Graphical rule of transforming continuous-variable graph states by local homodyne detection

TL;DR

This paper presents a graphical rule for transforming continuous-variable graph states via local homodyne detection, enabling easier design and construction of such states through simple graphical operations.

Contribution

It introduces a unified graphical rule for any single-mode quadrature measurement on CV graph states, simplifying their manipulation and design.

Findings

Graphical rule for CV graph state transformation established

Single-mode quadrature measurement corresponds to local complement and vertex deletion

Facilitates easy design of CV weighted graph states from larger states

Abstract

Graphical rule, describing that any single-mode homodyne detection turns a given continuous-variable (CV) graph state into a new one, is presented. Employing two simple graphical rules: local complement operation and vertex deletion (single quadrature-amplitude $\hat{x}$ measurement), the graphical rule for any single-mode quadrature component measurement can be obtained. The shape of CV weighted graph state may be designed and constructed easily from a given larger graph state by applying this graphical rule.