Quantum control and the Strocchi map

Identifying the real and imaginary parts of wave functions with coordinates and momenta, quantum evolution may be mapped onto a classical Hamiltonian system. In addition to the symplectic form, quantum mechanics also has a positive-definite real inner product that provides a geometrical interpretation of the measurement process. Together they endow the quantum Hilbert space with the structure of a Kaeller manifold. Quantum control is discussed in this setting. Quantum time evolution corresponds to smooth Hamiltonian dynamics and measurements to jumps in the phase space. This adds additional power to quantum control, nonunitarily controllable systems becoming controllable by 'measurement plus evolution'. A picture of quantum evolution as the Hamiltonian dynamics in a classical-like phase space is the appropriate setting to carry over techniques from classical to quantum control. This is illustrated by a discussion on optimal control and sliding mode techniques.