Quantum channels, wavelets, dilations and representations of \(\mathcal O_ n\). (English) Zbl 1051.46046

Kribs, David W.

Proc. Edinb. Math. Soc., II. Ser. 46, No. 2, 421-433 (2003).

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.)

Keywords:

operator; completely positive map; quantum channel; orthogonal wavelet; isometric dilation; Cuntz algebra