Stabilizer Rényi Entropy in Quantum States

• Stabilizer Rényi entropy is a precise measure of quantum nonstabilizerness (magic) defined via moments of the Pauli spectrum and Weyl-Heisenberg operators.
• It serves as a computable monotone in magic-state resource theory, guiding fault-tolerant quantum computation and classical simulation methods.
• Efficient numerical techniques, including tensor networks, Monte Carlo, and machine learning, make SRE accessible for studying many-body quantum dynamics.

The stabilizer entropy, specifically the stabilizer Rényi entropy (SRE), is a fundamental information-theoretic measure quantifying nonstabilizerness—also called “magic”—in quantum states and processes. SRE serves as a computable, experimentally accessible monotone within magic-state resource theories and has broad applicability in the characterization of quantum computational power, universality, simulability, and the universal properties of many-body quantum systems. Mathematically, SRE is defined via the moments of the Pauli spectrum or the Weyl-Heisenberg displacement operator expectation values, describing the spread of a quantum state in the stabilizer (Clifford orbit) manifold and providing rigorous operational connections to resource conversion, complexity, and topological properties.

1. Formal Definition and Mathematical Structure

Let $|\psi\rangle$ be an %%%%1%%%%-qubit pure state in Hilbert space dimension $d=2^n$, and $\mathcal{P}_n$ the group of $n$-qubit Pauli strings. The stabilizer Rényi entropy of order $\alpha$ is defined by

$$M_\alpha(\psi) = \frac{1}{1-\alpha} \log \left( \frac{1}{2^n} \sum_{P \in \mathcal{P}_n} |\langle\psi|P|\psi\rangle|^{2\alpha} \right)$$

For $\alpha=2$ (the most commonly analyzed case),

$$M_2(\psi) = -\log \left( \frac{1}{2^n} \sum_{P \in \mathcal{P}_n} |\langle\psi|P|\psi\rangle|^4 \right)$$

The linear stabilizer entropy (“second-order Tsallis” version) is

$$M_{\text{lin}}(\psi) = 1 - \frac{1}{2^n} \sum_{P \in \mathcal{P}_n} |\langle\psi|P|\psi\rangle|^4$$

For mixed states $\rho$, replace $|\psi\rangle\langle\psi|$ by $\rho$ in the above formulas. For systems of local dimension $d$, replace the Pauli group by the Weyl-Heisenberg displacement operators $\{D_a\}$ and $2^n$ by $d^n$ (Cepollaro et al., 28 Dec 2025).

The SRE is zero if and only if the state is a stabilizer state, is invariant under Clifford unitaries, additive on tensor products, and monotone under pure-state and mixed-state stabilizer protocols for $\alpha \geq 2$ (Leone et al., 2024, Maity et al., 11 Nov 2025). The SRE can be efficiently computed for low-rank states and is accessible via randomized measurement protocols (Leone et al., 2021).

2. Operational and Physical Interpretation

Stabilizer entropy is a bona fide magic monotone in resource theory, quantifying the distance from efficient classical simulation (the stabilizer manifold) and strictly increasing with the addition of non-Clifford resources. Key operational results include:

• The Clifford orbit of a state $|\psi\rangle$ forms an approximate state $k$-design with an error $\Theta(\exp(-M_2(\psi)))$ (Bittel et al., 30 Jul 2025).
• The optimal probability of distinguishing $|\psi\rangle$ from a stabilizer state using $k$ copies is governed by $M_3(\psi)$.
• The conversion rate between magic states is bounded by the SRE ratio, $$r[|R_1\rangle \rightarrow |R_2\rangle] \leq M_\alpha(|R_1\rangle)/M_\alpha(|R_2\rangle)$$ (Leone et al., 2024).

SRE is connected to the complexity of classical simulation (exponential in $M_2$), diagnostic for fault-tolerant quantum computation, and is related to out-of-time-order correlators necessary for quantum chaos (Leone et al., 2021).

3. Monotonicity and Comparison to Other Magic Measures

Stabilizer entropies $M_\alpha$ with $\alpha \geq 2$ are strong monotones under stabilizer protocols, including Clifford unitaries, measurements, and discarding qubits; they remain monotonic under convex roof extensions for mixed states (Leone et al., 2024). For $\alpha < 2$, monotonicity (and strong monotonicity) fails in general (Haug et al., 2023). SREs are bounded above by twice the log-robustness and, for $n > 1$, by $2n/(n-1)$ times the min-relative entropy of magic. However, no $N$-independent lower bound in terms of these measures exists for general $n$ (Haug et al., 2023, Leone et al., 2021).

4. Direct Computation and Numerical Methods

Computing the SRE for generic $n$-qubit states requires $O(4^n)$ operations—a complexity which becomes intractable for large $n$ (Lipardi et al., 20 Sep 2025). For structured states such as translation-invariant matrix product states (MPS) or subspaces of SU(2) invariant spin networks, exact and efficient tensor-contraction algorithms for the SRE exist (Liu et al., 5 Aug 2025, Cepollaro et al., 2024).

Monte Carlo and machine learning techniques can estimate SRE from circuit-level features and classical shadows. Support vector regressors (SVR) outperform random forest regressors (RFR) on structured datasets, but generalization out-of-distribution remains poor on random circuits (Lipardi et al., 20 Sep 2025). Non-equilibrium quantum Monte Carlo algorithms offer polynomial cost in system size for sign-problem-free Hamiltonians (Liu et al., 2024).

5. SRE in Many-body Physics and Quantum Dynamics

The SRE reveals universal features in quantum critical chains, conformal field theory scaling, and dynamical phase transitions:

• In Gaussian quadratic fermion models, SRE is exactly mapped to Shannon-Rényi entropy on a doubled system (Rajabpour, 12 Sep 2025).
• In 1D critical states, SRE contains size-independent universal terms controlled by the boundary $g$-factor and logarithmic scaling of mutual SRE dictated by the scaling dimension of boundary-condition-changing operators (Hoshino et al., 17 Mar 2025, Hoshino et al., 14 Jul 2025).
• SRE encodes fusion rules of topological defects in the Ising model, reflecting noninvertible symmetry algebra (Hoshino et al., 14 Jul 2025).

Under quantum quenches, SRE equilibrates in times scaling linearly with subsystem size, spreads ballistically, and exhibits light-cone constrained growth and phase transitions in its localization properties (Rattacaso et al., 2023, Leone et al., 2023, Odavić et al., 2024, Maity et al., 11 Nov 2025).

6. Embedding, Subspaces, and Resource Cost

The stabilizer entropy of subspaces quantifies the "magic gap" between a subsystem and its embedding in a larger Hilbert space (Cepollaro et al., 28 Dec 2025). Both zero and negative gaps are achievable via specific choices of stabilizer codes and symmetry-induced subspaces, enabling resource-efficient quantum simulations. Haar-average formulas give the expected magic cost for arbitrary embeddings, and explicit conditions for negative gaps exist for certain code families.

7. Universal, Statistical, and Topological Properties

For Haar-random pure states, the probability density function of SRE shows Van Hove-type singularities—logarithmic divergences at special “magic states” like $|H\rangle$ for a single qubit. These singularities vanish for $d \geq 3$ (Iannotti et al., 25 Oct 2025). SRE integrally quantifies partial incompatibility in quantum measurements, directly linking the measure to core features of quantum structure and nonlocality.

SRE vectors in multipartite stabilizer states satisfy classical subadditivity and strong subadditivity, along with linear rank inequalities such as the Ingleton inequality, exactly characterizing the quantum entropy cone of stabilizer states (Linden et al., 2013). The reachability graphs of stabilizers reveal sharply structured transitions between holographic and non-holographic entropy regions as qubit number increases (Keeler et al., 2022).

---

Summary Table: Stabilizer Rényi Entropy Key Properties

Property	Mathematical Statement	Reference
Definition	$$M_\alpha(\psi) = \frac{1}{1-\alpha} \log \frac{1}{2^n} \sum_P |\langle\psi|P|\psi\rangle|^{2\alpha}$$	(Cepollaro et al., 28 Dec 2025, Leone et al., 2024)
Faithfulness	$M_\alpha(\psi) = 0 \iff \psi$ stabilizer	(Leone et al., 2024, Maity et al., 11 Nov 2025)
Clifford-invariance	$M_\alpha(C \psi) = M_\alpha(\psi)$	(Maity et al., 11 Nov 2025, Hoshino et al., 14 Jul 2025)
Additivity	$$M_\alpha(\psi \otimes \phi) = M_\alpha(\psi) + M_\alpha(\phi)$$	(Leone et al., 2024, Maity et al., 11 Nov 2025)
Magic monotonicity ($\alpha \geq 2$)	$M_\alpha(\mathcal{E}(\psi)) \leq M_\alpha(\psi)$	(Leone et al., 2024, Maity et al., 11 Nov 2025)
Computability	Efficient for low-rank, MPS, ML, QMC methods; #P-hard in general	(Lipardi et al., 20 Sep 2025, Liu et al., 2024)
Physical meaning	Quantifies nonstabilizerness, resource for quantum computation	(Maity et al., 11 Nov 2025, Leone et al., 2021)
Topological/Universal terms	$c_\alpha$ governed by BCFT g-factor, fusion rules, Ising CFT	(Hoshino et al., 17 Mar 2025, Hoshino et al., 14 Jul 2025)

---

Stabilizer entropy, embodied in the family of stabilizer Rényi entropies, provides a rigorous, computable, and operationally meaningful quantifier of quantum magic with direct implications for quantum computing power, simulation complexity, resource theory, phase transitions, conformal and topological properties, and universal quantum phenomena. Its mathematical structure and numerically tractable algorithms underlie much of the recent progress on nonstabilizerness in quantum information science and many-body theory.