Question: A referee picks a value i {1, . . . , N} uniformly at random, prepares a quantum system in the state i, and sends

A referee picks a value i {1, . . . , N} uniformly at random, prepares a quantum system in the state i, and sends it to a player, Alice, asking her to discover what is the value of i. To find out the value of i, Alice performs a quantum measurement described by a POVM (P1, . . . , PN ). If here measurement gives outcome j, then she guesses that the label is j. On average, the probability that Alice correctly guesses the label of the state is

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