Quantum response of dephasing open systems

We develop a theory of adiabatic response for open systems governed by Lindblad evolutions. The theory determines the dependence of the response coefficients on the dephasing rates and allows for residual dissipation even when the ground state is protected by a spectral gap. We give quantum response a geometric interpretation in terms of Hilbert space projections: For a two level system and, more generally, for systems with suitable functional form of the dephasing, the dissipative and non-dissipative parts of the response are linked to a metric and to a symplectic form. The metric is the Fubini-Study metric and the symplectic form is the adiabatic curvature. When the metric and symplectic structures are compatible the non-dissipative part of the inverse matrix of response coefficients turns out to be immune to dephasing. We give three examples of physical systems whose quantum states induce compatible metric and symplectic structures on control space: The qubit, coherent states and a model of the integer quantum Hall effect.Comment: Article rewritten, two appendices added. 16 pages, 2 figure

Optimal parametrizations of adiabatic paths

The parametrization of adiabatic paths is optimal when tunneling is minimized. Hamiltonian evolutions do not have unique optimizers. However, dephasing Lindblad evolutions do. The optimizers are simply characterized by an Euler-Lagrange equation and have a constant tunneling rate along the path irrespective of the gap. Application to quantum search algorithms recovers the Grover result for appropriate scaling of the dephasing. Dephasing rates that beat Grover imply hidden resources in Lindblad operators.Comment: 4 pages, 2 figures; To prevent from misunderstanding, we clarified the discussion of an apparent speedup in the Grover algorithm; figures improved + minor change

Adiabatic theorems for generators of contracting evolutions

We develop an adiabatic theory for generators of contracting evolution on Banach spaces. This provides a uniform framework for a host of adiabatic theorems ranging from unitary quantum evolutions through quantum evolutions of open systems generated by Lindbladians all the way to classically driven stochastic systems. In all these cases the adiabatic evolution approximates, to lowest order, the natural notion of parallel transport in the manifold of instantaneous stationary states. The dynamics in the manifold of instantaneous stationary states and transversal to it have distinct characteristics: The former is irreversible and the latter is transient in a sense that we explain. Both the gapped and gapless cases are considered. Some applications are discussed.Comment: 31 pages, 4 figures, replaced by the version accepted for publication in CM

Optimal time-schedule for adiabatic evolution

We show that, provided dephasing is taken into account, there is a unique timetable which maximizes the fidelity with a target state in adiabatic evolutions. The optimum has constant tunneling rate along the path. Application to quantum search algorithms recovers the Grover result for appropriate scaling of the dephasing with the size of the database. Moreover, the Grover bound imposes constraints on the dephasing rates of systems coupled to a universal Markovian bath. while having a built-in protection from decoherence associated with the exchange of energy with the environment. This protection comes from the (assumed) energy gap of the quantum system. The simplicity and physical character of the model led to a resurgence of interest in adiabatic control of both isolated Dephasing is a special case of decoherence which leads to a loss of information that does not depend on exchange of energy. The role of dephasing in adiabatic quantum computation is, at present, less well understood than that of decoherence in general The time-scheduling problem is to determine the optimal time parametrization of a given path of Hamiltonians. In the absence of dephasing, there is no unique optimizer-there are plenty of them. Dephasing singles out a unique optimizer. The optimizer turns out to have a "local" characterization: It has a fixed tunneling rate along the path. This means that monitoring the tunneling rate (or, equivalently, the purity of the state) allows one to adhere to an optimal time schedule. No a priori knowledge about the governing dynamics is required. As an application we derive relations between Lindblad operators Let us now describe the setting and results in more detail. Let H q , q ∈ [0,1], be a path in the space of Hamiltonians, e.g., a linear interpolation, We are interested in the optimal parametrization of the interpolating path. That is, a timetable q(s), which optimizes the fidelity of the state, initially in the ground state of H 0 , with the ground state of the target Hamiltonian at the end time T . Here s = εt ∈ [0,1] is the slow time parametrization and ε = 1/T the adiabaticity parameter. For the sake of simplicity we assume that the Hilbert space has a dimension N (finite) and that H q is a self-adjoint matrixvalued function of q with ordered simple eigenvalues e a (q), so that P a (q) = |ψ a (q) ψ a (q)| are the corresponding spectral projections. The cost function is the tunneling T q,ε (1) at the end point defined by T q,ε (s) = 1 − tr[P 0 (q)ρ q,ε (s)]. ρ q,ε (s) is the quantum state at slow time s which has evolved from the initial condition ρ q,ε (0) = P 0 (0). We consider the quantum evolution generated by a Lindbladia