Implementation of group-covariant positive operator valued measures by orthogonal measurements

Decker T., Janzing D.

We consider group-covariant positive operator valued measures (POVMs) on afinite dimensional quantum system. Following Neumark’s theorem a POVM can beimplemented by an orthogonal measurement on a larger system. Accordingly, ourgoal is to find a quantum circuit implementation of a given group-covariant POVMwhich uses the symmetry of the POVM. Based on representation theory of thesymmetry group we develop a general approach for the implementation of groupcovariantPOVMs which consist of rank-one operators. The construction relies on amethod to decompose matrices that intertwine two representations of a finite group.We give several examples for which the resulting quantum circuits are efficient. Inparticular, we obtain efficient quantum circuits for a class of POVMs generated byWeyl-Heisenberg groups. These circuits allow to implement an approximative simultaneousmeasurement of the position and crystal momentum of a particle movingon a cyclic chain.