Uncertainty relations in implementation of unitary operations

TL;DR

This paper explores the fundamental quantum limitations on implementing precise unitary operations, revealing a trade-off between accuracy and energy fluctuation in the external apparatus involved.

Contribution

It introduces uncertainty relations that quantify the trade-off between implementation accuracy and energy fluctuation during unitary operations.

Findings

Accurate unitary operations require large energy fluctuations.

Quantum fluctuations impose fundamental limits on implementation precision.

Trade-offs are characterized by new uncertainty relations.

Abstract

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 a good 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 large energy fluctuation inside the apparatus originated from quantum fluctuation.