State Tomography?

1. Oct 27, 2008

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?

2. Oct 28, 2008

Dragonfall

Is there even enough information to determine a general U?

3. Oct 30, 2008

Anyone?

4. Nov 1, 2008

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)

5. Nov 1, 2008

Dragonfall

But for a specific U, like a permutation?

6. Nov 1, 2008

borgwal

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

7. Nov 1, 2008

Dragonfall

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

8. Nov 1, 2008

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.