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arxiv: 2507.02540 · v2 · submitted 2025-07-03 · 🪐 quant-ph

An Algorithm for Estimating α-Stabilizer R\'enyi Entropies via Purity

Benjamin Stratton This is my paper

Pith reviewed 2026-05-19 06:42 UTC · model grok-4.3

classification 🪐 quant-ph
keywords stabilizer Renyi entropynon-stabilizernessquantum magicpurity estimationquantum resource theoryquantum algorithmsentanglement
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The pith

A quantum channel on alpha copies of a state encodes its stabilizer Renyi entropy directly into the purity of the output state.

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

This paper shows how to measure the alpha-Stabilizer Renyi Entropy of an unknown quantum state by first applying a specific channel to alpha copies of the state. The resulting state has a purity that equals a simple function of the desired entropy value. Existing algorithms for estimating purity can therefore be reused to obtain the entropy for any integer alpha greater than one. This matters for quantum computing because the entropy quantifies non-stabilizerness, a resource needed for universal fault-tolerant computation beyond classical simulation. The construction also reveals a link between non-stabilizerness and entanglement during the measurement process.

Core claim

There exists a state produced by the action of a channel on alpha copies of some state rho that encodes the alpha-Stabilizer Renyi Entropy of rho into its purity. This encoding permits an algorithm for measuring the entropies for all integers alpha greater than one by calling existing purity estimators. The algorithm is benchmarked on qubits with resource counts compared to prior methods, a non-stabilizerness/entanglement relationship is exhibited, and an instance of resource hiding is identified.

What carries the argument

A quantum channel applied to alpha copies of rho that produces an output state whose purity equals 2 to the power of minus the alpha-Stabilizer Renyi Entropy of rho.

If this is right

  • The entropies become measurable for any integer alpha greater than one using standard purity estimators.
  • Resource requirements for qubits can be directly compared with those of earlier algorithms.
  • A concrete relationship between non-stabilizerness and entanglement appears inside the measurement routine.
  • Cases of resource hiding can be exhibited within the same framework.

Where Pith is reading between the lines

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

  • Similar channel constructions might reduce sample complexity for estimating magic on near-term hardware.
  • The purity-encoding idea could be adapted to quantify other quantum resources if analogous channels exist.
  • The demonstrated non-stabilizerness-entanglement link may motivate joint resource measures in hybrid protocols.

Load-bearing premise

The channel can be implemented on alpha copies of rho in a manner compatible with efficient purity measurement without hidden overheads that would break the direct encoding.

What would settle it

Apply the channel to alpha copies of a known state such as the T-state, estimate the output purity, and check whether it exactly matches the value predicted from the state's known alpha-Stabilizer Renyi Entropy.

Figures

Figures reproduced from arXiv: 2507.02540 by Benjamin Stratton.

Figure 1
Figure 1. Figure 1: FIG. 1. A circuit diagram for the coherent algorithm for mea [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

Non-stabilizerness, or magic, is a resource for universal quantum computation in most fault-tolerant architectures; access to states with non-stabilizerness allows for non-classically simulable quantum computation to be performed. Quantifying this resource for unknown states is therefore essential to assessing their utility in quantum computation. The Stabilizer R\'enyi Entropies have emerged as a leading tool for achieving this, having already enabled one efficient algorithm for measuring non-stabilizerness. In addition, the Stabilizer R\'enyi Entropies have proven useful in developing connections between non-stabilizerness and other quantum phenomena. In this work, we introduce an alternative algorithm for measuring the Stabilizer R\'enyi Entropies of an unknown quantum state. Firstly, we show the existence of a state, produced from the action of a channel on $\alpha$ copies of some \ben{state $\rho$}, that encodes the $\alpha$-Stabilizer R\'enyi Entropy of $\rho$ into its purity. We detail several methods of applying this channel, and then, by employing existing purity-measuring algorithms, provide an algorithm for measuring the $\alpha$-Stabilizer R\'enyi Entropies for all integers $\alpha>1$. This algorithm is benchmarked for qubits and the resource requirements compared to other known algorithms. Finally, a non-stabilizerness/entanglement relationship is shown to exist in the algorithm, demonstrating a novel relationship between the two resources, before an instance of resource hiding is found.

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

1 major / 2 minor

Summary. The paper claims to introduce an algorithm for estimating the α-Stabilizer Rényi Entropies of an unknown state ρ for integer α > 1. It first establishes the existence of a quantum channel Λ acting on α copies of ρ such that the purity Tr(σ²) of the output state σ = Λ(ρ⊗α) directly encodes the α-SRE of ρ. Several concrete methods for implementing this channel are described, after which existing purity-estimation routines are invoked to obtain the entropy values. The work includes qubit benchmarking, resource comparisons with prior algorithms, a demonstrated relationship between non-stabilizerness and entanglement within the algorithm, and an example of resource hiding.

Significance. If the purity-encoding identity survives the concrete channel implementations without hidden corrections or post-selection overheads, the algorithm would constitute a useful alternative route to quantifying magic, with potentially distinct sampling or circuit-depth trade-offs relative to existing SRE estimators. The explicit benchmarking and the non-stabilizerness–entanglement link would further strengthen the contribution, provided they rest on the same verified encoding.

major comments (1)
  1. [Section on channel application] Section on channel application: the central claim requires that Tr(σ²) for σ = Λ(ρ⊗α) equals (or is a simple function of) the α-SRE of ρ. The manuscript must explicitly verify that each listed implementation of Λ preserves this identity. If any method introduces ancillary qubits that are traced out or employs post-selection, the reduced-system purity generally acquires extra factors Tr(σ_anc²) and cross terms; these must be shown to equal unity or to be corrected by a known, efficiently computable multiplier. Without such a derivation or explicit statement that no tracing/post-selection occurs, the direct-encoding step remains unverified and load-bearing for the entire algorithm.
minor comments (2)
  1. [Benchmarking section] The abstract states that the algorithm is 'benchmarked for qubits'; the corresponding section should include a table or plot of sample complexity versus α and system size, together with direct numerical comparison to the prior SRE algorithm referenced in the introduction.
  2. [Throughout] Notation for the channel Λ and the output state σ should be introduced once in a dedicated paragraph and used consistently; currently the abstract and later sections appear to employ slightly different symbols for the same objects.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit verification of the central purity-encoding identity. We address the major comment below and have revised the manuscript to incorporate the requested derivations and clarifications.

read point-by-point responses

  1. Referee: Section on channel application: the central claim requires that Tr(σ²) for σ = Λ(ρ⊗α) equals (or is a simple function of) the α-SRE of ρ. The manuscript must explicitly verify that each listed implementation of Λ preserves this identity. If any method introduces ancillary qubits that are traced out or employs post-selection, the reduced-system purity generally acquires extra factors Tr(σ_anc²) and cross terms; these must be shown to equal unity or to be corrected by a known, efficiently computable multiplier. Without such a derivation or explicit statement that no tracing/post-selection occurs, the direct-encoding step remains unverified and load-bearing for the entire algorithm.

    Authors: We agree that explicit verification of the encoding identity is necessary to rigorously establish the algorithm. While the original manuscript demonstrated the existence of the channel Λ and outlined several concrete implementations, we acknowledge that the preservation of Tr(σ²) = f(α-SRE(ρ)) was not derived in full detail for each case. In the revised manuscript we have added a dedicated subsection that supplies step-by-step calculations for every listed implementation. These derivations show that each implementation acts unitarily on the α copies of ρ without ancillary qubits that are traced out and without post-selection; consequently, no extraneous factors Tr(σ_anc²) or cross terms appear, and the output purity directly encodes the α-SRE. We have also inserted an explicit statement confirming the absence of such overheads. This revision directly addresses the referee’s concern and strengthens the load-bearing step of the algorithm. revision: yes

Circularity Check

0 steps flagged

No circularity: construction uses independent purity estimators on a new channel encoding

full rationale

The paper defines a channel Λ acting on α copies of ρ such that the purity of the output state directly encodes the α-SRE of ρ. It then invokes existing, separately developed purity estimation algorithms (e.g., those based on randomized measurements or swap tests) as black-box components to extract that purity. No parameter is fitted to the target SRE value, no self-citation supplies a uniqueness theorem that forces the channel form, and the encoding identity is presented as a derived mathematical fact rather than a renaming or tautology. The derivation chain therefore remains self-contained against external benchmarks and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard quantum information assumptions about density operators, channels, and purity; no free parameters, new axioms, or invented entities are introduced in the abstract description.

axioms (1)
  • domain assumption Standard quantum mechanics: density operators represent states and channels are completely positive trace-preserving maps
    Invoked throughout the channel construction and purity encoding (abstract description of the state produced from α copies).

pith-pipeline@v0.9.0 · 5808 in / 1246 out tokens · 46241 ms · 2026-05-19T06:42:42.497079+00:00 · methodology

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Reference graph

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    Aα(|ψ⟩) from the purity of EP (|ψ⟩⟨ψ|⊗α)’s output Here, we provide the proof of how Aα(|ψ⟩) is encoded into the purity of the state EP |ψ⟩⟨ψ|⊗α = d−2 d2−1 ∑ j=0 Pj|ψ⟩⟨ψ|Pj ⊗α. (B1) The purity of this state is given by tr EP |ψ⟩⟨ψ|⊗α 2 = d−4 d2−1 ∑ j,k=0 tr Pj|ψ⟩⟨ψ|PjPk|ψ⟩⟨ψ|Pk α = d−4 d2−1 ∑ j,k=0 ⟨ψ| PjPk|ψ⟩⟨ψ|PkPj |ψ⟩ α. (B2) It is then noted that PjPk ...

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