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arXiv:2510.26696

Witnessing Short- and Long-Range Nonstabilizerness via the Information Lattice

Yuliya Bilinskaya, Miguel F. Martínez, Soumi Ghosh, Thomas Klein Kvorning, Claudia Artiaco, Jens H. Bardarson

quant-phcond-mat.stat-mechcond-mat.str-elSubmitted Oct 30, 2025• Announced Oct 31, 2025• Revised Oct 30, 2026 (v2)
v1 · Oct 30, 2025v2 · Oct 30, 2026
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Abstract

We study nonstabilizerness on the information lattice and demonstrate that noninteger local information directly indicates nonstabilizerness. For states with a clear separation of short- and large-scale information, a noninteger total information at large scales $Γ$ serves as a witness of long-range nonstabilizerness. We propose a folding procedure to separate the global and edge-to-edge contributions to $Γ$. As an example we show that the ferromagnetic ground state of the spin-1/2 three-state Potts model has long-range nonstabilizerness originating from global correlations, while the paramagnetic ground state has at most short-range nonstabilizerness.

Comments
9 pages, 3 figures, 1 end matter

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We study nonstabilizerness on the information lattice and demonstrate that noninteger local information directly indicates nonstabilizerness. For states with a clear separation of short- and large-scale information, a noninteger total information at large scales ΓΓΓ serves as a witness of long-range nonstabilizerness. We propose a folding procedure to separate the global and edge-to-edge contributions to ΓΓΓ. As an example we show that the ferromagnetic ground state of the spin-1/2 three-state Potts model has long-range nonstabilizerness originating from global correlations, while the paramagnetic ground state has at most short-range nonstabilizerness.
We study long-range nonstabilizerness (LRN), namely the obstruction to remove nonstabilizerness with shallow-depth local quantum circuits. In one-dimensional settings, the mutual information between disconnected spatial regions has proven to be a powerful tool to diagnose LRN. In this work, we focus on encoded states of two-dimensional topologically-ordered systems, and explore the ability of the mutual information to serve as a diagnostic of LRN. Focusing on the concrete setting of lattice models defined on a torus, we show that information about LRN can be gained from the analysis of the mutual information between non-overlapping regions containing non-contractible loops, and of the change of such mutual information under modular real-space transformations. We exemplify this idea in the toric code and the non-abelian string-net model with doubled Fibonacci topological order. In the former case, we show that the mutual information provides a full classification, certifying LRN for all encoded non-stabilizer states. In the latter case, instead, our approach does not lead to a full classification, as it detects LRN for all states except from a finite subset with special transformation properties under the modular group. Finally, we discuss how our results on LRN constrain the logical gates that can be implemented fault-tolerantly on the torus.
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Developing a local description of quantum information is a central problem in quantum many-body physics: the goal is to identify how correlations are organized across positions and scales without choosing specific observables. We introduce a higher-dimensional information lattice, an inclusion-exclusion construction that decomposes the von Neumann information of reduced density matrices into contributions labeled by subsystem position and scale. Above one dimension, overlapping subsystems can form loops, allowing information associated with one subsystem to be inferred nonlocally from the density matrices of others. The sign of the inclusion-exclusion information resolves such overlap redundancies in position and scale: positive values mark positions and scales with net information gain, while negative values occur where redundant contributions dominate new ones. We implement the construction in two dimensions and apply it to many-body ground states with distinct entanglement structures. For localized states, such as disordered-localized states or gapped ground states of local Hamiltonians, the local information decays exponentially along different directions, defining direction-dependent information length scales. In critical states with a Fermi surface, decay becomes algebraic, with dominant directions tied to the velocity of gapless modes. In the toric code, topological order manifests as a negative information value at the largest lattice scale, interpreted as global information redundancy between the smallest loop-enclosing overlapping subsystems. We also isolate the universal information carried by edge states and locate the signatures of non-Abelian fusion channels in position and scale. Our results establish the higher-dimensional information lattice as a framework for resolving universal correlation structures through the spatial and scale distribution of quantum information.
The classification of mixed-state phases requires criteria beyond two-point correlation functions, such as the decay of the mutual information (MI) and the conditional mutual information (CMI), with the latter encapsulated in the notion of Markov length. Here we show how such static properties of the fixed point of a Lindbladian follow from natural dynamical features of its generators: rapid mixing and frustration-freeness. We focus on systems with long-range interactions, and prove (i) that local Lindbladians satisfying (global) rapid mixing and frustration-freeness have fixed-points whose CMI decays with the shielding distance, and (ii) that (local) rapid mixing together with primitivity and regularity implies global decay of MI. For long-range interactions decaying with a power law with rate ααα, both quantities decay polynomially rather than exponentially, in contrast to the finite- and short-range regimes where exponential decay (a finite Markov length) is expected within a phase. We further show that Gibbs states of long-range, non-commuting Hamiltonians satisfy a local Markov property at any temperatures, extending the recent results (Chen--Rouzé, 2025) for short-range systems to the long-range regime relevant to a variety of experimental platforms. As a numerical example, we study the long-range Ising model both with and without a transverse field. We find regimes in which the polynomial decay of the CMI holds, in accordance with the bounds proven.
Quantum entanglement and quantum nonstabilizerness are fundamental resources that characterize distinct aspects of a quantum state: Entanglement reflects nonlocal correlations, while nonstabilizerness quantifies the deviation from stabilizer states. A quantum state becomes a valuable resource for applications like universal quantum computation only when both quantities are present. Here, we propose that quantum nonlocal nonstabilizerness (NN) serves as an effective measure of this combined resource, incorporating both entanglement and nonstabilizerness. We demonstrate that NN can be precisely computed for two-qubit pure states, establishing a direct relationship with the entanglement spectrum. We then extend the definition of NN to mixed states and explore its presence in many-body quantum systems. We reveal that the two-point NN decays according to a power law in critical states, indicating that it can serve as an effective quantum information probe for phase transitions. Furthermore, we explore measurement-induced NN and uncover an intriguing phenomenon termed ``nonstabilizerness swapping,'' wherein post-measurement NN decays more slowly than any pre-measurement correlations. The occurrence of this phenomenon provides a sufficient condition for both entanglement swapping and the presence of a nontrivial sign structure. Our results thus represent a pivotal step toward accurately quantifying the ``quantumness'' of a state and bring insights from multiple perspectives to the property of NN.
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