Duality transformations reveal unexpected equivalences between seemingly distinct models. We introduce a concept of global duality, in which a duality operator intertwines two one-dimensional local Hamiltonians via generalised exchange relations, in contrast to local dualities associated with generalised symmetries. We introduce an out-of-equilibrium generalisation of matrix product operators to implement such global duality transformations in one-dimensional boundary-driven Markov processes on lattices. We construct these operators exactly for the symmetric simple exclusion process with distinct out-of-equilibrium boundaries. In this case, out-of-equilibrium boundaries are dual to equilibrium boundaries satisfying Liggett's condition, implying that the Gibbs–Boltzmann measure captures out-of-equilibrium physics when leveraging the duality operator. In addition, we show the close connection between the global duality operator and quantum integrability, a powerful framework for the exact characterisation of one-dimensional strongly correlated systems. Using solutions of the boundary Yang–Baxter equations, integrable Hamiltonians with boundaries and their transfer matrices can be constructed within the quantum inverse scattering method. We study the Heisenberg XXX quantum spin chain with triangular boundaries and consider additional Yang–Baxter defect equations. We show that the solution to the defect equations has a matrix product operator structure and implements the global duality between different triangular boundaries.
