Optimal simulation of two-qubit Hamiltonians using general local operations

We consider the simulation of the dynamics of one nonlocal Hamiltonian by another, allowing arbitrary local resources but no entanglement nor classical communication. We characterize notions of simulation, and proceed to focus on deterministic simulation involving one copy of the system. More specifically, two otherwise isolated systems $A$ and $B$ interact by a nonlocal Hamiltonian $H \neq H_A+H_B$. We consider the achievable space of Hamiltonians $H'$ such that the evolution $e^{-iH't}$ can be simulated by the interaction $H$ interspersed with local operations. For any dimensions of $A$ and $B$, and any nonlocal Hamiltonians $H$ and $H'$, there exists a scale factor $s$ such that for all times $t$ the evolution $e^{-iH'st}$ can be simulated by $H$ acting for time $t$ interspersed with local operations. For 2-qubit Hamiltonians $H$ and $H'$, we calculate the optimal $s$ and give protocols achieving it. The optimal protocols do not require local ancillas, and can be understood geometrically in terms of a polyhedron defined by a partial order on the set of 2-qubit Hamiltonians.