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arxiv: 2504.18098 · v1 · submitted 2025-04-25 · 🪐 quant-ph · cond-mat.stat-mech· cs.CC

Efficient witnessing and testing of magic in mixed quantum states

Tobias Haug , Poetri Sonya Tarabunga This is my paper

Pith reviewed 2026-05-22 18:39 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcs.CC
keywords magicnonstabilizernessstabilizer Rényi entropymixed quantum statesproperty testingnoisy quantum circuitsquantum resources
0
0 comments X

The pith

Stabilizer Rényi entropy provides efficient witnesses that detect and quantify magic in mixed quantum states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops efficient witnesses for nonstabilizerness, or magic, in mixed quantum states by using the stabilizer Rényi entropy. These witnesses detect the presence of magic and give quantitative estimates of magic monotones. Efficient property testing algorithms distinguish states with high versus low magic when the entropy is bounded. The approach certifies the number of noisy T-gates under various noise models and was verified experimentally on the IonQ processor, where magic remained robust even under exponentially strong noise. The methods also apply to matrix product states and carry implications for hiding magic in cryptographic settings.

Core claim

Stabilizer Rényi entropy serves as an efficient witness for magic in mixed states that robustly indicates its presence, quantitatively estimates magic monotones, and supports property testing algorithms to distinguish high-magic from low-magic states under a bounded-entropy assumption.

What carries the argument

Stabilizer Rényi entropy, used as a witness that measures nonstabilizerness and enables both detection and quantitative estimation in mixed states.

If this is right

  • The number of noisy T-gates can be certified under a wide class of noise models.
  • Magic persists in noisy random circuits even under exponentially strong noise.
  • Subsystems of many-body states can contain extensive magic despite entanglement.
  • Mimicking high-magic states with little actual magic requires an extensive amount of entropy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The observed noise robustness suggests magic resources may survive in realistic quantum devices more readily than expected.
  • The witnesses could be combined with existing quantum verification protocols to certify computational resources without full state tomography.
  • Similar entropy-based witnesses might extend to other nonclassical resources such as contextuality.

Load-bearing premise

Property testing reliably separates high-magic from low-magic states only when the stabilizer Rényi entropy remains bounded.

What would settle it

An explicit mixed state known to contain high magic, with bounded stabilizer Rényi entropy, that the testing algorithm nonetheless classifies as low-magic.

Figures

Figures reproduced from arXiv: 2504.18098 by Poetri Sonya Tarabunga, Tobias Haug.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic representation of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Magic of random local circuit of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. 2-SRE magic witness [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

Nonstabilizerness or `magic' is a crucial resource for quantum computers which can be distilled from noisy quantum states. However, determining the magic of mixed quantum has been a notoriously difficult task. Here, we provide efficient witnesses of magic based on the stabilizer R\'enyi entropy which robustly indicate the presence of magic and quantitatively estimate magic monotones. We also design efficient property testing algorithms to reliably distinguish states with high and low magic, assuming the entropy is bounded. We apply our methods to certify the number of noisy T-gates under a wide class of noise models. Additionally, using the IonQ quantum computer, we experimentally verify the magic of noisy random quantum circuits. Surprisingly, we find that magic is highly robust, persisting even under exponentially strong noise. Our witnesses can also be efficiently computed for matrix product states, revealing that subsystems of many-body quantum states can contain extensive magic despite entanglement. Finally, our work also has direct implications for cryptography and pseudomagic: To mimic high magic states with as little magic as possible, one requires an extensive amount of entropy. This implies that entropy is a necessary resource to hide magic from eavesdroppers. Our work uncovers powerful tools to verify and study the complexity of noisy quantum systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to develop efficient witnesses for nonstabilizerness (magic) in mixed quantum states based on the stabilizer Rényi entropy; these witnesses indicate the presence of magic and quantitatively estimate magic monotones. It further designs efficient property testing algorithms to distinguish high- from low-magic states under a bounded-entropy assumption. The methods are applied to certify the number of noisy T-gates under various noise models, experimentally verified on the IonQ quantum computer (showing magic persists under exponentially strong noise), extended to matrix product states (revealing extensive magic in subsystems despite entanglement), and used to derive cryptographic implications that entropy is required to hide magic from eavesdroppers.

Significance. If the central constructions and experimental claims hold after clarification, the work would supply practical, efficiently computable tools for verifying magic in noisy quantum systems, directly relevant to resource theories, fault-tolerant quantum computing, and many-body physics. The reported robustness of magic and the MPS results would be notable strengths; the cryptographic link provides a broader implication for pseudomagic and entropy as a hiding resource.

major comments (2)
  1. [Property testing algorithms] Property testing algorithms section: the claim of efficient distinction between high- and low-magic mixed states is conditioned on the assumption that the stabilizer Rényi entropy is bounded. For generic noisy states this bound is not known a priori; the manuscript does not supply an efficient subroutine that both enforces and verifies the bound internally, rendering the 'efficient' label conditional on an external promise rather than unconditionally guaranteed.
  2. [Experimental verification] Experimental verification section: the IonQ experiments are presented as verifying magic in noisy random circuits and demonstrating robustness under strong noise, yet the manuscript provides neither full derivations of the witness quantities, detailed error analysis, nor the underlying data tables. This absence prevents independent verification of the quantitative robustness claim.
minor comments (1)
  1. [Abstract] Abstract: the qualifier 'assuming the entropy is bounded' is stated without a quantitative definition or reference to the precise bound used in the algorithms; a brief clarification would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which have helped us clarify several points. We address each major comment below and indicate the revisions made.

read point-by-point responses

  1. Referee: [Property testing algorithms] Property testing algorithms section: the claim of efficient distinction between high- and low-magic mixed states is conditioned on the assumption that the stabilizer Rényi entropy is bounded. For generic noisy states this bound is not known a priori; the manuscript does not supply an efficient subroutine that both enforces and verifies the bound internally, rendering the 'efficient' label conditional on an external promise rather than unconditionally guaranteed.

    Authors: We agree that the efficiency of the property testing algorithms is conditional on the bounded stabilizer Rényi entropy assumption, which is explicitly stated in the relevant section of the manuscript. This is a standard formulation in property testing, where the promise enables efficient distinction between high- and low-magic states. We do not claim an unconditional efficient algorithm that works for arbitrary mixed states without any promise on the entropy. The manuscript does not include an internal subroutine to enforce or verify the bound because such a subroutine would generally require additional assumptions on state preparation and is outside the scope of the current algorithms. We have added a clarifying paragraph in the revised version to emphasize the conditional nature of the efficiency claim and to discuss how the bound might be estimated in practice under specific noise models. revision: partial

  2. Referee: [Experimental verification] Experimental verification section: the IonQ experiments are presented as verifying magic in noisy random circuits and demonstrating robustness under strong noise, yet the manuscript provides neither full derivations of the witness quantities, detailed error analysis, nor the underlying data tables. This absence prevents independent verification of the quantitative robustness claim.

    Authors: We appreciate the referee's point that additional details are needed for independent verification of the experimental results. In the revised manuscript and supplementary material, we now include the full derivations of the witness quantities computed from the IonQ data, a detailed error analysis accounting for the dominant noise sources, and the underlying data tables. These additions allow readers to reproduce the quantitative claims regarding the persistence of magic under strong noise. revision: yes

Circularity Check

0 steps flagged

No circularity: witnesses and testers built on established entropy definition under explicit assumption

full rationale

The paper defines efficient witnesses directly from the stabilizer Rényi entropy (an externally established monotone) and presents property-testing algorithms whose efficiency is conditioned on a stated bounded-entropy promise. No equation or algorithm reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The central claims retain independent algorithmic content once the assumption is granted; verification of the bound itself is outside the claimed scope and does not create a hidden circular loop.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The approach rests on the prior definition of stabilizer Rényi entropy and the explicit assumption that entropy is bounded for the property-testing guarantees; no new free parameters, axioms, or invented entities are introduced in the abstract.

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Forward citations

Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Triangle Criterion: a mixed-state magic criterion with applications in distillation and detection

    quant-ph 2025-12 unverdicted novelty 8.0

    The Triangle Criterion detects mixed-state magic, proves multi-qubit distillation is strictly stronger than single-qubit schemes, and identifies a purity bound plus undetectable unfaithful magic states.

  2. Operational interpretation of the Stabilizer Entropy

    quant-ph 2025-07 unverdicted novelty 7.0

    The stabilizer Rényi entropy governs the exponential rate at which Clifford orbits become indistinguishable from Haar-random states and sets the optimal distinguishability from stabilizer states in property testing.

  3. Rise and fall of nonstabilizerness via random measurements

    quant-ph 2025-07 conditional novelty 7.0

    Analytical and numerical study of stabilizer nullity and Rényi entropies in monitored Clifford circuits shows quantized decay for computational measurements and size-dependent relaxation to a non-trivial steady state ...

  4. An Algorithm for Estimating $\alpha$-Stabilizer R\'enyi Entropies via Purity

    quant-ph 2025-07 unverdicted novelty 7.0

    Introduces a purity-encoding algorithm for estimating α-Stabilizer Rényi Entropies of unknown quantum states for integer α > 1, with benchmarks and a non-stabilizerness/entanglement link.

  5. Nonstabilizerness and Error Resilience in Noisy Quantum Circuits

    quant-ph 2025-06 unverdicted novelty 6.0

    Amplitude damping generates nonstabilizerness in qubit systems unlike depolarizing noise, with local injection washed out collectively after encoding, decoding, and postselection.

  6. Optimal quantum reservoir learning in proximity to universality

    quant-ph 2025-10 unverdicted novelty 5.0

    A tunable mixing parameter p in random quantum circuits controls the transition from classically simulable to expressive quantum reservoir dynamics via entanglement and nonstabilizer content.

  7. Quantum Complexity and New Directions in Nuclear Physics and High-Energy Physics Phenomenology

    quant-ph 2026-04 unverdicted novelty 2.0

    A review of how quantum information science is expected to provide new tools and insights for nuclear and high-energy physics phenomenology and quantum simulations.

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