Implementation of group-covariant POVMs by orthogonal measurements

Details Implementation of group-covariant POVMs by orthogonal measurements Implementation of group-covariant POVMs by orthogonal measurements

2004-07-07

arxiv.org/abs/quant-ph/0407054v1

Physics

Quantum Physics

latex, 25 pages, 3 figures

Scientific paper

10.1063/1.1827924

We consider group-covariant positive operator valued measures (POVMs) on a finite dimensional quantum system. Following Neumark's theorem a POVM can be implemented by an orthogonal measurement on a larger system. Accordingly, our goal is to find an implementation of a given group-covariant POVM by a quantum circuit using its symmetry. Based on representation theory of the symmetry group we develop a general approach for the implementation of group-covariant POVMs which consist of rank-one operators. The construction relies on a method to decompose matrices that intertwine two representations of a finite group. We give several examples for which the resulting quantum circuits are efficient. In particular, we obtain efficient quantum circuits for a class of POVMs generated by Weyl-Heisenberg groups. These circuits allow to implement an approximative simultaneous measurement of the position and crystal momentum of a particle moving on a cyclic chain.

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