Like the ground state of a Hamiltonian, the steady state of a Lindblad master equation is the extremal eigenvector of a matrix. However, while ground states of gapped Hamiltonians have been classified into phases, the analogous classification problem for steady states is challenging even to state precisely. I will discuss some progress in this direction, focusing on the key role played by the non-orthogonality of the eigenstates of the Lindbladian [1]. I will introduce a class of error-corrected memories that can be locally stabilized against erasure noise as examples of nontrivial steady-state phases of local Lindbladians [2]. 

[1] T. Rakovszky, SG, C. W. Von Keyserlingk, PRX 14, 041031 (2024)

[2] S. Chirame, F. J. Burnell, SG, A. Prem, PRL 134, 010403 (2025)