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 positive operator valued measures 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 d2. Since the decomposition is not unique, we show how to tailor our algorithm to provide particular types of decompositions that exhibit some desired property. © 2013 IOP Publishing Ltd.