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arxiv: 2410.23721 · v5 · submitted 2024-10-31 · 🪐 quant-ph

Assessing non-Gaussian quantum state conversion with the stellar rank

Oliver Hahn , Maxime Garnier , Giulia Ferrini , Alessandro Ferraro , Ulysse Chabaud This is my paper

Pith reviewed 2026-05-23 19:04 UTC · model grok-4.3

classification 🪐 quant-ph
keywords approximate stellar ranknon-GaussianityGaussian operationsstate conversioncontinuous-variable quantum systemsno-go theoremsquantum resource theorybosonic codes
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The pith

The approximate stellar rank serves as an operational measure of non-Gaussianity for bounding approximate Gaussian state conversions.

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

This paper introduces a framework to evaluate approximate conversions between quantum states using only Gaussian operations by defining an approximate version of the stellar rank. The stellar rank originally measures the degree of non-Gaussianity in continuous-variable systems, and its approximate extension allows for practical analysis when exact conversions are not feasible. By deriving bounds under approximate and probabilistic conditions, the work yields new no-go results that prohibit certain non-Gaussian state preparations. This approach also provides a way to assess how well existing Gaussian conversion protocols perform in realistic settings. An open-source Python library is provided to compute these quantities.

Core claim

The paper claims that extending the stellar rank to the approximate stellar rank provides an operational measure of non-Gaussianity, enabling the derivation of bounds for Gaussian state conversion and distillation under approximate and probabilistic conditions, which in turn yield new no-go results for non-Gaussian state preparation.

What carries the argument

The approximate stellar rank, an extension of the stellar rank that quantifies non-Gaussianity in approximate settings and acts as a faithful measure for deriving conversion bounds.

Load-bearing premise

That the approximate stellar rank maintains its operational meaning as a faithful measure of non-Gaussianity capable of producing valid bounds on state conversions.

What would settle it

Finding a specific pair of states where the approximate stellar rank difference allows a conversion that the derived bound prohibits, or vice versa, would disprove the framework's validity.

Figures

Figures reproduced from arXiv: 2410.23721 by Alessandro Ferraro, Giulia Ferrini, Maxime Garnier, Oliver Hahn, Ulysse Chabaud.

Figure 1
Figure 1. Figure 1: The set of quantum states, partitioned following the multimode stellar hierarchy (extended [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Stellar fidelities (in black) and approx [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Lower bounds on the number of copies of odd cate states required to achieve a given trace distance precision with a GKP state with ∆ = 0.1, via deterministic Gaussian conversion, for various cat state amplitudes α. single-copy to single-copy conversions, our corre￾sponding bounds from Theorem 4 are computed as δ ≥ 0.01 and δ ≥ 0.05, respectively. This indi￾cates that either stronger bounds may be derived f… view at source ↗
Figure 4
Figure 4. Figure 4: Lower bounds on the number of copies of any state of stellar rank 1 required to achieve a given trace distance precision with an even cat state via probabilistic Gaussian conversion, for various cat state amplitudes α. The blue cross depicts the performance of the protocol reported in [46, Figure 2b]. Using Theorem 4, we obtain lower bounds on the number of copies of any state of stellar rank 1 (including … view at source ↗
read the original abstract

State conversion is a fundamental task in quantum information processing. Quantum resource theories allow for analyzing and bounding conversions that use restricted sets of operations. In the context of continuous-variable systems, state conversions restricted to Gaussian operations are crucial for both fundamental and practical reasons, particularly in state preparation and quantum computing with bosonic codes. However, previous analysis did not consider the relevant case of approximate state conversion. In this work, we introduce a framework for assessing approximate Gaussian state conversion by extending the stellar rank to the approximate stellar rank, which serves as an operational measure of non-Gaussianity. We derive bounds for Gaussian state conversion and distillation under approximate and probabilistic conditions, yielding new no-go results for non-Gaussian state preparation and enabling a reliable assessment of the performance of Gaussian conversion protocols. We also provide an open-source Python library to compute stellar-rank-related quantities and to assess Gaussian conversion.

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

0 major / 3 minor

Summary. The paper extends the stellar rank to an approximate stellar rank as an operational measure of non-Gaussianity for assessing approximate Gaussian state conversions in continuous-variable systems. It derives bounds on Gaussian conversion and distillation under approximate and probabilistic conditions, obtains new no-go results for non-Gaussian state preparation, and supplies an open-source Python library for computing stellar-rank quantities and evaluating conversion protocols.

Significance. If the approximate stellar rank preserves its operational meaning and the derived bounds hold, the framework supplies a concrete tool for quantifying non-Gaussian resources in bosonic state preparation and quantum computing, with the open-source library enabling reproducible assessment of protocols.

minor comments (3)
  1. The abstract states that the approximate stellar rank 'serves as an operational measure of non-Gaussianity,' but the manuscript should explicitly verify in §3 or §4 that the extension inherits the faithfulness and monotonicity properties of the exact stellar rank under the restricted operations.
  2. The Python library is mentioned as open-source; the manuscript should include a dedicated subsection or appendix with installation instructions, example usage for the new bounds, and a link to the repository to ensure immediate usability.
  3. Notation for the approximate stellar rank (e.g., how the approximation parameter enters the definition) should be introduced once and used consistently; current usage in the abstract and introduction risks ambiguity for readers unfamiliar with the exact stellar rank.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript, recognition of its significance, and recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces an extension of the stellar rank to the approximate stellar rank as a new operational measure of non-Gaussianity for assessing approximate Gaussian state conversions. The provided abstract and context present this extension directly as the foundation for deriving bounds and no-go results, without any quoted equations or steps that reduce a claimed result to a fitted parameter, self-citation chain, or definitional equivalence by construction. The derivation is self-contained against external benchmarks, as the new measure is defined and applied within the work rather than presupposing its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on the definition of the approximate stellar rank and standard assumptions of continuous-variable quantum resource theory; no free parameters or new physical entities are mentioned in the abstract.

axioms (1)
  • domain assumption Gaussian operations form a restricted set that cannot increase non-Gaussianity
    Implicit in the resource-theory framing of the abstract.
invented entities (1)
  • approximate stellar rank no independent evidence
    purpose: Operational measure of non-Gaussianity for approximate state conversions
    Newly defined extension introduced in the paper; no independent evidence supplied in abstract.

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Reference graph

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