Higher-order Spatially Separated Correlations and Quantum Magic in the Long-range Transverse Field Ising Model

Technical Summary: Higher-order Spatially Separated Correlations and Quantum Magic in the Long-range Transverse Field Ising Model

Problem Statement
Long-range interacting quantum systems, particularly those with power-law decaying interactions ($\sim 1/r^\alpha$), exhibit phenomena distinct from nearest-neighbor models. While two-body correlations in such systems are known to decay algebraically, the structure of higher-order multipartite correlations and their relationship to quantum resources like "quantum magic" (non-stabiliserness) remain less understood. Specifically, there is a need to characterize how the interaction exponent $\alpha$ influences the spatial scaling of basis-independent correlations and the classical simulability of the ground state in the Long-range Transverse Field Ising Model (LITF).

Methodology
The authors investigate the ground state of a one-dimensional antiferromagnetic LITF Hamiltonian with power-law interactions. The study employs a combination of numerical techniques:

1. Density Matrix Renormalization Group (DMRG): Implemented via the ITensor library in Julia to simulate systems of size $L=50, 100, 200$ for determining critical points and spatial correlation scaling. Bond dimensions ($\chi$) were chosen to ensure truncation errors below $10^{-16}$.
2. Exact Diagonalization (ED): Used for smaller systems ($L \le 25$) to validate DMRG results and compute quantum magic directly.
3. Tensor Cross Interpolation: Applied to ground state Matrix Product States (MPS) to evaluate quantum magic for systems up to $L=25$, overcoming the exponential scaling of correlation tensor elements.
4. Analytical Approximation: The limit of large coupling ($\lambda \gg 1$) was analyzed to derive the ground state structure and its corresponding magic.

Key observables include:

• Sector Lengths ($S_k$): Basis-independent measures of $k$-body correlations defined as the sum of squared correlation tensor elements. These are used to detect entanglement ($S_L > 1$) and test for local realistic models ($S_L|_{(x,y)} < 1$).
• Stabilizer Rényi Entropy (SRE): Specifically the second-order Rényi entropy ($M_2$), used as a measure of quantum magic (non-stabiliserness).
• Cumulative Magic: The integral of $M_2$ over the coupling parameter $\lambda$ up to a specific point, used to compare total resource content across different $\alpha$.

Key Results

1. Critical Points: Pseudo-critical points $\lambda_c(\alpha)$ were determined for $\alpha \in \{5/3, 2, 7/3, 8/3, 3\}$ using half-chain entanglement entropy, consistent with previous literature.
2. Spatial Scaling of Correlations: For a fixed system size $L=100$, the basis-independent sector lengths $S_k$ for equidistant particle configurations decay algebraically near the critical point, following a form $(a + b \cdot r)^{-\gamma_k}$.
   • The pseudo-scaling exponents $\gamma_k(\alpha)$ are non-monotonic with respect to $\alpha$.
   • For $k=2$, the fastest decay occurs at $\alpha=2$. For $k>2$, the fastest decay occurs at $\alpha=8/3$.
   • Generally, the slowest decay of correlations (smallest $\gamma_k$) is observed at the smallest interaction exponent $\alpha=5/3$.
3. Entanglement and Non-locality:
   • Two-body reduced states are Bell-diagonal and do not violate separability conditions.
   • Entanglement can be detected via sector lengths for $k=5$ (all $\alpha$) and $k=7$ ($\alpha > 5/3$) when particles are nearest neighbors ($S_k > 1$).
   • However, for all studied correlation functions, the condition $S_L|_{(x,y)} < 1$ holds, indicating that a local hidden variable model exists for two-setting correlation scenarios. No Bell non-locality is violated.
4. Quantum Magic and Simulability:
   • The Rényi stabilizer entropy $M_2$ peaks at the critical point $\lambda_c$ for all $\alpha$.
   • Cumulative Magic: A key finding is that the cumulative magic, integrated from $\lambda=0$ up to the critical point, increases monotonically with the interaction exponent $\alpha$. That is, systems with faster decaying interactions (higher $\alpha$) possess a greater total amount of quantum magic in the pre-critical regime.
   • In the limit of large $\lambda$, the ground state approaches a superposition of Néel states. The magic vanishes for equal superpositions (GHZ-like) but is non-zero for asymmetric superpositions, reaching a maximum of $\log(4/3)$.

Significance and Claims
The paper claims to provide new insights into the interplay between long-range interactions, observable non-classical phenomena, and classical simulability.

• Correlation Structure: The work demonstrates that basis-independent correlations in long-range systems exhibit complex, non-monotonic scaling behaviors dependent on $\alpha$, differing from standard intuitions derived from short-range models.
• Quantum Resources: The authors establish that the "quantum magic" content of the LITF ground state is highly sensitive to the rate of interaction decay. Specifically, the finding that cumulative magic increases with $\alpha$ suggests that long-range systems with faster decaying couplings may contain more resources for quantum computation (in terms of non-stabiliserness) than those with slower decaying couplings, at least up to the critical point.
• Classical Simulability: By quantifying magic, the results offer a perspective on the difficulty of classically simulating these models, linking the interaction range directly to the computational hardness of the ground state.

The study remains modest regarding future implications, focusing strictly on the characterization of the ground state properties within the defined parameter space and numerical limits, without proposing new experimental protocols or applications beyond the scope of the analyzed model.