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Is it possible to construct a quantum circuit, using any unitary and measurement operations, that can distinguish $|0 \rangle$ and $\frac{1}{\sqrt{2}} (|0 \rangle + |1 \rangle)$?

In my estimation, the answer should be no, because the two states are not orthogonal, and any unitary operation applied to the two states preserves their (non-zero) inner product. We could however obtain some information as to which state we have by measuring the output: if $1$, then we would know the state to have been $\frac{1}{\sqrt{2}} (|0 \rangle + |1 \rangle)$; if $0$, then we cannot tell with certainty which of the two original states collapsed to $0$. Is there something missing in this argument?

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  • $\begingroup$ See also here and many the links provided therein. I think this particular example is used by N+C. $\endgroup$ Commented Apr 25 at 15:03

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Yes, your reasoning is sound. The outcome ``0'' is an ambiguous result and so you can only guess which state you were given if you measure this outcome.

If you want to make this more rigorous and general, you can use the following result (modifed from [1], Theorem 9.2.1):

Given one of two possible states with equal probability, $|\psi_z\rangle$ with either $z=1$ or $z=2$, any procedure that outputs a guess $\hat{z}\in \{1, 2\}$ will be correct (outputs $\hat{z}=z$) with probability at most $$ \frac{1}{2} + \frac{1}{2} \sqrt{1-|\langle \psi_1 |\psi_2\rangle|^2}. $$

This task is an example of "quantum state discrimination" which can be generalized in various ways (minimizing different types of errors, different prior probabilities of being given a state, etc.).


[1] Kaye, P., Laflamme, R., & Mosca, M. (2006). An introduction to quantum computing.

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