Uncertainty Relations in Implementation of Unitary Control*

We study the underlying mechanism in the implementation of unitary control on a system with an experimental apparatus. We regard the unitary time evolution in the system as a physical phenomenon that results from the interaction between the system and the external system. We consider the conditions required to approximate the dynamics of the reduced density matrix of the system by the desired unitary time evolution U_S. Then, we derive fundamental trade-off relations to implement the unitary dynamics: δ_Uδ_E≧‖[U_S,H_S]‖/40. Here, H_S is the Hamiltonian of the system, δ_U is the implementation error of the desired unitary U_S, and δ_E is the energy fluctuation of the external system. We also show that the energy fluctuation δ_E should be caused by a quantum superposition. Namely, precise unitary control in the system requires large quantum fluctuation of energy in the external system.
Because of the generality of our results, we can apply them to many objects, e.g., quantum heat engines. Our uncertainty relation says that when the dynamics of the heat engine and baths becomes close to unitary, the energy fluctuation of the work storage diverges.