how to even construct a unitary operator that does the associated mapping.

Other sources consider the states [itex] | \Psi_x \rangle |\Psi_x \rangle, | \Psi_y \rangle |\Psi_y \rangle [/itex], or just talk about accessible information rather than optimal measurements.

If anyone could shed some light on how to construct the Unitary Map necessary for the procedure in Kaye et al. or any other insight into this, it would be much appreciated.

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# Distinguishing Quantum States

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