General theory of measurement with two copies of a quantum state

Abstract

We analyze the possible results of the most general measurement on two copies of a quantum state. We show that $\mu$ can label a set of outcomes of such measurement if and only if there is a family of completely co--positive (ccP) maps $C_\mu$ such that the probability of occurrence $Prob(\mu)$ is the fidelity of the map $C_\mu$, i.e. $Prob(\mu)= Tr(\rho C_\mu(\rho))$ which must add up to the fully depolarizing map. This implies that a POVM on two copies induces a measure on the set of ccP maps (i.e., a ccPMVM). We present examples of ccPMVM's and discuss their tomographic applications showing that two copies of a state provide an exponential improvement in the efficiency of quantum state tomography. This enables the existence of an efficient universal detector.