# POVM advantage in state discrimination

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?

• 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? Mar 11, 2013 at 20:59
• 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. Mar 12, 2013 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. Mar 12, 2013 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. Mar 12, 2013 at 19:04
• Well a trivial answer could be " if the states are orthogonal " :) May 28, 2014 at 23:22