Statement
Consider the finite-dimensional control system
$$\dot x = A x + \sum_{i=1}^{p} u_i(t)\, B_i x, \qquad x \in \mathbb C^n,$$
with skew-adjoint matrices \(A, B_1, \dots, B_p\), unbounded controls \(u_i \in L^\infty\), and the time unit fixed by \(\lVert A\rVert = 1\) (operator norm). The state moves on the unit sphere \(S^{2n-1} \subset \mathbb C^n\). The minimum time of the system is the worst case, over pairs of states, of the shortest time needed to connect them:
$$T(\Sigma) = \sup_{X, Y \in S^{2n-1}}\ \inf\big\{T \ge 0 : \text{a trajectory goes from } X \text{ to } Y \text{ in time } T\big\}.$$
Conjecture (Agrachev). There is a universal \(\delta > 0\) such that, whatever the dimension \(n\) and the system, either \(T(\Sigma) = 0\) or \(T(\Sigma) \ge \delta\) [1].
Why it matters
\(T(\Sigma)\) is the quantum speed limit of the whole control system: the time below which some state-to-state task becomes impossible, however strong the controls. If the controls alone generate a group that acts transitively on the sphere, every task can be done arbitrarily fast and \(T(\Sigma) = 0\). Otherwise the drift must be used, and the question is whether the minimal time can be arbitrarily small — for instance in large systems — or is bounded below in units of the drift. Estimates of the controllability time had been obtained for particular classes, for example single-input systems on compact Lie groups [2], and the 2015 European roadmap lists a universal estimate of the control time for finite-dimensional systems with drift as an open challenge [3]. The gap answers part of it: a lower bound, in units of the drift, that holds for every system that cannot be controlled arbitrarily fast.
Known results
Reduction to geometry. Let \(G \subset U(n)\) be the connected group generated by the control directions \(B_i\), and let
$$D(G) = \sup_{X, Y \in S^{2n-1}}\ \inf_{g, h \in G} \lVert gX - hY\rVert$$
be the largest distance between two of its orbits on the sphere. If \(G\) is transitive, \(D(G) = T(\Sigma) = 0\). Otherwise, since the drift moves the state at speed at most \(\lVert A\rVert = 1\) while the controls only move it along orbits, Gauthier and Rossi show
$$\sin\!\Big(\frac{T(\Sigma)}{2}\Big) \;\ge\; \frac{D(G)}{2},$$
so a universal gap for the orbit-space diameter \(D(G)\) gives one for \(T(\Sigma)\) [1].
All cases except spin representations. Gauthier and Rossi proved that diameter gap for every unitary action except the basic spinor actions (built from basic spin representations of spin groups), which proves the conjecture for all other systems. If the dimension \(n\) is neither a power of 2 nor in an explicit finite list of exceptions, they obtain explicitly: either \(T(\Sigma) = 0\) or \(\sin\big(T(\Sigma)/2\big) \ge \tfrac12\sqrt{2 - \sqrt 2}\) [1].
The diameter gap in general. Gorodski, Lange, Lytchak and Mendes proved that for any group acting by isometries on a unit sphere of dimension at least 2, the quotient space has diameter 0 or at least a universal constant \(\varepsilon > 0\), independent of the dimension. Unitary actions on \(S^{2n-1}\) are isometric, so the spin cases are covered; the authors refer to Gauthier and Rossi for the relevance of the problem to control theory [4].
Status
Solved. Together, the inequality above [1] and the general diameter gap [4] give the conjecture for every finite-dimensional system. We have not found a paper that states this combination as a separate theorem; it is our reading of the two papers and is marked for expert review. If you know a reference that states it, please tell us.