Fig. 2: Topological rejection of noise.
a We create a bi-photon entangled state, with photon A in the spatial degree of freedom and photon B in the polarization degree of freedom, for a non-local skyrmionic topology as a shared emergent property of the two photons, shown as a stereographic projection. b The Skyrmion number is a digitized observable, shown here for integer values from N = −2 to N = 2, remaining intact along with the topology after the isotropic noise even while the entanglement decreases, except in the case when the entanglement reaches its minimum, indicating a non-entangled state. c In the geometric picture of photon B's Hilbert space as expressed by Stokes vectors (Sx, Sy, Sz), the change of state purity due to noise is a sphere of diminishing size, which for partially mixed states can always be renormalized while leaving the topology unaltered. Once the state is maximally-mixed, the Hilbert space of photon B has zero radius, no entanglement, and thus only a trivial topology (N = 0).
