A rigorous analysis is presented for the entanglement spectrum of quantum many-body states possessing a higher-form group-representation symmetry generated by topological Wilson loops, which is generally noninvertible. A general framework based on elementary algebraic topology and category theory is developed to determine the block structure of reduced density matrices for arbitrary bipartite manifolds on which the states are defined. Within this framework, we scrutinize the impact of topology on the entanglement structure for low-dimensional manifolds, including especially the torus, the Klein bottle, and lens spaces. Unlike in the case of 0-form symmetries, the reduced density matrix fragments into area-law-order blocks. By further incorporating gauge invariance, we refine our framework to determine the entanglement structure for topological gauge theories in arbitrary dimensions. In particular, in two dimensions, it is shown for the Kitaev quantum double model that not only the topological entanglement entropy can be reproduced, but also the Li-Haldane conjecture concerning the full entanglement spectrum holds exactly. It demonstrates that the block size of the density matrix corresponds to the number of ways to fuse anyons in the split manifold, and that the degeneracy of the entanglement spectrum is given by the product of the quantum dimensions of all anyons connecting the two manifolds and area-law entanglement spectrum fragmentations.

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• Entanglement entropy
• Mathematical physics
• Quantum entanglement
• Topological field theories
• Topological order