E'tudes on Infinite Tensor Products in Quantum Contexts

In this talk, I will present original mathematical constructions designed to describe two fundamental types of quantum systems: (1) an infinite array of qubits, exemplified by a quantum computer with an infinite register, and (2) an infinite array of boson sites.

The first ́problem addresses the challenge of overcoming the so-called curse of complexity in quantum engineering. Using multi-scale methods tied to the infinite group Z2 × Z2 × Z2 × . . ., we provide a complete solution to certain types of quantum dynamics, even when involving an infinite number of qubits. This framework offers an alternative approach to quantum information processing that, while equivalent to the canonical framework, proves advantageous in an- alyzing some types of problems. This is joint work with Mandana Bidarvand, our recent PhD graduate.

The second ́etude focuses on an infinite array of boson sites. I will highlight the utility of the multiplicative group of positive rationals Q+ and its associ- ated generalized Fourier transform in analyzing such systems. This approach leads to the formulation of nonlocal coherent states (NCS) in the bosonic Fock space, enabling rigorous calculations for quantum systems with infinitely many degrees of freedom. This research has been conducted in collaboration with Jonas Fransson from the Department of Physics and Astronomy at Uppsala University.

These results stem from close collaboration between mathematicians and physicists, with conceptual foundations rooted in harmonic analysis, multi-scale methods, and analytic number theory. The applications extend from quantum theory to post-quantum cryptography and information science.