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Quantum channels, wavelets, dilations and representations of \(\mathcal O_ n\). (English) Zbl 1051.46046
Summary: We show that the representations of the Cuntz \(C^*\)-algebras \(\mathcal{O}_n\) which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finite-dimensional space. Based on this analysis, we find an application in quantum information theory; namely, a structure theorem for the fixed-point set of a unital quantum channel. We also include some open problems motivated by this work.

MSC:
46L60 Applications of selfadjoint operator algebras to physics
47A20 Dilations, extensions, compressions of linear operators
81P15 Quantum measurement theory, state operations, state preparations
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
81P68 Quantum computation
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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