General Theory of Measurement with Two Copies of a Quantum State

We analyze the results of the most general measurement on two copies of a quantum state. We show that by using two copies of a quantum state it is possible to achieve an exponential improvement with respect to known methods for quantum state tomography. We demonstrate that mu can label a set of outcomes of a measurement on two copies if and only if there is a family of maps C(mu) such that the probability Prob (mu) is the fidelity of each map, i.e., Prob (mu) = Tr[(rho)C(mu)(rho)]. Here, the map C(mu) must be completely positive after being composed with the transposition ( these are called completely copositive, or CCP, maps) and must add up to the fully depolarizing map. This implies that a positive operator valued measure on two copies induces a measure on the set of CCP maps (i.e., a CCP map valued measure).