Finite-Temperature Stabilizer Rényi Entropy

• Finite-temperature stabilizer Rényi entropy is a measure that extends the pure-state stabilizer Rényi entropy to Gibbs states by combining participation-type string evaluations with ordinary thermal Rényi entropy.
• It isolates nontrivial string-structure through replica and path-integral constructions, revealing phase transitions invisible to conventional thermodynamic observables.
• Different formulations in spin systems, coupled SYK models, and open Ising chains demonstrate its dual role as a diagnostic for both informational order and mixed-state magic.

Searching arXiv for papers on finite-temperature stabilizer Rényi entropy and related formulations. Finite-temperature stabilizer Rényi entropy extends stabilizer Rényi entropy from pure states to Gibbs states $\rho(\beta)=e^{-\beta H}/\mathrm{Tr}\,e^{-\beta H}$ by combining a participation-type Rényi entropy of Pauli-string or Majorana-string expectation values with the ordinary Rényi entropy of the thermal density matrix. Across recent work, the object appears in several closely related notational conventions: for spin systems, $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$ for Rényi index $n$ (Liu et al., 2024); for Gibbs states at $\alpha=2$, $\widetilde M_2(\rho)=M_2(\rho)-S_2(\rho)$ (Ding et al., 21 Jan 2025); for coupled SYK, the second-Rényi SRE is defined from the Majorana-string expansion of $\rho_\beta$ (Zhang et al., 22 Sep 2025); and for the open critical transverse-field Ising chain, the mixed-state stabilizer Rényi-$\tfrac12$ entropy is written $S_{\rm stab}^{(1/2)}(T)=M_{1/2,L}(\beta)$ (Khasseh et al., 7 Jun 2026). In all cases, the finite-temperature quantity is designed to isolate nontrivial string-structure beyond ordinary thermodynamics, although papers draw sharply different conclusions about its status as a mixed-state magic measure.

1. Definitions and operator-string formulations

For a general $N$-qubit density matrix $\rho$, one definition used in spin systems is

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$0

with $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$1, and for Gibbs states one extends the construction by subtracting the usual Rényi entropy. In particular, for $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$2,

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$3

Introducing $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$4 and $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$5, one obtains

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$6

This representation separates ordinary partition functions from a generalized four-replica object $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$7 (Ding et al., 21 Jan 2025).

In the sign-problem-free spin-Hamiltonian framework, the $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$8-th finite-temperature stabilizer Rényi entropy is written

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$9

where

$n$0

Within that convention, $n$1, vanishes if and only if $n$2 is a stabilizer (Clifford) state, and is additive under tensor-product states (Liu et al., 2024).

For the coupled SYK model, Zhang et al. specialize to the second-Rényi case and define the participation Rényi entropy of the Majorana-string expansion of the thermal density matrix by

$n$3

with $n$4 an orthonormal operator basis of Majorana strings. The stabilizer Rényi entropy is then

$n$5

Equivalently, writing $n$6, one has the exact identity

$n$7

with $n$8 (Zhang et al., 22 Sep 2025).

A distinct but related formulation appears at Rényi index $n$9 for the open critical transverse-field Ising chain: $\alpha=2$0 Here the numerator is the first moment of thermal Pauli-string expectation values, and Wick’s theorem reduces it to a sum over absolute values of all square minors of the finite-temperature correlation matrix $\alpha=2$1 (Khasseh et al., 7 Jun 2026).

2. Replica and path-integral constructions

A common structural feature is replica enlargement. In the generic spin-system construction, one introduces $\alpha=2$2 replicas and uses the identity

$\alpha=2$3

where $\alpha=2$4 is a site-local connection tensor of rank $\alpha=2$5. For any subset $\alpha=2$6, the interpolating operators

$\alpha=2$7

define partition functions

$\alpha=2$8

Then $\alpha=2$9 gives the string moment and $\widetilde M_2(\rho)=M_2(\rho)-S_2(\rho)$0, so that

$\widetilde M_2(\rho)=M_2(\rho)-S_2(\rho)$1

An interpolation $\widetilde M_2(\rho)=M_2(\rho)-S_2(\rho)$2 converts the problem into a free-energy difference, and Jarzynski’s equality yields $\widetilde M_2(\rho)=M_2(\rho)-S_2(\rho)$3 (Liu et al., 2024).

For $\widetilde M_2(\rho)=M_2(\rho)-S_2(\rho)$4 in spin models, Ding, Wang, and Yan formulate the four-replica quantity $\widetilde M_2(\rho)=M_2(\rho)-S_2(\rho)$5 in stochastic-series-expansion form and identify the source of sign cancellations in direct Pauli-string sampling. They replace $\widetilde M_2(\rho)=M_2(\rho)-S_2(\rho)$6 by $\widetilde M_2(\rho)=M_2(\rho)-S_2(\rho)$7 and restrict the sum to a reduced configuration space in which, at each site and at the imaginary-time boundary, an even number of replicas carry a spin-up. This yields a reduced Pauli-string set $\widetilde M_2(\rho)=M_2(\rho)-S_2(\rho)$8 and the exact rewriting

$\widetilde M_2(\rho)=M_2(\rho)-S_2(\rho)$9

with all weights strictly nonnegative (Ding et al., 21 Jan 2025).

In the coupled SYK model, the finite-temperature SRE is represented by four replicas $\rho_\beta$0 and a quartic insertion at imaginary time $\rho_\beta$1,

$\rho_\beta$2

Equivalently, one decouples the insertion by auxiliary Ising spins $\rho_\beta$3 and site-local SWAP operators

$\rho_\beta$4

leading to

$\rho_\beta$5

After disorder averaging and bilocal Hubbard-Stratonovich fields, the large-$\rho_\beta$6 limit becomes a self-consistent saddle-point problem (Zhang et al., 22 Sep 2025).

The open-chain Ising analysis replaces stochastic sampling or large-$\rho_\beta$7 saddle methods by an exact algebraic reduction. The exponentially large sum of absolute values of all square minors of the correlation matrix is exactly reducible to a single Pfaffian, which in the open-chain basis can be rewritten as a finite-size block Toeplitz–Hankel Pfaffian. This exact reformulation places the stabilizer quantity within a class of Pfaffian determinants amenable to asymptotic methods (Khasseh et al., 7 Jun 2026).

3. Large-$\rho_\beta$8 finite-temperature SRE in the coupled SYK model

The coupled SYK analysis establishes a general framework for analyzing the SRE in solvable SYK models in the large-$\rho_\beta$9 limit, enabling the application of the saddle-point approximation (Zhang et al., 22 Sep 2025). After disorder averaging in the Maldacena–Qi model and introducing bilocal fields $\tfrac12$0 and $\tfrac12$1, with $\tfrac12$2 labeling the two SYK clusters, the saddle equations are

$\tfrac12$3

together with

$\tfrac12$4

where the fixed-boundary Green’s functions $\tfrac12$5 are defined by inverting

$\tfrac12$6

subject to twisted anti-periodic boundary conditions $\tfrac12$7 on each cluster. The saddle action is

$\tfrac12$8

Together with the usual SYK free energy $\tfrac12$9, this yields $S_{\rm stab}^{(1/2)}(T)=M_{1/2,L}(\beta)$0 and hence $S_{\rm stab}^{(1/2)}(T)=M_{1/2,L}(\beta)$1 (Zhang et al., 22 Sep 2025).

The analysis identifies controlled limits. At high temperature, $S_{\rm stab}^{(1/2)}(T)=M_{1/2,L}(\beta)$2 and $S_{\rm stab}^{(1/2)}(T)=M_{1/2,L}(\beta)$3, one finds perturbatively

$S_{\rm stab}^{(1/2)}(T)=M_{1/2,L}(\beta)$4

so the SRE starts at zero at $S_{\rm stab}^{(1/2)}(T)=M_{1/2,L}(\beta)$5 and grows quadratically. At low temperature, in the decoupled limit $S_{\rm stab}^{(1/2)}(T)=M_{1/2,L}(\beta)$6, one exactly shows $S_{\rm stab}^{(1/2)}(T)=M_{1/2,L}(\beta)$7 for all $S_{\rm stab}^{(1/2)}(T)=M_{1/2,L}(\beta)$8. For $S_{\rm stab}^{(1/2)}(T)=M_{1/2,L}(\beta)$9 small and $N$0,

$N$1

where $N$2 is the SYK zero-$N$3 entropy density, so $N$4 saturates to a constant less than $N$5 (Zhang et al., 22 Sep 2025).

For $N$6, numerical solution of the saddle-point equations yields three first-order transitions in $N$7, and hence in SRE, as temperature is tuned. The first is $N$8, described as a “half” Hawking–Page transition and inherited from a discontinuity in $N$9 at $\rho$0. The second is $\rho$1, an “intrinsic” SRE transition with no counterpart in the thermal free energy, arising from a change in the dominant $\rho$2-boundary-twist sector. The third is $\rho$3, the usual wormhole$\rho$4black-hole transition in $\rho$5, which also feeds into $\rho$6 through the $\rho$7 term. In each case, the order parameter is the dominant connectivity of the four replicas, and the critical point $\rho$8 is characterized by two coexisting large-$\rho$9 saddles of equal action $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$00 but different replica-connectivity pattern (Zhang et al., 22 Sep 2025).

A central point is that $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$01 is invisible to thermodynamics: neither the thermal free energy $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$02 nor any standard correlator shows a singularity there, yet $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$03 and thus $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$04 jump. The paper therefore states that SRE serves as a genuine order parameter for a new class of finite-$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$05 transitions which cannot be detected by conventional thermodynamic observables, and describes the change of SWAP-twist boundary conditions as a switch from a “disconnected” to a “connected” phase, analogous to replica-wormhole saddles in gravitational path integrals (Zhang et al., 22 Sep 2025).

4. Exact finite-size scaling and hidden boundary data in the open critical Ising chain

For the open critical transverse-field Ising chain, the mixed-state stabilizer Rényi-$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$06 entropy admits an exact finite-size treatment (Khasseh et al., 7 Jun 2026). Writing

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$07

the numerator becomes the sum of absolute values of all square minors of the $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$08 Majorana correlation matrix $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$09. Introducing the $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$10 antisymmetric lift $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$11 and the selector matrix $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$12, with a staggered Jordan–Wigner gauge conjugation by

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$13

one obtains the exact Pfaffian identity

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$14

After block-wise site reordering, this becomes

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$15

a finite-size block Toeplitz–Hankel Pfaffian (Khasseh et al., 7 Jun 2026).

The exact reformulation yields asymptotics in four thermal regimes. For fixed $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$16 and $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$17,

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$18

with

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$19

The purity part has an analogous expansion $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$20, so

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$21

In the high-$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$22 regime, $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$23 at fixed $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$24,

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$25

with

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$26

In the saturated low-$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$27 regime, $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$28 with $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$29,

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$30

where

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$31

and the $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$32 term is the boundary Fisher–Hartwig logarithm (Khasseh et al., 7 Jun 2026).

In the finite-size thermal crossover $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$33, the stabilizer quantity factorizes: $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$34 or equivalently

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$35

where $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$36. The universal factor is

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$37

a level-eight eta quotient, rather than the ordinary free-boundary Majorana thermal factor

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$38

As $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$39,

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$40

which the paper interprets in terms of an effective “Pauli-weight depletion central charge”

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$41

The reported conclusion is that finite-temperature stabilizer entropy reveals hidden defect-like conformal data invisible to ordinary thermodynamic probes (Khasseh et al., 7 Jun 2026).

5. Quantum Monte Carlo evaluation in spin systems

Two complementary Monte Carlo strategies have been developed for many-body SRE calculations in sign-problem-free spin systems. The first is a non-equilibrium QMC algorithm based on the path integral of the work between two partition-function ensembles, implemented within stochastic-series-expansion QMC and applicable to all spatial dimensions and temperatures (Liu et al., 2024). The second is a reduced-Pauli-string sampling method that treats $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$42-SRE as partition-function ratios and eliminates the sign problem in the imaginary-time path integral by sampling reduced Pauli strings within a reduced configuration space (Ding et al., 21 Jan 2025).

In the non-equilibrium construction, one thermalizes $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$43 independent replicas at $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$44, evolves through a schedule of $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$45-values while updating connection topology and SSE operator configurations, accumulates the work

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$46

and finally estimates

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$47

The reported time per path is $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$48, with number of $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$49-steps chosen $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$50 independent of $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$51. Empirically, the signal-to-noise ratio for $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$52 scales as $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$53 with $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$54, so $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$55 samples suffice and the total CPU cost is $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$56 (Liu et al., 2024).

Benchmarks in the transverse-field Ising model show quantitative agreement with tensor-network-based algorithms. In a 1D ring with $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$57 and periodic boundary conditions, ground-state SRE $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$58 peaks near $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$59, and QMC agrees to within $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$60 of exact MPS-contraction data. In a 2D cylinder $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$61, ground-state and finite-$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$62 SRE $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$63 versus $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$64 or $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$65 agree with MPS with bond dimension $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$66 up to $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$67 to within $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$68. In a 2D square $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$69 with $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$70 up to $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$71, the finite-$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$72 SRE density at $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$73 versus $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$74 displays two extremal points at $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$75 and $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$76, and a feature near the critical $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$77 (Liu et al., 2024).

The reduced-configuration-space approach of Ding, Wang, and Yan addresses a different bottleneck. By replacing direct sampling of $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$78 strings with a reduced set $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$79 and valid SSE configurations obeying the even-up constraint across replicas, it renders the four-replica generalized partition function $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$80 sign-problem free and enables efficient classical computations of $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$81-SRE and its derivatives in previously inaccessible $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$82D and higher-dimensional systems (Ding et al., 21 Jan 2025). This paper emphasizes that the method gives scalable access not only to the full $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$83 but also to its separate free-energy and characteristic-function contributions.

6. Interpretation, limitations, and points of tension

A central tension in the literature concerns whether finite-temperature stabilizer Rényi entropy should be regarded as a mixed-state magic measure. Ding, Wang, and Yan state that $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$84-SRE fails to characterize magic in mixed states, yielding nonphysical results (Ding et al., 21 Jan 2025). In the 2D transverse-field Ising test at $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$85, the true finite-$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$86 critical point is $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$87, but the measured $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$88 shows a clear singularity at a lower “ghost” temperature $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$89. Their decomposition of

$$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$90

shows that the $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$91-part diverges exactly at $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$92, the $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$93-part diverges at $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$94, and the $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$95-part diverges at $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$96. Because $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$97 mixes these three contributions with fixed coefficients $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$98, the total exhibits a nonphysical singularity at $$S_n^{(\mathrm{stab})}(\beta)\equiv M_n(\rho)-S_n(\rho)$$99. The same paper concludes that no meaningful volume-law expansion survives at nonzero $n$00, and that the finite-$n$01 $n$02 neither scales simply nor locates phase transitions correctly (Ding et al., 21 Jan 2025).

By contrast, the coupled SYK work and the open-chain Ising work assign finite-temperature stabilizer observables a different role. In the coupled SYK model, SRE is presented as an order parameter for a new class of finite-$n$03 transitions which cannot be detected by conventional thermodynamic observables, including an intrinsic SRE-only jump at $n$04 (Zhang et al., 22 Sep 2025). In the open critical Ising chain, stabilizer Rényi-$n$05 entropy reveals hidden defect-like conformal data invisible to ordinary thermodynamic probes, encoded in a level-eight eta quotient rather than the ordinary free-boundary Majorana factor (Khasseh et al., 7 Jun 2026).

Taken together, these results distinguish two uses of finite-temperature stabilizer entropy. One use is as a mixed-state magic monotone, where the criticism of $n$06 for Gibbs states is explicit (Ding et al., 21 Jan 2025). The other use is as an informational order parameter or probe of replica structure, boundary data, and saddle reorganization beyond standard thermodynamics (Zhang et al., 22 Sep 2025, Khasseh et al., 7 Jun 2026). This suggests that the phrase “finite-temperature stabilizer Rényi entropy” now covers a mathematically coherent family of replica observables whose physical interpretation depends strongly on context, Rényi index, and the distinction between resource-theoretic monotonicity and diagnostic sensitivity.

7. Relation to entanglement, thermodynamics, and many-body structure

The motivation for finite-temperature SRE is rooted in the separation between quantum entanglement and quantum magic as distinct resources (Zhang et al., 22 Sep 2025). In the coupled SYK analysis, entanglement-type thermodynamic quantities can remain smooth while the SRE jumps, so the stabilizer observable isolates structure in the replicated boundary-twist sector that ordinary free energy does not register. In the open critical Ising chain, the relevant extra information is boundary or defect-like conformal data hidden from standard thermal factors. In spin-model Monte Carlo studies, the decomposition into $n$07, $n$08, and $n$09 likewise separates a characteristic-function contribution from ordinary free-energy terms (Ding et al., 21 Jan 2025).

At zero temperature, Ding, Wang, and Yan fit the SRE on an $n$10 cluster as

$n$11

and report that the subleading constant $n$12, the “volume-law correction,” jumps discontinuously as $n$13 crosses the critical point $n$14. They interpret $n$15 as genuine nonlocal many-body magic in correlations and propose that $n$16 is a sharper diagnostic of criticality than the full SRE density $n$17 (Ding et al., 21 Jan 2025). For finite temperature, however, that paper rejects the same observable as a faithful magic measure, whereas the SYK and conformal-boundary analyses emphasize precisely its ability to register structures invisible to ordinary thermodynamics.

The resulting research program is therefore two-sided. On one side are exact and large-$n$18 results showing that finite-temperature stabilizer observables can be computed analytically, reduced to Pfaffians, or solved by saddle points, and that they detect replica-connectivity changes or hidden boundary data (Zhang et al., 22 Sep 2025, Khasseh et al., 7 Jun 2026). On the other side are quantum Monte Carlo results showing both that such quantities can be evaluated efficiently in large sign-problem-free systems and that, for Gibbs states, the resulting number need not behave as a physically faithful mixed-state magic measure (Liu et al., 2024, Ding et al., 21 Jan 2025). Within current literature, finite-temperature stabilizer Rényi entropy is therefore best understood as a technically rich family of replica-based observables whose interpretive status remains model- and purpose-dependent.