⚛️ quantum physics

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

This paper demonstrates that noninteger local and total information on the information lattice serve as direct witnesses for short- and long-range nonstabilizerness, respectively, and applies this framework to show that the ferromagnetic ground state of the spin-1/2 three-state Potts model exhibits long-range nonstabilizerness driven by global correlations, unlike its paramagnetic counterpart.

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

Published 2026-11-02

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CC BY 4.0

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

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, the way particles are connected to one another is the key to understanding how matter behaves. Physicists have long known that some quantum states are relatively simple to describe, while others are so tangled that they seem to require infinite detail to capture. A major tool for sorting these states has been the study of entropy, a measure of how much information is hidden in the connections between different parts of a system. For many years, scientists focused on whether these connections were local, affecting only nearby neighbors, or global, stretching across the entire system. However, a newer and more subtle property has emerged as a critical factor: the complexity of the quantum state itself. This complexity, often called "nonstabilizerness," measures how far a state is from a class of simple, easily simulated configurations. If a state is too complex, it cannot be efficiently modeled by classical computers, making the detection of this complexity vital for understanding the limits of simulation and the nature of quantum matter.

A team of researchers at the KTH Royal Institute of Technology in Stockholm has now developed a new way to spot this complexity, distinguishing between connections that are merely local and those that stretch across the entire system. They utilized a conceptual map known as the information lattice, which organizes a quantum system by the size of the chunks of information it contains. In this framework, the researchers discovered a simple rule: if the information contained in a specific region of the map is not a whole number, the system possesses a form of quantum complexity that cannot be easily removed. This finding allows them to identify "nonstabilizerness" directly, without needing to perform difficult calculations or search for specific operations that might simplify the state.

The researchers focused their attention on a specific type of quantum system called a localized state, where information is concentrated at very small scales and very large scales, with a noticeable gap in between. In such systems, the total amount of information found at the largest scales acts as a witness. If this total value is a whole number, the system's complexity is likely confined to short distances. However, if the value is a fraction, it proves that the system contains long-range complexity that cannot be erased by simple, local adjustments. To understand where this complexity comes from, the team introduced a "folding" procedure. Imagine taking a long chain of quantum particles and folding it in half, pairing the ends together. This process turns connections between the far ends of the original chain into local connections in the new, shorter chain. By comparing the information before and after this folding, the researchers could separate out the complexity that comes from global correlations—those that link the entire system together—from those that simply link the edges.

To test their method, the team applied it to the ground state of a spin-1/2 three-state Potts model, a theoretical system that can exist in different phases depending on the strength of a magnetic field. In the paramagnetic phase, where the magnetic field is strong, the system behaves simply. The researchers found that the total information at large scales was zero, indicating that any complexity present was limited to short distances. In contrast, in the ferromagnetic phase, where the magnetic field is weak, the system exhibited a different behavior. The total information at large scales was approximately 1.58, a non-integer value that served as a clear signature of long-range complexity. This fractional value arose because the ground state of the system is threefold degenerate, meaning there are three distinct, equally stable configurations. The information required to describe the relative phases between these three states creates a global complexity that persists across the entire system.

The study confirms that the information lattice provides a direct and efficient way to identify these deep quantum connections. Unlike previous methods that required searching through countless possible operations to see if a state could be simplified, this approach looks at the state's inherent structure. The researchers demonstrated that for localized states, a non-integer value at large scales is a definitive sign that the system cannot be transformed into a simple state by any shallow, local circuit. This distinction is crucial because it separates systems that are merely complex in a local sense from those that possess a fundamental, global complexity. The findings suggest that this method can be applied to experimental data, as the necessary information can be reconstructed from measurements of the quantum state. By mapping out where complexity lives, from the smallest scales to the largest, this work offers a clearer picture of the intricate architecture of quantum matter.

Technical Summary: Witnessing Short- and Long-Range Nonstabilizerness via the Information Lattice

Problem Statement
Quantum information theory seeks to classify quantum states beyond traditional correlation measures like von Neumann entropy. While area-law and volume-law classifications describe correlation scaling, they do not fully capture "nonstabilizerness" (or "magic"), a resource quantifying the distance of a state from stabilizer states. Stabilizer states, which admit efficient classical representation and are closed under Clifford operations, serve as a baseline; generic condensed matter states require non-Clifford unitaries (e.g., T-gates) to be reached. A critical challenge in the field is distinguishing between short-range nonstabilizerness (removable by shallow local unitary circuits) and long-range nonstabilizerness (requiring circuits scaling with system size). Existing monotones often measure the total amount of nonstabilizerness but fail to decompose it by scale or locality without complex minimization procedures over all possible local circuits.

Methodology
The authors utilize the information lattice, a hierarchical structure that decomposes a quantum state's total information into local information $i^\ell_n$ at specific spatial locations $n$ and scales $\ell$ .

Local Information Definition: For a subsystem $C^\ell_n$ of $\ell+1$ contiguous sites, the local information $i^\ell_n$ is defined as the mutual information of its two largest sub-subsystems conditioned on their overlap. Formally, $i^\ell_n = I(\rho^\ell_n) - I(\rho^{\ell-1}_{n-1/2}) - I(\rho^{\ell-1}_{n+1/2}) + I(\rho^{\ell-2}_n)$ .
Stabilizer Characterization: The authors establish that for any stabilizer state, the local information $i^\ell_n$ is strictly an integer. This arises because stabilizer states possess a specific entanglement structure where information is encoded in independent Pauli generators, yielding binary measurement outcomes.
Witnessing Nonstabilizerness: The paper posits that any non-integer value of $i^\ell_n$ is a sufficient condition for nonstabilizerness.
Distinguishing Range: For "localized" states (those with an extensive amount of information at small scales and an "information gap" at intermediate scales), the authors define the total information at large scales, $\Gamma = \sum_{\ell=\lfloor L/2 \rfloor}^{L-1} I_\ell$ $Γ = \sum_{ℓ = ⌊ L /2 ⌋}^{L - 1} I_{ℓ}$ .
- Since local unitary circuits cannot redistribute information across an extensive information gap, $\Gamma$ is invariant under shallow circuits.
- Stabilizer states always have integer $\Gamma$ . Therefore, a non-integer $\Gamma$ in a localized state serves as a witness for long-range nonstabilizerness.
Folding Procedure: To distinguish whether long-range nonstabilizerness originates from global correlations or edge-to-edge correlations, the authors propose a "folding" procedure. This combines sites $n$ and $L-n-1$ , effectively halving the chain length while doubling the local Hilbert space dimension. This procedure renders edge-to-edge correlations local, allowing the remaining large-scale information ( $\Gamma_{\text{folded}}$ ) to estimate the contribution from global correlations ( $\gamma_{\text{global}}$ ).

Key Results

Integer vs. Non-Integer Information: The authors demonstrate that non-integer local information is a direct indicator of nonstabilizerness. They show examples where stabilizer states (e.g., Bell pairs, GHZ states, and random Clifford circuits) exhibit strictly integer local information, while non-stabilizer states (e.g., T-doped Clifford circuits) exhibit non-integer values.
Long-Range Witness: In localized states, a non-integer $\Gamma$ proves the presence of long-range nonstabilizerness. The paper shows that a localized state with $\Gamma = 0$ possesses at most short-range nonstabilizerness.
Application to the Potts Model: The method is applied to the spin-1/2 three-state Potts model:
- Ferromagnetic Phase ( $h \ll 1/3$ ): The symmetric ground state exhibits an information gap and a non-integer $\Gamma \approx \log_2(3)$ . The folding procedure confirms $\Gamma_{\text{folded}} \approx \Gamma$ , indicating the nonstabilizerness originates from global correlations associated with the threefold degeneracy of the ground state.
- Paramagnetic Phase ( $h \gg 1/3$ ): The ground state has $\Gamma \approx 0$ , indicating at most short-range nonstabilizerness.
- Critical Point: Near the phase transition ( $h \approx 1/3$ ), the information gap closes, $\Gamma$ scales with system size, and the witness is no longer applicable as the state is no longer localized.

Significance and Claims
The paper claims to provide a direct and computationally efficient method for witnessing nonstabilizerness without the need for minimization procedures over local unitary circuits (which are computationally expensive).

State Property: The approach relies solely on the properties of the quantum state (via the information lattice) and is independent of the parent Hamiltonian or the protocol used to generate the state.
Efficiency: For localized states, calculating $\Gamma$ is computationally less expensive than brute-force minimization of nonstabilizerness monotones, as it avoids diagonalizing density matrices at all intermediate scales.
Experimental Accessibility: The authors note that since the information lattice relies on the density matrix, these quantities are potentially accessible via quantum state tomography.
Scope: The work focuses on one-dimensional systems but suggests the framework can be extended to higher dimensions via quasi-1D coarse-graining or higher-dimensional local decompositions, and potentially to mixed-state phases.

The authors maintain a modest tone, noting that while non-integer local information is a sufficient condition for nonstabilizerness, it is not a necessary one (citing a fine-tuned example of a non-stabilizer state with integer local information). The primary contribution is the ability to distinguish the range of nonstabilizerness in localized phases through the invariant $\Gamma$ .

Technical Summary: Witnessing Short- and Long-Range Nonstabilizerness via the Information Lattice

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