Stabilizer Rényi Entropy in Quantum Magic

• Stabilizer Rényi Entropy is a resource measure defined via the 2α-th stabilizer purity that quantifies nonstabilizerness and vanishes for stabilizer states.
• It governs the exponential indistinguishability of Clifford orbits from Haar randomness and sets criteria for forming k-designs in quantum circuits.
• SRE regulates optimal discrimination from stabilizer states, underpinning its use in randomness testing and magic certification in quantum resource theory.

The stabilizer Rényi entropy (SRE) is a central quantitative measure of nonstabilizerness—or "magic"—in quantum states, operationalizing the resource character of states outside the stabilizer (Clifford) framework. SRE is of fundamental interest in resource theory, quantum property testing, the simulation complexity of quantum systems, and the characterization of randomness generation in quantum circuits. Its practical significance is anchored in its computability, additivity, and invariance properties, but until recently, its operational interpretation had remained unresolved. The SRE now admits a rigorous operational perspective by connecting quantum magic to property-testing scenarios, where it precisely governs both the indistinguishability of Clifford orbits from Haar randomness and the optimal discrimination from stabilizer states (Bittel et al., 30 Jul 2025).

1. Formal Definition and Magic Monotone Properties

The α-stabilizer Rényi Entropy (SRE) for a pure $n$-qubit state $|\psi\rangle$ is defined using the $2\alpha$-th stabilizer purity: $$P_{2\alpha}(\psi) = \frac{1}{d} \sum_{P\in\mathbb{P}_n} \langle\psi|P|\psi\rangle^{2\alpha}, \quad d = 2^n$$ The order-$\alpha$ SRE is then

$$M_\alpha(\psi) = -\frac{1}{1-\alpha} \log_2 P_{2\alpha}(\psi)$$

For $\alpha=2$: $$M_2(\psi) = -\log_2\left( \frac{1}{d} \sum_P \langle\psi|P|\psi\rangle^4 \right)$$ Key properties making $M_\alpha$ a valid magic monotone:

• $M_\alpha(\sigma)=0$ iff $\sigma$ is a stabilizer state.
• Invariant under Clifford unitaries: $M_\alpha(C\psi) = M_\alpha(\psi)$ for any Clifford $C$.
• Non-increasing under all stabilizer operations.
• Additive under tensor product: $$M_\alpha(\psi\otimes\phi) = M_\alpha(\psi) + M_\alpha(\phi)$$.

2. Clifford Orbit Indistinguishability from Haar Randomness

The SRE supplies a quantitative control on the exponential indistinguishability between the Clifford orbit of a quantum state and Haar-random states. Let $\mathcal{E}_\psi = \{C|\psi⟩ : C \in \Cl_n\}$ be the Clifford orbit. In the hypothesis testing problem of distinguishing $k$ copies of a state sampled from either $\mathcal{E}_\psi$ or Haar, the optimal success probability is

$$q_{\mathrm{succ}}^{(k)} = \frac12 + \Theta(2^{-M_2(\psi)})$$

The distinguishing bias is computed via the trace norm distance between the $k$-fold Clifford-twirled and Haar-twirled states. Bounding all generalized stabilizer purities as $P_\Omega(\psi) \leq 2^{-M_2(\psi)}$ shows that the trace-norm difference, and thus the bias, decays exponentially with $M_2(\psi)$. This result establishes that a large $M_2(\psi)$ makes the Clifford orbit exponentially hard to distinguish from Haar randomness with a constant number of copies.

3. Approximate $k$-Designs and Magic Certification

The SRE controls convergence to state $k$-designs. The Clifford orbit forms an $\varepsilon$-approximate state $k$-design in trace norm when

$$M_{2}(\psi) \gtrsim \frac{k^2}{2} + \log_2 \left( \frac{1}{\varepsilon} \right)$$

Precisely, the 1-norm distance between the $k$-fold average over the orbit and the $k$-fold Haar average is upper-bounded by $\varepsilon$ when this condition is met. Conversely, if $M_3(\psi) = O(\log_2(1/\varepsilon))$, then the Clifford orbit does not form a good $k$-design. Increasing the SRE ensures more randomness is generated by Clifford twirling, making the state more Haar-like in higher tensor powers.

4. Optimal Discrimination from Stabilizer States

SRE also governs the optimal success probability in distinguishing a given state from the set of stabilizer states. With $k=6$ copies, the optimal success probability for the discrimination task is

$$p_{\mathrm{succ}}^{(6)} = \frac12 + \frac14 [1 - 2^{-2 M_3(\psi)}]$$

The optimal measurement projects onto the +1 eigenspace of a Hermitian invariant in the Clifford commutant ($\Omega_6$), effectively measuring the sixth stabilizer purity. For $k>6$ one can boost the success exponentially by using tensor powers of the six-copy test, and the amplification rate is controlled by $M_3(\psi)$. The upper and lower bounds on $p_{\mathrm{succ}}^{(k)}$ show exponential amplification, underscoring $M_3(\psi)$ as the relevant magic monotone for certification against stabilizer states.

5. Resource Transitions and Operational Meaning

The stabilizer Rényi entropy thus quantifies the transition from free (zero SRE) stabilizer states to universal magic states (large SRE):

• Larger $M_2(\psi)$ implies the Clifford orbit is harder to distinguish from random and approximates a $k$-design more closely.
• Larger $M_3(\psi)$ makes it easier to certify magic by discrimination from stabilizer states.
• SRE, as a one-parameter family indexed by $\alpha$, interpolates between these operational tasks.

This operational interpretation assigns SRE as the most robust measurable magic monotone: it simultaneously governs the rates for randomness-testing (equilibration to Haar) and stabilizer-testing (certification of nonstabilizerness). As such, $M_\alpha(\psi)$ has a clear experimental signature: it sets the exponential rate at which the Clifford orbit of a state becomes indistinguishable from Haar random ($M_2$), and the rate at which it remains distinguishable from the stabilizer polytope ($M_3$).

6. Summary Table of Key Quantities

Quantity	Formula	Operational Role
$\alpha$-SRE $M_\alpha(\psi)$	$$- \frac{1}{1-\alpha} \log_2 \left[ \frac{1}{d} \sum_P \langle \psi|P|\psi\rangle^{2\alpha} \right]$$	Magic monotone; vanishes on stabilizer states
Haar distinguishability	$$q_{\mathrm{succ}}^{(k)} = 1/2 + \Theta(2^{-M_2(\psi)})$$	Confidence in distinguishing orbit from random
$k$-Design criterion	$M_2(\psi) \gtrsim k^2/2 + \log_2(1/\varepsilon)$	When Clifford orbit forms an $\varepsilon$-design
Stabilizer discrimination	$$p_{\mathrm{succ}}^{(6)} = 1/2 + (1/4)[1-2^{-2 M_3(\psi)}]$$	Certifying magic by optimal measurement

The quantitative structure of SRE as a resource monotone and its tight connection to randomized benchmarking, $k$-design formation, and hypothesis testing tasks makes it pivotal for both theoretical and experimental advances in quantum resource theory, circuit complexity, and the classical simulation of quantum circuits.