Does a classical speed limit survive quantization?

Statement

Let \(M\) be a connected, complete Riemannian manifold, and consider the controlled Schrödinger equation and its classical counterpart [1]

$$i\,\partial_t\psi = \Big(-\tfrac12\Delta + V(x) + u(t)\,W(x)\Big)\psi, \qquad \psi \in L^2(M),$$

$$H(t,\xi) = \tfrac12\lVert p\rVert^2 + V(x) + u(t)\,W(x), \qquad \xi = (p,x) \in T^*M,$$

with real potentials \(V, W\) such that \(-\tfrac12\Delta + V + uW\) is essentially self-adjoint for every \(u \in \mathbb R\), and piecewise-constant controls \(u\). Let \(\mathcal A^q_T(\psi_0)\) be the set of states reachable from \(\psi_0\) in time at most \(T\), and \(\mathcal A^c_T(\xi_0)\) the same for the classical system. The controllability diameters are

$$T_q = \sup_{\psi_0}\, \inf\Big\{T \ge 0 : \overline{\mathcal A^q_T(\psi_0)} = S(M)\Big\},$$

$$T_c = \inf\Big\{T \ge 0 : \exists\, \Omega \subset T^*M \text{ dense},\ \forall \xi_0 \in \Omega:\ \overline{\mathcal A^c_T(\xi_0)} = T^*M\Big\},$$

where \(S(M)\) is the unit sphere of \(L^2(M)\). A system is small-time controllable when its diameter is zero.

Conjecture (Beschastnyi, Boscain, Sigalotti). For every Hamiltonian of this form, $$T_c > 0 \;\Longrightarrow\; T_q > 0.$$

In words: the quantum effects that come from the non-locality of the Schrödinger equation have a limited effect at small times.

Why it matters

A positive controllability diameter is a speed limit for control: no field, however strong, can perform every task faster. In finite dimension \(n > 1\) there always is one: for \(i\dot\psi = (A + u(t)B)\psi\), the overlap of the state with an eigenvector of \(B\) cannot change faster than \(\lVert A\rVert\) allows, so \(T_q > 0\) [2, 1]. In infinite dimensions the drift is unbounded and this argument fails, so whether a speed limit exists becomes a real question — important whenever a system must be controlled much faster than it decoheres [1]. A semiclassical criterion would let one decide it from classical mechanics.

The conjecture goes in one direction only. The converse is false: the harmonic oscillator driven by an electric field is controllable as a classical system but not approximately controllable as a quantum one; and tunnelling can make a quantum system controllable when its classical counterpart is not [1].

Known results

Quantum speed limits proved directly. For a particle in a controlled electric field, Gaussian wave packets stay localized along classical trajectories for a time that does not depend on the control, which forces \(T_q > 0\) [3, 4].

The conjecture in a class of systems. If \(M = N_1 \times N_2\) and, on an open subset \(\Omega \subset N_1\), the control potential \(W\) does not depend on the \(N_1\) coordinate and the \(N_1\)-gradient of \(V\) is bounded by a Lipschitz function, then \(T_c > 0\) — and the quantum system also has \(T_q > 0\). The simplest case is a \(W\) that is constant on an open set, where the classical dynamics does not feel the control. The quantum proof uses the WKB method to show that tunnelling cannot empty such a region in arbitrarily small time [1].

Systems without a speed limit. Small-time controllability is possible in infinite dimensions. An early example was given for a fractional Schrödinger-type equation [5] (see the discussion in [1]). Small-time control of phases was proved for nonlinear Schrödinger equations on tori [6] and on \(\mathbb R^N\) with electromagnetic controls [7], and small-time control between particular eigenfunctions on closed manifolds, with an application to rotating linear molecules [8]. The first bilinear Schrödinger equations that are globally approximately controllable in arbitrarily small time have two controls, one of them on the frequency of a quadratic potential [9]. A second method, based on controlling flows of diffeomorphisms, covers equations on tori and on \(\mathbb R^d\) [10] and extends to a logarithmic nonlinear Schrödinger equation [11]. These results use several controls, while the conjecture is stated for one.

Status

Partial progress. The conjecture is proved for the class of systems above and open in general. Its authors note that most questions about small-time controllability are open [1], and more recent work calls the characterization of small-time controllable bilinear Schrödinger equations an ambitious open problem [10].