Quantum Entanglement of Topological Charge: Mechanism, Monogamy, and Why a Hopf Soliton's Spin Entanglement Is Not Topological

Abstract

Two Hopf solitons with disjoint cores can be linked, and their preimage curves are quantised vortex lines carrying an integer charge. This paper derives the mechanism by which linking structures the entanglement of that charge, and delimits what it can and cannot reach. Entanglement itself arrives with quantisation, not with topology. A region's boundary torus carries two cycles. The meridian reads the region's own charge; the 0-framed longitude is homologous, in the complement, to the partners' meridians weighted by their linking numbers. An operator confined to the region therefore reads its partners' charges, and reads nothing when the linking vanishes. Monogamy follows from counting cycles alone — one longitude means one readout channel however many partners, so linking to a second removes the pairwise entanglement rather than reducing it, and the three-party state is of Greenberger–Horne–Zeilinger type. For a charge valued in a finite cyclic group the entropy across any cut is the rank of the corresponding block of the linking matrix. Because the charge is an integer, decoherence is a topological transition rather than a gradual decay, and the entanglement ends at a finite time rather than approaching zero asymptotically. The mechanism does not reach spin. Every readout it affords is a sum of topological invariants and so cannot depend on a soliton's orientation, which is where its spin state lives. This is a theorem rather than an observation about the cases tried, and it holds for any gauge group: a rotation is an ambient isotopy, and a readout factoring through the topology of the core complement is a function on a character variety, which quotients by exactly the conjugation that isotopy leaves free. What does reach orientation is an ordinary force — the R⁻³ interaction of the dipole pairs the solitons' asymptotic fields carry. Extended here to arbitrary relative orientation in closed form, and validated against published channel energies, it projects onto the collective coordinates as a pure tensor coupling with no central part. Its scale is about 6% of a soliton's rest energy at three core radii against a rotational gap of 7.5%, so the two-rotor description holds only beyond roughly 2.8 soliton radii; inside that the orientations are locked, which is the same boundary reached independently from the merged-soliton argument. That coupling annihilates the spin singlet exactly, and the second-order term, computed here rather than estimated, is proportional to the identity on spin. Since the s-wave lies lower and Fermi statistics ties it to the singlet, the ground state of two unit-charge hopfions is the spin singlet — maximally entangled, with that entanglement supplied by antisymmetrisation and by no mechanism in the framework.