Decomposition of any quantum measurement into extremals

We design an efficient and constructive algorithm to decompose any generalized quantum measurement into a convex combination of extremal measurements. We show that if one allows for a classical post-processing step only extremal rank-1 POVMs are needed. For a measurement with $N$ elements on a $d$-dimensional space, our algorithm will decompose it into at most $(N-1)d+1$ extremals, whereas the best previously known upper bound scaled as $d^2$. Since the decomposition is not unique, we show how to tailor our algorithm to provide particular types of decompositions that exhibit some desired property.