Reduced State Stabilizer Rényi Entropy as a Probe of Quantum Phase Transitions in Frustrated J_1-J_2 Spin Models

Abstract: We investigate whether the second-order purity-corrected stabilizer Rényi entropy (SRE) of reduced two-qubit density matrices can serve as a reliable local indicator of quantum phase transitions (QPTs) in frustrated quantum spin systems. We consider the one-dimensional isotropic (J_1-J_2) Heisenberg model, the one-dimensional XXZ (J_1-J_2) model, and the two-dimensional (J_1-J_2) Heisenberg model on a (4\times4) square lattice. Unlike several previously studied quantum information measures, which fail to detect QPTs in the ground state of these frustrated systems, the reduced ground-state purity-corrected SRE successfully identifies most transitions. For the remaining cases, we consider a low temperature subjacent state, modeled as a statistical mixture of the ground and first excited states with a Maxwell--Boltzmann-type occupation probability. For the 1D isotropic model, the subjacent-state SRE shows a discontinuity at the critical point, yielding (α_c(\infty)=0.24116), in excellent agreement with established values; the ground-state SRE shows a point of inflection, yielding (α_c(\infty)=0.2681). For the 1D XXZ model, the subjacent-state SRE reproduces the full anisotropy dependent phase diagram, while the ground-state SRE captures transitions only at low anisotropy. For the 2D model, the subjacent-state SRE detects two transitions, at (α_c(4\times4)=0.40781) and (0.6208), while the ground-state SRE identifies the second at (0.6230). Compared with conventional two-qubit entanglement, purity-corrected SRE shows a clear advantage in revealing otherwise-invisible phase transitions, establishing it as a robust, efficient, local probe of frustrated quantum criticality.

• The paper demonstrates that purity-corrected stabilizer Rényi entropy (SRE) of nearest-neighbor two-qubit states detects transitions that ground-state concurrence and negativity overlook, using exact diagonalization of one- and two-dimensional J₁–J₂ models.
• The subjacent-state SRE accurately estimates the isotropic-chain critical point at αc = 0.24116 and reconstructs the full V-shaped phase boundary of the XXZ J₁–J₂ model, while the ground-state signal is less reliable away from the isotropic limit.
• In the 4×4 square-lattice model, SRE identifies transitions near α ≈ 0.408 and 0.621, including the second transition that entanglement measures miss, though finite-size results and mixed-state interpretation remain limitations.

The paper investigates whether the second-order purity-corrected stabilizer Rényi entropy (SRE) of reduced two-qubit density matrices can serve as a local indicator of quantum phase transitions (QPTs) in frustrated spin systems (2608.17313). The authors study three models: the one-dimensional isotropic $J_1$–$J_2$ Heisenberg chain, the one-dimensional XXZ $J_1$–$J_2$ chain, and the two-dimensional $J_1$–$J_2$ Heisenberg model on a $4\times4$ square lattice. Their central claim is that reduced-state SRE detects phase transitions that conventional bipartite entanglement measures—concurrence, negativity, and related quantities evaluated on the ground state—fail to reveal.

Motivation and method

Nonstabilizerness ("magic") quantifies the obstruction to classical simulation within the stabilizer formalism, and the stabilizer Rényi entropy is among the most computationally accessible magic monotones. The authors evaluate the second-order purity-corrected SRE of nearest-neighbor two-qubit reduced density matrices,

$$\widetilde{M_2}(\rho_{ab}) = -\log_2\!\left(\frac{1}{4\,\mathrm{Tr}(\rho_{ab}^2)}\sum_{P\in\mathcal{P}_2}\left[\mathrm{Tr}(\rho_{ab}P)\right]^4\right),$$

where the $-\log_2\mathrm{Tr}(\rho_{ab}^2)$ term removes entanglement-induced mixedness. This correction is essential because reduced states are mixed; notably, the authors concede that for mixed states $\widetilde{M_2}$ is not a valid magic monotone but rather measures non-flatness of Pauli expectation values among states of comparable mixedness. The single-qubit reduced state vanishes identically (the ground state lies in the total-spin singlet sector, so $J_2$0), making the two-qubit reduction the minimal informative choice; the Pauli sum then contains only 16 terms, keeping the computation tractable.

Because QPTs are strictly zero-temperature phenomena yet realistic systems cannot be prepared exactly at zero temperature, the authors also evaluate the "subjacent state," a mixture of the ground state and the degenerate first excited level with Maxwell–Boltzmann-type weight fixed at $J_2$1. All eigenstates are obtained by exact diagonalization.

One-dimensional isotropic model

For the ground state, $J_2$2 varies smoothly with frustration $J_2$3 but exhibits a change in curvature near the critical point: concave growth through the spin-fluid phase, an inflection, a maximum, and decay into the dimerized phase. Finite-size scaling over $J_2$4 yields $J_2$5, which deviates from the established value $J_2$6. For the subjacent state, however, $J_2$7 shows a clear discontinuity, and finite-size scaling gives $J_2$8 with scaling exponent $J_2$9—in excellent agreement with the accepted critical point. Thus the subjacent-state SRE matches the accuracy of entanglement-based indicators while the ground-state SRE provides a signature (curvature change) that concurrence and negativity entirely lack.

One-dimensional XXZ model

In the anisotropic chain, the ground-state SRE retains its inflection-point signature only near the isotropic point $J_1$0; away from it the signature disappears, so the ground-state quantity cannot be regarded as reliable across the full phase diagram. The subjacent-state SRE, by contrast, reproduces the entire known V-shaped phase boundary in the $J_1$1 plane: discontinuities trace the spin-fluid–dimer boundary for $J_1$2, rising from $J_1$3 at $J_1$4 to roughly $J_1$5 at both $J_1$6 and $J_1$7, while a sharp kink at $J_1$8 captures the spin-fluid–Néel transition for $J_1$9. This constitutes the strongest result of the paper: a local two-qubit quantity reconstructing the full anisotropy-dependent phase diagram from exact-diagonalization data at $J_2$0.

Two-dimensional model

On the $J_2$1 lattice, the subjacent-state SRE exhibits a discontinuity at $J_2$2 (Néel-to-intermediate transition) and a curvature change at $J_2$3 (intermediate-to-collinear Néel). The comparison with entanglement is decisive here: concurrence and negativity of the subjacent state detect only the first transition and vanish identically for $J_2$4, whereas SRE remains informative throughout. Moreover, even the ground-state SRE identifies the second transition via curvature change at $J_2$5, while ground-state bipartite entanglement signals neither transition. The paper notes that the precise locations of the 2D phase boundaries remain under active investigation in the literature, and that the nature of the intermediate phase (plaquette/dimer order versus spin liquid) is unsettled, so the extracted values should be read as consistent with, not definitive for, the thermodynamic phase diagram.

Limitations and open questions

Several caveats are explicit in the paper. First, the ground-state estimate $J_2$6 for the 1D isotropic model overshoots the accepted value by about 11%, so only the subjacent-state indicator is quantitatively accurate there. Second, the purity-corrected SRE is not a magic monotone for mixed states, limiting its resource-theoretic interpretation. Third, the microscopic mechanism linking dimerization or Néel ordering to the redistribution of local nonstabilizer resources remains unexplained—the authors flag this as an open problem. Fourth, all results rest on small systems ($J_2$7 in 1D, $J_2$8 in 2D) treated by exact diagonalization, and the specific mixing probability $J_2$9 is a representative choice whose robustness across temperatures is not systematically explored here.

Conclusion

This work establishes the reduced two-qubit purity-corrected SRE as a computationally cheap, local probe that recovers frustrated quantum criticality inaccessible to standard entanglement diagnostics—most strikingly the second transition of the 2D $J_1$0–$J_1$1 model and the full XXZ phase diagram from the subjacent state. The open questions left by the paper are concrete: whether the ground-state inflection signature can be connected analytically to dimer order, whether the approach extends to larger lattices via tensor-network methods, and how it compares against other correlation measures across quantum phase boundaries.