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I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators.

So my question is something like this:

Let $i \partial_t \psi(x,t) = H_0(x)\psi(x,t) + u(t)H_1(x)\psi(x,t)$ be a Schrödinger equation. $H_0,H_1$ are nice operators (as nice as there is theory available, so for instance: $H_0$ has compact resolvent and $H_1$ is bounded, both on a finite interval with dirichlet conditions).

now we want to solve the problem whether we can move from a state $\psi_0$ to a state $\psi_1$ and want to minimize the function $u$ with respect to some norm.

Is there any theory available?

I already read that in an infinite dimensional space, we can only expect approximate controllability in the sense that $\psi_0$ can be only moved to $\psi_1$ up to an error of $\epsilon.$

So all this is fine, but I would really like to know if there is any literature on when this problem has a solution?

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The earliest reference is On the controllability of quantum‐mechanical systems (1983). For recent developments, see On the problem of quantum control in infinite dimensions (2011) and Finite Controllability of Infinite-Dimensional Quantum Systems (2006):

For some quantum systems, such as spin systems, the quantum evolution equation (the Schrödinger equation) is finite-dimensional and old results on controllability of systems defined on on Lie groups and quotient spaces provide most of what is needed insofar as controllability of non-dissipative systems is concerned. However, in an infinite-dimensional setting, controlling the evolution of quantum systems presents both conceptual and technical difficulties, in essence because the set of all attainable states has dense complement in the sphere. We analyze controllability for infinite-dimensional bilinear systems under assumptions that make controllability possible using trajectories lying in a nested family of pre-defined subspaces. This result provides a set of sufficient conditions for controllability in an infinite-dimensional setting.

Treatments of the optimality of quantum control are mainly restricted to finite-dimensional systems, but they do include the effects of coupling to the environment (which adds a non-unitary time-dependence, instead of the Schrödinger equation you would have the Lindblad equation). This is the topic of the Ph.D. thesis Optimal Control of Quantum Systems by Symeon Grivopoulos (2005). (The reference list contains many other contributions, in particular Refs. 30-40.) There is also this 2007 overview, Quantum Optimal Control Theory .

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  • $\begingroup$ thank you. all these papers are concerned with the question whether it is possible to control the equation. Despite, do you know of any reference where the optimality question is discussed? $\endgroup$ – QuantumTheory Sep 13 '15 at 22:00
  • $\begingroup$ added pointers to the literature on optimal quantum control $\endgroup$ – Carlo Beenakker Sep 14 '15 at 6:30
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This is not quite what you are asking, but I addressed time optimality in my paper "Time-optimal control of finite quantum systems", J. Math. Phys. 41 (2000), 5262–5269. We have results like: if any control function reaches a desired target then there is one which does so in minimal time; a target is reached by a unique trajectory at time $t$ if it is an extreme point of the set of reachable targets at time $t$; any target reachable at time $t$ is reachable by a bang-bang control. The important qualifiers are that we work in the Born approximation, so the results are only valid over small times or energies, and we require the Hilbert space to be finite dimensional.

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In addition to the literature indicated by Carlo Beenakker: in this 2015 review http://arxiv.org/abs/1508.00442 (Training Schrödinger's cat: quantum optimal control, by S.J. Glaser et al.) the authors "address key challenges and sketch a roadmap to future developments".

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