Block entanglement bounds distribution of regionally localized entanglement

TL;DR

This paper introduces the concept of regionally localized entanglement in multi-qubit systems and establishes bounds on it using localizable block entanglement, with implications for quantum network robustness and noise resilience.

Contribution

The paper defines regionally localized entanglement and proves bounds for it based on localizable block entanglement, including numerical validation under noise channels.

Findings

Bounds hold for permutation-symmetric states and general pure states.

Bounds remain valid under local phase-flip channels for large systems.

Distinct bounds observed for states with specific magnetization sectors.

Abstract

In quantum networks, eliminating connections between nodes is crucial to mitigate the effects of decoherence, often achieved by performing measurements on nodes that are idle, or vulnerable to noise. To characterize the entanglement content of the resulting smaller network, we introduce the notion of ``regionally localized entanglement", defined as the average entanglement concentrated over a two-qubit region in a multi-qubit system. Hence, the total regionally localized entanglement can be obtained by considering all two-qubit regions sharing a common qubit, referred to as the ``hub". We prove that the total regionally localized entanglement corresponding to a specific hub is bounded above and below via the localizable block entanglement shared between the hub and the rest of the multi-qubit system for a number of paradigmatic pure quantum states, including permutation-symmetric states and arbitrary superposition of states from a specific magnetization sector. Numerical simulations confirm that the bounds for permutation-symmetric pure states remain valid even for Haar-uniformly generated pure states, and when each of the qubits is sent through local phase-flip channels of Markovian and non-Markovian types, except when the system-size is small. On the other hand, arbitrary states from a particular magnetization sector yield bounds on total regionally localized entanglement that are distinct from the permutation-symmetric states, highlighting the structurally unique entanglement properties of the former.