Implementation of group-covariant POVMs by orthogonal measurements

Abstract

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.

latex, 25 pages, 3 figures

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• On Quantum A/D and D/A Conversion
• The block-coherence measures and the coherence measures based on positive-operator-valued measures
• On quantum channels generated by covariant positive operator-valued measures on a locally compact group
• Efficient quantum algorithms for testing symmetries of open quantum systems
• Minimally-disturbing Heisenberg-Weyl symmetric measurements using hard-core collisions of Schrödinger particles
• Bounds on positive operator-valued measure based coherence of superposition
• Symmetric measurements attaining the accessible information
• Frame-dependent coherence of a quantum state