Quantum states: an analysis via the orthogonality relation

Shengyang Zhong

Rachel Bailey

The aim of this review is to discuss some applications of orthogonal polynomials in quantum information processing. The hope is to keep the paper self-contained so that someone wanting a brief introduction to the theory of orthogonal polynomials and continuous time quantum walks on graphs may find it in one place. In particular, we focus on the associated Jacobi operators and discuss how these can be used to detect perfect state transfer (PST). We also discuss how orthogonal polynomials have been used to give results which are analogous to those given by Karlin and McGregor when studying classical birth and death processes. Finally, we show how these ideas have been extended to quantum walks with more than nearest-neighbour interactions using exceptional orthogonal polynomials (XOPs). We also provide a (non-exhaustive) list of related open questions.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.

Ryuji Ukai

BERNHARD BAUMGARTNER, HEIDE NARNHOFER

Each semigroup describing time evolution of an open quantum system on a finite dimensional Hilbert space is related to a special structure of this space. It is shown how the space can be decomposed into orthogonal subspaces: One part is related to decay, some subspaces of the other subspace are ranges of the stationary states. Specialities are highlighted where the complete positivity of evolutions is actually needed for analysis, mainly for evolution of coherence. Decompositions are done the same way for discrete as for continuous time evolutions, but they may show differences: Only for discrete semigroups there may appear cases of sudden decay and of perpetual oscillation. Concluding the analysis, we identify the relation of the state space structure to the processes of decay, decoherence, dissipation and dephasing.

S. Malin

Gary E. Bowman