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In quantum information science, quantum state discrimination refers to the task of inferring the quantum state that produced the observed measurement probabilities.
More precisely, in its standard formulation, the problem involves performing some POVM i {\displaystyle _{i}} on a given unknown state ρ {\displaystyle \rho } , under the promise that the state received is an element of a collection of states { σ i } i {\displaystyle \{\sigma _{i}\}_{i}} , with σ i {\displaystyle \sigma _{i}} occurring with probability p i {\displaystyle p_{i}} , that is, ρ = ∑ i p i σ i {\displaystyle \rho =\sum _{i}p_{i}\sigma _{i}}. The task is then to find the probability of the POVM i {\displaystyle _{i}} correctly guessing which state was received. Since the probability of the POVM returning the i {\displaystyle i} -th outcome when the given state was σ j {\displaystyle \sigma _{j}} has the form Prob = tr {\displaystyle {\text{Prob}}=\operatorname {tr} } , it follows that the probability of successfully determining the correct state is P s u c c e s s = ∑ i p i tr {\displaystyle P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} }.