Graphical description of local Gaussian operations for continuous-variable weighted graph states

TL;DR

This paper introduces a local Gaussian operation framework for continuous-variable weighted graph states, revealing their stabilizer formalism and graph-theoretic evolution, advancing the understanding of their local equivalence classes.

Contribution

It presents a novel local Gaussian operation for CV weighted graph states and characterizes their evolution and equivalence classes using graph theory and stabilizer formalism.

Findings

Defines local Gaussian operations for CV weighted graph states

Provides a graph rule for state evolution under these operations

Shows the potential to generate infinite LC equivalent states

Abstract

The form of a local Clifford (LC, also called local Gaussian (LG)) operation for the continuous-variable (CV) weighted graph states is presented in this paper, which is the counterpart of the LC operation of local complementation for qubit graph states. The novel property of the CV weighted graph states is shown, which can be expressed by the stabilizer formalism. It is distinctively different from the qubit weighted graph states, which can not be expressed by the stabilizer formalism. The corresponding graph rule, stated in purely graph theoretical terms, is described, which completely characterizes the evolution of CV weighted graph states under this LC operation. This LC operation may be applied repeatedly on a CV weighted graph state, which can generate the infinite LC equivalent graph states of this graph state. This work is an important step to characterize the LC equivalence class of CV weighted graph states.