Limitations of Quantum Measurements and Operations of Scattering Type under the Energy Conservation Law

TL;DR

This paper investigates the fundamental limitations imposed by the energy conservation law on quantum measurements and operations, providing bounds on measurement errors and gate fidelities in scattering processes.

Contribution

It extends the Wigner-Araki-Yanase theorem to energy conservation, deriving bounds and conditions for quantum measurement accuracy and gate implementation.

Findings

Lower bound for measurement error under energy conservation.

Conditions for zero-error controlled unitaries with scattering processes.

Relationship between gate fidelity and energy fluctuation in qubit systems.

Abstract

It is important to improve the accuracy of quantum measurements and operations both in engineering and fundamental physics. It is known, however, that the achievable accuracy of measurements and unitary operations are generally limited by conservation laws according to the Wigner-Araki-Yanase theorem (WAY theorem) and its generalizations. Although many researches have extended the WAY theorem quantitatively, most of them, as well as the original WAY theorem, concern only additive conservation laws like the angular momentum conservation law. In this paper, we explore the limitation incurred by the energy conservation law, which is universal but is one of the non-additive conservation laws. We present a lower bound for the error of a quantum measurement using a scattering process satisfying the energy conservation law. We obtain conditions that a control system Hamiltonian must fulfill in order to implement a controlled unitary gate with zero error when a scattering process is considered. We also show the quantitative relationship between the upper bound of the gate fidelity of a controlled unitary gate and the energy fluctuation of systems when a target system and a control system are both one qubit.