For example, suppose Alice prepares a system in either the state ρ ρ or σ σ , each with probability 1 2 and sends it to Bob who has to discriminate between the two states using a binary measurement. Let Bob assign the measurement outcome 0 and a POVM element P 0 such as the outcome 1 and a POVM element P 1 = 1 − − P 0 to identify the state ρ ρ or σ σ , respectively. His expected probability of correctly identifying the incoming state is then given by