On the Measurement attaining the Quantum Guesswork

The guesswork quantifies the minimum cost incurred in guessing the state of an ensemble, when only one state can be queried at a time. In the classical case, it is well known that the optimal strategy trivially consists of querying the states in their non-increasing order of posterior probability. In the quantum case, on the other hand, the most general strategy to obtain the optimal ordering in which to perform the queries consist of a quantum measurement. Here, we solve such an optimization problem by deriving the quantum measurement attaining the guesswork for a broad class of ensembles and cost functions.