---
title: 2-Stabilizer Rényi Entropy (2-SRE)
url: https://www.emergentmind.com/topics/2-stabilizer-renyi-entropy-2-sre
type: topic
---

# 2-Stabilizer Rényi Entropy (2-SRE)

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 [2106.12587][2602.23687][2503.13599].

## 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 [2602.23687].

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 \(p_P=(1/2^n)\Tr(\rho P)\) or the Pauli distribution \(\Xi_P\), 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 [2106.12587][2406.19457][2502.03093].

For mixed states, several extensions appear. In the dual-unitary XXZ work one defines
\[
\zeta_1(\rho)=2^{-N}\sum_P \Tr(\rho P)^2=\Tr(\rho^2),\qquad
\zeta_2(\rho)=2^{-N}\sum_P \Tr(\rho P)^4,
\]
and the mixed-state 2-SRE
\[
\widetilde M_2(\rho)=-\log\!\bigl[\zeta_2(\rho)/\zeta_1(\rho)\bigr].
\]
An equivalent “purity-corrected” form used in reduced-state studies is
\[
\widetilde M_2(\rho)=M_2(\rho)-S_2(\rho),\qquad
S_2(\rho)=-\log_2\Tr(\rho^2),
\]
which subtracts the ordinary second Rényi entropy and is intended to remove the trivial contribution of mixedness [2405.04448][2605.18391].

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 [2503.13599].

## 2. Resource-theoretic structure and derived variants

For pure states, 2-SRE is nonnegative and faithful: \(S_2(|\psi\rangle)\ge 0\) for all \(|\psi\rangle\), and \(S_2(|\psi\rangle)=0\) if and only if \(|\psi\rangle\) 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 \(S_2(|\psi\rangle)\le \nu(|\psi\rangle)\), where \(\nu\) is stabilizer nullity, and \(S_2(|\psi\rangle)\le 2\log R(|\psi\rangle)\), where \(R\) is the robustness measure appearing in the stabilizer decomposition problem [2106.12587].

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,
\[
M^{NL}(\rho)=\min_{U_A\otimes U_B}M\!\bigl((U_A\otimes U_B)\rho(U_A\otimes U_B)^\dagger\bigr),
\]
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 [2412.11918].

Reduced-state and local versions are now standard probes. For a single-qubit reduced density matrix \(\rho_i\), the full Pauli spectrum contains only four coefficients \(c_1,c_X,c_Y,c_Z\), yielding
\[
M^{(2)}(\rho_i)=-\log\!\Bigl[\frac{|c_1|^4+|c_X|^4+|c_Y|^4+|c_Z|^4}{|c_1|^2+|c_X|^2+|c_Y|^2+|c_Z|^2}\Bigr].
\]
The maximum single-qubit value is \(\log(4/3)\), attained for the \(T\)-state, and this motivates the normalized local quantity
\[
\widetilde M_i(t)=\frac{M^{(2)}(\rho_i(t))}{\log(4/3)},\qquad 0\le \widetilde M_i\le 1.
\]
This normalization is particularly useful in operator-spreading and hydrodynamic analyses of local magic transport [2511.07769].

A more aggressive localization is the two-point SRE, defined as the 2-SRE of the two-site reduced density matrix \(\rho_{ij}\). 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 [2601.13314].

## 3. Exact results in solvable states and circuits

A major exact advance is the Kagamihara–Tsuchiya theorem for 3-uniform hypergraph states
\[
|\psi\rangle=\prod_{(i,j,k)\in E_3}CCZ_{i,j,k}|+\rangle^{\otimes N}.
\]
For each \(x\in\{0,1\}^N\), define the upper-triangular \(GF(2)\) matrix \(C(x)\) by
\[
C_{i,j}(x)=\sum_{k:(i,j,k)\in E_3}x_k\pmod 2.
\]
Then
\[
m_\alpha(|\psi\rangle)=\frac{1}{2^N}\sum_{x\in\{0,1\}^N}2^{(1-\alpha)\,2h(x)},
\qquad
2h(x)=\operatorname{rank}_{GF(2)}[C(x)+C(x)^T].
\]
In particular,
\[
m_2=2^{-N}\sum_x2^{-2h(x)},\qquad
S_2=N-\log_2\!\Bigl[\sum_x2^{-2h(x)}\Bigr].
\]
This converts a naive \(\mathcal O(2^{3N})\) computation into \(\mathcal O}(N^3 2^N)\). For the one-dimensional chain hypergraph with hyperedges \((0,1,2),(1,2,3),\ldots,(N-3,N-2,N-1)\), an exact recurrence solution gives the asymptotic law
\[
S_2(N)\simeq 0.6637\,N-0.9125,
\]
while Monte Carlo for large 2D lattices yields \(S_2\approx 0.61N\) on the Union Jack lattice and \(S_2\approx 0.68N\) on the triangular lattice, both below the subadditivity upper bound \(0.9129N\) [2602.23687].

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 \(U_o(J)\) acting on \(|\phi^+\rangle^{\otimes N/2}\), the thermodynamic-limit 2-SRE density is
\[
\lim_{N\to\infty}\frac{1}{N}M_2
=
-\frac{1}{2}\log\!\Bigl(\frac{1+\cos^4(2J)+\sin^4(2J)}{2}\Bigr).
\]
For a special long-range bipartition \(B_0\), one obtains a closed form for
\[
L_2=\widetilde M_2(\rho_{AB})
\]
in terms of \(f_2(J)=\cos^4(2J)+\sin^4(2J)\) and a further polynomial expression \(g_2(J_o,J_e)\). In the symmetric slice \(J_o=J_e=J\), \(L_2\to 2\log 2\) for generic \(J\) as \(T\to\infty\), but \(L_2=0\) at the Clifford points \(J=0,\pi/4\) [2405.04448].

Generalized \(W\)-states furnish another rare closed-form family. For
\[
|W_p\rangle=\frac{1}{\sqrt L}\sum_{j=1}^L e^{ipj}\sigma_j^z|-\rangle^{\otimes L},
\qquad p=2\pi\ell/L,
\]
the finite-\(L\) 2-SRE is
\[
\mathcal M_2(p,L)
=
-\log_2\!\Biggl[
-\frac{11-12L+\sin((2-4L)p)/\sin(2p)}{2L^3}
\Biggr].
\]
At \(p=0\),
\[
\mathcal M_2(0,L)=3\log_2L-\log_2(7L-6),
\]
while for any nonzero quantized momentum \(p=2\pi\ell/L\) with \(\ell\neq 0\),
\[
\mathcal M_2\Bigl(\frac{2\pi\ell}{L},L\Bigr)
=
\mathcal M_2(0,L)+\log_2\!\Bigl(\frac{7L-6}{6L-6}\Bigr).
\]
The resulting magic gap
\[
\Delta\mathcal M_2(L)=\log_2\!\frac{7L-6}{6L-6}
\]
is independent of \(\ell\) and tends to \(\log_2(7/6)\approx 0.2224\) as \(L\to\infty\) [2406.19457].

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

A field-theoretic framework now exists for 2-SRE in \((1+1)\)-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 \(\Gamma_1\). The universal size-independent term is determined by the Affleck–Ludwig \(g\)-factor:
\[
M_2(\psi)=m_2L-c_2+o(1),\qquad c_2=\ln g_1.
\]
For Ising criticality one finds \(g_1=\sqrt 2\), hence \(c_2=\ln\sqrt 2\). For the mutual 2-SRE of adjacent intervals,
\[
W_2(l)=\frac{1}{4}\ln l_c +(\text{non-univ.}),
\]
with the coefficient fixed by the boundary-condition-changing operator dimension \(\Delta_4=1/16\) [2503.13599].

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
\[
S_{\rm stab}^{(2)}(L)=m_2L-\frac{1}{4}\ln L+O(1).
\]
For a single topological defect \(A\),
\[
S_{\rm stab}^{(2)}(L)=m_2L-c_2^A+o(1),\qquad c_2^A=\ln g_2^A.
\]
With multiple defects, the constant shifts encode fusion rules:
\[
c_2^{A\otimes B}=\min_i c_2^{C_i}
\quad\text{if}\quad
A\otimes B=\bigoplus_i C_i.
\]
Critical Ising numerics give
\[
c_2^{\,1}=\ln\sqrt2\approx 0.3466,\qquad
c_2^{\,\eta}=0.755(1),\qquad
c_2^{\,\mathcal D}=0.020(3),
\]
and for two duality lines, \(c_2^{\mathcal D\otimes\mathcal D}=c_2^1\), in agreement with \(\mathcal D\otimes\mathcal D=1\oplus \eta\) [2507.10656].

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,
\[
S_{\rm stab}^{(2),\rm PBC}(L)
=
2L\ln2-\ln\Phi(L),
\qquad
\Phi(L)=\frac{(2L)!}{L!\,L^L}.
\]
The asymptotic CFT-type expansion is
\[
S_{\rm stab}^{(2)}(L)=a_2L+b_2\ln L+c_2+o(1),
\]
with \(a_2=1\), \(b_2=0\), \(c_2=\ln2/2\) for periodic boundaries, and \(b_2=-1/4\) for open boundaries. These exact Gaussian results are consistent with the BCFT logarithmic correction and constant-term structure described above [2509.10700].

## 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 | \(GF(2)\) rank of \(C(x)+C(x)^T\) | Exact \(S_2\) in \(\mathcal O}(N^3 2^N)\) [2602.23687] |
| ZX-calculus contractions | Replica-\(\Lambda^{(2)}\) tensor simplification | Closed-form 2-SRE in dual-unitary XXZ [2405.04448] |
| Nonequilibrium QMC | Jarzynski evaluation of replicated partition-function ratios | Overall cost \(\sim N^{2.2}\) up to logs [2405.19577] |
| Reduced-Pauli SSE QMC | Sign-free sampling in reduced configuration space | Polynomial-cost \(\alpha\)-SRE in higher dimensions [2501.12146] |
| bond-DMRG for iMPS | Dominant eigenvalue of \(\chi^8\times\chi^8\) transfer operator \(D\) | Stable thermodynamic-limit SRE density [2508.03534] |
| Randomized Clifford measurements | Four-copy correlator after Clifford twirling | Additive-\(\epsilon\) estimation with \(O(1/\epsilon^2)\) random Cliffords [2106.12587] |

The nonequilibrium QMC method rewrites \(M_2(\rho)\) as \(-\ln(\mathcal Z_{[N]}/\mathcal Z_\emptyset)\), introduces the interpolating ensemble \(\mathcal Z(\lambda)\), 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 \(O(\beta N/\Delta\lambda)\), the signal-to-noise ratio scales empirically as \(N^{-1.2}\), and the overall cost is therefore polynomial. The reported benchmarks include 1D rings up to \(N=20\)–\(64\), 2D square lattices up to \(16\times 16\), and ground-state walkers on \(16\times16\) costing \(\lesssim 4\) CPU-hours each, with \(\lesssim 200\) walkers sufficient for sub-percent precision [2405.19577].

The reduced-Pauli-string SSE scheme isolates the free-energy and characteristic-function parts of purity-corrected 2-SRE,
\[
\widetilde M_2=-\ln Q+2\ln Z+\ln Z_2+N\ln2,
\]
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 \(\widetilde M_2\) but also to its first and second derivatives, as well as to subleading volume-law corrections [2501.12146].

For translation-invariant matrix product states, the infinite-size 2-SRE density can be reduced to the leading eigenvalue \(\lambda_0(D)\) of
\[
D=\sum_{t=0}^3(E^t)^{\otimes 4},
\]
where the \(E^t\) are Pauli-twirled transfer matrices built from the canonical MPS tensor. The thermodynamic-limit density is
\[
m_2=-\log\lambda_0(D)-\log2.
\]
The bond-DMRG algorithm treats \(D\) as an MPO and variationally extracts \(\lambda_0(D)\); in the Ising ground state this yields a stable peak of \(m_2(h_x)\) at the critical point \(h_x/J=1\) [2508.03534].

A separate machine-learning line treats \(S_2\) estimation as supervised regression. On random quantum circuits and transverse-Ising-model circuit datasets with \(n=2\ldots 6\) qubits, Support Vector Regressors outperform Random Forest Regressors, and the best interpolation errors are obtained by combined feature sets with test MSE \(0.04\) on random circuits and \(0.02\) on TIM circuits. Out-of-distribution generalization remains poor on random circuits but is substantially better on the structured TIM family [2509.16799].

## 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 \(T\)-state at site \(m\), the single-qubit \(M_i(t)\) is nonzero only inside the causal cone \(|i-m|\le t\). The total single-qubit SRE
\[
\mathcal M(t)=\sum_i\langle M_i(t)\rangle
\]
decays as \(e^{-\Gamma t}\) with \(\Gamma\approx 0.44\), while the normalized profile
\[
a_i(t)=\langle M_i(t)\rangle/\mathcal M(t)
\]
obeys
\[
a_i(t+1)=\tfrac12[a_{i-1}(t)+a_{i+1}(t)]
\]
inside the cone, so that the continuum limit has diffusion constant \(D=1/2\) and ballistic front velocity \(v=1\). Under a restricted CNOT-based Clifford ensemble, the profile becomes superdiffusive with width exponent \(\beta\approx 0.65\) [2511.07769].

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 \(W\)-states yields a jump \(\Delta\mathcal M_2\to \log_2(7/6)\) when the ground-state momentum changes from \(0\) to \(\pm p\neq 0\), 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 [2406.19457]. 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 \(J_x/J_z=1\) [2605.18391].

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 \(t\)-\(V\) 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
\[
V_c(L)=2+\frac{\pi^4/4}{[\ln L-\ln L_0]^2},
\]
with fits yielding \(L_0\approx 1.7\). In determinant QMC on the 2D honeycomb model, finite-size scaling of the same observable gives \(\eta\approx 0.423\), consistent with Gross–Neveu–Ising universality [2601.13314].

In SYK-related systems, 2-SRE separates chaotic from integrable behavior. For the SYK\(_4\)+SYK\(_2\) interpolation, exact diagonalization plus perfect Pauli-string sampling shows that the SYK\(_4\) ground-state 2-SRE is close to the Haar reference:
\[
\langle \mathcal M_2\rangle \approx -2.4+0.95\,\frac{N}{2},
\qquad
\mathcal M_2^{\rm Haar}=-2+\frac{N}{2},
\]
while \(\mathcal M_2\) decreases sharply as the model approaches the SYK\(_2\) limit [2502.03093]. In the coupled Maldacena–Qi SYK model, large-\(N\) saddle-point equations reveal three first-order jumps in \(M_2(\beta)\), including an intrinsic transition at \(\beta^*_{\rm SRE}\) that is invisible to ordinary thermodynamic quantities because it is controlled by the replica-connectivity structure of the SWAP-decorated path integral [2509.17417].

2-SRE has also entered quantum metrology and relativistic quantum information. In one-axis twisting, the best-squeezed regime satisfies
\[
\mathcal M_2[|\psi(t_s)\rangle]\sim 2\ln(N/2)+\text{const},
\]
whereas later “kitten” regimes have \(N\)-independent \(\mathcal M_2\sim O(\ln n)\), even as many-body Bell correlations become large. In the extreme GHZ limit the large-\(N\) 2-SRE vanishes, showing a separation between magic and multipartite nonlocality [2510.01380]. 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 [2412.11918].

## 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 \(W\)-states and the frustrated XYZ chain, the half-chain 2-Rényi entanglement is asymptotically \(p\)-independent while 2-SRE exhibits a finite discontinuity \(\log_2(7/6)\). This establishes that momentum-sensitive nonstabilizerness can remain visible when standard bipartite entanglement is effectively blind [2406.19457].

The distinction also appears in dynamics. In the dual-unitary XXZ model, the building block \(f_2(J)=\cos^4(2J)+\sin^4(2J)\) controls both short-time and long-range behavior: generically \(f_2(J)<1\) implies extensive one-step SRE growth, whereas the long-range 2-SRE saturates to an \(O(1)\) constant, \(2\log 2\), rather than exhibiting unbounded growth. This shows that local or global extensive magic generation does not automatically imply long-range magic accumulation [2405.04448].

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 \(\widetilde M_2\) fails to characterize Gibbs-state magic and can display nonphysical singularities: in the 2D model the \(Q\)-part diverges at an unphysical \(\beta^*<\beta_c\), while the \(Z\)- and \(Z_2\)-parts diverge at \(\beta_c\) and \(\beta_c/2\), respectively. The conclusion drawn there is that \(\widetilde M_2\) is not a bona fide mixed-state magic monotone for Gibbs states or finite-temperature phase transitions [2501.12146]. 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 [2605.18391].

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 \(\lambda_1^r\). This indicates that short-range injective tensor-network states can carry local magic density without sustaining asymptotic mutual magic between distant sites [2508.03534].

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 \(g\)-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.

Source: https://www.emergentmind.com/topics/2-stabilizer-renyi-entropy-2-sre