No quantum system is perfectly isolated. Coupling to an environment causes decoherence and dissipation: loss of phase information, relaxation of populations and leakage of entanglement. Open-system control asks how to reach targets despite this, and sometimes because of it. It also changes what control can do. Hamiltonian controls alone can never purify a mixed state, while engineered dissipation and measurement can.

Dynamics

For weak, memoryless coupling the density matrix obeys a master equation of Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) form [1, 2],

$$\dot\rho = -\frac{i}{\hbar}\big[H(t), \rho\big] + \sum_k \gamma_k\Big(L_k\rho L_k^\dagger - \tfrac12\big\{L_k^\dagger L_k, \rho\big\}\Big),$$

with jump operators \(L_k\) and rates \(\gamma_k\). When the environment has memory, the dynamics is non-Markovian: information flows back from the environment, and time-local Lindblad forms no longer suffice [3].

Optimal control with dissipation

  • Fighting decoherence. Optimal control finds pulses that complete a task before decoherence takes over, or that steer through weakly decaying subspaces. When the relaxation rates are known, exploiting them can beat simply minimizing time [4]. Robust gates for interacting qubits have been optimized in both Markovian and non-Markovian environments [5].
  • Numerics in Liouville space. Density matrices have \(d^2\) entries and superoperators act on them, so open-system control is expensive. For gates, three well-chosen states suffice instead of a full Liouville-space basis [6]. Quantum trajectories with automatic differentiation reduce cost further [7]. Distributed-memory codes target larger open systems [8]. See Numerical propagation & simulation and Problem formulation & figures of merit.
  • Time-optimal control with relaxation. Even a single dissipative spin has optimal solutions with singular arcs, realized experimentally in NMR [9]; see Geometric & time-optimal control.

Using the environment

Dissipation is not only an enemy:

  • Cooling and reset. Optimal control of relaxing systems can speed up or slow down their approach to the steady state [10]. It has also been used to design laser cooling of internal molecular degrees of freedom [11]. For qubit reset under non-Markovian dynamics, initial system–environment correlations allow faster reset and smaller errors [12].
  • Non-Markovian back-flow. Amplitude and phase returning from the environment can be exploited to perform control tasks that would be impossible with a memoryless bath [13].
  • Incoherent control. A suitably prepared, time-dependent environment can itself be the control that steers a system to its target [14].
  • Engineered dissipation. Designing the coupling to an environment so that the target is the steady state is the subject of Reservoir engineering & dissipative preparation.

Limits

Controllability notions change for open systems [15]; see Quantum controllability. Speed limits must be generalized to non-unitary dynamics; see Quantum speed limits. A review of the tools, achievements and limitations of open-system optimal control is [16]. Measurement adds another resource, real-time information; see Measurement-based feedback.