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In this paper, we study $(2+1)$-dimensional quantum anomalous Hall states,i.e. band insulators with quantized Hall conductance, using the exactholographic mapping. The exact holographic mapping is an approach toholographic duality which maps the quantum anomalous Hall state to a differentstate living in $(3+1)$-dimensional hyperbolic space. By studying topologicalresponse properties and the entanglement spectrum, we demonstrate that theholographic dual theory of a quantum anomalous Hall state is a$(3+1)$-dimensional topological insulator. The dual description enables a newcharacterization of topological properties of a system by the quantumentanglement between degrees of freedom at different length scales.

Holographic duality between $(2+1)$-d quantum anomalous Hall state and $(3+1)$-d topological insulators