Anyon condensation forms a mechanism that allows us to relate different topological phases. We study anyon condensation in the framework of projected entangled pair states (PEPS) where topological order is characterized through local symmetries of the entanglement. We show that anyon condensation is in one-to-one correspondence to the behavior of the virtual entanglement state at the boundary (i.e., the entanglement spectrum) under those symmetries, which encompasses both symmetry breaking and symmetry protected (SPT) order, and we use this to characterize all anyon condensations for Abelian double models through the structure of their entanglement spectrum. We illustrate our findings with the Z(4) double model, which can give rise to both toric code and doubled semion order through condensation, distinguished by the SPT structure of their entanglement. Using the ability of our framework to directly measure order parameters for condensation and deconfinement, we numerically study the phase diagram of the model, including direct phase transitions between the doubled semion and the toric code phase, which are not described by anyon condensation.
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