Localizing multipartite entanglement with local and global measurements

TL;DR

This work develops a unified framework to localize multipartite entanglement onto a subsystem via measurements on the complement, introducing MEA and LME relative to seed measures such as the $n$-tangle $\tau_n$, GME-concurrence $C_{\mathrm{GME}}$, and concentratable entanglement $C(|\psi\rangle;s)$. The authors derive easily computable upper and lower bounds, prove Lipschitz-continuity, and establish measure-concentration results for Haar-random states, enabling insight into typical localization behavior without heavy optimization. They apply the framework to graph states, providing a polynomial-time matrix-equation criterion for feasible transformations and extending to weighted graphs to analyze GHZ extraction protocols; they also connect the localization framework to phase transitions in the spin-half TFIM. Together, these results offer practical benchmarks for entanglement-localization protocols, reveal when local measurements suffice, and show how entanglement localization can illuminate critical phenomena in many-body systems. The work thus advances both foundational understanding and operational tools for distributing and transforming multipartite entanglement in quantum information tasks.

Abstract

We study the task of localizing multipartite entanglement in pure quantum states onto a subsystem by measuring the remaining systems. To this end, we fix a multipartite entanglement measure and consider two quantities: the multipartite entanglement of assistance (MEA), defined as the entanglement measure averaged over the post-measurement states and maximized over arbitrary measurements; and the localizable multipartite entanglement (LME), defined in the same way but restricted to only local single-system measurements. We choose the n-tangle, the genuine multipartite entanglement concurrence and the concentratable entanglement (CE) as the underlying seed measure, and discuss the resulting MEA and LME quantities. First, we prove easily computable upper and lower bounds on MEA and LME and establish Lipschitz-continuity for the n-tangle and CE-based LME and MEA. Using these bounds we investigate the typical behavior of entanglement localization by deriving concentration inequalities for the MEA evaluated on Haar-random states and performing numerical studies for small tractable system sizes. We then turn our attention to protocols that transform graph states. We give a simple criterion based on a matrix equation to decide whether states with a specified n-tangle value can be obtained from a given graph state, providing no-go theorems for a broad class of such graph state transformations beyond the usual local Clifford plus local Pauli measurement framework. We generalize this analysis to weighted graph states and show that our entanglement localization framework certifies the near-optimality of recently discussed local-measurement protocols to transform uniformly weighted line graph states into GHZ states. Finally, we demonstrate how our MEA and LEA quantities can be used to detect phase transitions in transversal field Ising models.

Figures (11)

• Figure 1: Each red point indicates the localizable entanglement $\mathcal{L}^\tau$ averaged across $10^4$ numerically sampled Haar-random states for $N_A = 2,3$ and across $10^3$ states for $N_A = 4,5,6$, all with $N_B = 4$. The blue points indicate the entanglement of assistance $\mathcal{A}^\tau$ averaged across the same samples. Error bars represent one standard deviation for each sample.
• Figure 2: We compare values of the LME to the values of our upper bounds from Section \ref{['sec:bounds']} for Haar-random states. Each color corresponds to a sample size of 2500 states. The dashed black lines are plots of the line $y=x$. Tightness of the bounds from Theorems \ref{['Th:n-tangle-upper-bound']}, \ref{['Th:cgme-upper-bound']}, and \ref{['Th:CE-upper-bound']} is judged by proximity of the sampled data to this line. (a) Data obtained for the LME defined by the $n$-tangle with $N_B = 4$ held fixed. Values of $\mathcal{A}^\tau(|\Psi\rangle)$ were obtained by using the fact that $\mathcal{A}^\tau(|\Psi\rangle) = F(\Psi_B, \tilde{\Psi}_B)$ from Theorem \ref{['Th:n-tangle-upper-bound']}. (b) A visualization of the LME defined by the GME-concurrence with $N_B = 3$ held fixed. (c) Values of the LME defined by the CE for varying sizes of $s$ and $N_A = 1, N_B = 3$ held constant. Colored dashed lines represent simple upper bounds on $\mathcal{L}^{\mathrm{CE}}(|\Psi\rangle)$ obtained by assuming that every state in the post-measurement ensemble is absolutely maximally entangled - that is, each marginal obtained by tracing out at least half of the subsystems in $B$ is maximally mixed. Note that these bounds coincide for $|s| = 2$ and $|s| = 3$, as shown by the overlapping orange and blue lines. (d) The situation is essentially the same as in (c), with the only difference being that $N_A = 2$ is held fixed.
• Figure 3: Two graphs on six vertices divided into qubits $A$ (red) and $B$ (blue).
• Figure 4: A graph on eight vertices divided into qubits $A$ (red) and $B$ (blue).
• Figure 5: An example where AME state extraction via LC + LPM is not feasible. We consider the subsystem $A$ to be some subset of the unlabeled qubits of the cluster state (above).

Theorems & Definitions (44)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6
• Corollary 7
• Theorem 8
• Theorem 9
• Lemma 10