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Scalable Certification of Entanglement in Quantum Networks

Jing-Tao Qiu, D. M. Tong, Xiao-Dong Yu

TL;DR

A family of sub-symmetric witnesses (SSWs) is proposed, which are tractable both theoretically and experimentally and can be evaluated via local measurements, with resource requirements independent of the local dimension in general, and even independent of the overall network size in many practical networks.

Abstract

Quantum networks form the backbone of long-distance quantum information processing. Genuine multipartite entanglement (GME) serves as a key indicator of network performance and overall state quality. However, the widely used methods for certifying GME suffer from a major drawback that they either detect only a limited range of states or are applicable only to systems with a small number of parties. To overcome these limitations, we propose a family of sub-symmetric witnesses (SSWs), which are tractable both theoretically and experimentally. Analytically, we establish a connection between SSWs and the cut space of graph theory, enabling several powerful detection criteria tailored to practical quantum networks. Numerically, we show that the optimal detection can be formulated as a linear program, offering a significant efficiency advantage over the semidefinite programs commonly employed in quantum certification. Experimentally, SSWs can be evaluated via local measurements, with resource requirements independent of the local dimension in general, and even independent of the overall network size in many practical networks.

Scalable Certification of Entanglement in Quantum Networks

TL;DR

A family of sub-symmetric witnesses (SSWs) is proposed, which are tractable both theoretically and experimentally and can be evaluated via local measurements, with resource requirements independent of the local dimension in general, and even independent of the overall network size in many practical networks.

Abstract

Quantum networks form the backbone of long-distance quantum information processing. Genuine multipartite entanglement (GME) serves as a key indicator of network performance and overall state quality. However, the widely used methods for certifying GME suffer from a major drawback that they either detect only a limited range of states or are applicable only to systems with a small number of parties. To overcome these limitations, we propose a family of sub-symmetric witnesses (SSWs), which are tractable both theoretically and experimentally. Analytically, we establish a connection between SSWs and the cut space of graph theory, enabling several powerful detection criteria tailored to practical quantum networks. Numerically, we show that the optimal detection can be formulated as a linear program, offering a significant efficiency advantage over the semidefinite programs commonly employed in quantum certification. Experimentally, SSWs can be evaluated via local measurements, with resource requirements independent of the local dimension in general, and even independent of the overall network size in many practical networks.
Paper Structure (11 sections, 1 theorem, 95 equations, 5 figures)

This paper contains 11 sections, 1 theorem, 95 equations, 5 figures.

Table of Contents

  1. GME negativity
  2. Proof of Observation \ref{['obs:lp']}
  3. Network topology and the cut space
  4. $E$-SSWs and $\mathcal{B}(G)$-SSWs
  5. SSWs for permutation-invariant states in cactus networks
  6. SSWs for IPEN states in different networks
  7. Tree networks
  8. Biseparability of IPEN states in the bilocal network
  9. Biseparability of IPEN states in the triangle network
  10. Highly connected networks
  11. Complexity of estimation

Key Result

Lemma 1

For an SSW $W$, if $W \succeq Q_S^{\Gamma_S}$ for some $Q_S \succeq 0$, then there exists $\tilde{Q}_S \succeq 0$ in the form of such that $W \succeq \tilde{Q}_S^{\Gamma_S}$ and $\tilde{Q}_S \succeq 0$. Moreover, the constraint $\tilde{Q}_S\preceq\mathbb{I}$ is also satisfied whenever $Q_S \preceq \mathbb{I}$ holds.

Figures (5)

  • Figure 1: Networks with different topologies. (a)-(c) are tree networks, (d) and (e) are ring networks.
  • Figure 2: Five distinct topologies for a graph with three different edges $\rho_1,\rho_2,\rho_3$.
  • Figure 3: Average visibilities for random graphs obtained using SSWs and fidelity-based entanglement witnesses. The solid lines and dashed lines denote the SSWs and the fidelity-based witnesses, respectively. The visibility $p$ represents the lower bound above which the network state is certified as GME by the corresponding witness. The graph density $D$ denotes the ratio of the number of edges $\abs{E}$ to the maximum possible number of edges, i.e., $D=\abs{E}/\binom{\abs{V}}{2}$.
  • Figure 4: An example of a graph, where the blue walk $u_1e_2u_2e_3u_3e_6u_1$ characterizes a cycle, and the set $\{e_1, e_4\}$ of the orange edges is the cut for the bipartition $u_1u_2u_3|u_4u_5$.
  • Figure 5: Examples of cactus networks. The edges of the same color are equivalent for permutation invariant states.

Theorems & Definitions (7)

  • proof
  • Lemma 1
  • proof
  • proof
  • proof
  • proof
  • proof