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Suppose we are given the task of discriminating, with minimum error, between a set of states $\{|\psi_1\rangle,|\psi_2\rangle,\ldots,|\psi_N\rangle\}$. In other words, we are given an unknown state form this set and our task is to output which of the states we were given.

Let $P_{err}$ be the probability of incorrectly identifying the state. Under what circumstances is this probability minimized by using a projective measurement?

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This is either enough like a homework question that you should read the homework policy, or too broad to be answered in this format. What exactly is your question? –  Emilio Pisanty Mar 11 '13 at 20:59
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I'm willing to bet there is a nice succinct answer to this question. Someone who works in quantum state discrimination could easily answer it while it might take you or I many hours of literature searching to find it. Seems like the perfect kind of question to ask in my opinion. –  Chris Ferrie Mar 12 '13 at 4:17
    
Maybe you should quantify "advantage". To discriminate $\left|0\right>$ from $\left|+\right>$ there is a POVM that will tell either "surely $\left|0\right>$", "surely $\left|+\right>$", or "I don't know". This cannot be done with projective measurement. The "surely $\left|0\right>$" POVM element is proportional to $\left|-\right>\left<-\right|$ and you can probably piece together the rest from there. –  Dan Stahlke Mar 12 '13 at 12:46
    
By advantage I mean that the error probability is decreased. I realize now that I should not have included the unambiguous case. I will edit the question accordingly. However, I must agree with Chris that this is the type of questions that may find a concise answer from an expert. –  Juan Miguel Arrazola Mar 12 '13 at 19:04
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