State Tomography: Least Pairs Needed?

[Dragonfall]

I don't know where this question belongs:

Given many pairs of \left|\Psi\right> and U\left|\Psi\right>, for some unitary U, is it possible to identify U without completely determining the two states independently? I mean what is the least possible number of pairs needed (to be x% certain), and is it less than simply determining the two states?

 

[Dragonfall]

Is there even enough information to determine a general U?

 

[Dragonfall]

Anyone?

 

[borgwal]

No, there's not enough info to determine U from what you propose: even if you know |psi> and U|\psi, you really know only one column vector of U, not the whole U (take |\psi> as your first basis vector in Hilbert space)

 

[Dragonfall]

But for a specific U, like a permutation?

 

[borgwal]

Same answer, you only learn one column of the matrix.

 

[Dragonfall]

What if \left|\Psi\right> were a tensor state of n qubits?

 

[borgwal]

If the tensor product consists of many different qubit states, and if the big U is a tensor product of identical U_2s on each qubit, then of course you can learn everything about U_2.