Bilayer construction for mixed state phenomena with strong, weak symmetries and symmetry breakings

We introduce the bilayer construction, as a specific purification scheme for a general mixed state, where each mixed state has a one-to-one correspondence with a bilayer pure state with two constraints: non-negativity of the bilayer wavefunction; and the presence of an anti-unitary layer-exchange symmetry T. Different from the Choi-Jamio{\l}kowski isomorphism, any mixed state can be realized as the monolayer reduced density matrix of a bilayer pure state, and its physical properties can be experimentally realized and detected in non-magnetic bilayer 2D materials with a layer-exchange mirror symmetry. We study a variety of mixed state phenomena in the bilayer construction: (1) strong and weak symmetries, their explicit and spontaneous breakings in mixed states can be understood as usual Landau-type symmetry breakings in the bilayer pure state, and their criteria can be derived accordingly; (2) decoherence of a pure state by local errors can be mapped to quantum quench dynamics of the bilayer pure states; (3) mixed symmetry protected topological (SPT) states and mixed state topological orders can be classified, characterized and realized as pure state SPTs and topological orders in the bilayer. We further study examples of strong-to-weak spontaneous symmetry breaking (SWSSB) and their critical scalings at the SWSSB transition in the bilayer construction.