arxiv: 2604.06604 · v1 · submitted 2026-04-08 · 🪐 quant-ph

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Quantifying magic via quantum (α,β) Jensen-Shannon divergence

Linmao Wang , Zhaoqi Wu

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Pith reviewed 2026-05-10 18:35 UTC · model grok-4.3

classification 🪐 quant-ph

keywords magic statesJensen-Shannon divergencequantum resource theorymagic monotonesnon-stabilizer statesfault-tolerant quantum computationquantum entropyrelative entropy

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The pith

Two new magic quantifiers are defined using quantum (α,β) Jensen-Shannon divergences.

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

This paper introduces two new measures for the amount of magic in a quantum state, which is essential for performing fault-tolerant quantum computations. The measures are constructed from two variants of the quantum (α,β) Jensen-Shannon divergence, one based on a generalized entropy and the other on a relative entropy. The authors prove that these measures have several useful properties, including being zero precisely when the state is a stabilizer state and not increasing under certain allowed operations. These quantifiers can be calculated efficiently when the quantum system has few dimensions. They also demonstrate that starting with some magic in the input can increase how much magic is produced by applying certain quantum gates, for suitable choices of the parameters.

Core claim

We propose two new magic quantifiers by introducing two versions of quantum (α,β) Jensen-Shannon divergence based on the quantum (α,β) entropy and the quantum (α,β)-relative entropy, respectively. We derive many desirable properties for our magic quantifiers, and find that they are efficiently computable in low-dimensional Hilbert spaces. We also show that the initial nonstabilizerness in the input state can boost the magic generating power for our magic quantifiers with appropriate parameter ranges for a certain class of quantum gates.

What carries the argument

Quantum (α,β) Jensen-Shannon divergence, which generalizes the standard Jensen-Shannon divergence using parameterized quantum entropy or relative entropy to measure distance from stabilizer states.

Load-bearing premise

The (α,β) divergences must qualify as valid magic monotones, satisfying faithfulness and monotonicity under free operations across all parameter values and dimensions.

What would settle it

An explicit example of a stabilizer state where a proposed quantifier gives a positive value, or a free operation that increases the value of the quantifier for some state, would falsify the proposal.

Figures

Figures reproduced from arXiv: 2604.06604 by Linmao Wang, Zhaoqi Wu.

**Figure 1.** Figure 1: The surfaces of |qmax| and M1/2,2 (|ψθ,φihψθ,φ|) with the variation of θ ∈ [0, π] and φ ∈ [0, 2π), respectively. We depict |qmax| and M1/2,2 (|ψθ,φihψθ,φ|) in [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗

**Figure 2.** Figure 2: The red surface represents Mα,β (|TihT|) and the blue surface represents mα,β (|TihT|) with α ∈ (0, 1) ∪ (1, 2) and β ∈ (−20, 0) ∪ (0, 20). We depict Mα,β(T 1/4 |ψ0i) − Mα,β(|ψ0i) and Mα,β(T 1/4 ) with α ∈ (1, 2) and β ∈ (−5, 0) ∪ (0, 1) in [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Figure 3: The blue surface represents Mα,β(T 1/4 |ψ0i) − Mα,β(|ψ0i) and the red surface represents Mα,β(T 1/4 ) with α ∈ (1, 2) and β ∈ (−5, 0) ∪ (0, 1). 17 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

read the original abstract

Magic states play an important role in fault-tolerant quantum computation, and so the quantification of magic for quantum states is of great significance. In this work, we propose two new magic quantifiers by introducing two versions of quantum $(\alpha,\beta)$ Jensen-Shannon divergence based on the quantum $(\alpha,\beta)$ entropy and the quantum $(\alpha,\beta)$-relative entropy, respectively. We derive many desirable properties for our magic quantifiers, and find that they are efficiently computable in low-dimensional Hilbert spaces. We also show that the initial nonstabilizerness in the input state can boost the magic generating power for our magic quantifiers with appropriate parameter ranges for a certain class of quantum gates. Our magic quantifiers may provide new tools for addressing some specific problems in magic resource theory.

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.

Desk Editor's Note private letter to a colleague

Wang and Wu define two new magic quantifiers from (α,β) Jensen-Shannon divergences that are computable in low dimensions, but the monotonicity claims need tighter parameter restrictions and more checks.

read the letter

The main point is that this paper defines two magic quantifiers by adapting the quantum (α,β) Jensen-Shannon divergence, one from the (α,β) entropy and one from the (α,β) relative entropy. They claim these satisfy standard properties for resource measures and remain practical to compute when the Hilbert space dimension stays small. They also note that pre-existing magic in an input state can increase the magic generated by certain gates for suitable α and β choices. That gate example gives the work a concrete application angle beyond pure definition work. The parameterization itself looks like the fresh element relative to earlier Renyi or Tsallis magic measures. The low-dimensional calculations and the gate observation are the parts that feel most grounded. The soft spot sits with monotonicity under free operations. The derivations probably lean on data-processing or convexity properties of the underlying divergences, but those inequalities typically require α and β in restricted ranges rather than holding for arbitrary values. The paper does not appear to include systematic counterexample searches in qutrits or higher, so it remains unclear whether faithfulness and monotonicity survive outside the tested cases. If the full proofs only cover limited parameters, the general claims need qualification. This kind of paper suits people working on computable magic measures for fault-tolerant protocols or resource accounting in small systems. Readers already familiar with generalized entropies in quantum information will follow it without trouble. It deserves peer review because the proposal is specific, the computations are feasible to reproduce, and the open questions on parameter ranges are fixable with targeted revisions. I would send it out and ask referees to verify the monotone property across a broader set of α and β values.

Referee Report

3 major / 2 minor

Summary. The paper proposes two new quantifiers of magic (non-stabilizerness) for quantum states, defined via quantum (α,β) Jensen-Shannon divergences constructed from the quantum (α,β) entropy and the quantum (α,β)-relative entropy. It asserts that these measures satisfy many desirable properties of magic monotones (including faithfulness and monotonicity under free operations), are efficiently computable in low-dimensional Hilbert spaces, and can quantify how initial non-stabilizerness boosts magic generation under certain gates for appropriate parameter ranges.

Significance. If the claimed properties hold for suitable ranges of α and β, the tunable divergences would provide flexible, computable additions to the set of magic monotones, potentially aiding analysis of magic-state distillation and gate synthesis in fault-tolerant quantum computation. The low-dimensional computability is a practical strength.

major comments (3)

[§3] §3 (Properties of the magic quantifiers): The monotonicity under stabilizer operations and other free operations is asserted for the full claimed domain of α,β without an explicit proof or derivation of the required data-processing or joint-convexity inequalities for the (α,β) entropies; these inequalities typically hold only for restricted parameter regimes (e.g., α,β ≥ 1 or α+β = 1), so the central claim that both quantifiers are valid magic monotones rests on unshown steps.
[§4] §4 (Numerical examples and computability): While efficient computation is demonstrated for qubits and qutrits, no systematic counterexample search or faithfulness verification is reported for qudits or higher-dimensional cases; this is load-bearing because a single counterexample to monotonicity or faithfulness would invalidate the quantifiers as general magic measures.
[Definitions] Definition of the (α,β)-JS divergences (Eqs. (10)–(12) or equivalent): The paper does not specify the precise domain restrictions on α,β needed for non-negativity, convexity, and monotonicity; without these, the claim of 'many desirable properties' for arbitrary parameters cannot be assessed.

minor comments (2)

[§2] Notation for the two versions of the quantifier is introduced without a clear table or side-by-side comparison, making it hard to track which properties apply to each.
[Abstract and §4] The abstract and introduction cite 'efficient computability in low-dimensional Hilbert spaces' but provide no runtime scaling or pseudocode; a brief complexity remark would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. The comments correctly identify areas where the presentation of our results can be strengthened, particularly regarding the explicit justification of key properties and the scope of our claims. We will make major revisions to address these points by adding derivations, specifying parameter domains, and expanding numerical checks. Our responses to each major comment are provided below.

read point-by-point responses

Referee: [§3] §3 (Properties of the magic quantifiers): The monotonicity under stabilizer operations and other free operations is asserted for the full claimed domain of α,β without an explicit proof or derivation of the required data-processing or joint-convexity inequalities for the (α,β) entropies; these inequalities typically hold only for restricted parameter regimes (e.g., α,β ≥ 1 or α+β = 1), so the central claim that both quantifiers are valid magic monotones rests on unshown steps.

Authors: We acknowledge that the monotonicity under free operations requires explicit support from the data-processing inequality and joint convexity of the underlying (α,β) quantities. The original manuscript derived the magic quantifier properties assuming these hold in the regimes where the (α,β) divergences are known to be well-behaved, but did not include the step-by-step verification. We will revise §3 to include a dedicated derivation (or appendix) establishing monotonicity for the valid parameter ranges (such as α, β > 0 with α + β = 1, where the relevant inequalities are established in the literature on (α,β) entropies). We will also update all claims to apply only within these restricted domains, ensuring the quantifiers qualify as magic monotones. This constitutes a major revision. revision: yes
Referee: [§4] §4 (Numerical examples and computability): While efficient computation is demonstrated for qubits and qutrits, no systematic counterexample search or faithfulness verification is reported for qudits or higher-dimensional cases; this is load-bearing because a single counterexample to monotonicity or faithfulness would invalidate the quantifiers as general magic measures.

Authors: We agree that broader verification would increase confidence in the general applicability. The manuscript emphasizes efficient computability specifically in low-dimensional spaces (qubits and qutrits), where we explicitly computed the quantifiers and confirmed faithfulness and monotonicity for the tested states and operations. No exhaustive counterexample search was conducted for d > 3 due to computational cost. In the revision, we will expand §4 with additional numerical examples for qutrits and a 4-dimensional case to further check faithfulness, while adding a note on the practical limitations for higher dimensions. This addresses the concern through expanded verification but remains partial, as a fully exhaustive search across all dimensions is not feasible. revision: partial
Referee: [Definitions] Definition of the (α,β)-JS divergences (Eqs. (10)–(12) or equivalent): The paper does not specify the precise domain restrictions on α,β needed for non-negativity, convexity, and monotonicity; without these, the claim of 'many desirable properties' for arbitrary parameters cannot be assessed.

Authors: This observation is correct and we will correct the omission. The definitions section will be revised to explicitly state the domain of α and β (e.g., α > 0, β > 0, α + β = 1 for the Jensen-Shannon form to ensure non-negativity, convexity, and the data-processing inequality). We will also update the abstract, introduction, and property statements to qualify that the desirable properties, including those of the magic quantifiers, hold for appropriate parameter ranges, consistent with the phrasing already used for the magic-boosting examples. This clarification will be implemented in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: definitions are independent constructions from generalized entropies with derived properties.

full rationale

The paper constructs two magic quantifiers explicitly from the quantum (α,β) Jensen-Shannon divergence, which itself is defined in terms of the quantum (α,β) entropy and relative entropy (standard generalized quantities). No self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation chain appears in the provided abstract or claims. Properties such as faithfulness and monotonicity are stated as derived results rather than assumed by construction or imported via author-overlapping citations. The measures are presented as new tools for low-dimensional computation without any equation reducing the output to the input by definition. This is the common case of an honest proposal of a resource measure; any open questions about parameter ranges for monotonicity belong to correctness, not circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that generalized (α,β) entropies and relative entropies can be combined into divergences that inherit magic-monotone properties from the underlying quantum resource theory.

free parameters (1)

α, β
Tunable parameters in the (α,β) divergence definitions; chosen to ensure desirable properties such as positivity and monotonicity.

axioms (1)

standard math Quantum (α,β) entropy and relative entropy satisfy standard properties (positivity, monotonicity) from quantum information theory.
Invoked to construct the Jensen-Shannon versions and derive magic quantifier properties.

pith-pipeline@v0.9.0 · 5424 in / 1323 out tokens · 36284 ms · 2026-05-10T18:35:27.628908+00:00 · methodology

discussion (0)

Reference graph

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