Multipartite Non-local Magic and SYK Model

TL;DR

This paper introduces a multipartite non-local magic measure to analyze quantum non-stabilizerness in disordered fermionic systems, revealing complex global correlations and differences between microstates and thermal states, especially in SYK models.

Contribution

It develops a new multipartite non-local magic functional to diagnose global quantum magic and applies it to SYK models, uncovering hidden complexities in black hole microstates.

Findings

Thermal pure states differ from thermal density matrices in magic complexity.

Multipartite measure reveals sector-dependent correlations in supersymmetric SYK.

Microstructure of black hole states encodes computational hardness.

Abstract

We investigate the structure of quantum magic in interacting disordered fermionic systems, quantifying non-stabilizerness via the fermionic stabilizer R\'enyi entropy (SRE). To resolve the distribution of magic across different scales, we introduce a multipartite non-local magic functional, constructed from an inclusion-exclusion combination of subsystem contributions. This measure serves as a fine-grained diagnostic, isolating genuinely global contributions and revealing nontrivial interactions between local and collective supports of magic. We illustrate the measure on paradigmatic multipartite states and apply these diagnostics to the Sachdev-Ye-Kitaev model and its variants. Crucially, for thermal/typical ensembles, we observe a marked disparity between Thermal Pure Quantum (TPQ) states and the thermal density matrix. This reveals a concealed complexity: the immense computational hardness characterizing the unitary evolution is encoded in the specific microstructure of the black hole microstates, while being washed out in the coarse-grained thermodynamic description. Furthermore, in $\mathcal N=2$ supersymmetric SYK, we show that while fortuitous BPS states exhibit intermediate stabilizer complexity, the multipartite measure unveils a rich, sector-dependent pattern of global correlations, distinguishing them from generic chaotic states.