For the past two decades, there have been great advances in our understanding of the complexity of quantum Hamiltonian systems. Tools like Lieb-Robinson bounds, tensor-networks, computational complexity reductions, and entanglement theory have helped us to answer questions like what the complexity of approximating the ground states of various classes of local Hamiltonians is, or whether or not there exists an efficient classical representation for such states.

 

Most physical systems, however, are open, and are often governed by local Lindbladians rather than local Hamiltonians. It is therefore natural to ask if we could use the same tools to study the complexity of such systems, and in particular the complexity of their steady states. The biggest obstacle in bridging these two worlds is Hermiticity: while Hamiltonians are Hermitian and induce unitary dynamics, Lindbladians are not, and their dynamics is dissipative. 

 

In this talk, I will use the quantum detailed-balance condition to overcome this problem. I will present a mapping between local Lindbladians that satisfy the quantum detailed-balance condition to local Hamiltonians. This will enable me to identify sufficient conditions under which the steady states of these systems satisfies exponential decay of correlations, satisfies an area law, and can be efficiently represented by a tensor-network. I will also discuss the implications of this mapping for numerical simulations, as well as to the complexity of the local Hamiltonian problem.