A bstract We develop a topological framework for preparing families of non-stabilizer states, and computing their entanglement entropies, in SU (2) 1 Chern-Simons theory. Using the Kac-Moody algebra, we construct Pauli and Clifford operators as path integrals over 3-manifolds with Wilson loop insertions, enabling an explicit topological realization of W n and Dicke states, as well as their entanglement properties. For the genus-1 case, the modular SL (2 , ℤ) generators on a single torus boundary realize the standard Clifford generators. We further conjecture a higher-genus extension of this correspondence specifically for the Clifford orbits of W and Dicke states, in which the orbit of W n is represented by a sum over genus- g handlebodies with Dehn twists applied to each torus. Our results extend the known correspondence between Clifford structures and modular transformations, and demonstrate that non-stabilizer states can be understood as controlled deformations within the SU (2) 1 modular tensor category, providing a concrete example of how non-stabilizer quantum resources emerge from purely topological data.
The way in which geometry encodes entanglement is a topic of much recent interest in quantum many-body physics and the AdS/CFT duality. This relation is particularly pronounced in the case of topological quantum field theories, where topology alone determines the quantum states of the theory. In this work, we study the set of quantum states that can be prepared by the Euclidean path integral in three-dimensional Chern-Simons theory. Specifically, we consider arbitrary three-manifolds with a fixed number of torus boundaries in both Abelian
U(1) and non-Abelian
SO(3) Chern-Simons theory. For the Abelian theory, we find that the states that can be prepared coincide precisely with the set of stabilizer states from quantum information theory. This constrains the multipartite entanglement present in this theory, but it also reveals that stabilizer states can be described by topology. In particular, we find an explicit expression for the entanglement entropy of a many-torus subsystem using only a single replica, as well as a concrete formula for the number of GHZ states that can be distilled from a tripartite state prepared through path integration. For the non-Abelian theory, we find a notion of ``state universality,'' namely that any state can be prepared to an arbitrarily good approximation. The manifolds we consider can also be viewed as toy models of multiboundary wormholes in AdS/CFT.
Abstract Topological entanglement structure amongst disjoint torus boundaries of three manifolds have already been studied within the context of Chern-Simons theory. In this work, we study the topological entanglement due to interaction between the quasiparticles inside three-manifolds with one or more disjoint S 2 boundaries in SU(N) Chern-Simons theory. We focus on the world-lines of quasiparticles (Wilson lines), carrying SU(N) representations, creating four punctures on every S 2. We compute the entanglement entropy by partial tracing some of the boundaries. In fact, the entanglement entropy depends on the SU(N) representations on these four-punctured S 2 boundaries. Further, we observe interesting features on the GHZ-like and W-like entanglement structures. Such a distinction crucially depends on the multiplicity of the irreducible representations in the tensor product of SU(N) representations.
Entanglement entropies of multipartite pure states satisfy linear inequalities; for stabilizer states, the backbone of quantum error correction, the achievable entropy vectors generate a convex cone. We show that, after a linear lift, the closure of this cone is exactly the self-dual slice of a classical object on one more element than there are parties: the closed cone generated by the rank functions of arrangements of subspaces over
GF(d) for a prime local dimension
d, and of subgroups of abelian groups of exponent dividing
d in general. This correspondence brings two features of rank inequalities into play. First, the purifier acts as an extra variable: already for five parties the stabilizer cone is strictly smaller than the quantum linear rank cone, with which earlier evidence suggested it might coincide; an explicit six-variable inequality witnesses the gap. Second, the characteristic of the field enters, so the cone depends on the local dimension. It does not for up to four parties. With enough parties, the cone of
Zd qudits is contained in that of
Zd′ qudits if and only if
d divides
d′, and already for six parties the logical zero state of Steane's code lies outside the stabilizer cone of every odd local dimension. For this last result we show that the seven-variable characteristic-dependent inequalities we checked all follow from Shannon inequalities on the self-dual slice, and instead transfer one of them through a family of subgroups derived from the Fano plane.