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On quantum channels: special constructions οf random and optimally unitary channel maps

Varikou Erasmia

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Summary

The study of quantum channels constitutes a main area of Quantum Information Science. Quantum channel maps represent any dynamic changes and/or erroneous modifications affecting quantum signals in the course of their processing within the context of various computational or communicational algorithms. In the colloquial language of quantum information it is said that "a quantum channel acts on the quantum signal", or stated in precise mathematical terms, we have that "a positive and completely positive trace preserving map acts on the Hermitian, positive and trace one state operator". The representation theory of such channel maps provides the so called "operator sum representation" for them, in terms of the so called Kraus generators. This Thesis puts forward a construction technique for some new families of particular channels of the type of random and optimally unitary channels. This is done by working in the space of particular classes of circulant matrices acting on finite dimensional Hilbert spaces. The resulting channels are featuring unital maps which act on state matrices of signals via some convex combinations of the adjoint action of their unitary Kraus generators. The effect of these channels on quantum signals is further investigated by their induced action on the spectrum of their associated density matrices. This task is carried out for finite dimensional signals by determining the bi-stochastic matrices associated with the constructed channels. Basic convex geometric properties of bi-stochastic matrices (Birkhoff's theorem) provide means for studying the effects on the probabilistic eigenvalues of quantum signals, hence to account for e.g. entropic transformations exercised by the new maps upon quantum signals.

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