Linear Stabilizer Entropy

• Linear stabilizer entropy is a metric that quantifies the magic or non-stabilizerness of quantum states using a linearized Rényi entropy framework, applicable to both pure and mixed states.
• It serves as a faithful, strongly monotonic resource monotone in quantum resource theory, ensuring that magic does not increase under stabilizer operations and scales additively in composite systems.
• Efficient computational techniques and experimental protocols, such as randomized Pauli measurements and machine learning approaches, facilitate its practical application in quantum simulation and many-body physics.

Linear stabilizer entropy is a pivotal concept in quantum information theory and condensed matter physics, providing an analytically tractable, physically meaningful, and computationally efficient quantifier of "magic," i.e., the non-stabilizerness of quantum states. It connects resource theory, information inequalities, quantum measurement, many-body and topological order, and diverse operational tasks ranging from magic-state conversion to property testing, quantum simulation, and robust entanglement distribution.

1. Definition and General Formalism

Linear stabilizer entropy, typically denoted $M^{\mathrm{lin}}(\psi)$ for a pure state $|\psi\rangle$, is a canonical monotone measuring the degree of non-stabilizerness (magic) relative to the Clifford/stabilizer reference. For $n$ qubits/qudits, define the characteristic "Pauli spectrum" probabilities: $$p(P) := d^{-1} |\langle \psi | P | \psi \rangle|^2,\quad P\in \mathcal{P}_n,\, d=2^n$$ where $\mathcal{P}_n$ is the Pauli group.

The stabilizer Rényi entropies are: $$M_\alpha(|\psi\rangle) = \frac{1}{1-\alpha} \log_2\left(\sum_{P} p(P)^\alpha\right)$$ The linear stabilizer entropy (LSE) is the linearized ($\alpha=2$) case: $$M^{\mathrm{lin}}(|\psi\rangle) = 1 - \sum_{P} p(P)^2$$ or equivalently, $M^{\mathrm{lin}}(\psi) = 1 - P_2(\psi)$, where $P_2$ is the stabilizer purity. For single qubits, $p(P) = \frac{1 + \|\vec n\|_4^4}{2}$ with Bloch vector $\vec n$. For mixed states, the convex roof extension is used: $$M^{\mathrm{lin}}_{\rm mix}(\rho) = \inf_{p_j,\, |\psi_j\rangle}\left[ \sum_j p_j M^{\mathrm{lin}}(|\psi_j\rangle) \right]$$

2. Resource-Theoretic Properties and Monotonicity

Linear stabilizer entropy is a faithful, strongly monotonic resource monotone for magic-state resource theory on both pure and mixed states. Explicitly:

• Faithfulness: $M^{\mathrm{lin}}(\psi) = 0$ iff $|\psi\rangle$ is a stabilizer state.
• Strong monotonicity (Theorem 2 (Leone et al., 17 Apr 2024)): Non-increasing on average under all stabilizer protocols, including probabilistic outcomes.
• Additivity: $$M^{\mathrm{lin}}(\psi \otimes \phi) = M^{\mathrm{lin}}(\psi) + M^{\mathrm{lin}}(\phi)$$.
• Clifford invariance: $$M^{\mathrm{lin}}(U|\psi\rangle) = M^{\mathrm{lin}}(|\psi\rangle)$$ for all Clifford $U$.
• Computational/experimental efficiency: $M^{\mathrm{lin}}$ can be estimated efficiently (in polynomial time for small $\alpha$) via randomized Pauli measurements (Leone et al., 17 Apr 2024, Leone et al., 2021).

Compared to standard Rényi stabilizer entropies $M_\alpha$, LSE is the unique known additive, strongly monotonic, and experimentally measurable magic monotone for all $\alpha \geq 2$ ((Leone et al., 17 Apr 2024), Table I).

3. Operational Interpretation and Property Testing

Linear stabilizer entropy has direct operational meaning for distinguishing quantum resources:

• Clifford orbit indistinguishability: The probability to distinguish the Clifford orbit $\mathcal{E}_\psi$ from Haar-random states decays exponentially with $M_2(\psi)$,

$$q_{\rm succ}^{(k)} = \frac{1}{2} + \Theta(\exp(-M_2(\psi)))$$

• Stabilizer testing: The optimal probability to distinguish a state from stabilizer states is governed by $M_3(\psi)$,

$$p_{\rm succ}^{(6)} = \frac{1}{2} + \frac{1}{4}(1 - 2^{-2M_3(\psi)})$$

• Magic-state conversion bounds: The maximal probabilistic success probability for $|\psi\rangle \to |\phi\rangle$ under stabilizer protocols satisfies

$$p_{\max} \leq \frac{M^{\mathrm{lin}}(|\psi\rangle)}{M^{\mathrm{lin}}(|\phi\rangle)}$$

Thus, LSE precisely quantifies the operational crossover between stabilizerness and universal quantum resources (Bittel et al., 30 Jul 2025, Leone et al., 17 Apr 2024).

4. Physical, Geometric, and Measurement-Theoretic Significance

Linear stabilizer entropy exhibits deep physical and information-theoretic connections:

• Partial incompatibility: For single qubits, $M^{\mathrm{lin}}_2(\psi)$ quantifies the incompatibility deficit with respect to Pauli operators; stabilization is maximal incompatibility (Iannotti et al., 25 Oct 2025):

$$M^{\mathrm{lin}}_2(\psi) = 1 - \tfrac{1}{4}\Gamma_2(\psi)$$

where $\Gamma_2$ quantifies incompatibility via Schatten $p$-norm.

• Distribution and Van Hove singularities: For Haar-random qubits, the probability density of LSE displays a logarithmic divergence at the $|H\rangle$-state, analogous to Van Hove singularities; this signals the statistical prevalence and geometric significance of these states (Iannotti et al., 25 Oct 2025).
• Connection to OTOCs and chaos: Linear stabilizer entropy controls the nonstabilizing power of unitaries and is directly connected to the fourth power of out-of-time-order correlators (OTOCs); maximal $M^{\mathrm{lin}}$ is necessary for quantum chaos (Leone et al., 2021).

5. Linear Inequalities, Entropy Cones, and Topological/Entanglement Structure

Linear stabilizer entropy features prominently in global information inequalities:

• Entropy cone characterization: For stabilizer states, the set of achievable von Neumann/Rényi-2 entropy vectors is completely characterized by classical balanced information inequalities, including strong subadditivity, weak monotonicity, and the Ingleton inequality (Linden et al., 2013, Gross et al., 2013).
• Higher-party entanglement: For $n=4$, Ingleton uniquely constrains the cone; for $n\geq 5$, all quantum linear rank inequalities bound stabilizer cones, and there exist explicit separating inequalities with hypergraph models (Bao et al., 2020).
• Topological entanglement and fracton order: In 3D fracton stabilizer codes, the universal, robust linear-scaling term in topological entanglement entropy ($S_{\text{topo}} \sim \alpha R$) is branded as "linear stabilizer entropy," distinguishing fracton topological order from 2D TQFT-type order (Ma et al., 2017).
• Noisy stabilizer codes and syndrome entropy: In noisy settings, the syndrome entropy—essentially a linear Shannon entropy over error syndromes—governs the residual mixedness and entanglement in stabilizer code states, providing a classical-quantum bridge in networked scenarios (Goodenough et al., 4 Jun 2024).

6. Computation, Experimental Estimation, and Numerical Algorithms

Linear stabilizer entropy is uniquely suited to scalable computation and experimental diagnostics:

• Machine learning estimation: ML regressors (SVR, RFR) can efficiently estimate the SRE ($\alpha=2$, i.e., linear stabilizer entropy) from circuit data, outperforming traditional sampling for moderate system sizes (Lipardi et al., 20 Sep 2025).
• Non-equilibrium QMC: Path-integral QMC methods enable efficient, polynomial-cost calculation of the (linear) stabilizer Rényi entropy for large spin systems at finite and zero temperature (Liu et al., 29 May 2024).
• Tensor network/MPS contractions: For many-body states, tensor-network and perfect MPS sampling methods provide tractable evaluation of linear (or von Neumann) stabilizer entropy even for large bond-dimension systems (Haug et al., 2023).
• Randomized measurement protocols: Direct estimation via randomized Clifford/Pauli measurements is feasible, requiring only polynomial resources (Leone et al., 2021, Leone et al., 17 Apr 2024).

Computational cost is polynomial for MPS, QMC, or ML on relevant classes of states and circuits, enabling LSE to operate as a practical, scalable diagnostic and benchmark in theory and experiment.

7. Universal Features and Many-body Physics

LSE captures universal behavior in quantum phases and non-equilibrium dynamics:

• Many-body criticality and conformal field theory: In $(1+1)d$ critical systems, the linear stabilizer entropy admits a field-theoretic description, exhibiting universal, system-size-independent subleading terms given by BCFT boundary $g$-factors and logarithmic scaling in mutual LSE governed by BCCO scaling dimensions. These features hold smoothly in the $\alpha\to 1$ (linear/von Neumann) limit (Hoshino et al., 17 Mar 2025).
• Diagnostic of complexity and phase transitions: LSE, together with antiflatness and entanglement spectrum diagnostics, consistently peaks at quantum phase transitions in spin models, distinguishing phases by computational complexity not captured by entanglement entropy alone (Viscardi et al., 11 Mar 2025).
• Dynamics and quantum chaos: Under quenches, LSE spreads ballistically with subsystem size and time, equilibrates on a short ($O(L)$) timescale, and fully saturates in non-integrable (chaotic) systems, distinguishing complex, non-classically simulable dynamics (Odavić et al., 13 Dec 2024, Rattacaso et al., 2023).

8. Measurement, Non-locality, and Fundamental Limits

LSE rigorously characterizes the resource content harvested from quantum fields and links to quantum measurement theory:

• Resource harvesting from quantum fields: LSE (specifically, the non-local linear SRE) quantifies the genuinely non-local magic that can be extracted by Unruh-DeWitt detectors from the quantum vacuum. While entanglement and non-local magic can be harvested, CHSH-type Bell non-locality remains unachievable under realistic protocol conditions (Cepollaro et al., 16 Dec 2024).
• Connection to incompatibility and quantum foundations: For qubits, $M^{\mathrm{lin}}$ encodes the degree to which a state fails to exhibit complete incompatibility with respect to $X$, $Y$, $Z$, connecting resource theory of magic to foundational quantum measurement properties (Iannotti et al., 25 Oct 2025).

9. Summary Table: Key Mathematical and Operational Properties

Property/Task	Linear Stabilizer Entropy $M^{\mathrm{lin}}$	Relevance/Scaling
Faithfulness	$=0$ iff stabilizer state	Quantifies non-stabilizerness
Additivity	Yes	Sensible for resource extensivity
Strong monotonicity (resource theory)	Yes, for all integer $\alpha\geq 2$	Operational conversion/probabilistic bounds
Experimental measurability	Yes (randomized Pauli/Clifford measurements)	Efficient practical benchmark
Computational tractability	Yes (QMC, ML, tensor networks...)	Efficient for moderate/large system sizes
Robust operational quantification	Yes: governs property testing, resource conversion rates	Both classicality/randomness limit and universality
Universal signature in many-body/field theory	Yes: universal subleading, logarithmic mutual scaling	Diagnostic for criticality, phases, and complexity
Geometric/measurement-theoretic significance	Yes: Van Hove singularities, incompatibility deficit	Fundamental in single/multi-qubit quantum structure

10. Concluding Perspective

Linear stabilizer entropy serves as a mathematically rigorous, operationally meaningful, and experimentally/logically accessible magic monotone. It unifies magic-state resource theory, the structure of entropy cones, quantum measurement incompatibility, topological order, computational complexity measures, and foundational aspects of quantum information. As the unique, strongly monotonic, and scalable entropic magic monotone to date, it is an indispensable tool for theoretical characterization and experimental benchmarking of quantum resources beyond the stabilizer formalism.