South China Normal University utilised min-relative nonlocal magic (NLM) to characterise the crossover from ergodicity to many-body localisation (MBL) in the random-field XXZ chain. Nonlocal magic quantifies the minimum nonstabilizerness attainable under independent local unitary transformations on the two subsystems. Unlike entanglement entropy, NLM probes how entanglement is organised through the distance of the Schmidt spectrum from dyadic-flat stabilizer spectra. From weak to intermediate disorder, NLM evolves from an O Haar-like value into a size-enhanced dome, while entanglement remains volume-law. This reveals a spectral characteristic during the transition between ergodic and localised phases.

Nonlocal magic quantifies spectral changes undetectable by standard entanglement measures

Min-relative nonlocal magic (NLM), quantifying quantum nonstabilizerness, can detect subtle changes in spectral organization missed by standard entanglement measurements according to work and Frontier Research Institute. Mean NLM values decay from approximately W−1 to W−2 with increasing disorder within many-body localized systems; this level of detail was previously unattainable using only von Neumann entropy. This discovery allows for finer characterisation of the transition between ergodic behaviour and many-body localization in disordered spin chains.

The research and Frontier Research Institute found that NLM values initially increase from around an order one value to form a size-enhanced peak as disorder increases, while conventional entanglement measures remained consistent during this spectral change. Within many-body localized systems, mean NLM decays proportionally to W−1 whilst median NLM decreases by W−2, a distinction not detectable through von Neumann entropy alone.

A two-level cut-hybridization model successfully predicted the nearly binary Schmidt spectrum observed deep inside the MBL regime, explaining these specific decay rates for both average and typical nonstabilizerness. Following sudden changes to system parameters, termed product-state quenches, NLM exhibited overshoot followed by relaxation in ergodic conditions but grew logarithmically at high levels of disorder.

Detecting reorganised entanglement arrangements signals phase transitions in localised quantum matter

Long sought after are methods that fully map the breakdown of predictable behaviour within complex quantum systems; understanding how order dissolves into chaos is vital for advancements across materials science and future technologies. Discerning subtle changes proves difficult when relying solely on established metrics like entanglement entropy, a measure quantifying interconnectedness between particles. The work reveals that total entanglement may remain consistent during transitions towards many-body localization, yet the *arrangement* of this entanglement undergoes sharp reorganization detectable by min-relative nonlocal magic.

Quantifying how far quantum states deviate from easily described ones, min-relative nonlocal magic reveals previously hidden details about transitions between ordered and chaotic behaviours. This approach examines *how* entangled particles are arranged within a system, unlike standard methods focusing only on overall entanglement; it detects reorganization even with constant levels of overall entanglement. By pinpointing subtle shifts in arrangement as systems move toward many-body localization, where they resist spreading energy, researchers have identified an additional way to characterise ergodicity breaking beyond conventional measurements.

The research demonstrated that min-relative nonlocal magic successfully resolves Schmidt spectrum structure not visible using entanglement entropy alone. It shows how the organisation of quantum entanglement changes when moving from ergodic behaviour towards many-body localisation, despite total entanglement remaining consistent. Specifically, mean nonlocal magic decayed proportionally to W−1 and median values by W−2 within localised systems; this distinction was undetectable through von Neumann entropy. The authors characterised these transitions via analysis of a two-level cut-hybridization model and product-state quenches, providing a complementary probe for ergodicity breaking.

👉 More information
🗞 Nonlocal Magic across the Many-Body Localization Crossover
✍️ Shan-Zhong Li and Zhi Li
🧠 ArXiv: https://arxiv.org/abs/2609.16935

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