Stabilizer Rényi Entropies: Quantum Magic Measure

• Stabilizer Rényi Entropies are a measure of nonstabilizerness that quantifies quantum magic using Rényi-type entropies over squared Pauli-string expectation values.
• They are computed efficiently in matrix product states through replica methods and transfer matrices, with algorithmic scaling polynomial in system size and bond dimension.
• SRE provides practical insights into quantum phase transitions and criticality, underpinning operational tasks like magic-state distillation and simulation hardness in quantum circuits.

A stabilizer Rényi entropy (SRE) is a faithful, Clifford-invariant, additive measure of nonstabilizerness ("magic") in pure and mixed quantum states, quantifying the resource cost and complexity associated with non-Clifford operations in many-body quantum systems. For matrix product states (MPS), SREs are efficiently computable, providing both analytic and numerical access to the scaling and the structural features of quantum magic in regimes ranging from symmetry-protected phases to critical points. SREs are defined via Rényi-type entropies over the distribution of squared Pauli-string expectation values, with order-$n$ SRE given by $$M^{(n)}(|\psi\rangle) = (1-n)^{-1} \ln \left[ 2^{-N} \sum_{P} \langle\psi|P|\psi\rangle^{2n} \right]$$. In translation-invariant MPS, the SRE density is determined by the spectrum of a local transfer matrix, while in generic MPS and higher tensor networks, efficient algorithms scale polynomially in system size and bond dimension.

1. Definition and Resource-Theoretic Properties

Let $|\Psi_N\rangle$ be a pure $N$-qubit state, and let $\mathcal{P}_N$ denote the set of all $N$-qubit Pauli strings. For integer Rényi order $n > 1$, the stabilizer Rényi entropy is

$$M^{(n)}(|\Psi_N\rangle) = (1-n)^{-1} \ln\left[ 2^{-N} \sum_{P \in \mathcal{P}_N} \langle\Psi_N|P|\Psi_N\rangle^{2n} \right].$$

Equivalently, by introducing a stabilizer norm tensor $\rho_\psi^{(n)}$, $$M^{(n)}(|\psi\rangle) = (1-n)^{-1}\ln\operatorname{Tr}\rho_\psi^{(n)}$$ (Haug et al., 2022).

Key properties:

• Faithfulness: $M^{(n)} = 0$ iff $|\Psi_N\rangle$ is a stabilizer (Clifford) state.
• Clifford invariance: Unaffected by Clifford unitaries.
• Additivity: $$M^{(n)}(|\psi\rangle \otimes |\phi\rangle) = M^{(n)}(|\psi\rangle) + M^{(n)}(|\phi\rangle)$$.
• Extensive scaling: For TI MPS, SRE density emerges as $$m^{(n)} = \lim_{N\to\infty} M^{(n)}(|\Psi_N\rangle)/N$$ (Haug et al., 2022, Liu et al., 5 Aug 2025).

2. Replica-MPS and Efficient Algorithms

The SRE for MPS is computable via the norm of a replica MPS of bond dimension $\chi^{2n}$: $$2^{-N} \sum_{P} \langle\Psi_N|P|\Psi_N\rangle^{2n} = \langle\Phi_N^{(n)}|\Phi_N^{(n)}\rangle,$$ where $|\Phi_N^{(n)}\rangle$ is constructed from an explicit sitewise tensor product involving the original MPS tensors and a representation of the local Pauli basis (Haug et al., 2022).

Algorithmic complexity:

• Generic MPS: $O(N\,2^{2(n-1)}\,\chi^{6n})$; for $n=2$, symmetry can be exploited to achieve $O(N\chi^6)$.
• Translation-invariant MPS: SRE density extracted from the leading eigenvalue $\lambda_0^{(n)}$ of a $2n$-replica transfer matrix, $m^{(n)} = (1-n)^{-1}\ln\lambda_0^{(n)}$ (Haug et al., 2022, Liu et al., 5 Aug 2025).

3. Scaling, Criticality, and Locality in Quantum Ising Chains

Stabilizer Rényi entropy is a sensitive diagnostic of quantum phase transitions and criticality:

• For the transverse-field Ising model, SRE density peaks near the quantum critical point $h/J=1$.
• Finite-size scaling leads to the extraction of universal critical exponents and collapse of $m(h,N)$ data under appropriate scaling transformations (Haug et al., 2022).
• Basis dependence: SRE is not invariant under single-qubit rotations; the critical peak shifts depending on the computational basis, but the subleading term develops sharp extremum at $h_c$ independent of basis (Haug et al., 2022).

Long-range (irreducible) magic at criticality remains robust under basis optimization, evidencing its origin in genuinely non-local many-body quantum correlations.

4. Comparison to Other Magic Monotones and Operational Interpretation

SRE upper-bounds other quantifiers of nonstabilizerness:

• For $n \geq 1/2$: $M^{(n)} \leq 2\, \log R$, where $R$ is the robustness of magic (Haug et al., 2023, Leone et al., 2021).
• For $n > 1$: $M^{(n)} \leq (2n/(n-1))\, D_{\min}$, with $D_{\min}$ the min-relative entropy of magic.

Operational connections:

• SRE governs the rate at which the Clifford orbit of a state becomes indistinguishable from Haar-random states, quantifying magic-state dilution and simulation hardness (Bittel et al., 30 Jul 2025).
• The optimal probability of distinguishing a quantum state from the set of stabilizer states scales exponentially with the SRE, giving rise to operational thresholds for state and circuit certification (Bittel et al., 30 Jul 2025).
• SRE appears in magic-state distillation bounds and fault-tolerant circuit cost estimates.

5. Numerical and Sampling Methods

Efficient contraction and sampling are possible:

• Perfect MPS sampling: Enables computation of the von Neumann stabilizer entropy $SE_1$ for large bond dimensions by exact sampling of the Pauli-spectrum probabilities via progressive transfer-matrix contraction (Haug et al., 2023).
• Randomized measurements: For $n=2$, experimental protocols use Clifford twirling and computational-basis measurement to estimate SRE (Leone et al., 2021, Haug et al., 2023). Complexity scales as $O(n N)$ for copy and classical computational resources.
• Quantum Monte Carlo: SRE for spin systems with sign-problem-free Hamiltonians is efficiently accessible via nonequilibrium QMC, mapping the SRE to the ratio of replica partition functions connected by interpolating protocols; complexity is polynomial in system size (Liu et al., 2024).

6. Physical Interpretation, Limitations, and Open Problems

SRE represents the “magic cost” or the minimal number of non-Clifford resources required to synthesize a state:

• Much ground-state SRE is short-ranged and can be reduced by local basis changes, yet a nonzero long-range component persists and peaks at criticality, manifesting as irreducible magic (Haug et al., 2022).
• Main limitations: computational cost grows rapidly for large $n$, only integer orders efficiently accessible, and degeneracies in transfer matrix spectra require additional care.
• Open directions: extension to higher-dimensional tensor networks (PEPS, MERA), analytical study of replica transfer-matrix spectra, improvement of MPS approximation error bounds, and development of experimental measurement protocols (Haug et al., 2022).

SREs thus provide a tractable, information-rich, and physically meaningful tool for quantifying nonstabilizerness in large many-body systems, tightly connecting tensor-network methods, resource theory, and quantum simulation complexity.