Spectral signatures of nonstabilizerness and criticality in infinite matrix product states

1. Problem Statement

Nonstabilizerness (often called "magic") is a crucial resource for universal quantum computation, distinguishing quantum states that can be efficiently simulated classically (stabilizer states) from those that cannot. While entanglement is well-understood as a probe for quantum criticality and phase transitions, the behavior of nonstabilizerness in many-body systems remains poorly characterized.

Key open questions addressed in this work include:

• Does nonstabilizerness exhibit universal scaling properties near critical points similar to entanglement entropy?
• Why do different models realizing the same universality class (e.g., the Ising model vs. Rydberg-atom models) sometimes show conflicting behaviors regarding nonstabilizerness (non-analytic vs. smooth)?
• Can a rigorous framework be developed to extract universal information about criticality from nonstabilizerness measures in the thermodynamic limit?

2. Methodology

The authors develop a spectral transfer-matrix framework specifically for the Stabilizer Rényi Entropy (SRE) within infinite Matrix Product States (iMPS).

• SRE Definition: They utilize the SRE of order $n$, defined for a pure state $|\psi\rangle$ as $M^{(n)} = (1-n)^{-1} \log \sum_{P} \langle \psi | P | \psi \rangle^{2n} / 2^L$, where the sum runs over Pauli strings. For mixed states (subsystems), they use the mixed-state SRE $\tilde{M}^{(n)} = M^{(n)} - S^{(n)}$, where $S^{(n)}$ is the standard Rényi entropy.
• Replica Transfer Matrix: They construct a $2n$-fold replica transfer matrix, $\mathcal{E}$, derived from the MPS tensors. The eigenvalues $\mu_i$ and eigenvectors of this matrix govern the behavior of the SRE.
• Spectral Decomposition: By analyzing the eigenspectrum of $\mathcal{E}$, they decompose the SRE of a finite subsystem of size $N$ into distinct contributions based on the dominant and subleading eigenvalues.
• Models Studied:
  • Analytical: The $\chi=2$ "MPS skeleton" of the Cluster-Ising model, which allows for exact closed-form solutions.
  • Numerical: The full Cluster-Ising model phase diagram using the Variational Uniform Matrix Product State (VUMPS) algorithm with bond dimensions up to $\chi=50$ and truncated Pauli-MPS dimensions $\chi_t=60$.

3. Key Contributions and Theoretical Framework

A. Three-Term Decomposition of SRE

The central theoretical result is the decomposition of the mixed-state SRE density for a subsystem of size $N$ into three distinct terms (Eq. 19):

1. Extensive Term: Proportional to $N$, determined by the dominant eigenvalue $\mu_1$ of the SRE transfer matrix. This represents the bulk nonstabilizerness density but is generally non-universal (model-dependent).
2. Boundary Term (Mutual SRE): An $O(1)$ term determined by the overlap of the dominant eigenvectors of the SRE transfer matrix and the standard MPS transfer matrix. This term is identified as the Mutual SRE ($L^{(n)}_\infty$) between two adjacent semi-infinite subsystems.
3. Subleading Corrections: Exponentially decaying terms governed by the ratio of subleading eigenvalues to the dominant one.

B. The SRE Correlation Length ($\xi^{(n)}_{SRE}$)

The authors define a new length scale, the SRE correlation length:
$$\xi^{(n)}_{SRE} = \frac{-1}{\log(|\mu_2/\mu_1|)}$$
Unlike the standard MPS correlation length $\xi$ (governed by the second eigenvalue of the standard transfer matrix), $\xi^{(n)}_{SRE}$ is governed by the spectrum of the replica transfer matrix.

• Significance: This length scale dictates the spatial decay of nonstabilizerness correlations. It diverges at continuous phase transitions, serving as a robust diagnostic for criticality.
• Distinctness: The paper demonstrates that $\xi^{(n)}_{SRE}$ can diverge with a different critical exponent than the standard $\xi$, indicating that nonstabilizerness probes operator content beyond standard two-point functions.

C. Response to Local Perturbations

The framework is applied to analyze how the SRE changes under local unitary perturbations. The authors show that the change in SRE due to two spatially separated unitary operations decays exponentially with distance $r$ according to $\xi^{(n)}_{SRE}$. This confirms that $\xi^{(n)}_{SRE}$ quantifies the range of nonstabilizer correlations induced by local perturbations.

4. Key Results

A. Analytical Results on the MPS Skeleton

For the $\chi=2$ Cluster-Ising skeleton:

• Critical Point: At the multicritical point ($g=0$), the standard correlation length diverges as $\xi \sim |g|^{-1}$, while the SRE correlation length diverges faster, as $\xi^{(2)}_{SRE} \sim |g|^{-2}$.
• Magic Peak: The SRE density exhibits a peak at $g^* = \pm(3-2\sqrt{2})$, corresponding to the point where the circuit preparing the state uses maximally non-Clifford gates.
• Mutual SRE: The mutual SRE $L^{(2)}_\infty$ is positive in the product phase and negative in the Symmetry Protected Topological (SPT) phase, reflecting the dominance of entanglement in the SPT phase.

B. Numerical Results on the Full Cluster-Ising Model

• Phase Diagram: The SRE density $m^{(2)}$ clearly separates the three phases (Trivial, Ising, SPT). However, $m^{(2)}$ itself shows smooth behavior across some critical lines, failing to act as a sharp indicator of criticality on its own.
• Universal Scaling: The Mutual SRE ($W^{(n)}_\infty$) exhibits universal logarithmic scaling with the correlation length, consistent with Boundary Conformal Field Theory (BCFT) predictions: $W^{(n)}_\infty \propto \Delta_{2n} \log \xi^{(n)}_{SRE}$.
• Divergence: The SRE correlation length $\xi^{(2)}_{SRE}$ diverges along both critical lines (Ising and SPT-Ising boundaries) and at the multicritical point, confirming its utility as a universal diagnostic.
• Pre-asymptotic Regime: The authors observe that extracting the universal scaling slope (related to the scaling dimension $\Delta_{2n}$) requires very large bond dimensions. For intermediate parameters, the system exhibits a "pre-asymptotic" crossover where the slope overshoots the universal value, a phenomenon also observed in exact diagonalization of finite systems.

5. Significance and Conclusion

• New Lens on Criticality: The paper establishes that while the density of nonstabilizerness may not always be a sharp indicator of phase transitions (due to non-universal bulk contributions), the SRE correlation length and Mutual SRE provide robust, universal signatures of criticality.
• Beyond Entanglement: The discovery that $\xi^{(n)}_{SRE}$ and $\xi$ can have different critical exponents implies that nonstabilizerness captures distinct, non-local correlations that are invisible to standard entanglement measures.
• Theoretical Bridge: The work bridges the gap between resource theories of quantum computation (magic) and condensed matter physics (critical phenomena), showing that the spectral properties of the SRE transfer matrix encode universal data analogous to BCFT boundary constants.
• Future Directions: The authors suggest extending this spectral framework to higher-dimensional tensor networks (PEPS), dynamical settings (time-evolving iMPS), and other critical models (e.g., XY chains, frustrated systems).

In summary, this work provides a rigorous analytical and numerical framework to understand nonstabilizerness in the thermodynamic limit, identifying the SRE correlation length as a fundamental quantity that diverges at criticality and governs the spatial response of quantum computational resources in many-body systems.