Bounds to the speed of evolution of a quantum system are of fundamental interest in quantum metrology, quantum chemical dynamics and quantum computation. We derive a time-energy uncertainty relation for open quantum systems undergoing a general, completely positive and trace preserving (CPT) evolution which provides a bound to the quantum speed limit. When the evolution is of the Lindblad form, the bound is analogous to the Mandelstam-Tamm relation which applies in the unitary case, with the role of the Hamiltonian being played by the adjoint of the generator of the dynamical semigroup. The utility of the new bound is exemplified in different scenarios, ranging from the estimation of the passage time to the determination of precision limits for quantum metrology in the presence of dephasing noise. 03.65.Yz, 03.67.Lx How fast can a quantum system evolve? Quantum mechanics acts as a legislative body imposing speed limits to the evolution of quantum systems. While these limits are both ultimate and fundamental, at the same time, their existence is at the center of a surge of activity, as a result of their manifold applications, including the identification of precision bounds in quantum metrology [1], the formulation of computational limits of physical systems [2], and the development of quantum optimal control algorithms [3].Bounds on the speed of evolution are intimately related to the concept of the passage time τ P , which is the required time for a given pure state |χ to become orthogonal to itself under unitary dynamics [4]. One of the early answers to this problem was provided by Mandelstam and Tamm (MT), who showed that the passage time can be lower-bounded by the inverse of the variance in the energy of the system so thatwhere ∆H = ( H 2 − H 2 ) 1/2 , whenever the dynamics under study is governed by an Hermitian Hamiltonian H [5][6][7][8][9][10][11][12][13].A simple geometric interpretation of this result was provided by Brody using the Fubini-Study metric in the Hilbert space spanned by the initial state and its orthogonal complement [14]. Indeed, the passage time problem can be posed as a quantum brachistochrone problem. From this perspective, a particularly exciting result was found: whenever the Hamiltonian is non-Hermitian PT-symmetric, the passage time can be made arbitrarily small without violating the time-energy uncertainty principle [15,16]. A second bound, due to Margolus and Levitin (ML), takes the simpler form τ ≥ πwhere the zero of energy is generally shifted to the ground state energy so that E 0 = 0 [17]. This bound has been applied to ascertain fundamental computational limits in nature [2,18]. Despite the growing body of literature on the subject, the analysis has almost exclusively been focused on unitary dynamics of isolated quantum systems. An analogous bound for open quantum systems is highly desirable, since ultimately all systems are coupled to an environment [19,20]. As an example, such a bound on the evolution of an open system would help to address the robustness of quantum s...
