On the stochastic control of quantum ensembles (Q2722574)

This publication shows how to apply control theory to Nuclear Magnetic Resonance (NMR).NEWLINENEWLINENEWLINEFirst, the physics of NMR is recalled. One considers an ensemble of spin systems; for example a set of \(n\) nuclei which can occupy each finitely many spin states (\(2^n\) in total) in a quantum mechanical way. The evolution of the states \(\psi_j\) is governed by Schrödinger's equations but the quantity used is \(\rho_j= \psi_j\psi^\dag_j\) which after ensemble averaging (over the nuclei) produces the density matrix operator \(\rho\) which satisfies a Lax type equation \(\dot\rho= [H,\rho]\). Observer that the Hamiltonian \(H\) is a tensor product of matrices with \(n\) components and that it is composed of several terms: a drift term (expressing the influence of the external fixed component of the magnetic field and the mutual magnetic interactions of the nuclei), a control term (so that we have a bilinear control system) and possibly noise terms so that the equilibrium probability density is shaped in a convenient way (cf. the publication of the first author [Birkhäuser. Prog. Syst. Control Theory 22, 75-100 (1997; Zbl 0886.93058)]). The two stochastic effects should not be confused here. One of them is ``existential'' arising from quantum mechanics while the other perturbing the densities is a ``technical'' tool.NEWLINENEWLINENEWLINEThe complete controllability of the bilinear system evolving in \(U(2^n)\) is established. The result is due to \textit{T. Schulte-Herbrüggen} [Ph.D. thesis, ETH Zürich (1998)].NEWLINENEWLINENEWLINENext, an optimal observation problem is investigated (in order to minimize the effect of the noise). Here, \(\text{tr}(A^\dag\rho)\) is maximized and \(A\) is an observable. If \(A\) is skew symmetric, one can generate using the preceding controllability result a constant solution in \(U(2^n)\) but in other cases (which occur in NMR with simultaneous observations and related obstructions due to noncommutativity), there are many optimal solutions and it is suggested to took for them using simulated annealing.NEWLINENEWLINENEWLINEA last section makes a connection with quantum computation where now one attempts to embed the manipulation of bits in a (quantum) computer in the frame of pulse controlled evolution equations.NEWLINENEWLINENEWLINEThere are many typographical mistakes and the English is improper sometimes. The title is badly chosen because the ensemble averaging is just an arithmetic mean over intensities (whereby if one had averaged over the \(\psi_j\), additional interference terms would have appeared) even if advantages in filtering are gained. On the other hand the stochastic effects are a weak point of this publication: the noise should shape the ultimate probability distribution, should help find global optimal solutions via simulated annealing and should be filtered away while performing optimal observation so that an assessment of its respective influences should clarify this confusing situation. But a meaningful path (taking some roots in several previous works of the first author) is drawn and it suggests ramifications for future creative works which are urged in the conclusion.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00036].