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I have a collection of possible states that are not necessarily orthogonal to each other -- suppose $A_1, A_2, ... A_N, B_1, B_2... B_N$. I get a new state $C$, and I want to determine whether its in the set $A$ or the set $B$.

I am not allowed to make an error, but I am allowed to return that I don't know the answer.

Is there any measurement that can do this for me?

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  • $\begingroup$ How much do you know about $A, B$ and $C$? $\endgroup$ Jul 27, 2023 at 19:29
  • $\begingroup$ We are given the exact classical descriptions for A and B. C is known to be one among A/B. A/B are linearly dependent though so we cannot run unambiguous discrimination in general though. $\endgroup$ Jul 27, 2023 at 20:22

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In general, I think not. You need to be more restrictive on the properties of your sets. To see this, let's define 3 measurement operators: $M_A$, $M_B$ and $M_U$, corresponding to the answers A (definitely not B), B (definitely not A) and not sure. We require $$ \text{Tr}(B_iM_A)=0\qquad\forall i. $$ Now, since the $\{B_i\}$ could be anything, there are certainly cases where $\sum_iB_i\propto I$. If that's the case, $$ \sum_i\text{Tr}(B_iM_A)=\text{Tr}(M_A)=0\qquad\forall i. $$ Now since $M_A$ must be positive semi-definite, this proves that $M_A=0$. You will never get that answer. By symmetry, we can get the same for $M_B$. So, there definitely exist sets of possible states for which the measurement is trivial, and must just answer "unsure" every time.

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