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Entanglement in Directed Graph States

Lucio De Simone, Roberto Franzosi

TL;DR

This work addresses how entanglement distributes in multi-qubit states associated with directed graphs by introducing the Entanglement Distance (ED) derived from the Fubini-Study metric. It analyzes ordering-free graph states with uniform edge unitaries, deriving the ED per qubit as $E(\theta;\{d(i)\})=1-\dfrac{1}{M}\sum_{i\in V} [\cos(\theta)]^{2d(i)}$, where $d(i)$ is the total vertex degree $d(i)=d_\rightarrow(i)+d_\leftarrow(i)$. The key finding is that ED depends solely on the vertex degree distribution, making entanglement topology-driven and invariant under vertex relabeling and edge orientation. This provides a geometric interpretation of quantum correlations in networks and suggests design principles for topology-aware quantum information processing and communication systems.

Abstract

We investigate a family of quantum states defined by directed graphs, where the oriented edges represent interactions between ordered qubits. As a measure of entanglement, we adopt the Entanglement Distance - a quantity derived from the Fubini - Study metric on the system's projective Hilbert space. We demonstrate that this measure is entirely determined by the vertex degree distribution and remains invariant under vertex relabeling, underscoring its topological nature. Consequently, the entanglement depends solely on the total degree of each vertex, making it insensitive to the distinction between incoming and outgoing edges. These findings offer a geometric interpretation of quantum correlations and entanglement in complex systems, with promising implications for the design and analysis of quantum networks.

Entanglement in Directed Graph States

TL;DR

This work addresses how entanglement distributes in multi-qubit states associated with directed graphs by introducing the Entanglement Distance (ED) derived from the Fubini-Study metric. It analyzes ordering-free graph states with uniform edge unitaries, deriving the ED per qubit as , where is the total vertex degree . The key finding is that ED depends solely on the vertex degree distribution, making entanglement topology-driven and invariant under vertex relabeling and edge orientation. This provides a geometric interpretation of quantum correlations in networks and suggests design principles for topology-aware quantum information processing and communication systems.

Abstract

We investigate a family of quantum states defined by directed graphs, where the oriented edges represent interactions between ordered qubits. As a measure of entanglement, we adopt the Entanglement Distance - a quantity derived from the Fubini - Study metric on the system's projective Hilbert space. We demonstrate that this measure is entirely determined by the vertex degree distribution and remains invariant under vertex relabeling, underscoring its topological nature. Consequently, the entanglement depends solely on the total degree of each vertex, making it insensitive to the distinction between incoming and outgoing edges. These findings offer a geometric interpretation of quantum correlations and entanglement in complex systems, with promising implications for the design and analysis of quantum networks.
Paper Structure (4 sections, 1 theorem, 23 equations, 3 figures)

This paper contains 4 sections, 1 theorem, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Entanglement in General Graph Configurations
  3. Conclusion
  4. Acknowledgements

Key Result

Theorem 1

Let $\ket{G}$ be the graph state associated with the ordering-free graph $G(V,L)$. Let the unitary operator $U_{ab}$ be given in a controlled-$\bar{U}$ form as in Eq. oper. Then, the Entanglement Distance per qubit ent_dist is given by where $E(\theta;\{d(i)\}):=E(\ket{G})$ and $d(i)$ is the degree of the $i$-th qubit.

Figures (3)

  • Figure 1: This is an example with $\Gamma_\rightarrow(i)=\{j_1,\ldots,j_{d_{\rightarrow}(i)}\}$, $\Gamma_\leftarrow(i)=\emptyset$, and $U_{tot}=\prod^{d_{\rightarrow}(i)}_{k=1} {U}_{ij_k}$.
  • Figure 2: This is an example with $\Gamma_{\leftarrow}(i)=\{j_1,\ldots,j_{d_{\leftarrow}(i)}\}$ and $U_{tot}=\prod^{d_{\leftarrow}(i)}_{k=1} {U}_{j_k i}$.
  • Figure 3: This is an example with $\Gamma_{\rightarrow}(i)=\{j_1,\ldots,j_{d_{\rightarrow}(i)}\}$, $\Gamma_{\leftarrow}(i)=\{m_1,\ldots,m_{d_{\leftarrow}(i)}\}$ and $U_{tot}=\prod^{d_{\rightarrow}(i)}_{k=1} {U}_{ij_k}\prod^{d_{\leftarrow}(i)}_{p=1}{U}_{m_p i}$.

Theorems & Definitions (7)

  • Remark 1
  • Definition 1: Graph State
  • Remark 2
  • Definition 2: Ordering-free graph state
  • Theorem 1
  • Remark 3
  • Proof 1