We classify strong subadditivity for the canonical topological Rényi functional at finite positive orders on an unpunctured torus in the topological quantum field theory of any finite unitary modular tensor category. Universal validity means validity for all pure ground states and all regular region pairs with transverse or disjoint boundaries and compatible corner rounding. At von Neumann order one, the inequality holds universally. For each fixed order strictly between zero and one, universal validity is equivalent to pointedness, meaning that every simple anyon type has quantum dimension one. For every nonpointed category and each finite positive order other than one, we construct a violating pure state and an allowed pair. For a fixed pointed category and fixed order above one, universal validity is equivalent to a same-order modular Fourier entropy bound for every pure state. Global homological constraints on actual unions and intersections and established weighted modular uncertainty prove the order-one theorem. For parallel annuli at fixed flux distribution, the order-one functional is affine in their positive count. At other orders, unequal quantum dimensions on the occupied support induce strict curvature that suitable overlapping regions convert into negative deficits. Under the boundary-charge cutting prescription, the von Neumann theorem extends to arbitrary logical density matrices in the same ground space.
We classify strong subadditivity for the canonical topological Rényi functional at finite positive orders on an unpunctured torus in the topological quantum field theory of any finite unitary modular tensor category. Universal validity means validity for all pure ground states and all regular region pairs with transverse or disjoint boundaries and compatible corner rounding. At von Neumann order one, the inequality holds universally. For each fixed order strictly between zero and one, universal validity is equivalent to pointedness, meaning that every simple anyon type has quantum dimension one. For every nonpointed category and each finite positive order other than one, we construct a violating pure state and an allowed pair. For a fixed pointed category and fixed order above one, universal validity is equivalent to a same-order modular Fourier entropy bound for every pure state. Global homological constraints on actual unions and intersections and established weighted modular uncertainty prove the order-one theorem. For parallel annuli at fixed flux distribution, the order-one functional is affine in their positive count. At other orders, unequal quantum dimensions on the occupied support induce strict curvature that suitable overlapping regions convert into negative deficits. Under the boundary-charge cutting prescription, the von Neumann theorem extends to arbitrary logical density matrices in the same ground space.
Topological entanglement entropy (TEE) represents an intrinsic contribution to the entanglement entropy (EE) in topologically ordered systems. In quantum information theory, strong subadditivity (SSA) is a fundamental property of EE, reflecting the non-negativity of conditional mutual information. TEE was originally believed to be a universal correction to the area law of EE, suggesting that its SSA would directly follow from the SSA of EE. However, due to spurious contributions, the correction term is not universal; consequently, the value predicted by topological quantum field theory (TQFT) provides only a lower bound. In this work, we present a topological analysis showing that the SSA for TEE is equivalent to a specific inequality within the TQFT framework. We further verify that this inequality holds for all known unitary modular tensor categories (UMTCs) up to rank 11, supporting the conjecture that SSA holds universally in the TQFT framework. Conversely, assuming the validity of the SSA condition, the inequality can be interpreted as a consistency criterion for candidate UMTCs.
Abstract We study the multi-boundary entanglement structure of the state associated with the torus link complement S3\Tp,q in the set-up of three-dimensional SU(2)k Chern-Simons theory. The focal point of this work is the asymptotic behavior of the Rényi entropies, including the entanglement entropy, in the semiclassical limit of k → ∞. We present a detailed analysis of several torus links and observe that the entropies converge to a finite value in the semiclassical limit. We further propose that the large k limiting value of the Rényi entropy of torus links of type Tp,pn is the sum of two parts: (i) the universal part which is independent of n, and (ii) the non-universal or the linking part which explicitly depends on the linking number n. Using the analytic techniques, we show that the universal part comprises of Riemann zeta functions and can be written in terms of the partition functions of two-dimensional topological Yang-Mills theory. More precisely, it is equal to the Rényi entropy of certain states prepared in topological 2d Yang-Mills theory with SU(2) gauge group. Further, the universal parts appearing in the large k limits of the entanglement entropy and the minimum Rényi entropy for torus links Tp,pn can be interpreted in terms of the volume of the moduli space of flat connections on certain Riemann surfaces. We also analyze the Rényi entropies of Tp,pn link in the double scaling limit of k → ∞ and n → ∞ and propose that the entropies converge in the double limit as well.
We introduce a nonperturbative approach to calculate the Rényi entropy of a single interval on the torus for single-character (meromorphic) conformal field theories. Our prescription uses the Wrońskian method of Mathur, Mukhi, and Sen [], in which we construct differential equations for torus conformal blocks of the twist two-point function. As an illustrative example, we provide a detailed calculation of the second Rényi entropy for the E 8 , 1 Wess-Zumino-Witten (WZW) model. We find that the Z 2 cyclic orbifold of a meromorphic conformal field theory (CFT) results in a four-character CFT, which realizes the toric code modular tensor category. The Z 2 cyclic orbifold of the E 8 , 1 WZW model, however, yields a three-character CFT since two of the characters coincide. We then compute the torus conformal blocks and find that the twist two-point function, and therefore the Rényi entropy, is two-periodic along each cycle of the torus. The second Rényi entropy for a single interval of the E 8 , 1 WZW model has the universal logarithmic divergent behavior in the decompactification limit of the torus, as expected as well as the interval approaches the size of the cycle of the torus. Furthermore, we see that the q -expansion is UV finite, apart from the leading universal logarithmic divergence. We also find that there is a divergence as the size of the entangling interval approaches the cycle of the torus, suggesting that gluing two tori along an interval the size of a cycle is a singular limit.
We generalize the topological entanglement entropy to a family of topological Rényi entropies parametrized by a parameter alpha, in an attempt to find new invariants for distinguishing topologically ordered phases. We show that, surprisingly, all topological Rényi entropies are the same, independent of alpha for all nonchiral topological phases. This independence shows that topologically ordered ground-state wave functions have reduced density matrices with a certain simple structure, and no additional universal information can be extracted from the entanglement spectrum.
Abstract We use the topological entanglement entropy (TEE) as an efficient tool to fully characterize the Abelian phase of a Z 2 × Z 2 spin liquid emerging as the ground state of topological color code (TCC), which is a class of stabilizer states on the honeycomb lattice. We provide the fusion rules of the quasiparticle (QP) excitations of the model by introducing single- or two-body operators on physical spins for each fusion process which justify the corresponding fusion outcome. Besides this, we extract the TEE from Renyi entanglement entropy (EE) of the TCC, analytically and numerically by finite size exact diagonalization on the disk shape regions with contractible boundaries. We obtain that the EE has a local contribution, which scales linearly with the boundary length in addition to a topological term, i.e. the TEE, arising from the condensation of closed strings in the ground state. We further investigate the ground state dependence of the TEE on regions with non-contractible boundaries, i.e. by cutting the torus to half cylinders, from which we further identify multiple independent minimum entropy states (MES) of the TCC and then extract the U and S modular matrices of the system, which contain the self and mutual statistics of the anyonic QPs and fully characterize the topological phase of the TCC. Eventually, we show that, in spite of the lack of a local order parameter, TEE and other physical quantities obtained from the ground state wave function such as the entanglement spectrum (ES) and ground state fidelity are sensitive probes to study the robustness of a topological phase. We find that the topological order in the presence of a magnetic field persists until the vicinity of the transition point, where the TEE and fidelity drops to zero and the ES splits severely, signaling breakdown of the topological phase of the TCC.
We compute the Entanglement Entropy (EE) of a bipartition in 2D pure non-abelian U(N) gauge theory. We obtain a general expression for EE on an arbitrary Riemann surface. We find that due to area-preserving diffeomorphism symmetry EE does not depend on the size of the subsystem, but only on the number of disjoint intervals defining the bipartition. In the strong coupling limit on a torus we find that the scaling of the EE at small temperature is given by S(T)−S(0)=O(mgapTe−mgapT), which is similar to the scaling for the matter fields recently derived in literature. In the large N limit we compute all of the Renyi entropies and identify the Douglas–Kazakov phase transition.