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Assume we know two different mixed states, p and q, and an efficient (quantum) algorithm for creating such two. Does it follow that there exists a computationally efficient method/measurement for optimally(that is, according to the trace norm distance) distinguishing between the two?

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For one copy of each state you can in principle distinguish between them, but I don't know if it can be done efficiently. Since you have the circuit you may want to make many copies of the states, in which case is relevant. This gives hope even when the one-copy case has low trace distance. They claim that the collective measurement is efficient (on the bottom of page 2) but I don't know whether this assumes an efficient measurement for the one-copy case. – Dan Stahlke Aug 21 '12 at 12:39
Given two states is straight forward to figure out how well to distinguish them (trace norm distance), but yeah, I'm lost on efficency. I cannot really make many copies, as it is one specific state I'm given, that I want to check. Maybe being concrete will help my question, I want to make sure that I have the following state (or something very close): \rho = \sum_{r_0,r_1 \in {0,1}^n} |\psi_{r_0,r_1}> <\psi_{r_0,r_1}| where |\psi_{r_0,r_1}> = |(|0>|r_0> + |1>|r_1>) I know how to create this mixed state, but how do I, efficiently check it? – Thomas Hansen Aug 21 '12 at 13:41

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