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Operational interpretation of the Stabilizer Entropy

Lennart Bittel, Lorenzo Leone

Abstract

Magic-state resource theory is a fundamental framework with far-reaching applications in quantum error correction and the classical simulation of quantum systems. Recent advances have significantly deepened our understanding of magic as a resource across diverse domains, including many-body physics, nuclear and particle physics, and quantum chemistry. Central to this progress is the stabilizer Rényi entropy, a computable and experimentally accessible magic monotone. Despite its widespread adoption, a rigorous operational interpretation of the stabilizer entropy has remained an open problem. In this work, we provide such an interpretation in the context of quantum property testing. By showing that the stabilizer entropy is the most robust measurable magic monotone, we demonstrate that the Clifford orbit of a quantum state becomes exponentially indistinguishable from Haar-random states, at a rate governed by the stabilizer entropy $M_α(ψ)$ and the number of available copies. This implies that the Clifford orbit forms an approximate state $k$-design, with an approximation error $\exp(-Θ(M_α(ψ)))$ for $α\ge2$. Conversely, we establish that the optimal probability of distinguishing a given quantum state from the set of stabilizer states is also governed by its stabilizer entropy. These results reveal that the stabilizer entropy quantitatively characterizes the transition from stabilizer states to universal quantum states, thereby offering a comprehensive operational perspective of the stabilizer entropy as a quantum resource.

Operational interpretation of the Stabilizer Entropy

Abstract

Magic-state resource theory is a fundamental framework with far-reaching applications in quantum error correction and the classical simulation of quantum systems. Recent advances have significantly deepened our understanding of magic as a resource across diverse domains, including many-body physics, nuclear and particle physics, and quantum chemistry. Central to this progress is the stabilizer Rényi entropy, a computable and experimentally accessible magic monotone. Despite its widespread adoption, a rigorous operational interpretation of the stabilizer entropy has remained an open problem. In this work, we provide such an interpretation in the context of quantum property testing. By showing that the stabilizer entropy is the most robust measurable magic monotone, we demonstrate that the Clifford orbit of a quantum state becomes exponentially indistinguishable from Haar-random states, at a rate governed by the stabilizer entropy and the number of available copies. This implies that the Clifford orbit forms an approximate state -design, with an approximation error for . Conversely, we establish that the optimal probability of distinguishing a given quantum state from the set of stabilizer states is also governed by its stabilizer entropy. These results reveal that the stabilizer entropy quantitatively characterizes the transition from stabilizer states to universal quantum states, thereby offering a comprehensive operational perspective of the stabilizer entropy as a quantum resource.

Paper Structure

This paper contains 23 sections, 21 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of the results
  3. Relation to prior work and main contributions
  4. Structure of the paper
  5. Magic-state resource theory and the Clifford commutant
  6. Magic state resource theory and magic monotones
  7. Stabilizer Rényi entropies
  8. The Clifford commutant and generalized stabilizer purities
  9. Measurable magic monotones
  10. Operational interpretation of stabilizer entropies
  11. State $k$-designs
  12. Stabilizer property testing
  13. Conclusion and open questions
  14. Preliminaries
  15. Haar average over the Clifford group
  16. ...and 8 more sections

Key Result

Lemma 1

All unbiased estimators on less than $n-1$ copies aiming at measuring stabilizer purities $P_{2\alpha}$ with $\alpha$ even up to error $\varepsilon$ requires $\Omega(d^2\varepsilon^{-2})$ sample access to $\psi$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Stabilizer entropies have a double-edge operational interpretation. In a symmetric property testing scenario, the probability of distinguishing the Clifford orbit of the state $\ket{\psi}$ from the set of stabilizer state increases exponentially with $M_3$ (left). Conversely, the probability of distinguishing the Clifford orbit of the state $\ket{\psi}$ from Haar random states converges exponentially to the probability of random guessing with $M_2$ (right).

Theorems & Definitions (42)

  • Definition 1: Stabilizer Monotones
  • Definition 2: Stabilizer entropies leone_stabilizer_2022
  • Lemma 1
  • Definition 3: Generalized stabilizer purities
  • Lemma 2
  • Theorem 1
  • Definition 4: Measurable stabilizer monotones
  • Lemma 3: Measurable magic monotones lie in the Clifford group commutant bittel2025completetheorycliffordcommutant
  • Lemma 4
  • Theorem 2: Stabilizer purity upper bounds generalized purities
  • ...and 32 more