# Distinguishable quantum states (“counterexample”)

In the book Quantum Computation and Quantum Information (Nielsen & Chuang) the section 2.2.4 discusses quantum distinguishability and arrives at the conclusion that if someone were to show me a set of states and give me one of them, I can only tell which one it is if the set is orthogonal. So I came up with a counterexample to this. Let's say we have these two states: \begin{align} |\psi_1\rangle & = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\\ |\psi_2\rangle & = |0\rangle \, . \end{align}

If I make a measurement on the unknown state and the result is the eigenvalue associated with $|1\rangle$, I know with certainty that the given state was $|\psi_1\rangle$, so I was able to distinguish them even though they are not orthogonal. Clearly there's something wrong with my thinking, but I can't tell what it is.

• what if you got $0$? – spaceisdarkgreen Sep 14 '17 at 0:57
• You get one shot at telling what the state is and you have to be guaranteed correct. – spaceisdarkgreen Sep 14 '17 at 1:08
• I'm not sure if this matters; but in this example you also destroy the state you started with; so you no longer have it – Joshua Lin Sep 14 '17 at 1:15
• I don't have the book in front of me, so can't say for sure what the authors were thinking, but if I had to bet, I'd bet that @spaceisdarkgreen is exactly right. – WillO Sep 14 '17 at 1:26
• @spaceisdarkgreen Why don't you turn this into an answer? – Norbert Schuch Sep 15 '17 at 6:04

This is why your protocol fails: if you're given $|\psi_1\rangle$, you have a 50% chance of getting the measurement outcome $|0\rangle$, in which case you don't know which one to choose and you're forced to provide an answer that might be wrong.
• You might be given a single copy of the system, and you're not allowed to make mistaken identifications, but you allow the protocol to fail, i.e., to refuse to give an answer (often called unambiguous discrimination). In this case your measurement is OK but it can be optimized, and in the optimal case you can provide definite answers that it must be $|\psi_1\rangle$ or that it must be $|\psi_2\rangle$, no matter how similar the two are ─ but the closer they get, the more often the protocol fails to produce an answer.