Uncertainty Relations in Implementation of Unitary Operations

The underlying mechanism in the implementation of unitary operation on a system with an external apparatus is studied. We implement the unitary time evolution in the system as a physical phenomenon that results from the interaction between the system and the apparatus. We investigate the fundamental limitation of an accurate implementation for the desired unitary time evolution. This limitation is manifested in the form of trade-off relations between the accuracy of the implementation and quantum fluctuation of energy in the external apparatus. Our relations clearly show that an accurate unitary operation requires a large energy fluctuation inside the apparatus originated from the quantum fluctuation.

Figure 1

Schematic examples of implementation of unitary operation by the experimental apparatus. (a) A qubit system controlled by the electromagnetic field. (b) A heat engine controlled by a moving piston. The picture is a general composite model ($S+E$) for studying the mechanism of unitary time evolution in the system ($S$).

Figure 2

Demonstration of the first uncertainty relation in the Jaynes-Cummings model. The inset shows ${\delta }_U$ as a function of the parameter $\alpha$. Parameters: $\epsilon =10$, $\alpha \lambda \tau =\pi /2$ from which one gets $\parallel [H_S,U_S]\parallel =2\epsilon$. The relation (7) is justified for $\alpha >{\alpha }_c$, where we estimate ${\alpha }_c\sim 500$ from the inset. We show the data for a numerically computable regime of $\alpha$, which is much smaller than ${\alpha }_c$. Nevertheless, the relation (7) is satisfied.