Knots and signal transmission in topological quantum devices

TL;DR

This paper explores how topological quantum devices, modeled using knot theory, can control signal transmission, revealing properties like self-averaging phases and potential for physical realization.

Contribution

It introduces a novel approach linking knot topology with quantum signal transmission, demonstrating how topology variations influence device filtering properties.

Findings

Transmission coefficients depend on knot topology.

Existence of self-averaging phases with topology-independent transmission.

Potential physical implementations of topological quantum devices.

Abstract

We discuss the basic problem of signal transmission in quantum mechanics in terms of topological theories. Using the analogy between knot diagrams and quantum amplitudes we calculate the transmission coefficients of the concept topological quantum devices that realize the knot topology. We observe that the problem is in different ways similar to that of transmission on quantum graphs. The desired transmission or filtering properties can be attained by the variation of the topology of the device, or an external parameter, which in our model controls the topological phase. One interesting property of the transmission coefficients is the existence of "self-averaging" phases, in which the value of the transmission coefficient is independent from all the representatives in a chosen family of knots. We briefly discuss physical realizations of the concept devices.