Implementation of group-covariant POVMs by orthogonal measurements
<|reference_start|>Implementation of group-covariant POVMs by orthogonal measurements: 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.<|reference_end|>
@article{decker2004implementation,
    title={Implementation of group-covariant POVMs by orthogonal measurements},
    author={Thomas Decker, Dominik Janzing, Martin Roetteler},
    journal={Journal of Mathematical Physics, 46:012104, 2005},
    year={2004},
    doi={10.1063/1.1827924},
    archivePrefix={arXiv},
    eprint={quant-ph/0407054},
    primaryClass={quant-ph cs.ET}
}