# Local Operations and Completely Positive Maps in Algebraic Quantum Field Theory

## TL;DR

This paper justifies the use of completely positive maps as local operations in algebraic quantum field theory and demonstrates their approximation via Kraus operators under the funnel property.

## Contribution

It provides a theoretical justification for modeling local operations with completely positive maps in algebraic quantum field theory, connecting operational separability with Kraus operator representations.

## Key findings

- Justifies using completely positive maps as local operations.
- Shows local operations can be approximated with Kraus operators.
- Connects operational separability with mathematical representations.

## Abstract

Einstein introduced the locality principle which states that all physical effect in some finite space-time region does not influence its space-like separated finite region. Recently, in algebraic quantum field theory, R\'{e}dei captured the idea of the locality principle by the notion of operational separability. The operation in operational separability is performed in some finite space-time region, and leaves unchanged the state in its space-like separated finite space-time region. This operation is defined with a completely positive map. In the present paper, we justify using a completely positive map as a local operation in algebraic quantum field theory, and show that this local operation can be approximately written with Kraus operators under the funnel property.