2-Stabilizer Rényi Entropy (2-SRE)

• 2-Stabilizer Rényi Entropy is a measure of nonstabilizerness (quantum magic) defined via the fourth moments of Pauli-string expectation values in quantum states.
• It is resource-theoretic and algebraically tractable, with extensions to mixed, reduced, and non-local variants for complex many-body systems.
• Recent studies demonstrate its use in deriving exact results, probing dynamics and phase transitions, and linking computational frameworks with conformal field theory.

2-Stabilizer Rényi Entropy (2-SRE) is the order-two member of the stabilizer Rényi entropy family, defined from fourth moments of Pauli-string expectation values and used to quantify nonstabilizerness, or quantum magic. In the post-2021 literature it has become a central observable for state complexity beyond the stabilizer formalism because it is simultaneously resource-theoretic, algebraically tractable in several exactly solvable settings, and amenable to experimental or numerical estimation. Recent work has extended 2-SRE from its original pure-state formulation to mixed-state, reduced-state, long-range, mutual, non-local, and two-point variants, and has connected it to hypergraph states, dual-unitary circuits, conformal defects, SYK physics, metrological protocols, and magic harvesting from quantum fields (Leone et al., 2021, Kagamihara et al., 27 Feb 2026, Hoshino et al., 17 Mar 2025).

1. Definitions and normalization conventions

For an $N$-qubit pure state $|\psi\rangle$, one standard convention defines the Pauli–Liouville $\alpha$-moment

$$m_\alpha(|\psi\rangle)\equiv \frac{1}{2^N}\sum_{x,z\in\{0,1\}^N}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^{2\alpha},$$

with $P^{x,z}\equiv (-i)^{x\cdot z}X^xZ^z$, and then sets

$$S_\alpha(|\psi\rangle)=\frac{1}{1-\alpha}\log_2 m_\alpha(|\psi\rangle).$$

At $\alpha=2$,

$$m_2=\frac{1}{2^N}\sum_{x,z}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^4,\qquad S_2(|\psi\rangle)=-\log_2 m_2.$$

This is the convention used, for example, in the analysis of 3-uniform hypergraph states (Kagamihara et al., 27 Feb 2026).

A closely related convention, common in resource-theoretic treatments, writes for a pure state $\Psi=|\Psi\rangle\langle\Psi|$

$\mathcal M_2(\Psi)=-\log_2\!\Bigl[d^{-1}\!\!\sum_{P\in\mathcal P_N}\bigl|\Tr(\Psi P)\bigr|^4\Bigr],\qquad d=2^N.$

Equivalent formulations use the Pauli coefficients $|\psi\rangle$0 or the Pauli distribution $|\psi\rangle$1, so that 2-SRE is the Rényi-2 entropy of a state-dependent distribution over Pauli strings. The coexistence of base-2 and natural-log conventions, and of normalized versus unnormalized Pauli sums, reflects differing normalization choices across the literature rather than different underlying observables (Leone et al., 2021, Catalano et al., 2024, Jasser et al., 5 Feb 2025).

For mixed states, several extensions appear. In the dual-unitary XXZ work one defines

$|\psi\rangle$2

and the mixed-state 2-SRE

$|\psi\rangle$3

An equivalent “purity-corrected” form used in reduced-state studies is

$|\psi\rangle$4

which subtracts the ordinary second Rényi entropy and is intended to remove the trivial contribution of mixedness (López et al., 2024, Sarkar et al., 18 May 2026).

A further structural reformulation identifies 2-SRE with a classical participation entropy in the Bell basis of a doubled Hilbert space. In that language the probabilities are Born probabilities of Bell-state measurements on two copies of the state, and the stabilizer Rényi entropy becomes a replicated partition-function ratio with an interlayer line defect (Hoshino et al., 17 Mar 2025).

2. Resource-theoretic structure and derived variants

For pure states, 2-SRE is nonnegative and faithful: $|\psi\rangle$5 for all $|\psi\rangle$6, and $|\psi\rangle$7 if and only if $|\psi\rangle$8 is a stabilizer state. It is invariant under Clifford unitaries and nonincreasing under free operations consisting of Cliffords plus computational-basis measurements and feedforward. The original resource-theoretic analysis also established bounds $|\psi\rangle$9, where $\alpha$0 is stabilizer nullity, and $\alpha$1, where $\alpha$2 is the robustness measure appearing in the stabilizer decomposition problem (Leone et al., 2021).

Because local and global nonstabilizerness need not coincide, several derived quantities have been introduced. For a bipartite mixed state, the non-local part can be defined by minimizing over local unitaries,

$\alpha$3

and in accelerated-detector protocols the harvested non-local 2-SRE is defined as the increase of this locally minimized quantity between initial and final detector states. This construction isolates the part of magic that cannot be erased by local basis changes (Cepollaro et al., 2024).

Reduced-state and local versions are now standard probes. For a single-qubit reduced density matrix $\alpha$4, the full Pauli spectrum contains only four coefficients $\alpha$5, yielding

$\alpha$6

The maximum single-qubit value is $\alpha$7, attained for the $\alpha$8-state, and this motivates the normalized local quantity

$\alpha$9

This normalization is particularly useful in operator-spreading and hydrodynamic analyses of local magic transport (Maity et al., 11 Nov 2025).

A more aggressive localization is the two-point SRE, defined as the 2-SRE of the two-site reduced density matrix $$m_\alpha(|\psi\rangle)\equiv \frac{1}{2^N}\sum_{x,z\in\{0,1\}^N}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^{2\alpha},$$0. In the interacting-fermion setting this quantity is explicitly a lower bound on the global SRE by stabilizer monotonicity under partial trace, and its mutual version subtracts the single-site contributions to isolate genuinely nonlocal two-site magic (Fang et al., 19 Jan 2026).

3. Exact results in solvable states and circuits

A major exact advance is the Kagamihara–Tsuchiya theorem for 3-uniform hypergraph states

$$m_\alpha(|\psi\rangle)\equiv \frac{1}{2^N}\sum_{x,z\in\{0,1\}^N}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^{2\alpha},$$1

For each $$m_\alpha(|\psi\rangle)\equiv \frac{1}{2^N}\sum_{x,z\in\{0,1\}^N}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^{2\alpha},$$2, define the upper-triangular $$m_\alpha(|\psi\rangle)\equiv \frac{1}{2^N}\sum_{x,z\in\{0,1\}^N}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^{2\alpha},$$3 matrix $$m_\alpha(|\psi\rangle)\equiv \frac{1}{2^N}\sum_{x,z\in\{0,1\}^N}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^{2\alpha},$$4 by

$$m_\alpha(|\psi\rangle)\equiv \frac{1}{2^N}\sum_{x,z\in\{0,1\}^N}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^{2\alpha},$$5

Then

$$m_\alpha(|\psi\rangle)\equiv \frac{1}{2^N}\sum_{x,z\in\{0,1\}^N}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^{2\alpha},$$6

In particular,

$$m_\alpha(|\psi\rangle)\equiv \frac{1}{2^N}\sum_{x,z\in\{0,1\}^N}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^{2\alpha},$$7

This converts a naive $$m_\alpha(|\psi\rangle)\equiv \frac{1}{2^N}\sum_{x,z\in\{0,1\}^N}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^{2\alpha},$$8 computation into $$m_\alpha(|\psi\rangle)\equiv \frac{1}{2^N}\sum_{x,z\in\{0,1\}^N}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^{2\alpha},$$9. For the one-dimensional chain hypergraph with hyperedges $P^{x,z}\equiv (-i)^{x\cdot z}X^xZ^z$0, an exact recurrence solution gives the asymptotic law

$P^{x,z}\equiv (-i)^{x\cdot z}X^xZ^z$1

while Monte Carlo for large 2D lattices yields $P^{x,z}\equiv (-i)^{x\cdot z}X^xZ^z$2 on the Union Jack lattice and $P^{x,z}\equiv (-i)^{x\cdot z}X^xZ^z$3 on the triangular lattice, both below the subadditivity upper bound $P^{x,z}\equiv (-i)^{x\cdot z}X^xZ^z$4 (Kagamihara et al., 27 Feb 2026).

In the dual-unitary XXZ Floquet model, ZX-calculus makes the four-replica tensor network for 2-SRE analytically reducible. For a half time-step with only one odd layer $P^{x,z}\equiv (-i)^{x\cdot z}X^xZ^z$5 acting on $P^{x,z}\equiv (-i)^{x\cdot z}X^xZ^z$6, the thermodynamic-limit 2-SRE density is

$P^{x,z}\equiv (-i)^{x\cdot z}X^xZ^z$7

For a special long-range bipartition $P^{x,z}\equiv (-i)^{x\cdot z}X^xZ^z$8, one obtains a closed form for

$P^{x,z}\equiv (-i)^{x\cdot z}X^xZ^z$9

in terms of $$S_\alpha(|\psi\rangle)=\frac{1}{1-\alpha}\log_2 m_\alpha(|\psi\rangle).$$0 and a further polynomial expression $$S_\alpha(|\psi\rangle)=\frac{1}{1-\alpha}\log_2 m_\alpha(|\psi\rangle).$$1. In the symmetric slice $$S_\alpha(|\psi\rangle)=\frac{1}{1-\alpha}\log_2 m_\alpha(|\psi\rangle).$$2, $$S_\alpha(|\psi\rangle)=\frac{1}{1-\alpha}\log_2 m_\alpha(|\psi\rangle).$$3 for generic $$S_\alpha(|\psi\rangle)=\frac{1}{1-\alpha}\log_2 m_\alpha(|\psi\rangle).$$4 as $$S_\alpha(|\psi\rangle)=\frac{1}{1-\alpha}\log_2 m_\alpha(|\psi\rangle).$$5, but $$S_\alpha(|\psi\rangle)=\frac{1}{1-\alpha}\log_2 m_\alpha(|\psi\rangle).$$6 at the Clifford points $$S_\alpha(|\psi\rangle)=\frac{1}{1-\alpha}\log_2 m_\alpha(|\psi\rangle).$$7 (López et al., 2024).

Generalized $$S_\alpha(|\psi\rangle)=\frac{1}{1-\alpha}\log_2 m_\alpha(|\psi\rangle).$$8-states furnish another rare closed-form family. For

$$S_\alpha(|\psi\rangle)=\frac{1}{1-\alpha}\log_2 m_\alpha(|\psi\rangle).$$9

the finite-$\alpha=2$0 2-SRE is

$\alpha=2$1

At $\alpha=2$2,

$\alpha=2$3

while for any nonzero quantized momentum $\alpha=2$4 with $\alpha=2$5,

$\alpha=2$6

The resulting magic gap

$\alpha=2$7

is independent of $\alpha=2$8 and tends to $\alpha=2$9 as $$m_2=\frac{1}{2^N}\sum_{x,z}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^4,\qquad S_2(|\psi\rangle)=-\log_2 m_2.$$0 (Catalano et al., 2024).

4. Universal critical behavior, conformal boundaries, and defect data

A field-theoretic framework now exists for 2-SRE in $$m_2=\frac{1}{2^N}\sum_{x,z}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^4,\qquad S_2(|\psi\rangle)=-\log_2 m_2.$$1-dimensional critical systems. In boundary conformal field theory, the replicated path integral with Bell-measurement defect identifies the full-chain SRE with a partition function on a cylinder carrying a special conformal boundary condition $$m_2=\frac{1}{2^N}\sum_{x,z}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^4,\qquad S_2(|\psi\rangle)=-\log_2 m_2.$$2. The universal size-independent term is determined by the Affleck–Ludwig $$m_2=\frac{1}{2^N}\sum_{x,z}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^4,\qquad S_2(|\psi\rangle)=-\log_2 m_2.$$3-factor: $$m_2=\frac{1}{2^N}\sum_{x,z}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^4,\qquad S_2(|\psi\rangle)=-\log_2 m_2.$$4 For Ising criticality one finds $$m_2=\frac{1}{2^N}\sum_{x,z}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^4,\qquad S_2(|\psi\rangle)=-\log_2 m_2.$$5, hence $$m_2=\frac{1}{2^N}\sum_{x,z}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^4,\qquad S_2(|\psi\rangle)=-\log_2 m_2.$$6. For the mutual 2-SRE of adjacent intervals,

$$m_2=\frac{1}{2^N}\sum_{x,z}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^4,\qquad S_2(|\psi\rangle)=-\log_2 m_2.$$7

with the coefficient fixed by the boundary-condition-changing operator dimension $$m_2=\frac{1}{2^N}\sum_{x,z}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^4,\qquad S_2(|\psi\rangle)=-\log_2 m_2.$$8 (Hoshino et al., 17 Mar 2025).

Open boundaries and topological defects modify these universal terms in different ways. For an open critical chain, the replica geometry produces a universal corner contribution, and for the Ising case the 2-SRE scales as

$$m_2=\frac{1}{2^N}\sum_{x,z}\bigl|\langle\psi|P^{x,z}|\psi\rangle\bigr|^4,\qquad S_2(|\psi\rangle)=-\log_2 m_2.$$9

For a single topological defect $\Psi=|\Psi\rangle\langle\Psi|$0,

$\Psi=|\Psi\rangle\langle\Psi|$1

With multiple defects, the constant shifts encode fusion rules: $\Psi=|\Psi\rangle\langle\Psi|$2 Critical Ising numerics give

$\Psi=|\Psi\rangle\langle\Psi|$3

and for two duality lines, $\Psi=|\Psi\rangle\langle\Psi|$4, in agreement with $\Psi=|\Psi\rangle\langle\Psi|$5 (Hoshino et al., 14 Jul 2025).

For free-fermion critical chains, the stabilizer–Shannon equivalence yields exact finite-size formulas. In the critical transverse-field Ising chain with periodic boundary conditions,

$\Psi=|\Psi\rangle\langle\Psi|$6

The asymptotic CFT-type expansion is

$\Psi=|\Psi\rangle\langle\Psi|$7

with $\Psi=|\Psi\rangle\langle\Psi|$8, $\Psi=|\Psi\rangle\langle\Psi|$9, $\mathcal M_2(\Psi)=-\log_2\!\Bigl[d^{-1}\!\!\sum_{P\in\mathcal P_N}\bigl|\Tr(\Psi P)\bigr|^4\Bigr],\qquad d=2^N.$0 for periodic boundaries, and $\mathcal M_2(\Psi)=-\log_2\!\Bigl[d^{-1}\!\!\sum_{P\in\mathcal P_N}\bigl|\Tr(\Psi P)\bigr|^4\Bigr],\qquad d=2^N.$1 for open boundaries. These exact Gaussian results are consistent with the BCFT logarithmic correction and constant-term structure described above (Rajabpour, 12 Sep 2025).

5. Computational frameworks and estimators

Several distinct algorithmic routes have been developed for 2-SRE, each exploiting a different representation of the fourth Pauli moment.

Framework	Core reduction	Representative claim
Hypergraph rank method	$\mathcal M_2(\Psi)=-\log_2\!\Bigl[d^{-1}\!\!\sum_{P\in\mathcal P_N}\bigl|\Tr(\Psi P)\bigr|^4\Bigr],\qquad d=2^N.$2 rank of $\mathcal M_2(\Psi)=-\log_2\!\Bigl[d^{-1}\!\!\sum_{P\in\mathcal P_N}\bigl|\Tr(\Psi P)\bigr|^4\Bigr],\qquad d=2^N.$3	Exact $\mathcal M_2(\Psi)=-\log_2\!\Bigl[d^{-1}\!\!\sum_{P\in\mathcal P_N}\bigl|\Tr(\Psi P)\bigr|^4\Bigr],\qquad d=2^N.$4 in $\mathcal M_2(\Psi)=-\log_2\!\Bigl[d^{-1}\!\!\sum_{P\in\mathcal P_N}\bigl|\Tr(\Psi P)\bigr|^4\Bigr],\qquad d=2^N.$5 (Kagamihara et al., 27 Feb 2026)
ZX-calculus contractions	Replica-$\mathcal M_2(\Psi)=-\log_2\!\Bigl[d^{-1}\!\!\sum_{P\in\mathcal P_N}\bigl|\Tr(\Psi P)\bigr|^4\Bigr],\qquad d=2^N.$6 tensor simplification	Closed-form 2-SRE in dual-unitary XXZ (López et al., 2024)
Nonequilibrium QMC	Jarzynski evaluation of replicated partition-function ratios	Overall cost $\mathcal M_2(\Psi)=-\log_2\!\Bigl[d^{-1}\!\!\sum_{P\in\mathcal P_N}\bigl|\Tr(\Psi P)\bigr|^4\Bigr],\qquad d=2^N.$7 up to logs (Liu et al., 2024)
Reduced-Pauli SSE QMC	Sign-free sampling in reduced configuration space	Polynomial-cost $\mathcal M_2(\Psi)=-\log_2\!\Bigl[d^{-1}\!\!\sum_{P\in\mathcal P_N}\bigl|\Tr(\Psi P)\bigr|^4\Bigr],\qquad d=2^N.$8-SRE in higher dimensions (Ding et al., 21 Jan 2025)
bond-DMRG for iMPS	Dominant eigenvalue of $\mathcal M_2(\Psi)=-\log_2\!\Bigl[d^{-1}\!\!\sum_{P\in\mathcal P_N}\bigl|\Tr(\Psi P)\bigr|^4\Bigr],\qquad d=2^N.$9 transfer operator $|\psi\rangle$00	Stable thermodynamic-limit SRE density (Liu et al., 5 Aug 2025)
Randomized Clifford measurements	Four-copy correlator after Clifford twirling	Additive-$|\psi\rangle$01 estimation with $|\psi\rangle$02 random Cliffords (Leone et al., 2021)

The nonequilibrium QMC method rewrites $|\psi\rangle$03 as $|\psi\rangle$04, introduces the interpolating ensemble $|\psi\rangle$05, and evaluates the free-energy difference by Jarzynski’s equality. In the sign-problem-free transverse-field Ising benchmarks, the time complexity per walker is $|\psi\rangle$06, the signal-to-noise ratio scales empirically as $|\psi\rangle$07, and the overall cost is therefore polynomial. The reported benchmarks include 1D rings up to $|\psi\rangle$08–$|\psi\rangle$09, 2D square lattices up to $|\psi\rangle$10, and ground-state walkers on $|\psi\rangle$11 costing $|\psi\rangle$12 CPU-hours each, with $|\psi\rangle$13 walkers sufficient for sub-percent precision (Liu et al., 2024).

The reduced-Pauli-string SSE scheme isolates the free-energy and characteristic-function parts of purity-corrected 2-SRE,

$|\psi\rangle$14

and removes the sign problem by grouping Pauli strings according to diagonal/off-diagonal patterns into a reduced configuration space with nonnegative weights. This allows direct access not only to $|\psi\rangle$15 but also to its first and second derivatives, as well as to subleading volume-law corrections (Ding et al., 21 Jan 2025).

For translation-invariant matrix product states, the infinite-size 2-SRE density can be reduced to the leading eigenvalue $|\psi\rangle$16 of

$|\psi\rangle$17

where the $|\psi\rangle$18 are Pauli-twirled transfer matrices built from the canonical MPS tensor. The thermodynamic-limit density is

$|\psi\rangle$19

The bond-DMRG algorithm treats $|\psi\rangle$20 as an MPO and variationally extracts $|\psi\rangle$21; in the Ising ground state this yields a stable peak of $|\psi\rangle$22 at the critical point $|\psi\rangle$23 (Liu et al., 5 Aug 2025).

A separate machine-learning line treats $|\psi\rangle$24 estimation as supervised regression. On random quantum circuits and transverse-Ising-model circuit datasets with $|\psi\rangle$25 qubits, Support Vector Regressors outperform Random Forest Regressors, and the best interpolation errors are obtained by combined feature sets with test MSE $|\psi\rangle$26 on random circuits and $|\psi\rangle$27 on TIM circuits. Out-of-distribution generalization remains poor on random circuits but is substantially better on the structured TIM family (Lipardi et al., 20 Sep 2025).

6. Dynamics, phase transitions, and physical applications

In brickwork random Clifford circuits, local 2-SRE spreading has a hydrodynamic structure despite global Clifford conservation laws. Starting from a product state with a single $|\psi\rangle$28-state at site $|\psi\rangle$29, the single-qubit $|\psi\rangle$30 is nonzero only inside the causal cone $|\psi\rangle$31. The total single-qubit SRE

$|\psi\rangle$32

decays as $|\psi\rangle$33 with $|\psi\rangle$34, while the normalized profile

$|\psi\rangle$35

obeys

$|\psi\rangle$36

inside the cone, so that the continuum limit has diffusion constant $|\psi\rangle$37 and ballistic front velocity $|\psi\rangle$38. Under a restricted CNOT-based Clifford ensemble, the profile becomes superdiffusive with width exponent $|\psi\rangle$39 (Maity et al., 11 Nov 2025).

Several many-body phase transitions are now characterized by 2-SRE rather than by entanglement alone. In the topologically frustrated XYZ chain, a Clifford mapping to generalized $|\psi\rangle$40-states yields a jump $|\psi\rangle$41 when the ground-state momentum changes from $|\psi\rangle$42 to $|\psi\rangle$43, while the bipartite 2-Rényi entanglement remains continuous in the thermodynamic limit. This identifies a transition whose order parameter is nonstabilizerness rather than entanglement or local order (Catalano et al., 2024). In the transverse ANNNI model, purity-corrected reduced-state 2-SRE detects the antiphase–floating transition in the high-frustration regime, whereas in the low-frustration regime the raw, purity-uncorrected reduced-state SRE better reproduces the ferromagnetic–paramagnetic phase boundary. In the quantum compass model, the same reduced-state diagnostic shows a pronounced dip in its derivative at the first-order point $|\psi\rangle$44 (Sarkar et al., 18 May 2026).

A local-fermionic version, the two-point SRE, has been used as a Monte Carlo–friendly proxy of magic in interacting fermion systems. In the 1D spinless $|\psi\rangle$45-$|\psi\rangle$46 model, the mutual two-point SRE decays algebraically in the Luttinger liquid and exponentially in the charge-density-wave phase; the finite-size shift of the susceptibility peak follows the BKT form

$|\psi\rangle$47

with fits yielding $|\psi\rangle$48. In determinant QMC on the 2D honeycomb model, finite-size scaling of the same observable gives $|\psi\rangle$49, consistent with Gross–Neveu–Ising universality (Fang et al., 19 Jan 2026).

In SYK-related systems, 2-SRE separates chaotic from integrable behavior. For the SYK$|\psi\rangle$50+SYK$|\psi\rangle$51 interpolation, exact diagonalization plus perfect Pauli-string sampling shows that the SYK$|\psi\rangle$52 ground-state 2-SRE is close to the Haar reference: $|\psi\rangle$53 while $|\psi\rangle$54 decreases sharply as the model approaches the SYK$|\psi\rangle$55 limit (Jasser et al., 5 Feb 2025). In the coupled Maldacena–Qi SYK model, large-$|\psi\rangle$56 saddle-point equations reveal three first-order jumps in $|\psi\rangle$57, including an intrinsic transition at $|\psi\rangle$58 that is invisible to ordinary thermodynamic quantities because it is controlled by the replica-connectivity structure of the SWAP-decorated path integral (Zhang et al., 22 Sep 2025).

2-SRE has also entered quantum metrology and relativistic quantum information. In one-axis twisting, the best-squeezed regime satisfies

$|\psi\rangle$59

whereas later “kitten” regimes have $|\psi\rangle$60-independent $|\psi\rangle$61, even as many-body Bell correlations become large. In the extreme GHZ limit the large-$|\psi\rangle$62 2-SRE vanishes, showing a separation between magic and multipartite nonlocality (Hernández-Yanes et al., 1 Oct 2025). In accelerated Unruh–DeWitt detector protocols, non-local 2-SRE can be harvested from the Minkowski vacuum; for stabilizer initial states all harvested magic is non-local, the acceleration dependence is non-monotonic, and no CHSH violation arises with Pauli measurements despite the simultaneous harvesting of entanglement and magic (Cepollaro et al., 2024).

7. Limitations, distinctions from entanglement, and open tensions

A persistent theme in the literature is that 2-SRE and entanglement are related but inequivalent diagnostics. In generalized $|\psi\rangle$63-states and the frustrated XYZ chain, the half-chain 2-Rényi entanglement is asymptotically $|\psi\rangle$64-independent while 2-SRE exhibits a finite discontinuity $|\psi\rangle$65. This establishes that momentum-sensitive nonstabilizerness can remain visible when standard bipartite entanglement is effectively blind (Catalano et al., 2024).

The distinction also appears in dynamics. In the dual-unitary XXZ model, the building block $|\psi\rangle$66 controls both short-time and long-range behavior: generically $|\psi\rangle$67 implies extensive one-step SRE growth, whereas the long-range 2-SRE saturates to an $|\psi\rangle$68 constant, $|\psi\rangle$69, rather than exhibiting unbounded growth. This shows that local or global extensive magic generation does not automatically imply long-range magic accumulation (López et al., 2024).

The status of 2-SRE for mixed states is more delicate. The reduced-Pauli-string QMC study of the transverse-field Ising model found that purity-corrected $|\psi\rangle$70 fails to characterize Gibbs-state magic and can display nonphysical singularities: in the 2D model the $|\psi\rangle$71-part diverges at an unphysical $|\psi\rangle$72, while the $|\psi\rangle$73- and $|\psi\rangle$74-parts diverge at $|\psi\rangle$75 and $|\psi\rangle$76, respectively. The conclusion drawn there is that $|\psi\rangle$77 is not a bona fide mixed-state magic monotone for Gibbs states or finite-temperature phase transitions (Ding et al., 21 Jan 2025). By contrast, reduced-state studies of TANNNI and QCM show that purity correction can nonetheless be useful as a local probe in specific ground-state settings, though even there the corrected and raw versions may diagnose different phase boundaries with different accuracy (Sarkar et al., 18 May 2026).

For translation-invariant injective MPS, nonlocal magic is tightly constrained by entanglement structure. The non-local 2-SRE density is bounded by a universal function of entanglement entropy, and the two-site mutual 2-SRE vanishes asymptotically with separation because the relevant transfer-matrix corrections decay exponentially with the subleading eigenvalue $|\psi\rangle$78. This indicates that short-range injective tensor-network states can carry local magic density without sustaining asymptotic mutual magic between distant sites (Liu et al., 5 Aug 2025).

These developments suggest a precise but still incomplete picture. 2-SRE is now an exact algebraic invariant in certain non-Clifford state families, a BCFT observable tied to $|\psi\rangle$79-factors and boundary-condition-changing operators, a practical estimator in several Monte Carlo and tensor-network settings, and a probe of dynamics, criticality, defects, and topology. At the same time, mixed-state generalizations remain unsettled, local and long-range versions can behave very differently, and the relation between magic, entanglement, and nonlocality is demonstrably nontrivial rather than hierarchical.