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Minimum-cost quantum measurements for quantum information

Knowing about optimal quantum measurements is important for many applications in quantum information and quantum communication. However, deriving optimal quantum measurements is often difficult. We present a collection of results for minimum-cost quantum measurements, and give examples of how they can be used. Among other results, we show that a minimum-cost measurement for a set of given pure states is formally equivalent to a minimum-error measurement for certain mixed states of those same pure states. For pure symmetric states it turns out that for a certain class of cost matrices, the minimum-cost measurement is the square-root measurement. That is, the optimal minimum-cost measurement is in this case the same as the minimum-error measurement. These results are in agreement with Nakahira and Usuda (2012 Phys. Rev. A 86 062305). Finally, we consider sequences of individual uncorrelated systems, and examine when the global minimum-cost measurement is a sequence of optimal local measurements. We consider an example where the global minimum-cost measurement is, perhaps counter-intuitively, not a sequence of local measurements, and discuss how this is related to the Pusey–Barrett–Rudolph argument for the nature of the wave function.