Quantum control without access to the controlling interaction

Abstract

In our model a fixed Hamiltonian acts on the joint Hilbert space of a quantum system and its controller. We show conditions under which measurements, state preparations, and unitary implementations on the system can be performed by quantum operations on the controller only. It turns out that a measurement of the observable $A$ and an implementation of the one-parameter group $\mathrm{exp}(\mathrm{iAr})$ can be performed by almost the same sequence of control operations. Furthermore, measurement procedures for $A+B,$ for $(AB+BA),$ and for $i\ensuremath{\lfloor}A,B\ensuremath{\rfloor}$ can be constructed from measurements of $A$ and $B.$ This shows that the algebraic structure of the set of observables can be explained by the Lie group structure of the unitary evolutions on the joint Hilbert space of the measuring device and the measured system. A spin chain model with nearest-neighborhood coupling shows that the border line between controller and system can be shifted consistently.