Gunther Dirr: Are infinite-dimensional closed quantum systems generically controllable?

It is well-known that bilinear control systems of the form

\(\dot X(t)=\left(u_1(t)B_1+u_2(t)B_2\right)X(t),\quad X(0)=I_n\)     (1)

are generically controllable on semi-simple Lie groups like SL_n(ℝ) or SL_n(ℂ). Loosely speaking, this means that for a randomly chosen tuple (B₁,B₂) system (1) will be controllable. This readily implies generic controllability for finite-dimensional closed quantum systems of the form

\(\dot U(t)=-\mathrm{i}\left(H+u_1(t)H_1\right)X(t),\quad U(0)=I_n\).     (2)

But what about infinite-dimensional quantum systems? Of course, solving this question in full generality is currently out of (our) reach. Therefore, in this talk we will focus on a certain ‘natural’ and rather large class of systems which often occurs in applications. Within this subclass we prove that the answer to the above raised question is ‘yes’.

This video was produced by the International Centre for Mathematical Sciences, as part of the workshop Mathematical Physics in Quantum Technology: From Finite to Infinite Dimensions.