Characterize OSEE beyond one-dimensional local dynamics

Characterize the behavior of operator-space entanglement entropy for non-local observables and for quantum dynamics beyond one spatial dimension, and determine its relation to notions of operator complexity such as operator-space non-stabilizerness and non-gaussianity.

Background

The paper motivates its study by noting that operator-space entanglement entropy (OSEE) is well understood in certain one-dimensional and integrable settings but is substantially less tractable for non-local observables and higher-dimensional dynamics. It also discusses emerging attempts to relate OSEE to measures of operator complexity, including operator-space non-stabilizerness and possible extensions involving non-gaussianity.

The authors identify these issues as unresolved limitations in the broader understanding of OSEE. The present work provides exact results for random fermionic Gaussian dynamics governed by a Brownian SYK2 model, but it does not resolve the behavior of OSEE for non-local observables or dynamics beyond one dimension, nor does it establish a general complexity-theoretic characterization.

References

Despite the importance of the OSEE, several questions remain open. For instance, for non-local observables and dynamics beyond 1D, the behavior of OSEE is generally less clear. In addition, recent work has started to explore the relation between the OSEE and different notions of complexity: notably, Ref. put an upper bound on the OSEE in terms of the quantum resource of non-stabilizerness in operator space.

— Operator-Space Entanglement Entropy in Random Fermionic Gaussian Dynamics  (2610.21723 - Bayona-Pena et al., 15 Oct 2026) in Section 1, Introduction

Our work raises several questions. Prominently, it would be interesting to obtain an alternative derivation of our results using replica calculation techniques. Another direction pertains to the extension of our approach to different quantities, including the out-of-time order correlators (OTOCs). We leave these questions for future work.

— Operator-Space Entanglement Entropy in Random Fermionic Gaussian Dynamics  (2610.21723 - Bayona-Pena et al., 15 Oct 2026) in Section 6, Conclusions