Catability as a metric for evaluating superposed coherent states

Jan Provazn\'ik; Petr Marek; Vojt\v{e}ch Kala; \v{S}imon Br\"auer

arxiv: 2505.19723 · v4 · submitted 2025-05-26 · 🪐 quant-ph

Catability as a metric for evaluating superposed coherent states

\v{S}imon Br\"auer , Jan Provazn\'ik , Vojt\v{e}ch Kala , Petr Marek This is my paper

Pith reviewed 2026-05-19 13:30 UTC · model grok-4.3

classification 🪐 quant-ph

keywords cat statesnonlinear squeezingquantum superpositionstate detectionfidelityphoton lossquantum opticsmultiheaded cats

0 comments

The pith

Catability offers a measurable way to spot cat-like quantum superpositions using nonlinear squeezing.

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

The paper introduces catability as a directly measurable criterion for detecting cat-like features in quantum states. This criterion draws from nonlinear squeezing and avoids the need to reconstruct the full quantum state. It addresses a practical problem because fidelity often misses the superposition structure, especially in lossy conditions common to real experiments. Simulations demonstrate that the metric stays reliable under photon loss and lends itself to laboratory use. The same approach extends to more complex cases such as multiheaded cat states.

Core claim

The authors define catability as a criterion rooted in nonlinear squeezing for detecting cat-like features in superposed coherent states. This metric bypasses full state tomography and reveals the desired structure in cases where fidelity to an ideal state fails. Numerical results confirm that the criterion remains effective under loss and supports direct experimental implementation, while naturally generalizing to multiheaded cat states.

What carries the argument

Catability, a metric based on nonlinear squeezing that quantifies cat-like superposition features through higher-order correlations without requiring complete state knowledge.

If this is right

Cat states can be identified in lossy environments without full state tomography.
The metric extends to multiheaded cat states and other exotic superpositions.
Experimental setups can implement the criterion directly using existing squeezing measurements.
It provides an alternative when fidelity measures fail to capture the superposition structure.

Where Pith is reading between the lines

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

Verification protocols for cat-state-based quantum error correction could become simpler and less resource-intensive.
The underlying squeezing measurement might adapt to detect other non-classical features in continuous-variable systems.
Tests in superconducting or optical platforms with engineered loss would directly check the metric's practical limits.
Links to standard squeezing witnesses could unify several detection methods in quantum optics.

Load-bearing premise

Nonlinear squeezing reliably signals the presence of cat-like superposition structure even when the state suffers loss and without comparison to an ideal reference.

What would settle it

Prepare a known cat state, apply controlled photon loss, and check whether catability stays high while fidelity to the ideal state drops sharply; or prepare a non-superposed state that shows strong nonlinear squeezing and verify whether catability remains low.

Figures

Figures reproduced from arXiv: 2505.19723 by Jan Provazn\'ik, Petr Marek, Vojt\v{e}ch Kala, \v{S}imon Br\"auer.

**Figure 2.** Figure 2: FIG. 2: Simulated direct measurement of the operator (12) for an [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

read the original abstract

Superposed coherent states are central to quantum technologies, yet their reliable identification remains a challenge, especially in noisy or resource-constrained settings. We introduce a novel, directly measurable criterion for detecting cat-like features in quantum states, rooted in the concept of nonlinear squeezing. This approach bypasses the need for full state tomography and reveals structure where fidelity fails. The numerical results confirm its robustness under loss and its potential for experimental implementation. The method naturally generalizes to more exotic superpositions, including multiheaded cat 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

Catability is a workable, loss-robust metric for spotting cat-like features via nonlinear squeezing that skips full tomography.

read the letter

The main point is that this paper defines catability from nonlinear squeezing as a direct, measurable way to identify cat-state structure in superposed coherent states, and the numerics indicate it holds up under loss where fidelity drops off. They also extend it to multi-headed cats without extra machinery. That combination is the useful part: a practical proxy that does not require complete state reconstruction and shows some resilience in noisy conditions. The numerical checks for robustness look internally consistent and address the obvious experimental worry in quantum optics. The tie to nonlinear squeezing gives it a concrete foundation rather than an ad-hoc definition. On the softer side, the paper could have included tighter comparisons against other established non-Gaussianity or squeezing quantifiers to rule out simple overlap with existing tools. The experimental implementation section stays at the level of feasibility rather than a worked protocol, which is fine for a first paper but leaves some details open. These are incremental gaps, not load-bearing problems. The work is aimed at people in continuous-variable quantum information and quantum optics labs who generate or characterize cat states and need something quicker than tomography. A reader who cares about loss-tolerant diagnostics for nonclassical states will find the concrete metric and the loss tests worth looking at. It deserves a serious referee because the central claim is grounded in the numerics and the approach is honest about its scope. I would send it out for review.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces catability, a metric based on nonlinear squeezing, as a novel way to evaluate superposed coherent states for cat-like features. It claims this approach is directly measurable, avoids full state tomography, identifies structure missed by fidelity, shows robustness to loss via numerical results, and extends to multi-headed cat states.

Significance. Should the claims hold, this metric offers a practical, experimentally implementable tool for detecting non-classical superposition in quantum states under realistic conditions, potentially advancing quantum technology applications by reducing the need for resource-intensive tomography. The numerical demonstrations of robustness and the generalization to multi-headed cats are noted strengths if the underlying simulations are reproducible.

major comments (1)

Numerical results section: The central claim of robustness under loss rests on numerical simulations, but the manuscript does not specify the loss channel (e.g., amplitude damping rates or photon-loss probabilities) or the quantitative thresholds defining 'robustness,' which undermines verification of the metric's practical advantage over fidelity.

minor comments (2)

Figure captions should explicitly state what quantities are plotted (e.g., catability vs. loss parameter) and reference the corresponding equations.
The introduction would benefit from a brief comparison table contrasting catability with fidelity and other known cat-state witnesses.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment. We address the point below and will revise the manuscript to improve clarity and reproducibility.

read point-by-point responses

Referee: Numerical results section: The central claim of robustness under loss rests on numerical simulations, but the manuscript does not specify the loss channel (e.g., amplitude damping rates or photon-loss probabilities) or the quantitative thresholds defining 'robustness,' which undermines verification of the metric's practical advantage over fidelity.

Authors: We thank the referee for identifying this gap. We agree that the numerical results section would be strengthened by explicitly specifying the loss channel and the quantitative criteria used to assess robustness. In the revised manuscript we will describe the precise loss model (a photon-loss channel with the range of loss probabilities considered) and define the thresholds that quantify robustness, together with a direct comparison to fidelity under the same conditions. These additions will allow readers to verify the claimed practical advantage. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper defines catability as a measurable criterion explicitly rooted in the external prior concept of nonlinear squeezing rather than in terms of itself or its own outputs. Numerical demonstrations of robustness to loss and generalization to multi-headed cats are presented as independent verifications that do not reduce to fitted inputs renamed as predictions or to self-citation chains. The central claim bypasses tomography by construction from the squeezing concept and does not invoke uniqueness theorems or ansatzes from the authors' prior work as load-bearing justifications. The derivation remains self-contained against external benchmarks with no steps that equate outputs to inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

invented entities (1)

catability no independent evidence
purpose: Directly measurable metric for cat-like features based on nonlinear squeezing
Introduced in the abstract as the central new concept; no independent evidence or falsifiable prediction outside the paper is mentioned.

pith-pipeline@v0.9.0 · 5619 in / 1039 out tokens · 33447 ms · 2026-05-19T13:30:55.497355+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

catability ξ(±)(α)=min_γ Tr[Ô(±)(α,γ) ρ] / min_G Tr[Ô ρ_G] with Ô=(a†²−α*²)(a²−α²)+γ(1∓Π)
IndisputableMonolith/Foundation/AlphaDerivationExplicit.lean phi_golden_ratio echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

ω=0.618 chosen to minimize expectation for α=1 (approximating small cats)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

74 extracted references · 74 canonical work pages

[1]

If𝜉 (±) (𝛼)=0, thenˆ𝜌is an eigenstate of the opera- tor ˆO(±) (𝛼, 𝛾)and represents a perfect cat state|𝛼,±⟩, which is necessarily non-Gaussian

work page
[2]

If0< 𝜉 (±) (𝛼)<1, thenˆ𝜌is an imperfect, approximate cat state with amplitude𝛼, but we can still confirm its non-Gaussian nature

work page
[3]

Itmightbethecorrectcat statewithtoomuchdecoherence,adifferentcatstate,or a different state altogether

If𝜉 (±) (𝛼) ≥1,wecannotsay. Itmightbethecorrectcat statewithtoomuchdecoherence,adifferentcatstate,or a different state altogether. Imperfect or approximate cat states are understood here as re- alisticquantumstatesthatmaybeexpressedassuperpositions or mixtures of ideal cat states and other components, arising, for example, from loss, decoherence, and othe...

work page
[4]

Schrödinger, Die Naturwissenschaften23, 807 (1935), URL http://dx.doi.org/10.1007/bf01491891

E. Schrödinger, Die Naturwissenschaften23, 807 (1935), URL http://dx.doi.org/10.1007/bf01491891

work page doi:10.1007/bf01491891 1935
[5]

Jeong and M

H. Jeong and M. S. Kim, Phys. Rev. A65, 042305 (2002)

work page 2002
[6]

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, Physical Review A68, 042319 (2003), URLhttp: //dx.doi.org/10.1103/physreva.68.042319

work page doi:10.1103/physreva.68.042319 2003
[7]

1103/physrevlett.100.030503

A.P.Lund,T.C.Ralph,andH.L.Haselgrove,PhysicalReview Letters100, 030503 (2008), URLhttp://dx.doi.org/10. 1103/physrevlett.100.030503

work page 2008
[8]

Lee and H

S.-W. Lee and H. Jeong, Phys. Rev. A87, 022326 (2013)

work page 2013
[9]

S.Omkar,Y.S.Teo,andH.Jeong,Phys.Rev.Lett.125,060501 (2020)

work page 2020
[10]

S.Omkar,Y.S.Teo,S.-W.Lee,andH.Jeong,Phys.Rev.A103, 032602 (2021)

work page 2021
[11]

Hastrup and U

J. Hastrup and U. L. Andersen, Physical Review Research 4, 043065 (2022), URLhttp://dx.doi.org/10.1103/ physrevresearch.4.043065

work page 2022
[12]

J. Lee, N. Kang, S.-H. Lee, H. Jeong, L. Jiang, and S.-W. Lee, PRX Quantum5, 030322 (2024), URLhttp://dx.doi.org/ 10.1103/prxquantum.5.030322

work page doi:10.1103/prxquantum.5.030322 2024
[13]

D. Su, I. Dhand, and T. C. Ralph, Physical Review A 106, 042614 (2022), URLhttp://dx.doi.org/10.1103/ physreva.106.042614

work page 2022
[14]

Leghtas, G

Z. Leghtas, G. Kirchmair, B. Vlastakis, R. J. Schoelkopf, M. H. Devoret, and M. Mirrahimi, Physical Review Letters 111, 120501 (2013), URLhttp://dx.doi.org/10.1103/ physrevlett.111.120501

work page 2013
[15]

Duivenvoorden, B

K. Duivenvoorden, B. M. Terhal, and D. Weigand, Physical ReviewA95,012305(2017),URLhttp://dx.doi.org/10. 1103/physreva.95.012305

work page 2017
[17]

A. P. Lund, H. Jeong, T. C. Ralph, and M. S. Kim, Physical ReviewA70,020101(2004),URLhttp://dx.doi.org/10. 1103/physreva.70.020101

work page 2004
[18]

D. V. Sychev, A. E. Ulanov, A. A. Pushkina, M. W. Richards, I. A. Fedorov, and A. I. Lvovsky, Nature Photonics11, 379 (2017), URLhttp://dx.doi.org/10.1038/nphoton. 2017.57

work page doi:10.1038/nphoton 2017
[19]

D.J.WeigandandB.M.Terhal,PhysicalReviewA97,022341 (2018),URLhttp://dx.doi.org/10.1103/physreva.97. 022341

work page doi:10.1103/physreva.97 2018
[20]

M.Ježek,A.Tipsmark,R.Dong,J.Fiurášek,L.Mišta,R.Filip, andU.L.Andersen,PhysicalReviewA86,043813(2012),URL http://dx.doi.org/10.1103/physreva.86.043813

work page doi:10.1103/physreva.86.043813 2012
[21]

Yukawa, K

M. Yukawa, K. Miyata, T. Mizuta, H. Yonezawa, P. Marek, R. Filip, and A. Furusawa, Optics Express21, 5529 (2013), URLhttp://dx.doi.org/10.1364/oe.21.005529

work page doi:10.1364/oe.21.005529 2013
[22]

Eaton, C

M. Eaton, C. González-Arciniegas, R. N. Alexander, N. C. Menicucci, and O. Pfister, Quantum6, 769 (2022), URL http://dx.doi.org/10.22331/q-2022-07-20-769

work page doi:10.22331/q-2022-07-20-769 2022
[23]

M. Endo, T. Nomura, T. Sonoyama, K. Takahashi, S. Takasu, D.Fukuda,T.Kashiwazaki,A.Inoue,T.Umeki,R.Nehra,etal. (2025), URLhttps://arxiv.org/abs/2502.08952

work page arXiv 2025
[24]

Vlastakis, G

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frun- zio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Science342, 607 (2013), URLhttp://dx.doi. org/10.1126/science.1243289

work page doi:10.1126/science.1243289 2013
[25]

N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B.Vlastakis,Y.Liu,L.Frunzio,S.M.Girvin,L.Jiang,etal.,Na- ture536, 441 (2016), URLhttp://dx.doi.org/10.1038/ nature18949

work page 2016
[26]

Roudsari, P

X.L.He,Y.Lu,D.Q.Bao,H.Xue,W.B.Jiang,Z.Wang,A.F. Roudsari, P. Delsing, J. S. Tsai, and Z. R. Lin, Nature Com- munications14,6358(2023),URLhttp://dx.doi.org/10. 1038/s41467-023-42057-0

work page 2023
[27]

C.Monroe,D.M.Meekhof,B.E.King,andD.J.Wineland,Sci- ence272,1131(1996),URLhttp://dx.doi.org/10.1126/ science.272.5265.1131

work page arXiv 1996
[28]

Kienzler, C

D. Kienzler, C. Flühmann, V. Negnevitsky, H.-Y. Lo, M. Marinelli, D. Nadlinger, and J. Home, Physical Review Letters116, 140402 (2016), URLhttp://dx.doi.org/10. 1103/physrevlett.116.140402

work page 2016
[29]

Omran, H

A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S.Ebadi,H.Bernien,A.S.Zibrov,H.Pichler,S.Choi,etal.,Sci- ence365, 570 (2019), URLhttp://dx.doi.org/10.1126/ science.aax9743

work page 2019
[30]

Hacker, S

B. Hacker, S. Welte, S. Daiss, A. Shaukat, S. Ritter, L. Li, and G. Rempe, Nature Photonics13, 110 (2019), URLhttp: //dx.doi.org/10.1038/s41566-018-0339-5

work page doi:10.1038/s41566-018-0339-5 2019
[31]

Shomroni, L

I. Shomroni, L. Qiu, and T. J. Kippenberg, Physical Review A101, 033812(2020), URLhttp://dx.doi.org/10.1103/ physreva.101.033812

work page 2020
[32]

Hauer, J

B. Hauer, J. Combes, and J. Teufel, Physical Review Letters 130, 213604 (2023), URLhttp://dx.doi.org/10.1103/ physrevlett.130.213604

work page 2023
[34]

Walschaers, PRX Quantum2, 030204 (2021), URLhttp: //dx.doi.org/10.1103/prxquantum.2.030204

M. Walschaers, PRX Quantum2, 030204 (2021), URLhttp: //dx.doi.org/10.1103/prxquantum.2.030204

work page doi:10.1103/prxquantum.2.030204 2021
[35]

Physica E: Low-dimensional Systems and Nanostructures106, 208–238 (2019) https://doi.org/10.1016/j

L. Lachman and R. Filip, Progress in Quantum Electronics 6 83, 100395 (2022), URLhttp://dx.doi.org/10.1016/j. pquantelec.2022.100395

work page doi:10.1016/j 2022
[36]

Lee and H

C.-W. Lee and H. Jeong, Phys. Rev. Lett.106, 220401 (2011)

work page 2011
[37]

Merkel, Jay M

J. Fiurášek and M. Ježek, Physical Review A87, 062115 (2013),URLhttp://dx.doi.org/10.1103/physreva.87. 062115

work page doi:10.1103/physreva.87 2013
[39]

H.Kwon,C.-Y.Park,K.C.Tan,andH.Jeong,NewJ.Phys.19, 043024 (2017), ISSN 1367-2630

work page 2017
[40]

Chu, Science380, 274 (2023), URLhttp://dx.doi

M.Bild,M.Fadel,Y.Yang,U.vonLüpke,P.Martin,A.Bruno, and Y. Chu, Science380, 274 (2023), URLhttp://dx.doi. org/10.1126/science.adf7553

work page doi:10.1126/science.adf7553 2023
[41]

M.G.Genoni,M.L.Palma,T.Tufarelli,S.Olivares,M.S.Kim, andM.G.A.Paris,PhysicalReviewA87,062104(2013),URL http://dx.doi.org/10.1103/physreva.87.062104

work page doi:10.1103/physreva.87.062104 2013
[42]

Stoler, Physical Review D1, 3217 (1970), URLhttp:// dx.doi.org/10.1103/physrevd.1.3217

D. Stoler, Physical Review D1, 3217 (1970), URLhttp:// dx.doi.org/10.1103/physrevd.1.3217

work page doi:10.1103/physrevd.1.3217 1970
[43]

E.Y.C.Lu,LettereAlNuovoCimentoSeries22,1241(1971), URLhttp://dx.doi.org/10.1007/bf02770161

work page doi:10.1007/bf02770161 1971
[44]

H. P. Yuen, Physical Review A13, 2226 (1976), URLhttp: //dx.doi.org/10.1103/physreva.13.2226

work page doi:10.1103/physreva.13.2226 1976
[45]

S. L. Braunstein, Physical Review A71, 055801 (2005), URL http://dx.doi.org/10.1103/physreva.71.055801

work page doi:10.1103/physreva.71.055801 2005
[46]

B.Yadin,F.C.Binder,J.Thompson,V.Narasimhachar,M.Gu, andM.Kim,PhysicalReviewX8,041038(2018),URLhttp: //dx.doi.org/10.1103/physrevx.8.041038

work page doi:10.1103/physrevx.8.041038 2018
[48]

W. Jia, V. Xu, K. Kuns, M. Nakano, L. Barsotti, M. Evans, N. Mavalvala, members of the LIGO Scientific Collaboration†, R. Abbott, I. Abouelfettouh, et al., Science385, 1318 (2024), URLhttp://dx.doi.org/10.1126/science.ado8069

work page doi:10.1126/science.ado8069 2024
[49]

Holevo and Reinhard F

D. Gottesman and J. Preskill, Physical Review A63, 022309 (2001),URLhttp://dx.doi.org/10.1103/physreva.63. 022309

work page doi:10.1103/physreva.63 2001
[50]

doi.org/10.1088/2058-9565/ada9c4

A.n.Oruganti,I.Derkach,R.Filip,andV.C.Usenko,Quantum ScienceandTechnology10,025023(2025),URLhttp://dx. doi.org/10.1088/2058-9565/ada9c4

work page doi:10.1088/2058-9565/ada9c4 2025
[51]

K. Park, C. Oh, R. Filip, and P. Marek, Physical Review Applied18, 014060 (2022), URLhttp://dx.doi.org/10. 1103/physrevapplied.18.014060

work page 2022
[52]

S. L. Braunstein and H. J. Kimble, Physical Review Let- ters80, 869 (1998), URLhttp://dx.doi.org/10.1103/ physrevlett.80.869

work page 1998
[53]

Furusawa, J

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science282, 706 (1998), URL http://dx.doi.org/10.1126/science.282.5389.706

work page doi:10.1126/science.282.5389.706 1998
[54]

Bräuer, T

Š. Bräuer, T. Opatrný, and P. Marek, Physical Review Re- search7, 033176 (2025), URLhttp://dx.doi.org/10. 1103/9cxn-t1vf

work page 2025
[55]

V. Kala, J. Fadrný, M. Neset, J. Bílek, P. Marek, and M. Ježek (2025), URLhttps://arxiv.org/abs/2503.07574

work page arXiv 2025
[56]

& Fan, H

K. Miyata, H. Ogawa, P. Marek, R. Filip, H. Yonezawa, J.-i. Yoshikawa, and A. Furusawa, Physical Review A93, 022301 (2016),URLhttp://dx.doi.org/10.1103/physreva.93. 022301

work page doi:10.1103/physreva.93 2016
[57]

Konno, A

S. Konno, A. Sakaguchi, W. Asavanant, H. Ogawa, M.Kobayashi,P.Marek,R.Filip,J.-i.Yoshikawa,andA.Furu- sawa,PhysicalReviewApplied15,024024(2021),URLhttp: //dx.doi.org/10.1103/physrevapplied.15.024024

work page doi:10.1103/physrevapplied.15.024024 2021
[58]

Š.BräuerandP.Marek,OpticsExpress29,22648(2021),URL http://dx.doi.org/10.1364/oe.427637

work page doi:10.1364/oe.427637 2021
[59]

V. Kala, R. Filip, and P. Marek, Optics Express30, 31456 (2022), URLhttp://dx.doi.org/10.1364/oe.464759

work page doi:10.1364/oe.464759 2022
[60]

Marek, Physical Review Letters132, 210601 (2024), URLhttp://dx.doi.org/10.1103/physrevlett.132

P. Marek, Physical Review Letters132, 210601 (2024), URLhttp://dx.doi.org/10.1103/physrevlett.132. 210601

work page doi:10.1103/physrevlett.132 2024
[61]

Š.Bräuer, J.Provazník, V.Kala, andP.Marek,Supplement ma- terial for: Catability as a metric for evaluating superposed co- herent states(2025), see supplementary material for additional figures, derivations, and footnotes

work page 2025
[62]

Bräuer, J

Š. Bräuer, J. Provazník, V. Kala, and P. Marek,Catability as a metric for evaluating superposed coherent states,GitHubrepos- itoryhttps://github.com/jan-provaznik/catability

work page
[63]

Le Jeannic, A

H. Le Jeannic, A. Cavaillès, K. Huang, R. Filip, and J. Laurat, Physical Review Letters120, 073603 (2018), URLhttp:// dx.doi.org/10.1103/physrevlett.120.073603

work page doi:10.1103/physrevlett.120.073603 2018
[64]

Optics Express31(3), 4041 (2023) https://doi.org/10.1364/OE

J. Provazník, P. Marek, J. Laurat, and R. Filip, Optics Express 33, 16520 (2025), URLhttp://dx.doi.org/10.1364/oe. 555180

work page doi:10.1364/oe 2025
[65]

J. S. Ivan, K. K. Sabapathy, and R. Simon, Physical Review A84, 042311 (2011), URLhttp://dx.doi.org/10.1103/ physreva.84.042311

work page 2011
[66]

E., & Bacon, D

A. Mari and J. Eisert, Physical Review Letters109, 230503 (2012),URLhttp://dx.doi.org/10.1103/physrevlett. 109.230503

work page doi:10.1103/physrevlett 2012
[67]

Takahashi, K

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, Physical Review Letters 101, 233605 (2008), URLhttp://dx.doi.org/10.1103/ physrevlett.101.233605

work page 2008
[68]

Takase, J.-i

K. Takase, J.-i. Yoshikawa, W. Asavanant, M. Endo, and A. Furusawa, Physical Review A103, 013710 (2021), URL http://dx.doi.org/10.1103/physreva.103.013710

work page doi:10.1103/physreva.103.013710 2021
[69]

Ourjoumtsev, H

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, Nature448, 784 (2007), ISSN 1476-4687

work page 2007
[70]

Commun.337, 96 (2015), ISSN 0030-4018

A.Laghaout,J.S.Neergaard-Nielsen,andU.L.Andersen,Opt. Commun.337, 96 (2015), ISSN 0030-4018

work page 2015
[71]

Chabaud, D

U. Chabaud, D. Markham, and F. Grosshans, Physical Review Letters124, 063605 (2020), URLhttp://dx.doi.org/10. 1103/physrevlett.124.063605

work page 2020
[72]

Gherardini, G

U. Chabaud, G. Roeland, M. Walschaers, F. Grosshans, V. Pa- rigi, D. Markham, and N. Treps, PRX Quantum2, 020333 (2021), URLhttp://dx.doi.org/10.1103/prxquantum. 2.020333

work page doi:10.1103/prxquantum 2021
[73]

Fiurášek, Optics Express30, 30630 (2022), URLhttp:// dx.doi.org/10.1364/oe.466175

J. Fiurášek, Optics Express30, 30630 (2022), URLhttp:// dx.doi.org/10.1364/oe.466175

work page doi:10.1364/oe.466175 2022
[74]

J.Cohen,W.C.Smith,M.H.Devoret,andM.Mirrahimi,Phys. Rev. Lett.119, 060503 (2017)

work page 2017
[75]

C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, et al., Nature585, 357 (2020), URLhttp://dx.doi. org/10.1038/s41586-020-2649-2

work page doi:10.1038/s41586-020-2649-2 2020
[76]

E., et al

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber- land, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, et al., Nature Methods17, 261 (2020), URLhttp://dx.doi.org/10.1038/s41592-019-0686-2

work page doi:10.1038/s41592-019-0686-2 2020
[77]

J.D.Hunter,ComputinginScience&Engineering9,90(2007), URLhttp://dx.doi.org/10.1109/mcse.2007.55. 7 Supplementary Material for: Catability as a metric for evaluating superposed coherent states Šimon Bräuer, Jan Provazník, Vojtěch Kala, Petr Marek Optics Department, Faculty of Science, Palacký University

work page doi:10.1109/mcse.2007.55 2007
[78]

This follows from the linearity of expectation values and the convexity of mixed states: any expectation value for a mixed state is a convex combination of those for pure states

Listopadu 12, 77146 Olomouc, Czech Republic (April 27, 2026) GAUSSIAN BENCHMARK FOR THE OPERATORˆO To determine the minimum expectation value of the operatorˆOover all Gaussian states, it is sufficient to consider onlypure Gaussian states. This follows from the linearity of expectation values and the convexity of mixed states: any expectation value for a ...

work page 2026

[1] [1]

If𝜉 (±) (𝛼)=0, thenˆ𝜌is an eigenstate of the opera- tor ˆO(±) (𝛼, 𝛾)and represents a perfect cat state|𝛼,±⟩, which is necessarily non-Gaussian

work page

[2] [2]

If0< 𝜉 (±) (𝛼)<1, thenˆ𝜌is an imperfect, approximate cat state with amplitude𝛼, but we can still confirm its non-Gaussian nature

work page

[3] [3]

Itmightbethecorrectcat statewithtoomuchdecoherence,adifferentcatstate,or a different state altogether

If𝜉 (±) (𝛼) ≥1,wecannotsay. Itmightbethecorrectcat statewithtoomuchdecoherence,adifferentcatstate,or a different state altogether. Imperfect or approximate cat states are understood here as re- alisticquantumstatesthatmaybeexpressedassuperpositions or mixtures of ideal cat states and other components, arising, for example, from loss, decoherence, and othe...

work page

[4] [4]

Schrödinger, Die Naturwissenschaften23, 807 (1935), URL http://dx.doi.org/10.1007/bf01491891

E. Schrödinger, Die Naturwissenschaften23, 807 (1935), URL http://dx.doi.org/10.1007/bf01491891

work page doi:10.1007/bf01491891 1935

[5] [5]

Jeong and M

H. Jeong and M. S. Kim, Phys. Rev. A65, 042305 (2002)

work page 2002

[6] [6]

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, Physical Review A68, 042319 (2003), URLhttp: //dx.doi.org/10.1103/physreva.68.042319

work page doi:10.1103/physreva.68.042319 2003

[7] [7]

1103/physrevlett.100.030503

A.P.Lund,T.C.Ralph,andH.L.Haselgrove,PhysicalReview Letters100, 030503 (2008), URLhttp://dx.doi.org/10. 1103/physrevlett.100.030503

work page 2008

[8] [8]

Lee and H

S.-W. Lee and H. Jeong, Phys. Rev. A87, 022326 (2013)

work page 2013

[9] [9]

S.Omkar,Y.S.Teo,andH.Jeong,Phys.Rev.Lett.125,060501 (2020)

work page 2020

[10] [10]

S.Omkar,Y.S.Teo,S.-W.Lee,andH.Jeong,Phys.Rev.A103, 032602 (2021)

work page 2021

[11] [11]

Hastrup and U

J. Hastrup and U. L. Andersen, Physical Review Research 4, 043065 (2022), URLhttp://dx.doi.org/10.1103/ physrevresearch.4.043065

work page 2022

[12] [12]

J. Lee, N. Kang, S.-H. Lee, H. Jeong, L. Jiang, and S.-W. Lee, PRX Quantum5, 030322 (2024), URLhttp://dx.doi.org/ 10.1103/prxquantum.5.030322

work page doi:10.1103/prxquantum.5.030322 2024

[13] [13]

D. Su, I. Dhand, and T. C. Ralph, Physical Review A 106, 042614 (2022), URLhttp://dx.doi.org/10.1103/ physreva.106.042614

work page 2022

[14] [14]

Leghtas, G

Z. Leghtas, G. Kirchmair, B. Vlastakis, R. J. Schoelkopf, M. H. Devoret, and M. Mirrahimi, Physical Review Letters 111, 120501 (2013), URLhttp://dx.doi.org/10.1103/ physrevlett.111.120501

work page 2013

[15] [15]

Duivenvoorden, B

K. Duivenvoorden, B. M. Terhal, and D. Weigand, Physical ReviewA95,012305(2017),URLhttp://dx.doi.org/10. 1103/physreva.95.012305

work page 2017

[16] [17]

A. P. Lund, H. Jeong, T. C. Ralph, and M. S. Kim, Physical ReviewA70,020101(2004),URLhttp://dx.doi.org/10. 1103/physreva.70.020101

work page 2004

[17] [18]

D. V. Sychev, A. E. Ulanov, A. A. Pushkina, M. W. Richards, I. A. Fedorov, and A. I. Lvovsky, Nature Photonics11, 379 (2017), URLhttp://dx.doi.org/10.1038/nphoton. 2017.57

work page doi:10.1038/nphoton 2017

[18] [19]

D.J.WeigandandB.M.Terhal,PhysicalReviewA97,022341 (2018),URLhttp://dx.doi.org/10.1103/physreva.97. 022341

work page doi:10.1103/physreva.97 2018

[19] [20]

M.Ježek,A.Tipsmark,R.Dong,J.Fiurášek,L.Mišta,R.Filip, andU.L.Andersen,PhysicalReviewA86,043813(2012),URL http://dx.doi.org/10.1103/physreva.86.043813

work page doi:10.1103/physreva.86.043813 2012

[20] [21]

Yukawa, K

M. Yukawa, K. Miyata, T. Mizuta, H. Yonezawa, P. Marek, R. Filip, and A. Furusawa, Optics Express21, 5529 (2013), URLhttp://dx.doi.org/10.1364/oe.21.005529

work page doi:10.1364/oe.21.005529 2013

[21] [22]

Eaton, C

M. Eaton, C. González-Arciniegas, R. N. Alexander, N. C. Menicucci, and O. Pfister, Quantum6, 769 (2022), URL http://dx.doi.org/10.22331/q-2022-07-20-769

work page doi:10.22331/q-2022-07-20-769 2022

[22] [23]

M. Endo, T. Nomura, T. Sonoyama, K. Takahashi, S. Takasu, D.Fukuda,T.Kashiwazaki,A.Inoue,T.Umeki,R.Nehra,etal. (2025), URLhttps://arxiv.org/abs/2502.08952

work page arXiv 2025

[23] [24]

Vlastakis, G

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frun- zio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Science342, 607 (2013), URLhttp://dx.doi. org/10.1126/science.1243289

work page doi:10.1126/science.1243289 2013

[24] [25]

N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B.Vlastakis,Y.Liu,L.Frunzio,S.M.Girvin,L.Jiang,etal.,Na- ture536, 441 (2016), URLhttp://dx.doi.org/10.1038/ nature18949

work page 2016

[25] [26]

Roudsari, P

X.L.He,Y.Lu,D.Q.Bao,H.Xue,W.B.Jiang,Z.Wang,A.F. Roudsari, P. Delsing, J. S. Tsai, and Z. R. Lin, Nature Com- munications14,6358(2023),URLhttp://dx.doi.org/10. 1038/s41467-023-42057-0

work page 2023

[26] [27]

C.Monroe,D.M.Meekhof,B.E.King,andD.J.Wineland,Sci- ence272,1131(1996),URLhttp://dx.doi.org/10.1126/ science.272.5265.1131

work page arXiv 1996

[27] [28]

Kienzler, C

D. Kienzler, C. Flühmann, V. Negnevitsky, H.-Y. Lo, M. Marinelli, D. Nadlinger, and J. Home, Physical Review Letters116, 140402 (2016), URLhttp://dx.doi.org/10. 1103/physrevlett.116.140402

work page 2016

[28] [29]

Omran, H

A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S.Ebadi,H.Bernien,A.S.Zibrov,H.Pichler,S.Choi,etal.,Sci- ence365, 570 (2019), URLhttp://dx.doi.org/10.1126/ science.aax9743

work page 2019

[29] [30]

Hacker, S

B. Hacker, S. Welte, S. Daiss, A. Shaukat, S. Ritter, L. Li, and G. Rempe, Nature Photonics13, 110 (2019), URLhttp: //dx.doi.org/10.1038/s41566-018-0339-5

work page doi:10.1038/s41566-018-0339-5 2019

[30] [31]

Shomroni, L

I. Shomroni, L. Qiu, and T. J. Kippenberg, Physical Review A101, 033812(2020), URLhttp://dx.doi.org/10.1103/ physreva.101.033812

work page 2020

[31] [32]

Hauer, J

B. Hauer, J. Combes, and J. Teufel, Physical Review Letters 130, 213604 (2023), URLhttp://dx.doi.org/10.1103/ physrevlett.130.213604

work page 2023

[32] [34]

Walschaers, PRX Quantum2, 030204 (2021), URLhttp: //dx.doi.org/10.1103/prxquantum.2.030204

M. Walschaers, PRX Quantum2, 030204 (2021), URLhttp: //dx.doi.org/10.1103/prxquantum.2.030204

work page doi:10.1103/prxquantum.2.030204 2021

[33] [35]

Physica E: Low-dimensional Systems and Nanostructures106, 208–238 (2019) https://doi.org/10.1016/j

L. Lachman and R. Filip, Progress in Quantum Electronics 6 83, 100395 (2022), URLhttp://dx.doi.org/10.1016/j. pquantelec.2022.100395

work page doi:10.1016/j 2022

[34] [36]

Lee and H

C.-W. Lee and H. Jeong, Phys. Rev. Lett.106, 220401 (2011)

work page 2011

[35] [37]

Merkel, Jay M

J. Fiurášek and M. Ježek, Physical Review A87, 062115 (2013),URLhttp://dx.doi.org/10.1103/physreva.87. 062115

work page doi:10.1103/physreva.87 2013

[36] [39]

H.Kwon,C.-Y.Park,K.C.Tan,andH.Jeong,NewJ.Phys.19, 043024 (2017), ISSN 1367-2630

work page 2017

[37] [40]

Chu, Science380, 274 (2023), URLhttp://dx.doi

M.Bild,M.Fadel,Y.Yang,U.vonLüpke,P.Martin,A.Bruno, and Y. Chu, Science380, 274 (2023), URLhttp://dx.doi. org/10.1126/science.adf7553

work page doi:10.1126/science.adf7553 2023

[38] [41]

M.G.Genoni,M.L.Palma,T.Tufarelli,S.Olivares,M.S.Kim, andM.G.A.Paris,PhysicalReviewA87,062104(2013),URL http://dx.doi.org/10.1103/physreva.87.062104

work page doi:10.1103/physreva.87.062104 2013

[39] [42]

Stoler, Physical Review D1, 3217 (1970), URLhttp:// dx.doi.org/10.1103/physrevd.1.3217

D. Stoler, Physical Review D1, 3217 (1970), URLhttp:// dx.doi.org/10.1103/physrevd.1.3217

work page doi:10.1103/physrevd.1.3217 1970

[40] [43]

E.Y.C.Lu,LettereAlNuovoCimentoSeries22,1241(1971), URLhttp://dx.doi.org/10.1007/bf02770161

work page doi:10.1007/bf02770161 1971

[41] [44]

H. P. Yuen, Physical Review A13, 2226 (1976), URLhttp: //dx.doi.org/10.1103/physreva.13.2226

work page doi:10.1103/physreva.13.2226 1976

[42] [45]

S. L. Braunstein, Physical Review A71, 055801 (2005), URL http://dx.doi.org/10.1103/physreva.71.055801

work page doi:10.1103/physreva.71.055801 2005

[43] [46]

B.Yadin,F.C.Binder,J.Thompson,V.Narasimhachar,M.Gu, andM.Kim,PhysicalReviewX8,041038(2018),URLhttp: //dx.doi.org/10.1103/physrevx.8.041038

work page doi:10.1103/physrevx.8.041038 2018

[44] [48]

W. Jia, V. Xu, K. Kuns, M. Nakano, L. Barsotti, M. Evans, N. Mavalvala, members of the LIGO Scientific Collaboration†, R. Abbott, I. Abouelfettouh, et al., Science385, 1318 (2024), URLhttp://dx.doi.org/10.1126/science.ado8069

work page doi:10.1126/science.ado8069 2024

[45] [49]

Holevo and Reinhard F

D. Gottesman and J. Preskill, Physical Review A63, 022309 (2001),URLhttp://dx.doi.org/10.1103/physreva.63. 022309

work page doi:10.1103/physreva.63 2001

[46] [50]

doi.org/10.1088/2058-9565/ada9c4

A.n.Oruganti,I.Derkach,R.Filip,andV.C.Usenko,Quantum ScienceandTechnology10,025023(2025),URLhttp://dx. doi.org/10.1088/2058-9565/ada9c4

work page doi:10.1088/2058-9565/ada9c4 2025

[47] [51]

K. Park, C. Oh, R. Filip, and P. Marek, Physical Review Applied18, 014060 (2022), URLhttp://dx.doi.org/10. 1103/physrevapplied.18.014060

work page 2022

[48] [52]

S. L. Braunstein and H. J. Kimble, Physical Review Let- ters80, 869 (1998), URLhttp://dx.doi.org/10.1103/ physrevlett.80.869

work page 1998

[49] [53]

Furusawa, J

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science282, 706 (1998), URL http://dx.doi.org/10.1126/science.282.5389.706

work page doi:10.1126/science.282.5389.706 1998

[50] [54]

Bräuer, T

Š. Bräuer, T. Opatrný, and P. Marek, Physical Review Re- search7, 033176 (2025), URLhttp://dx.doi.org/10. 1103/9cxn-t1vf

work page 2025

[51] [55]

V. Kala, J. Fadrný, M. Neset, J. Bílek, P. Marek, and M. Ježek (2025), URLhttps://arxiv.org/abs/2503.07574

work page arXiv 2025

[52] [56]

& Fan, H

K. Miyata, H. Ogawa, P. Marek, R. Filip, H. Yonezawa, J.-i. Yoshikawa, and A. Furusawa, Physical Review A93, 022301 (2016),URLhttp://dx.doi.org/10.1103/physreva.93. 022301

work page doi:10.1103/physreva.93 2016

[53] [57]

Konno, A

S. Konno, A. Sakaguchi, W. Asavanant, H. Ogawa, M.Kobayashi,P.Marek,R.Filip,J.-i.Yoshikawa,andA.Furu- sawa,PhysicalReviewApplied15,024024(2021),URLhttp: //dx.doi.org/10.1103/physrevapplied.15.024024

work page doi:10.1103/physrevapplied.15.024024 2021

[54] [58]

Š.BräuerandP.Marek,OpticsExpress29,22648(2021),URL http://dx.doi.org/10.1364/oe.427637

work page doi:10.1364/oe.427637 2021

[55] [59]

V. Kala, R. Filip, and P. Marek, Optics Express30, 31456 (2022), URLhttp://dx.doi.org/10.1364/oe.464759

work page doi:10.1364/oe.464759 2022

[56] [60]

Marek, Physical Review Letters132, 210601 (2024), URLhttp://dx.doi.org/10.1103/physrevlett.132

P. Marek, Physical Review Letters132, 210601 (2024), URLhttp://dx.doi.org/10.1103/physrevlett.132. 210601

work page doi:10.1103/physrevlett.132 2024

[57] [61]

Š.Bräuer, J.Provazník, V.Kala, andP.Marek,Supplement ma- terial for: Catability as a metric for evaluating superposed co- herent states(2025), see supplementary material for additional figures, derivations, and footnotes

work page 2025

[58] [62]

Bräuer, J

Š. Bräuer, J. Provazník, V. Kala, and P. Marek,Catability as a metric for evaluating superposed coherent states,GitHubrepos- itoryhttps://github.com/jan-provaznik/catability

work page

[59] [63]

Le Jeannic, A

H. Le Jeannic, A. Cavaillès, K. Huang, R. Filip, and J. Laurat, Physical Review Letters120, 073603 (2018), URLhttp:// dx.doi.org/10.1103/physrevlett.120.073603

work page doi:10.1103/physrevlett.120.073603 2018

[60] [64]

Optics Express31(3), 4041 (2023) https://doi.org/10.1364/OE

J. Provazník, P. Marek, J. Laurat, and R. Filip, Optics Express 33, 16520 (2025), URLhttp://dx.doi.org/10.1364/oe. 555180

work page doi:10.1364/oe 2025

[61] [65]

J. S. Ivan, K. K. Sabapathy, and R. Simon, Physical Review A84, 042311 (2011), URLhttp://dx.doi.org/10.1103/ physreva.84.042311

work page 2011

[62] [66]

E., & Bacon, D

A. Mari and J. Eisert, Physical Review Letters109, 230503 (2012),URLhttp://dx.doi.org/10.1103/physrevlett. 109.230503

work page doi:10.1103/physrevlett 2012

[63] [67]

Takahashi, K

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, Physical Review Letters 101, 233605 (2008), URLhttp://dx.doi.org/10.1103/ physrevlett.101.233605

work page 2008

[64] [68]

Takase, J.-i

K. Takase, J.-i. Yoshikawa, W. Asavanant, M. Endo, and A. Furusawa, Physical Review A103, 013710 (2021), URL http://dx.doi.org/10.1103/physreva.103.013710

work page doi:10.1103/physreva.103.013710 2021

[65] [69]

Ourjoumtsev, H

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, Nature448, 784 (2007), ISSN 1476-4687

work page 2007

[66] [70]

Commun.337, 96 (2015), ISSN 0030-4018

A.Laghaout,J.S.Neergaard-Nielsen,andU.L.Andersen,Opt. Commun.337, 96 (2015), ISSN 0030-4018

work page 2015

[67] [71]

Chabaud, D

U. Chabaud, D. Markham, and F. Grosshans, Physical Review Letters124, 063605 (2020), URLhttp://dx.doi.org/10. 1103/physrevlett.124.063605

work page 2020

[68] [72]

Gherardini, G

U. Chabaud, G. Roeland, M. Walschaers, F. Grosshans, V. Pa- rigi, D. Markham, and N. Treps, PRX Quantum2, 020333 (2021), URLhttp://dx.doi.org/10.1103/prxquantum. 2.020333

work page doi:10.1103/prxquantum 2021

[69] [73]

Fiurášek, Optics Express30, 30630 (2022), URLhttp:// dx.doi.org/10.1364/oe.466175

J. Fiurášek, Optics Express30, 30630 (2022), URLhttp:// dx.doi.org/10.1364/oe.466175

work page doi:10.1364/oe.466175 2022

[70] [74]

J.Cohen,W.C.Smith,M.H.Devoret,andM.Mirrahimi,Phys. Rev. Lett.119, 060503 (2017)

work page 2017

[71] [75]

C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, et al., Nature585, 357 (2020), URLhttp://dx.doi. org/10.1038/s41586-020-2649-2

work page doi:10.1038/s41586-020-2649-2 2020

[72] [76]

E., et al

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber- land, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, et al., Nature Methods17, 261 (2020), URLhttp://dx.doi.org/10.1038/s41592-019-0686-2

work page doi:10.1038/s41592-019-0686-2 2020

[73] [77]

J.D.Hunter,ComputinginScience&Engineering9,90(2007), URLhttp://dx.doi.org/10.1109/mcse.2007.55. 7 Supplementary Material for: Catability as a metric for evaluating superposed coherent states Šimon Bräuer, Jan Provazník, Vojtěch Kala, Petr Marek Optics Department, Faculty of Science, Palacký University

work page doi:10.1109/mcse.2007.55 2007

[74] [78]

This follows from the linearity of expectation values and the convexity of mixed states: any expectation value for a mixed state is a convex combination of those for pure states

Listopadu 12, 77146 Olomouc, Czech Republic (April 27, 2026) GAUSSIAN BENCHMARK FOR THE OPERATORˆO To determine the minimum expectation value of the operatorˆOover all Gaussian states, it is sufficient to consider onlypure Gaussian states. This follows from the linearity of expectation values and the convexity of mixed states: any expectation value for a ...

work page 2026