I am trying to solve a problem in the measurement and identification of quantum

Question

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?={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 jth 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.

Explanation / Answer

I think that it may be quite difficult to find the most efficient solution, which would involve collective measurements involving several of the inputs at once (although in your question you seem to rule out this possibility by mentioning measurements of single qubits at a time). Also, one should consider POVMs rather than projective measurements.

Assuming that you measure only one qubit at a time, I can answer your question of how to represent them. The first qubit is as you suspect the fully mixed state. After that you should use Bayes' rule to update your probability distributions for S (if S is not assumed to always be uniformly random) and S'. This is not my expertise, but this part of it is a classical problem not a quantum problem. Use these updated probability distributions to form the density operator of the next state you receive. You then need to decide which type of measurement is best given your current state of knowledge about the probabilities. I can't help you there.

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.