Quantum skyrmions in general quantum channels

Quantum skyrmions as topologically structured entangled states have the potential to be a pathway toward robustness against external perturbations, but so far no theoretical framework exists to validate this. Here, we introduce the notion of a new entanglement observable based on such topologies and develop a theoretical framework for studying its evolution in general quantum channels. Using photons entangled in orbital angular momentum and polarization as an example, we show that the noise affecting both photons can be recast as a position-dependent perturbation affecting only the photon in the polarization state, restricting the noise to a finite-dimensional Hilbert space. From this we predict complete resilience for both non-depolarizing and depolarizing noise, the former by rigorous arguments based on homotopic maps and the latter by numerical simulation. Finally, we identify sources of local noise that can destabilize the topology and suggest why this may be ignored in practical situations. Our work sets a solid foundational framework for understanding how and why topology enhances the resilience of such entanglement observables, with immediate relevance to the distribution of information through entanglement in noisy environments, such as quantum computers and quantum networks.