# Unambiguous State Discrimination

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?

• How much do you know about $A, B$ and $C$? Jul 27, 2023 at 19:29
• 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. Jul 27, 2023 at 20:22

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.