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Consider this problem in quantum cryptography:

We have two pure states $\phi_1,\phi_2$ as input and constants $0 \leq \alpha <\beta \leq 1 $, where "Yes instances" are those for which $$\left|\left<\phi_1,\phi_2\right>\right| \leq \alpha$$ and "No instances" are those for which $$\left|\left<\phi_1,\phi_2\right>\right|\geq \beta$$

I need to devise a quantum circuit that accepts "Yes instances" with probability at least $p$, and "No instances" with at most probability $q$ for some $q<p$.

How can I do this? Is there a reference in which this problem is solved?

Thanks in advance.

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    $\begingroup$ Next time, please try to give an informative title. $\endgroup$ – Tsuyoshi Ito Jun 1 '12 at 11:25
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A standard tool is the swap test, which receives two unentangled pure states |φ1⟩ and |φ2⟩ and answers “equal” with probability (1+|⟨φ1|φ2⟩|2)/2 and “not equal” with probability (1−|⟨φ1|φ2⟩|2)/2. See [BCWW01] (Figure 1 shows the circuit).

[BCWW01] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. Quantum fingerprinting. Physical Review Letters, 87, article 167902, Sept. 2001. DOI: 10.1103/PhysRevLett.87.167902. A preprint version: arXiv:quant-ph/0102001v1.

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