Localizing entanglement in high-dimensional states

TL;DR

The paper investigates how entanglement can be localized in large multipartite quantum states using $n$-tangle-based LE and EA, examining both graph states and Haar-random states. A polynomial-time test links EA to a linear matrix equation, enabling efficient screening of candidate source states for transforming into maximally entangled targets like GHZ states. The authors derive concentration inequalities showing a sharp transition: when the measured fraction is below 1/2, LE and EA concentrate near zero, and when the measured fraction exceeds 1/2, EA concentrates near one while LE remains near zero, evidencing a fundamental advantage of global measurements in large systems. They also prove that, for linear cluster states, broader operation classes (LU+GM+CC) do not asymptotically outperform the standard LC+LPM+CC approaches for GHZ extraction, strengthening existing results. Methodologically, they introduce $\oldsymbol\varepsilon$-nets for bases and states to enable these concentration bounds, with potential broad applicability to other entanglement measures and localization tasks.

Abstract

In this work, we study the asymptotic behavior of protocols that localize entanglement in large multi-qubit states onto a subset of qubits by measuring the remaining qubits. We use the maximal average n-tangle that can be generated on a fixed subsystem by measuring its complement -- either with local or global measurements -- as our key figure of merit. These quantities are known respectively as the localizable entanglement (LE) and the entanglement of assistance (EA). We build upon the work of [arXiv:2411.04080] that proposed a polynomial-time test, based on the EA, for whether it is possible to transform certain graph states into others using local measurements. We show, using properties of the EA, that this test is effective and useful in large systems for a wide range of sizes of the measured subsystem. In particular, we use this test to demonstrate the surprising result that general local unitaries and global measurements will typically not provide an advantage over the more experimentally feasible local Clifford unitaries and local Pauli measurements in transforming large linear cluster states into GHZ states. Finally, we derive concentration inequalities for the LE and EA over Haar-random states which indicate that the localized entanglement structure has a striking dependence on the locality of the measurement. In deriving these concentration inequalities, we develop several technical tools that may be of independent interest.

Figures (6)

• Figure 1: Table of notation used throughout the text.
• Figure 2: Different graph states can be equivalent for the purpose of entanglement localization when their underlying graphs are related by a graph isomorphism that preserves $A$ and $B$. In this figure, $A = \{1\}$ and $B = \{2, 3\}$.
• Figure 3: Probability $p_s({\mathcal{E}_{\mathrm{unif}}})$ of finding a solution for the uniformly weighted ensemble against $N_A$ in graphs with $N = 10, 15, 20$ and $25$ vertices. Data points along solid lines represent numerical estimates of $p_s({\mathcal{E}_{\mathrm{unif}}})$ obtained from samples of 1000 random graphs drawn from ${\mathcal{E}_{\mathrm{unif}}}$. The dashed lines indicate our approximation from equation \ref{['eq:pr-sol-approx']}.
• Figure 4: (a) Probability $p_s({\mathcal{E}_{\mathrm{unif}}})$ of finding a solution for the uniformly weighted ensemble against $N_A$ in graphs with $N = 10, 15, 20$ and $25$ vertices. Data points along solid lines represent numerical estimates of $p_s({\mathcal{E}_{\mathrm{unif}}})$ obtained from samples of 1000 random graphs drawn from ${\mathcal{E}_{\mathrm{unif}}}$. The dashed lines indicate our approximation from equation \ref{['eq:pr-sol-approx']}. (b) Probability of finding a solution against $N_A$ for graphs drawn from a uniform distributions over graph families with various structural properties. Here, $N = 16$. The dashed line is the approximation we calculate in \ref{['eq:pr-sol-approx']}. Note that the approximation is almost identical to the probability of getting a solution for a $4$-regular graph.
• Figure 5: Typical values of ${L^\tau_{\mathrm{global}}}$ and ${L^\tau_{\mathrm{local}}}$ for the two possible parameter regimes in the framework of localizable entanglement.

Theorems & Definitions (42)

• Theorem 1: Theorem 11, Vairogs2024
• Theorem 2
• Corollary 3
• Theorem 4
• Theorem 5
• Corollary 6
• Definition 1
• Definition 2
• Theorem 7
• Theorem 8