Applications of persymmetric CMV matrices (including perfect state transfer)

Identify concrete applications of finite persymmetric CMV matrices—unitary pentadiagonal CMV operators associated with mirror-symmetric (persymmetric) Verblunsky coefficients—and determine whether these matrices can enable perfect state transfer or other mechanisms for quantum information transfer analogous to the established role of persymmetric Jacobi matrices.

Background

The paper develops the theory of mirror-symmetric (persymmetric) orthogonal polynomials on the unit circle (OPUC) and the corresponding CMV matrices, establishing multiple characterizations and examples. It shows close analogies with the real-line (Jacobi) setting, including uniqueness in certain inverse spectral problems.

In contrast to persymmetric Jacobi matrices, persymmetric CMV matrices exhibit parity-dependent relations with quasi-reflection operators, making their structural properties more intricate. Given that persymmetric Jacobi matrices play a central role in perfect state transfer in quantum spin chains, the authors raise the question of whether persymmetric CMV matrices could similarly support quantum information transfer and, more broadly, what their applications might be.

References

The most intriguing question is: what are possible applications of the persymmetric CMV matrices? It is well known that persymmetric Jacobi matrices play a crucial role in the theory of the perfect state transfer , . One can expect that persymmetric CMV matrices could play similar role in transferring the quantum information as well. This is an interesting open problem.

— Mirror symmetric polynomials orthogonal on the unit circle  (2510.10447 - Zhedanov, 12 Oct 2025) in Conclusion