Geometry of local quantum dissipation and fundamental limits to local cooling

We geometrically characterize one-qubit dissipators of a Lindblad type. An efficient parametrization in terms of 6 linear parameters opens the way to various optimizations with local dissipation. As an example, we study maximal steady-state singlet fraction that can be achieved with an arbitrary local dissipation and two qubit Hamiltonian. We show that this singlet fraction has a discontinuity as one moves from unital to non-unital dissipators and demonstrate that the largest possible singlet fraction is approximately 0.654. This means that for systems with a sufficiently entangled ground state there is a fundamental quantum limit to the lowest attainable energy. With local dissipation one is unable to cool the system below some limiting non-zero temperature.