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Non-symmetric GHZ states: weighted hypergraph and controlled-unitary graph representations

Hrachya Zakaryan, Konstantinos-Rafail Revis, Zahra Raissi

TL;DR

Non-symmetric GHZ states $\ket{n\text{-GHZ}}_\alpha$ arise in experiments and lack stabilizer/graph representations. The authors prove LU equivalence to fully connected weighted hypergraphs and to star-shaped controlled-unitary (CU) graphs, providing a unified graph-theoretic view. They show a single ancilla suffices for stabilizing the hypergraph states and extend the construction to qudits using $X^\alpha$, $CZ_G^{−\alpha}$, and diagonal corrections $CP_e(\vec{\alpha})$. The framework supports efficient stabilization, analysis, and applications to quantum networks and error-correction protocols.

Abstract

Non-symmetric GHZ states ($n$-GHZ$_α$), defined by unequal superpositions of $|00...0>$ and $|11...1>$, naturally emerge in experiments due to decoherence, control errors, and state preparation imperfections. Despite their relevance in quantum communication, relativistic quantum information, and quantum teleportation, these states lack a stabilizer formalism and a graph representation, hindering their theoretical and experimental analysis. We establish a graph-theoretic framework for non-symmetric GHZ states, proving their local unitary (LU) equivalence to two structures: fully connected weighted hypergraphs with controlled-phase interactions and star-shaped controlled-unitary (CU) graphs. While weighted hypergraphs generally lack stabilizer descriptions, we demonstrate that non-symmetric GHZ states can be efficiently stabilized using local operations and a single ancilla, independent of system size. We extend this framework to qudit systems, constructing LU-equivalent weighted qudit hypergraphs and showing that general non-symmetric qudit GHZ states can be described as star-shaped CU graphs. Our results provide a systematic approach to characterizing and stabilizing non-symmetric multipartite entanglement in both qubit and qudit systems, with implications for quantum error correction and networked quantum protocols.

Non-symmetric GHZ states: weighted hypergraph and controlled-unitary graph representations

TL;DR

Non-symmetric GHZ states arise in experiments and lack stabilizer/graph representations. The authors prove LU equivalence to fully connected weighted hypergraphs and to star-shaped controlled-unitary (CU) graphs, providing a unified graph-theoretic view. They show a single ancilla suffices for stabilizing the hypergraph states and extend the construction to qudits using , , and diagonal corrections . The framework supports efficient stabilization, analysis, and applications to quantum networks and error-correction protocols.

Abstract

Non-symmetric GHZ states (-GHZ), defined by unequal superpositions of and , naturally emerge in experiments due to decoherence, control errors, and state preparation imperfections. Despite their relevance in quantum communication, relativistic quantum information, and quantum teleportation, these states lack a stabilizer formalism and a graph representation, hindering their theoretical and experimental analysis. We establish a graph-theoretic framework for non-symmetric GHZ states, proving their local unitary (LU) equivalence to two structures: fully connected weighted hypergraphs with controlled-phase interactions and star-shaped controlled-unitary (CU) graphs. While weighted hypergraphs generally lack stabilizer descriptions, we demonstrate that non-symmetric GHZ states can be efficiently stabilized using local operations and a single ancilla, independent of system size. We extend this framework to qudit systems, constructing LU-equivalent weighted qudit hypergraphs and showing that general non-symmetric qudit GHZ states can be described as star-shaped CU graphs. Our results provide a systematic approach to characterizing and stabilizing non-symmetric multipartite entanglement in both qubit and qudit systems, with implications for quantum error correction and networked quantum protocols.
Paper Structure (7 sections, 4 theorems, 85 equations, 4 figures, 1 table)

This paper contains 7 sections, 4 theorems, 85 equations, 4 figures, 1 table.

Table of Contents

  1. Supplementary Material: [Non-symmetric GHZ states: weighted hypergraph and controlled-unitary graph representations]
  2. Graph Representations of GHZ and Non-Symmetric GHZ States
  3. Proof of Proposition 1
  4. Proof of Proposition 2
  5. Proof of Proposition 3
  6. Qudit weighted hypergraph algorithm
  7. Proof of Proposition 4

Key Result

Proposition 1

For a set of vertices $V$, let $E_k$ denote the set of all $k$ element combinations from $V$. The non-symmetric GHZ state $\ket{n\text{-GHZ}_\alpha}$ is LU equivalent to the following fully connected weighted hypergraph state: where each hyperedge $\textbf{e}$ involving $k+1$ vertices carries a weight of $(-2)^{k}\alpha \pi$. This equivalence is achieved via the local transformation: where

Figures (4)

  • Figure 1: Representation of a weighted hypergraph state. A hyperedge of weight $\varphi_1$ connects three vertices on the left, while a separate hyperedge of weight $\varphi_2$ links four vertices on the right.
  • Figure 2: Controlled-Unitary (CU) Star-Shaped Graph. In this representation, the central vertex serves as the target, while all other vertices act as controls. The non-symmetric GHZ state, Eq. \ref{['eq:CU-GHZ']}, is local unitary (LU) equivalent to this CU graph.
  • Figure 3: LU-equivalent graph representations of GHZ states. (Left) A star-shaped graph where a central qubit is entangled with all others. (Right) A fully connected graph where each qubit is connected to every other qubit.
  • Figure 4: Fully connected weighted hypergraph representation of the five-qubit non-symmetric GHZ state. Each subfigure depicts hyperedges of different orders with corresponding phase weights.

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof