A test of holographic complexity = purification complexity.

A conjectured entry into the AdS/CFT dictionary relates the circuit complexity of a boundary state to either the action on the bulk Wheeler-DeWitt patch or a maximal volume bulk slice homologous to the boundary slice. If one or both conjectures are correct, there is a plausible generalization to a duality between the same bulk quantities computed on entanglement wedges of boundary subregions, and the mixed state complexity on those subregions. However, there is no unique extension of pure state complexity to mixed state complexity. One version of this conjecture uses the “purification complexity” as a measure of mixed state complexity. We test this conjecture in the context of multi-sided eternal black hole solutions, including three-dimensional black holes with arbitrary genus in the behind-the-horizon region. We find that neither the subregion action nor subregion volume behaves entirely as expected in all contexts, suggesting that the proposal should be modified or abandoned.