  We present a lower bound for the free energy of a quantum many-body system at
finite temperature. This lower bound is expressed as a convex optimization
problem with linear constraints, and is derived using strong subadditivity of
von Neumann entropy and a relaxation of the consistency condition of local
density operators. The dual to this minimization problem leads to a set of
quantum belief propagation equations, thus providing a firm theoretical
foundation to that approach. The minimization problem is numerically tractable,
and we find good agreement with quantum Monte Carlo for the spin-half
Heisenberg anti-ferromagnet in two dimensions. This lower bound complements
other variational upper bounds. We discuss applications to Hamiltonian
complexity theory and give a generalization of the structure theorem of Hayden,
Jozsa, Petz and Winter to trees in an appendix.
