Linked quantum states retain an arithmetic fingerprint

How can linked loops describe entanglement?

Topology studies properties that survive smooth deformation, such as whether loops remain linked. A theoretical preprint by Vikash Ganwariya, Simran Sain and Siddharth Dwivedi asks how this structure shapes quantum entanglement: correlations that prevent a joint pure state from being described as independent states of its parts.

The researchers study SU(2) Chern–Simons theory, a topological quantum field theory in two spatial dimensions and one time dimension. Its observables depend on topology rather than distances and angles. This makes it a controlled mathematical setting, not a laboratory experiment or a demonstration of quantum-computing hardware.

Their objects are torus links called T(d, dn): collections of d individually unknotted loops arranged on a torus, with each pair having linking number n. Removing a tubular neighborhood around each loop leaves a space with d torus-shaped boundaries. The theory assigns a quantum state to those boundaries, making each boundary one subsystem.

The question is how much entanglement remains when the theory’s positive integer level, k, becomes very large. This is the semiclassical limit considered in the paper; it does not mean that every quantum correlation must disappear.

A large state space, but two dominant ends

At level k, each boundary has k + 1 allowed basis labels. Although the full multipartite state space grows rapidly, these particular link states simplify after a change of basis performed separately on each subsystem. Only combinations carrying the same label on every subsystem appear.

That structure lets the authors calculate entanglement across a division into two nonempty groups of boundaries. For this family, the nonzero eigenvalues of the reduced density matrix do not depend on how many boundaries go into each group. The quantity studied is therefore bipartite entropy within a multipartite state, not a complete description of all its correlations.

The calculation tracks moments of the reduced density matrix—sums of powers of its eigenvalues. These yield Rényi entropies and, through a limiting procedure, the usual von Neumann entanglement entropy.

To obtain their large-k behavior, the authors use Poisson summation and stationary-phase analysis, methods for reorganizing discrete sums and estimating rapidly oscillating expressions. They distinguish labels near zero, labels near k, and labels whose ratio to k stays strictly between zero and one.

Their central result is that the two end regions supply the leading contributions to the moments, while the interior is subleading. These “edges” are ends of a representation-label range, not physical surfaces where an experiment has detected concentrated entanglement.

Summing the edge contributions produces expressions involving Hurwitz zeta functions, which package infinite sums of inverse powers into a standard mathematical function.

Why the remainder matters

The limiting entropy generally depends on k modulo n: the remainder when k is divided by the linking parameter n. Consequently, increasing k through all integers need not produce one common entropy limit. Instead, different fixed-remainder sequences can approach different values.

For two components with n = 5—the T(2, 10) link—the authors report three limiting entropies. Levels with remainder zero or one approach approximately 1.828; remainders two or four approach 2.439; remainder three approaches 1.341. These values use natural logarithms, so their units are nats rather than bits. They describe calculated entanglement entropies, not measurement uncertainties or experimental success rates.

The authors also increase the number of components after taking the large-k limit. Their asymptotic expression reduces to a binary entropy determined by normalized contributions from the two ends. Equal weights give ln 2, equivalent to one bit; a single surviving weight gives zero. Comparison with earlier work gives the same expression when the two limits are taken in reverse order, along the corresponding fixed-remainder sequences.

What remains unresolved

These results concern a specific link family and gauge theory. They do not establish edge dominance for arbitrary links, other gauge groups or physical materials.

Some additional patterns are numerical rather than established general limits. For example, calculations through n = 100 suggest a shared large-n entropy for several winding subsequences under a specified choice of remainder, but the paper leaves systematic analytical study of that behavior open.

The supplied version is an arXiv preprint, not evidence of peer-reviewed validation. Its contribution is a mathematical account of how topology-based quantum states can retain discrete arithmetic information even as their available state space becomes large.