Stabilizer Purity in Quantum States

• Stabilizer purity is defined via Pauli expectation values and quantifies how closely a quantum state resembles a stabilizer state, serving as a key measure of magic.
• Operational protocols use mixed-unitary Pauli channels and swap-test-based purity estimation to efficiently diagnose quantum state magic and noise resilience.
• The concept connects to Rényi entropies, measurement incompatibility, and ancilla–system entanglement, offering practical metrics for resource quantification in quantum computation.

Stabilizer purity quantifies the degree to which a quantum state resembles a stabilizer state, serving as a central monotone in the resource theory of non-stabilizerness (magic). It also underpins algorithms for magic certification, resource estimation, and purity diagnostics in quantum computation. The concept is operationalized via statistical properties of Pauli operator expectation values, relates mathematically to stabilizer Rényi entropies, and enjoys deep connections to both measurement incompatibility and entanglement structure.

1. Mathematical Definition of Stabilizer Purity

For a pure quantum state $\ket{\psi}$ on $n$ qubits ($d=2^n$), the stabilizer purity of order $\alpha$ is given by

$$\Xi_\alpha(\ket{\psi}) = \frac{1}{d} \sum_{P\in \mathbb{P}_n} |\langle \psi | P | \psi \rangle|^{2\alpha}$$

where $\mathbb{P}_n$ denotes the $n$-qubit Pauli group. This characteristic distribution $$\mathcal{D}(\ket{\psi}) = \left\{ d^{-1} |\langle\psi|P_j|\psi\rangle|^2 \right\}_{j=0}^{d^2-1}$$ underlies the computation of the $\alpha$-Stabilizer Rényi Entropy (SRE)

$$M_\alpha(\ket{\psi}) = \frac{1}{1-\alpha} \ln \Xi_\alpha(\ket{\psi})$$

with $\Xi_\alpha = A_\alpha(\ket{\psi})$ in the notation of (Stratton, 3 Jul 2025).

Stabilizer purity is maximized ($\Xi_\alpha=1$) for stabilizer states and reduced for non-stabilizer ("magic") states. The linear stabilizer entropy, $M^\mathrm{lin}_\alpha(\psi) = 1 - \Xi_\alpha(\psi)$, provides an operationally convenient, additive monotone. Both SRE and linear stabilizer entropy vanish if and only if $\ket{\psi}$ is a stabilizer state (Iannotti et al., 25 Oct 2025).

2. Operational Measurement and Purity-Estimation Protocols

A principal advance in stabilizer purity research is the reduction of SRE estimation to quantum purity estimation on a class of mixed-unitary encoded states. Specifically, for pure $\ket{\psi}$ and integer $\alpha > 1$, the action of the mixed-unitary Pauli channel $\mathcal{E}_\mathcal{P}$ on $\alpha$ copies produces

$$\rho_{\mathcal{P},\alpha} := \mathcal{E}_\mathcal{P}(\psi^{\otimes\alpha}) = \frac{1}{d^2} \sum_{j=0}^{d^2-1} (P_j \psi P_j)^{\otimes\alpha}$$

whose purity is directly related to stabilizer purity: $$\operatorname{tr}[\rho_{\mathcal{P},\alpha}^2] = d^{-1} \Xi_\alpha(\ket{\psi})$$ (Stratton, 3 Jul 2025). Thus, $M_\alpha$ can be estimated experimentally by preparing $\alpha$ copies, applying a uniformly-random Pauli, discarding the label, and performing purity measurements (e.g., via the swap test or randomized measurements).

Algorithmically:

• The required physical resources scale as $\mathcal{O}(\alpha d^2 \epsilon^{-2})$ copies for additive error $\epsilon$.
• Benchmarking against single-qubit states confirms agreement with theoretical values for all integer $\alpha > 1$.
• Compared to state tomography, this method is more efficient for even $\alpha$ and less so for odd $\alpha$; the state-of-the-art method ([Phys. Rev. Lett. 132, 240602 (2024)]) is strictly superior for all cases, but at higher circuit complexity (Stratton, 3 Jul 2025).

3. Statistical and Geometric Properties of Stabilizer Purity

For Haar-random pure states, the probability density function (PDF) of stabilizer purity exhibits non-analytic features. Notably:

• For a single qubit ($d=2$) and $\alpha \geq 2$, the PDF of $\Xi_\alpha$ displays a logarithmic divergence ("Van Hove singularity") at the value corresponding to the $|H\rangle$-magic state ($\Xi_\alpha = (1 + n_c)/2$, $n_c=1/2$) (Iannotti et al., 25 Oct 2025).
• The explicit PDF for $\alpha=2$,

$$P_{N_2}(n) = \begin{cases} \frac{4}{\pi} \int_{x_-}^{x_+} \frac{dx}{\sqrt{(1-x^2)^4-[3(1-x^2)^2 + 4x^4 - 4n]^2}}, & n \in [1/3,1/2) \ \cdots & n\in (1/2,1] \end{cases}$$

exhibits

$$P_{N_2}(n) \propto -\frac{3}{\sqrt{2}\pi} \ln | n - 1/2 |$$

as $n \to 1/2$. This universal behavior is absent for $d\geq 3$.

• The singularity reflects the fact that Haar-random pure states are statistically most likely to possess magic corresponding to $|H\rangle$.

This geometric effect is particular to stabilizer-based magic monotones; coherence and related quantities do not show similar singularities (Iannotti et al., 25 Oct 2025).

4. Fundamental Connections: Measurement Incompatibility and Entanglement

Stabilizer purity—especially its linearized form—has a direct operational interpretation in terms of quantum measurement incompatibility. For one-qubit pure states,

$$\Gamma_2(\psi) = 4\Xi_2(\psi) = 4(1 - M_2^{\text{lin}}(\psi))$$

gives the partial incompatibility with respect to Pauli $X$, $Y$, $Z$ measurements. Stabilizer states (Pauli eigenstates) exhibit maximal incompatibility, while magic states minimize it (Iannotti et al., 25 Oct 2025). This reveals a direct link: non-stabilizerness implies a deficit in measurement incompatibility, central to the structure of quantum theory.

Moreover, there exists a precise relationship between the non-stabilizerness of an original state and the entanglement generated across the ancilla–system bipartition in the SRE-encoding purifications: $$(1-\alpha) M_\alpha(\ket{\psi}) + E_2(\ket{\psi'^{(\alpha)}_{A\tilde{B}}}) = \ln d$$ where $E_2$ is the Rényi-2 entropy of entanglement across the split and $\ket{\psi'^{(\alpha)}_{A\tilde{B}}}$ is the coherently prepared, purity-encoding state (Stratton, 3 Jul 2025). Thus, higher magic in $\ket{\psi}$ yields more ancilla–system entanglement in this context.

5. Stabilizer Purity Under Noise and in Practical Certification Protocols

Stabilizer purity serves as a practical diagnostic for noise resilience:

• Under Pauli noise channels, stabilizer states evolve to mixed states, and their purity degrades in a tractable, efficiently simulable manner (Mor-Ruiz et al., 2022).
• The noisy stabilizer formalism allows linear-in-$n$ computation of the final mixed state's purity (and related quantities) after arbitrary Clifford evolution and Pauli (possibly correlated) noise, provided the final state supports a small number of qubits.

For experimental certification:

• Efficient, sample-optimal protocols for certifying purity/fidelity to stabilizer states are constructed from local Pauli measurement settings (Dangniam et al., 2020). For infidelity $\epsilon$ and significance $\delta$, the sample complexity is $N\sim (3/2)\ln(1/\delta)/\epsilon$, independent of qubit number.
• Passing all such protocol tests certifies purity $\geq 1-\epsilon$ with high confidence.

6. Numerical Estimation and Computational Aspects

Computable proxies for stabilizer purity—such as stabilizer extent $\xi(\psi)$ and stabilizer fidelity—quantify how efficiently a state can be decomposed into, or approximated by, stabilizer states (Hamaguchi et al., 24 Jun 2024).

• Efficient algorithms using column generation, branch-and-bound pruning, and canonical enumerations now allow exact computation of stabilizer extent or fidelity for generic pure states up to 9–10 qubits (for real amplitude states).
• Stabilizer purity, extent, and fidelity are tightly linked: low stabilizer fidelity or high extent implies high non-stabilizerness/purity.

For property testing, single-copy stabilizer purity tests leveraging Clifford randomization and computational difference sampling provide sample-complexity-optimal certification of the stabilizer property, with direct operational interpretations in terms of purity over subspaces (Hinsche et al., 10 Oct 2024).

7. Summary Table: Key Definitions and Algorithms

Quantity	Definition / Relation
Stabilizer purity	$$\Xi_\alpha(\ket{\psi}) = d^{-1} \sum_{P\in \mathbb{P}_n} |\langle \psi|P|\psi\rangle|^{2\alpha}$$
Stabilizer Rényi entropy	$$M_\alpha(\ket{\psi}) = (1-\alpha)^{-1} \ln \Xi_\alpha(\ket{\psi})$$
Linear stabilizer entropy	$M_{\text{lin}} = 1-\Xi_\alpha$
Purity–SRE operational link	$$\operatorname{tr}[\rho^2] = d^{-1} \Xi_\alpha(\ket{\psi})$$ for SRE-encoding state $\rho$
Swap-test resources	$\mathcal{O}(\alpha d^2 \epsilon^{-2})$ copies to estimate $M_\alpha$
Noisy stabilizer formalism	Linear-in-$n$ tracking of purity under Clifford+Pauli evolution (Mor-Ruiz et al., 2022)
Certification protocol	$N\sim (3/2)\ln(1/\delta)/\epsilon$ for $\geq 1-\epsilon$ purity (Dangniam et al., 2020)
Magic–entanglement relation	$(1-\alpha)M_\alpha + E_2 = \ln d$ for SRE-encoding state (Stratton, 3 Jul 2025)

• "Noisy Stabilizer Formalism" (Mor-Ruiz et al., 2022)
• "Van Hove singularities in stabilizer entropy densities" (Iannotti et al., 25 Oct 2025)
• "Optimal verification of stabilizer states" (Dangniam et al., 2020)
• "Phase transition in Stabilizer Entropy and efficient purity estimation" (Leone et al., 2023)
• "Single-copy stabilizer testing" (Hinsche et al., 10 Oct 2024)
• "Faster computation of nonstabilizerness" (Hamaguchi et al., 24 Jun 2024)
• "An Algorithm for Estimating $\alpha$-Stabilizer Rényi Entropies via Purity" (Stratton, 3 Jul 2025)