Time optimal quantum control of mixed states

We formulate a variational principle for finding the time-optimal quantum evolution of mixed states governed by the master equation, when the Hamiltonian H and the Lindblad operators L_j are subject to certain constraints. We show that the problem can be reduced to solving first a fundamental equation (the “quantum brachistochrone”) for H(t), which can be written down once the constraints are specified, and then solving the constraints and the master equation for the L_j(t)s and the density operator ρ(t). As an application of our formalism, we analytically solve a simple one qubit model where the optimal Lindblad operators correspond either to a continuous Markovian measurement or to a decoherence process by the environment.