I am trying to solve a problem in the measurement and identification of quantum states with a promise as to what states it could be. Here is the problem. Imagine a system that produces qubits in one of four states $S = \{a,b,c,d \}$, evenly distributed. In one shot, I can receive $K$ copies of a state in $S$. However, a random unitary, evenly distributed in $SU(2)$ has been applied, so, at the detectors I receive $S^{\prime} = \{Ha,Hb,Hc,Hd \}$. I have a detector system that includes $2M$ detectors and they represent the projection operators onto any pair of basis vectors in any basis (thus we can choose which $M$ bases we want to use the detectors in). All I want to do is decide if the state is $a,b,c,$ or $d$. How many copies of the state do I need (ie what is a minimum for $K$)? What is the smallest $M$?

Another, more general question is this: what should I use to represent the state before I do a measurement? Should it not be a density matrix which is an integral over all states? Since there is a random unitary applied, that means that I can receive, in a fixed basis, any state with equal probability. What would be the update rule for this density matrix after the $j^{th}$ measurement result?

I realize there are pieces missing in my question. I will revise it this post this evening, but hopefully it will give the general idea.

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