arxiv: 2604.20303 · v1 · submitted 2026-04-22 · 🪐 quant-ph

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Operational criterion for Wigner function negativity

Paolo Solinas , Beatrice Donelli , Stefano Gherardini

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Pith reviewed 2026-05-10 00:38 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Wigner functionnegativitycoherent statessuperpositionsquantum non-demolitionSchrödinger cat statespositivityphase space

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The pith

A quantum state has a positive Wigner function if it contains no coherent superpositions in the coherent-state basis.

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

The paper introduces an operational criterion that identifies Wigner function negativity through quantum non-demolition measurements performed in the coherent-state basis. It establishes that the lack of superpositions between distinct coherent states is enough to guarantee that the Wigner function stays non-negative for any quantum state. The same absence of superpositions becomes necessary as well as sufficient for the positivity of the Wigner function in the cases of Schrödinger-cat states and higher-order cat states on a circle in the dense limit. This connection turns the sign of the Wigner function into a directly measurable property tied to the presence or absence of measurable superpositions.

Core claim

The absence of coherent superpositions in the coherent-state basis constitutes a sufficient condition for the positivity of the Wigner function for an arbitrary quantum state. This condition is both necessary and sufficient for Schrödinger-cat states in full generality, and it is necessary for the positivity of the Wigner function of higher-order cat states on a circle when the number of coherent states becomes large and they are densely packed.

What carries the argument

The operational criterion realized by quantum non-demolition measurements in the coherent-state basis, which directly detects the presence or absence of coherent superpositions and thereby determines the sign of the Wigner function.

If this is right

Any quantum state lacking coherent superpositions in the coherent-state basis must have a non-negative Wigner function.
For Schrödinger-cat states the Wigner function is positive if and only if the state contains no coherent superpositions.
For higher-order cat states on a circle the Wigner function becomes positive in the limit of densely packed coherent states when superpositions are absent.
The sign of the Wigner function becomes accessible through direct, experimentally realizable non-demolition measurements without requiring full tomography.

Where Pith is reading between the lines

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

The criterion suggests that experiments can test for Wigner negativity by checking only for the presence of superpositions rather than reconstructing the full quasiprobability distribution.
The coherent-state basis acquires a distinguished role for determining Wigner-function properties that may extend to related phase-space distributions.
States engineered to avoid superpositions in this basis are expected to behave classically in the sense of positive Wigner functions.

Load-bearing premise

Quantum non-demolition measurements in the coherent-state basis can be performed to access the sign of the Wigner function for arbitrary states without state-dependent post-selection or additional assumptions that would limit the criterion.

What would settle it

An experimental preparation of an arbitrary quantum state with no coherent superpositions in the coherent basis whose Wigner function is nevertheless measured to take negative values in some region of phase space.

Figures

Figures reproduced from arXiv: 2604.20303 by Beatrice Donelli, Paolo Solinas, Stefano Gherardini.

**Figure 2.** Figure 2: FIG. 2: Higher-order cat states. a) Comparison of the radial [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

read the original abstract

We introduce an operational criterion to identify Wigner function (WF) negativity for an arbitrary quantum state within the framework of quantum non-demolition measurements. This criterion corresponds to experimentally accessible schemes that enable a direct measurement of the WF, and establishes the coherent-state basis as a privileged basis for determining when the WF exhibits negative regions. We show that the absence (presence) of coherent superpositions in the coherent-state basis provides direct information about the positivity (negativity) of the WF. In particular, the absence of such superpositions constitutes a sufficient condition for WF positivity. Although a general proof of necessity remains elusive, we demonstrate that this condition is also necessary in two relevant cases: Schr\"{o}dinger-cat states and higher-order cat states on a circle. More precisely, for Schr\"{o}dinger-cat states we establish a necessary and sufficient condition for the positivity of the WF in full generality, whereas for high-order cat states on a circle we derive an analogous condition in the limit of a large number of densely packed coherent states.

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

The paper shows absence of coherent superpositions suffices for Wigner positivity in general, with necessity proven only for cat states and circular higher-order cats.

read the letter

The paper's main result is that the lack of coherent superpositions in the coherent-state basis is a sufficient condition for the Wigner function to be positive everywhere. They show this holds generally, and then prove that the condition is also necessary for Schrödinger-cat states and for higher-order cat states on a circle in the large-N limit. This gives an operational way to check for Wigner negativity using quantum non-demolition measurements in that basis. What stands out is the explicit necessary-and-sufficient condition they derive for cat states. That part is concrete and could be tested in experiments with optical cats. The link to QND schemes makes the criterion feel practical rather than purely formal. They also note clearly that a general proof of necessity is still missing, which is the right way to present it. The limitation is exactly what they flag: without necessity in general, the presence of superpositions does not guarantee Wigner negativity for every possible state. So the criterion can confirm positivity when there are no superpositions, but it cannot always certify negativity when superpositions are detected. That makes the operational claim a bit one-sided for arbitrary states. The weakest part is probably how well the QND measurement works without extra assumptions for non-cat states. This paper is aimed at researchers in quantum optics and continuous-variable quantum information who want better handles on nonclassical resources. Someone studying Wigner negativity or designing experiments with cat states would find the specific cases helpful. I would send it for peer review. The sufficient condition and the cat-state proofs are solid enough to warrant referee input, even with the open question on the general case.

Referee Report

2 major / 2 minor

Summary. The paper proposes an operational criterion for detecting Wigner function (WF) negativity in arbitrary quantum states using quantum non-demolition (QND) measurements in the coherent-state basis. It establishes that the absence of coherent superpositions in this basis is a sufficient condition for WF positivity in general. Necessity is proven for Schrödinger-cat states (necessary and sufficient) and for higher-order cat states on a circle in the large-N limit, but a general necessity proof is stated to be elusive. The criterion is framed as directly linking the presence/absence of such superpositions to WF negativity/positivity via experimentally accessible QND schemes.

Significance. If the sufficient condition and the specific-case necessity results hold, the work provides a concrete link between coherent-state superpositions and WF negativity that could enable direct experimental probes of nonclassicality without full state tomography. The proofs for cat states and the large-N circle states are technically useful for those families. However, the absence of a general necessity result limits the operational utility for certifying negativity in arbitrary states, reducing the overall impact relative to a bidirectional criterion.

major comments (2)

[Abstract] Abstract: The claim of an 'operational criterion to identify Wigner function negativity' for arbitrary states is not fully supported, because the paper explicitly notes that a general proof of necessity (presence of coherent superpositions implying WF negativity) remains elusive and is only demonstrated for Schrödinger-cat states and higher-order cat states on a circle. This makes the criterion sufficient for positivity but only partially operational for detecting negativity in full generality.
[Abstract] Abstract and introduction (implied QND scheme): The operational claim assumes that QND measurements in the coherent-state basis can be implemented to access WF sign directly for arbitrary states without state-dependent post-selection or additional assumptions that would invalidate the measurement for general cases. This assumption is load-bearing for the 'direct measurement' aspect but is not shown to hold universally.

minor comments (2)

[Abstract] Abstract: The phrasing 'establishes the coherent-state basis as a privileged basis' could be clarified to specify in what sense it is privileged (e.g., for this particular criterion) to avoid overstatement.
[Abstract] Abstract: Typographical rendering of 'Schrödinger' with escaped quotes should be standardized in the final version.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the scope of our results. We agree that the abstract requires revision to avoid overstating the generality of the necessity direction. Our responses to the major comments are provided below, along with planned changes to the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: The claim of an 'operational criterion to identify Wigner function negativity' for arbitrary states is not fully supported, because the paper explicitly notes that a general proof of necessity (presence of coherent superpositions implying WF negativity) remains elusive and is only demonstrated for Schrödinger-cat states and higher-order cat states on a circle. This makes the criterion sufficient for positivity but only partially operational for detecting negativity in full generality.

Authors: We acknowledge that the current abstract phrasing may suggest a fully bidirectional operational criterion for arbitrary states, whereas the manuscript already states that necessity holds only for the specified families. The sufficient condition (absence of coherent superpositions implies WF positivity) is general and directly operational via QND measurements. For detecting negativity, the criterion is operational precisely in the cases where necessity is proven. We will revise the abstract to state explicitly that the criterion is sufficient for positivity in general and necessary and sufficient for Schrödinger-cat states (and in the large-N limit for higher-order cat states on a circle). This change aligns the abstract with the body of the paper without altering the technical claims. revision: yes
Referee: [Abstract] Abstract and introduction (implied QND scheme): The operational claim assumes that QND measurements in the coherent-state basis can be implemented to access WF sign directly for arbitrary states without state-dependent post-selection or additional assumptions that would invalidate the measurement for general cases. This assumption is load-bearing for the 'direct measurement' aspect but is not shown to hold universally.

Authors: The QND framework is invoked to render the criterion experimentally accessible by connecting the presence or absence of coherent-state superpositions to measurable outcomes, without requiring full tomography. While a universal, state-independent experimental protocol is not derived in the manuscript (as the focus is on the theoretical link), existing QND techniques for coherent-state projections are referenced as the basis for feasibility. We will add a clarifying paragraph in the introduction that discusses the assumptions, notes that implementations may require state-specific adaptations, and cites relevant experimental literature on QND measurements in the coherent-state basis. This addresses the concern without claiming universality of the experimental scheme. revision: partial

Circularity Check

0 steps flagged

No circularity: sufficient condition derived from standard QND framework; necessity acknowledged as partial and non-general.

full rationale

The paper's core derivation establishes that absence of coherent superpositions in the coherent-state basis is a sufficient condition for WF positivity via operational QND measurements, with explicit proofs supplied for Schrödinger-cat states (necessary and sufficient) and high-order cat states on a circle (large-N limit). It openly states that a general necessity proof remains elusive, avoiding any claim of full equivalence. No self-definitional loops, fitted parameters renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the provided derivation chain; the criterion rests on standard quantum measurement theory applied to specific state classes without reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on the established framework of quantum mechanics, Wigner functions, and non-demolition measurements without introducing new free parameters or entities.

axioms (1)

standard math Standard quantum mechanics including definition of Wigner function and quantum non-demolition measurements
The criterion is built directly on these established concepts.

pith-pipeline@v0.9.0 · 5475 in / 1187 out tokens · 42585 ms · 2026-05-10T00:38:14.979313+00:00 · methodology

discussion (0)

Reference graph

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