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Quantum-to-Classical Transition via Single-Shot Generalized Measurements

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Pith's one-line read A single round of an N-level coherent-state measurement erases all quasiprobability negativity in any finite-dimensional state.

desk verdict Solid math, honest citations, but the headline t_c < t_deco claim is definition-sensitive and should be framed as such. read the letter →

arxiv 2507.13174 v3 pith:HLI2N37T submitted 2025-07-17 quant-ph cond-mat.stat-mech

Zhenyu Xu This is my paper · ORCID
classification quant-phcond-mat.stat-mech MSC 81P1681P4081P6881S30 PACS 03.65.Ta03.65.Yz03.67.-a
keywords quasiprobabilitynegativitycoherent-statePOVMquantum-to-classicaltransitionWignerisotropicdepolarizingchanneldecoherencetimefinite-dimensionalphasespacesingle-shotmeasurement
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that in any finite-dimensional quantum system, a single round of an $N$-level coherent-state POVM removes all negativity from every $s$-parametrized Stratonovich-Weyl quasiprobability distribution, regardless of the initial state. It proves that $n$ measurement rounds produce exactly the same state as isotropic depolarization for time $t=2n/(\gamma N)\ln(1+N)$, so the measurement process and continuous decoherence are two views of one dynamics. It then shows that Wigner negativity disappears abruptly at a critical time $t_c$, and that for qubits, qutrits, and suitably mixed higher-dimensional states $t_c$ can be shorter than the conventional decoherence time. If true, this supplies a sharp operational picture of the quantum-to-classical transition in finite dimensions and warns that the conventional decoherence time can overestimate how long phase-space nonclassicality survives.

What carries the argument

The central object is the $N$-level coherent-state POVM, whose elements $E(\Omega)=N|\Omega\rangle\langle\Omega|$ integrate to the identity over the complex projective space $CP^{N-1}$. Its covariance under $SU(N)$ forces the induced channel to be a scalar on the traceless generators, and the Haar second-moment identity $\int d\mu(\Omega)|\Omega\rangle\langle\Omega|^{\otimes2}=(1^{\otimes2}+S)/[N(N+1)]$ fixes the scalar to $1/(N+1)$. This gives Eq. (2), and iterating gives the exact correspondence $t=2n/(\gamma N)\ln(1+N)$ with the depolarizing solution; the $s$-parametrized Stratonovich-Weyl kernel then converts the state map into the positivity statement and the critical-time formula.

What would settle it

For a pure qubit under the same isotropic depolarizing channel, replace the short-time slope by the exponential purity decay constant $t_{\rm deco}'=1/(\gamma N)$ and compute $t_c/t_{\rm deco}'=\ln3\approx1.10>1$; observing this would show that the paper's $t_c/t_{\rm deco}<1$ statement depends on the chosen convention. Alternatively, implement the ancilla-assisted POVM circuit on a qubit initialized in a state orthogonal to $|\Omega\rangle$ and directly measure the Wigner function: after one round it must be nonnegative everywhere.

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Extended reading notes

Core claim

Applying one round of the $N$-level coherent-state POVM with elements $E(\Omega)=N|\Omega\rangle\langle\Omega|$ maps any state to $\rho_1=(1_N+\rho_0)/(N+1)$. Because the Stratonovich-Weyl kernel yields $W^{(s)}_{\rho_0}(\Omega)\ge (1-r_s)/N$ for $s\in[-1,1]$, the transformed distribution obeys $W^{(s)}_{\rho_1}(\Omega)=(1+W^{(s)}_{\rho_0}(\Omega))/(N+1)\ge 0$, so a single round eliminates quasiprobability negativity in every finite dimension. Equivalently, under the isotropic depolarizing channel the same family of states arises, and the Wigner negativity volume drops discontinuously to zero at $t_c=\frac{2}{\gamma N}\ln\big[\sqrt{N+1}(1-N\lambda_{\min})\big]$. For $N=2,3$ and for certain mixed states in $N\ge4$ with purity between $N/(N^2-1)$ and the boundary value $P_0^b$ defined through the Lambert $W$ function, $t_c/t_{\rm deco}<1$, so the conventional decoherence time does not faithfully track the disappearance of nonclassicality.

Load-bearing premise

The load-bearing premise is that the conventional decoherence time should be identified with the short-time purity slope $t_{\rm deco}=P_0/[\gamma(N P_0-1)]$; if one instead uses the exponential purity decay time $1/(\gamma N)$, the claimed ordering $t_c<t_{\rm deco}$ for a pure qubit reverses.

Editorial extensions

If this is right

  • A single round of the POVM acts as a universal negativity eraser: no initial state or dimension $N$ can keep any Stratonovich-Weyl quasiprobability negative after one measurement.
  • Repeated POVM rounds and isotropic depolarization are operationally interchangeable through $t=2n/(\gamma N)\ln(1+N)$, so circuits that implement the POVM can simulate the continuous decoherence trajectory and vice versa.
  • The negative Wigner volume $P(\rho_t)$ vanishes abruptly at $t_c$ rather than decaying asymptotically; for $N=2,3$ this occurs before the conventional short-time decoherence time.
  • For mixed states in dimensions $N\ge4$ with purity in the window $N/(N^2-1)<P_0<P_0^b$, there exist states whose critical time is shorter than the decoherence time, so the conventional decoherence time can overestimate the lifetime of nonclassicality.
  • With current superconducting gate fidelities and coherence times, the proposed circuit remains implementable in the estimate up to about $N\simeq51$, enough to observe the single-shot negativity removal.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the paper's comparison inherits a convention; using the exponential purity decay time $1/(\gamma N)$ instead of the short-time slope would give $t_c/t_{\rm deco}'=\ln3>1$ for a pure qubit, reversing the $N=2$ 'overestimate' message, and this sensitivity is left implicit in the text.
  • Editorial inference: because Eq. (2) holds for every ordering parameter $s$, the same single-shot positivity applies to $P$ and $Q$ functions, so the result likely extends beyond Wigner negativity to other phase-space nonclassicality witnesses.
  • Editorial inference: the unraveling in terms of the coherent-state POVM suggests an experimental route to heralded coherent-state preparation, since conditioning on the measurement record turns depolarizing noise into pure trajectories, a possibility the paper discusses as open.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper establishes an exact correspondence between repeated rounds of an N-level coherent-state POVM and continuous isotropic depolarization: one round maps any state to (1_N + ρ0)/(N+1), so after n rounds (or time t = 2n/(γN) ln(1+N)) the state equals the depolarized evolution. From this the paper shows that a single round removes all negativity of the s-parametrized Stratonovich-Weyl quasiprobabilities for s in [-1,1], derives a critical time t_c at which Wigner negativity vanishes, compares t_c with a conventional decoherence time, and proposes circuit implementations and a heralded resource-extraction protocol. The core algebraic derivations (Eqs. (2), (7), (8), and the single-shot positivity statement) are sound and self-contained.

Significance. If the technical issues are fixed, the exact measurement-to-depolarization correspondence (Eq. (8)) and the single-shot negativity-elimination theorem are clean and useful results, with concrete experimental implications and a clear connection to magic-state resources. The paper is also commendable for stating its definitions transparently in the Supplemental Material and for including a heralded-resource-extraction discussion with an explicit impossibility statement for the unconditional channel. However, the negative-volume formula contains a sign/normalization error, and the headline t_c < t_deco comparison is sensitive to the chosen definition of decoherence time; both issues bear on advertised quantitative claims and need to be resolved before publication.

major comments (2)
  1. [Sudden vanishing of negative quasiprobability volume (Eq. (13); SM IV)] Equation (13) and its derivation in SM IV are not correct as written. For a pure qubit with eigenvalues (0,1), Eq. (13) gives P(ρ0) = (p_c - 0)/(0 - 1) = -p_c, whereas the correct value is p_c and the paper's own reduction 1-(1-p_c)^{N-1} gives p_c. The origin of the error is in SM Eq. (S41): the simplex integral ∫_Δ e^{-ζλ·x} dx is not the Laplace transform of the probability density f_p (it is normalized by the simplex volume), and the divided-difference denominator has the opposite sign. The correct cumulative expression is P(ρ0) = Σ_{λ_j<p_c} (p_c - λ_j)^{N-1}/∏_{k≠j}(λ_k - λ_j), with no 1/(N-1)! prefactor. This error does not affect the single-shot positivity theorem or the critical-time formula (Eq. (14)), but it invalidates the displayed negative-volume formula and the quantitative volumes in Fig. 2, and it must be corrected.
  2. [Critical time versus decoherence time (Eq. (15); SM IIIb)] The advertised comparison t_c < t_deco for N=2,3 is convention-dependent. For the same dynamics the purity is P_t = 1/N + (P0 - 1/N)e^{-γNt} (SM Eq. (S24)), whose natural 1/e timescale is t_deco' = 1/(γN). With this equally standard definition, a pure initial state gives t_c/t_deco' = ln(1+N), which is 1.099 for N=2 and 1.386 for N=3, reversing the 'overestimates' message in the abstract and Fig. 3. The paper states its short-time purity-slope convention in SM, but the main text presents the inequality as a physical finding. The claim should be qualified, or the comparison should be repeated for alternative standard decoherence timescales to show robustness.
minor comments (5)
  1. [Experimental proposal and feasibility (main text vs SM V)] The main text calls κ=1, r=0.1% a 'conservative analysis' and quotes N≲51, while SM V uses κ=1.5, r=0.2% and quotes N≲17 as the more conservative estimate; please harmonize the parameter sets and terminology.
  2. [Fig. 2 caption] The caption states that Wigner negativity disappears at t_c (n_c = 1/2); this n_c value holds for pure initial states (λ_min=0), while for mixed states t_c is smaller and the corresponding n_c is less than 1/2, so the caption should specify the initial-state assumption.
  3. [Fig. 2 caption (typo)] The phrase 'plotted against time or the number' appears as 'timetor'; please correct the typo.
  4. [Abstract and Introduction] The statement that a single measurement round 'eliminates quasiprobability negativity' refers to the unconditional, outcome-discarded channel; the discussion in SM VI correctly notes that conditional post-measurement states remain pure. Please make this distinction explicit in the main text to avoid a possible misreading.
  5. [SM VI] The environment pointer states |Ω>_E with delta-function orthonormality form a nonseparable Hilbert space; the paper notes that discretization is needed for physical implementation, but this idealization should be flagged more prominently in the main discussion of the dilation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central measurement-channel correspondence, single-shot positivity theorem, and critical-time formula are all derived from standard Haar, GKSL, and Stratonovich-Weyl identities, with no fitted parameters or load-bearing self-citations.

full rationale

The derivation chain is self-contained rather than circular. Equation (2) is proven in the text using the partial-trace identity and the Haar second-moment formula for coherent-state projectors, so the single-round state does not assume the conclusion. Equation (7) is the explicit solution of the GKSL equation with uniform su(N) Lindblad generators, and Eq. (8) follows by direct comparison of the two exponential forms; no parameter is fitted. The single-shot positivity statement W^{(s)}_{rho1}(Omega) >= 0 follows from Eqs. (10) and (11) and the bound W^{(s)}_{rho0}(Omega) >= -1 for s in [-1,1], again by algebra already in the paper. The critical time in Eq. (14)/(S31) is obtained by setting the minimal evolved Wigner function to zero, and the decoherence time in Eq. (S23) is a stated short-time purity-slope convention following Zurek; the ratio t_c/t_deco compares two independently derived quantities and is not a reduction of one to the other. The paper cites the author's own prior works ([19,20,67,84]), but these are used for background, for standard SW-kernel criteria, for decoherence-time expansion references, and for experimental circuit techniques; none carries the central derivation, and the SW-kernel criteria are standard properties with independent references such as [33]. The only substantive caveat is that the advertised 't_c < t_deco' message depends on the chosen purity-slope convention for t_deco; with the alternative exponential purity-decay constant 1/(gamma N), the pure-state ratio becomes ln(1+N) > 1 for N=2,3. That is a convention-sensitivity or robustness concern, not a circularity, because the paper does not define t_c in terms of t_deco or fit either quantity to the other.

Assumptions & free parameters 2 free parameters · 8 assumptions · 1 invented entities

The central derivations use standard Haar integration, SW kernel properties, and the GKSL equation; no free parameters are fitted to data. The only hand-chosen quantities are the uniform decoherence rate gamma (a model parameter) and the circuit-depth constant kappa used for the feasibility estimate. The environment pointer register is a standard dilation, not a new physical entity.

free parameters (2)
  • uniform decoherence rate gamma
    Model parameter in Eq. (6); cancels in the ratio t_c/t_deco, so the qualitative claims do not depend on its value.
  • circuit depth constant kappa = 1 or 1.5
    Hand-chosen for the feasibility estimates N less than about 17 or 51 in Section V; not used in the central derivations.
assumptions (8)
  • standard math Haar second-moment identity on CP^{N-1}: integral dmu(Omega) |Omega><Omega|^{x2} = (1_N^{x2}+S)/(N(N+1)).
    Used in the proof of Eq. (2); cited to Collins and Sniady [44].
  • standard math Stratonovich-Weyl kernels Delta^{(s)}(Omega) satisfy standardization, covariance, and trace preservation for s in [-1,1].
    Underpins the phase-space representation in Eqs. (9)-(11); cited to Refs. [19,33].
  • domain assumption The continuous decoherence process follows the GKSL/Lindblad master equation.
    Standard open-systems model; cited to Refs. [46-49].
  • domain assumption All su(N) Lindblad generators share a uniform decoherence rate gamma.
    Defines the generalized isotropic depolarizing channel in Eq. (6); a modeling choice.
  • domain assumption Decoherence time is defined by the short-time purity expansion P_t/P0 approx 1 - t/t_deco (Zurek convention).
    The comparison t_c/t_deco depends on this convention; see SM IIIa.
  • standard math For s in [-1,1], W^{(s)}_{rho0}(Omega) >= (1-r_s)/N >= -1.
    Follows from Eq. (11) and nonnegativity of <Omega|rho|Omega>; key to single-shot positivity.
  • standard math The coherent-state family spans the Hilbert space, so min_Omega <Omega|rho0|Omega> equals the smallest eigenvalue of rho0.
    Used in deriving the critical time Eq. (14); standard completeness of coherent states.
  • standard math Simplex integral identity for uniform sampling on the (N-1)-simplex (B-spline/truncated power form).
    Used in SM Section IV to derive Eq. (13); cited to de Boor [60].
invented entities (1)
  • Environment register with generalized pointer states |Omega>_E, normalized as <Omega|Omega'>_E = delta(Omega,Omega')
    purpose: Provides a classical measurement record so that the depolarizing channel can be unraveled into coherent-state trajectories, enabling heralded resource extraction.
    This is a standard Stinespring dilation of the coherent-state POVM, explicitly labeled as a record basis in SM VI; it is a mathematical construction, not a new physical entity, and carries no falsifiable prediction outside the paper.

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Pith. "Pith review of Quantum-to-Classical Transition via Single-Shot Generalized Measurements." pith.science (2026). https://pith.science/paper/HLI2N37T

@misc{pith2026250713174,
  author       = {Pith},
  title        = {Pith review of: Quantum-to-Classical Transition via Single-Shot Generalized Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HLI2N37T}},
  note         = {Machine review of arXiv:2507.13174}
}
read the original abstract

We establish an operational connection between discrete rounds of generalized measurements and continuous-time decoherence, with an explicit correspondence between the number of measurement rounds and the evolution time. Operationally, we show that a single round of such a generalized measurement eliminates quasiprobability negativity in finite-dimensional systems. From the decoherence perspective, this loss of negativity occurs abruptly at a critical time. In particular, this critical time can be shorter than the conventional decoherence time, indicating that the latter does not always faithfully track the disappearance of nonclassicality. Our results provide new insight into the quantum-to-classical transition in finite-dimensional systems from the viewpoint of phase-space quasiprobability, and suggest feasible quantum-circuit tests as well as possible heralded resource extraction from noise.

Figures

Figures reproduced from arXiv: 2507.13174 by the authors.

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Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
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Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Double-Bracket Master Equations: Phase-Space Representation and Classical Limit

    quant-ph 2026-01 conditional novelty 6.0 of 10

    The classical limit of double-commutator and double-anticommutator master equations is derived using Wigner-Weyl phase-space methods, yielding L+γL² dynamics and a nonlinear energy-shell cooling equation, respectively.

Reference graph

Works this paper leans on

97 extracted references · 56 canonical work pages · cited by 1 Pith paper

  1. [52]

    H. E. Haber, Useful relations among the generators in the defining and adjoint representations of SU(N), SciPost Phys. Lect. Notes , 21 (2021)

    work page 2021

  2. [26]

    D. C. Brody, E.-M. Graefe, and R. Melanathuru, Phase- space measurements, decoherence, and classicality, Phys. Rev. Lett.134, 120201 (2025)

    work page 2025

  3. [1]

    From the perspective of decoherence, similar calcula- tions show that for 0≤t≤t c,P(ρ t) takes the same form as Eq

    This result is consistent with the previous analysis, confirming that a single round of coherent-state POVM completely eliminates the negativity of the phase-space function. From the perspective of decoherence, similar calcula- tions show that for 0≤t≤t c,P(ρ t) takes the same form as Eq. (13), withp c replaced byp c(t) = [1−1/(Γ trs)]/N. Here,t c denotes...

  4. [2]

    W. H. Zurek, Decoherence, einselection, and the quan- tum origins of the classical, Rev. Mod. Phys.75, 715 (2003)

    2003

  5. [3]

    E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and I.-O. Stamatescu,Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd ed. (Springer Berlin Heidelberg, Berlin, Heidelberg, 2003)

    2003

  6. [4]

    Schlosshauer, Decoherence, the measurement prob- lem, and interpretations of quantum mechanics, Rev

    M. Schlosshauer, Decoherence, the measurement prob- lem, and interpretations of quantum mechanics, Rev. Mod. Phys.76, 1267 (2005)

    2005

  7. [5]

    Schlosshauer, Quantum decoherence, Physics Reports 831, 1 (2019)

    M. Schlosshauer, Quantum decoherence, Physics Reports 831, 1 (2019)

    2019

  8. [6]

    W. H. Zurek,Decoherence and Quantum Darwinism: From Quantum Foundations to Classical Reality(Cam- bridge University Press, 2025)

    2025

Show all 97 references
  1. [7]

    Yu and J

    T. Yu and J. H. Eberly, Sudden death of entanglement, Science323, 598 (2009)

    2009

  2. [8]

    K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Ve- dral, The classical-quantum boundary for correlations: Discord and related measures, Rev. Mod. Phys.84, 1655 (2012)

    2012

  3. [9]

    T. L. Curtright, D. B. Fairlie, and C. K. Zachos,A Concise Treatise on Quantum Mechanics in Phase Space (World Scientific, 2014)

    2014

  4. [10]

    Tilma, M

    T. Tilma, M. J. Everitt, J. H. Samson, W. J. Munro, and K. Nemoto, Wigner functions for arbitrary quantum systems, Phys. Rev. Lett.117, 180401 (2016)

    2016

  5. [11]

    Zhu, Quasiprobability representations of quantum mechanics with minimal negativity, Phys

    H. Zhu, Quasiprobability representations of quantum mechanics with minimal negativity, Phys. Rev. Lett. 117, 120404 (2016)

    2016

  6. [12]

    Deffner, Geometric quantum speed limits: a case for wigner phase space, New Journal of Physics19, 103018 (2017)

    S. Deffner, Geometric quantum speed limits: a case for wigner phase space, New Journal of Physics19, 103018 (2017)

    2017

  7. [13]

    Shanahan, A

    B. Shanahan, A. Chenu, N. Margolus, and A. del Campo, Quantum speed limits across the quantum-to-classical transition, Phys. Rev. Lett.120, 070401 (2018)

    2018

  8. [14]

    Le Jeannic, A

    H. Le Jeannic, A. Cavaill` es, K. Huang, R. Filip, and J. Laurat, Slowing quantum decoherence by squeezing in phase space, Phys. Rev. Lett.120, 073603 (2018)

    2018

  9. [15]

    J. E. Runeson and J. O. Richardson, Spin-mapping ap- proach for nonadiabatic molecular dynamics, The Jour- nal of Chemical Physics151, 044119 (2019)

    2019

  10. [16]

    Oliva and O

    M. Oliva and O. Steuernagel, Dynamic shear suppression in quantum phase space, Phys. Rev. Lett.122, 020401 (2019)

    2019

  11. [17]

    Bohmann, E

    M. Bohmann, E. Agudelo, and J. Sperling, Probing non- classicality with matrices of phase-space distributions, Quantum4, 343 (2020)

    2020

  12. [18]

    J. E. Runeson and J. O. Richardson, Quantum entangle- ment from classical trajectories, Phys. Rev. Lett.127, 250403 (2021)

    2021

  13. [19]

    Pedernales and M

    J. Pedernales and M. Plenio, On the origin of force sen- sitivity in tests of quantum gravity with delocalised me- chanical systems, Contemporary Physics64, 147 (2023)

    2023

  14. [20]

    Meng and Z

    W. Meng and Z. Xu, Quantum speed limits in arbitrary phase spaces, Phys. Rev. A107, 022212 (2023)

    2023

  15. [21]

    Zhang, J.-T

    J.-W. Zhang, J.-T. Bu, J. C. Li, W. Meng, W.-Q. Ding, B. Wang, W.-F. Yuan, H.-J. Du, G.-Y. Ding, W.-J. Chen, L. Chen, F. Zhou, Z. Xu, and M. Feng, Single-atom ver- ification of the optimal trade-off between speed and cost in shortcuts to adiabaticity, Phys. Rev. Lett.132, 2136...

    2024

  16. [22]

    Jorquera Riera and L

    M. Jorquera Riera and L. Loveridge, Uncertainty rela- tions relative to phase-space quantum reference frames, Phys. Rev. A111, L060201 (2025)

    2025

  17. [23]

    Holtzman, O

    R. Holtzman, O. Raz, and C. Jarzynski, Shortcuts to adiabaticity across a separatrix, Phys. Rev. Lett.134, 157201 (2025)

    2025

  18. [24]

    A. W. Shrestha, B. Bhattacharjee, and A. del Campo, Double-bracket master equations: Phase-space repre- sentation and classical limit (2026), arXiv:2601.20925 [quant-ph]

    2026 arXiv

  19. [25]

    Descamps, N

    E. Descamps, N. Fabre, A. Keller, and P. Milman, Quan- tum metrology using time-frequency as quantum contin- uous variables: Resources, sub-shot-noise precision and phase space representation, Phys. Rev. Lett.131, 030801 (2023)

    2023

  20. [27]

    D. C. Brody and R. Melanathuru, Decoherence from universal tomographic measurements (2025), arXiv:2511.07369 [quant-ph]

    2025

  21. [28]

    Wang and M

    Q. Wang and M. Robnik, Statistics of phase space local- ization measures and quantum chaos in the kicked top model, Phys. Rev. E107, 054213 (2023)

    2023

  22. [29]

    R. Basu, A. Ganguly, S. Nath, and O. Parrikar, Com- plexity growth and the krylov-wigner function, Journal of High Energy Physics2024, 264 (2024)

    2024

  23. [30]

    Pizzi, Quantum trails and memory effects in the phase space of chaotic quantum systems, Phys

    A. Pizzi, Quantum trails and memory effects in the phase space of chaotic quantum systems, Phys. Rev. Lett.134, 140402 (2025)

    2025

  24. [31]

    RouhbakhshNabati, D

    M. RouhbakhshNabati, D. Braun, and H. Schomerus, Semiclassical approach to quantum fisher information, Phys. Rev. Lett.135, 190202 (2025)

    2025

  25. [32]

    Huh, H.-S

    K.-B. Huh, H.-S. Jeong, L. A. Pando Zayas, and J. F. Pedraza, Krylov complexity in mixed phase space, Phys. Rev. D111, L121902 (2025)

    2025

  26. [33]

    R. P. Rundle and M. J. Everitt, Overview of the phase space formulation of quantum mechanics with applica- tion to quantum technologies, Advanced Quantum Tech- nologies4, 2100016 (2021)

    2021

  27. [34]

    J. E. Runeson and J. O. Richardson, Generalized spin 6 mapping for quantum-classical dynamics, The Journal of Chemical Physics152, 084110 (2020)

    2020

  28. [35]

    It is worth noting that, in contrast to the definition con- sidered here, related spin-coherent-state POVMs have been constructed for spin-Jsystems [36–39]

  29. [36]

    Zyczkowski and H.-J

    K. Zyczkowski and H.-J. Sommers, Induced measures in the space of mixed quantum states, Journal of Physics A: Mathematical and General34, 7111 (2001)

    2001

  30. [37]

    D. M. Appleby, Optimal measurements of spin direction, International Journal of Theoretical Physics39, 2231 (2000)

    2000

  31. [38]

    Kofler and C

    J. Kofler and C. Brukner, Conditions for quantum viola- tion of macroscopic realism, Phys. Rev. Lett.101, 090403 (2008)

    2008

  32. [39]

    Y. Yang, G. Chiribella, and G. Adesso, Certifying quan- tumness: Benchmarks for the optimal processing of gen- eralized coherent and squeezed states, Phys. Rev. A90, 042319 (2014)

    2014

  33. [40]

    Shojaee, C

    E. Shojaee, C. S. Jackson, C. A. Riofr ´ ıo, A. Kalev, and I. H. Deutsch, Optimal pure-state qubit tomography via sequential weak measurements, Phys. Rev. Lett.121, 130404 (2018)

    2018

  34. [41]

    E. B. Davies and J. T. Lewis, An operational approach to quantum probability, Communications in Mathematical Physics17, 239 (1970)

    1970

  35. [42]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information, 10th ed. (Cambridge Univer- sity Press, Cambridge, UK, 2010)

    2010

  36. [43]

    G¨ uhne, E

    O. G¨ uhne, E. Haapasalo, T. Kraft, J.-P. Pellonp¨ a¨ a, and R. Uola, Colloquium: Incompatible measurements in quantum information science, Rev. Mod. Phys.95, 011003 (2023)

    2023

  37. [44]

    DefineF= Tr 2 (1 N ⊗ A)(X⊗Y)

    LetX, Y, Aact onHand fix an orthonormal basis{|r⟩} for the second tensor factor. DefineF= Tr 2 (1 N ⊗ A)(X⊗Y) . Then, for any|i⟩and|j⟩,⟨i|F|j⟩=P r ⟨i, r|(1 N ⊗A)(X⊗Y)|j, r⟩= P r ⟨i|X|j⟩ ⟨r|AY|r⟩= ⟨i|X|j⟩Tr(AY).Since the equality holds for all matrix elements, which ends the pr...

  38. [45]

    Collins and P

    B. Collins and P. ´Sniady, Integration with respect to the haar measure on unitary, orthogonal and symplectic group, Communications in Mathematical Physics264, 773 (2006)

    2006

  39. [46]

    See Supplemental Material for details

  40. [47]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of n-level sys- tems, Journal of Mathematical Physics17, 821 (1976)

    1976

  41. [48]

    Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

    G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

    1976

  42. [49]

    Breuer and P

    H.-P. Breuer and P. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, Oxford, 2007)

    2007

  43. [50]

    Rivas and S

    A. Rivas and S. F. Huelga,Open Quantum Systems: An Introduction(Springer Berlin Heidelberg, 2012)

    2012

  44. [51]

    B. C. Hall,Lie Groups, Lie Algebras, and Representa- tions: An Elementary Introduction, Graduate Texts in Mathematics, Vol. 222 (Springer International Publish- ing, Cham, Switzerland, 2015)

    2015

  45. [53]

    A closely related result was derived in Ref. [26]

  46. [54]

    R. L. Stratonovich, On distributions in representation space, Sov. Phys. JETP4, 891 (1957)

    1957

  47. [55]

    W. K. Wootters, A wigner-function formulation of finite- state quantum mechanics, Annals of Physics176, 1 (1987)

    1987

  48. [56]

    Galetti and M

    D. Galetti and M. Ruzzi, Dynamics in discrete phase spaces and time interval operators, Physica A: Statistical Mechanics and its Applications264, 473 (1999)

    1999

  49. [57]

    K. E. Cahill and R. J. Glauber, Ordered expansions in boson amplitude operators, Phys. Rev.177, 1857 (1969)

    1969

  50. [58]

    K. E. Cahill and R. J. Glauber, Density operators and quasiprobability distributions, Phys. Rev.177, 1882 (1969)

    1969

  51. [59]

    In the remainder of this Letter, we focus on the widely used Wigner function withs= 0, unless explicitly stated otherwise

  52. [60]

    Hudson, When is the wigner quasi-probability density non-negative?, Reports on Mathematical Physics6, 249 (1974)

    R. Hudson, When is the wigner quasi-probability density non-negative?, Reports on Mathematical Physics6, 249 (1974)

    1974

  53. [61]

    de Boor,A Practical Guide to Splines, revised ed

    C. de Boor,A Practical Guide to Splines, revised ed. (Springer, New York, NY, 2001)

    2001

  54. [62]

    In addition, this is a threshold-crossing phe- nomenon for the negativity functional rather than a non- analyticity ofρ t

    This sudden behavior is distinct from the previously studied sudden death of entanglement [6] and instead concerns negative quasiprobability volume in phase space. In addition, this is a threshold-crossing phe- nomenon for the negativity functional rather than a non- analyticity ofρ t

  55. [63]

    D. A. Lidar, I. L. Chuang, and K. B. Whaley, Decoherence-free subspaces for quantum computation, Phys. Rev. Lett.81, 2594 (1998)

    1998

  56. [64]

    D. A. Lidar, A. Shabani, and R. Alicki, Conditions for strictly purity-decreasing quantum markovian dynamics, Chemical Physics322, 82 (2006)

    2006

  57. [65]

    Bedingham and J

    D. Bedingham and J. J. Halliwell, Classical limit of the quantum zeno effect by environmental decoherence, Phys. Rev. A89, 042116 (2014)

    2014

  58. [66]

    Chenu, M

    A. Chenu, M. Beau, J. Cao, and A. del Campo, Quantum simulation of generic many-body open system dynam- ics using classical noise, Phys. Rev. Lett.118, 140403 (2017)

    2017

  59. [67]

    M. Beau, J. Kiukas, I. L. Egusquiza, and A. del Campo, Nonexponential quantum decay under environmental de- coherence, Phys. Rev. Lett.119, 130401 (2017)

    2017

  60. [68]

    Z. Xu, L. P. Garc ´ ıa-Pintos, A. Chenu, and A. del Campo, Extreme decoherence and quantum chaos, Phys. Rev. Lett.122, 014103 (2019)

    2019

  61. [69]

    Clarke and F

    J. Clarke and F. K. Wilhelm, Superconducting quantum bits, Nature453, 1031 (2008)

    2008

  62. [70]

    Huang, D

    H.-L. Huang, D. Wu, D. Fan, and X. Zhu, Supercon- ducting quantum computing: a review, Science China Information Sciences63, 180501 (2020)

    2020

  63. [71]

    Kjaergaard, M

    M. Kjaergaard, M. Schwartz, J. Braum¨ uller, P. Krantz, J. J. Wang, S. Gustavsson, and W. Oliver, Supercon- ducting qubits: Current state of play, Annual Review of Condensed Matter Physics11, 369 (2020)

    2020

  64. [72]

    Jiang, C

    Y.-Y. Jiang, C. Deng, H. Fan, B.-Y. Li, L. Sun, X.-S. Tan, W. Wang, G.-M. Xue, F. Yan, H.-F. Yu, Y.-S. Zhang, Y.-R. Zhang, and C.-L. Zou, Advancements in superconducting quantum computing, National Science Review12, nwaf246 (2025)

    2025

  65. [73]

    Knill and R

    E. Knill and R. Laflamme, Power of one bit of quantum information, Phys. Rev. Lett.81, 5672 (1998)

    1998

  66. [74]

    Poulin, R

    D. Poulin, R. Laflamme, G. J. Milburn, and J. P. Paz, Testing integrability with a single bit of quantum infor- 7 mation, Phys. Rev. A68, 022302 (2003)

    2003

  67. [75]

    Swingle, G

    B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hay- den, Measuring the scrambling of quantum information, Phys. Rev. A94, 040302 (2016)

    2016

  68. [76]

    D. V. Vasilyev, A. Grankin, M. A. Baranov, L. M. Sieberer, and P. Zoller, Monitoring quantum simulators via quantum nondemolition couplings to atomic clock qubits, PRX Quantum1, 020302 (2020)

    2020

  69. [77]

    H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Decay of loschmidt echo enhanced by quantum crit- icality, Phys. Rev. Lett.96, 140604 (2006)

    2006

  70. [78]

    Zhang, X

    J. Zhang, X. Peng, N. Rajendran, and D. Suter, Detec- tion of quantum critical points by a probe qubit, Phys. Rev. Lett.100, 100501 (2008)

    2008

  71. [79]

    Dorner, S

    R. Dorner, S. R. Clark, L. Heaney, R. Fazio, J. Goold, and V. Vedral, Extracting quantum work statistics and fluctuation theorems by single-qubit interferometry, Phys. Rev. Lett.110, 230601 (2013)

    2013

  72. [80]

    Mazzola, G

    L. Mazzola, G. De Chiara, and M. Paternostro, Measur- ing the characteristic function of the work distribution, Phys. Rev. Lett.110, 230602 (2013)

    2013

  73. [81]

    T. B. Batalh˜ ao, A. M. Souza, L. Mazzola, R. Auccaise, R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra, Experimental recon- struction of work distribution and study of fluctuation relations in a closed quantum system, Phys. Rev. Lett. 1...

    2014

  74. [82]

    Wei and R.-B

    B.-B. Wei and R.-B. Liu, Lee-yang zeros and critical times in decoherence of a probe spin coupled to a bath, Phys. Rev. Lett.109, 185701 (2012)

    2012

  75. [83]

    X. Peng, H. Zhou, B.-B. Wei, J. Cui, J. Du, and R.-B. Liu, Experimental observation of lee-yang zeros, Phys. Rev. Lett.114, 010601 (2015)

    2015

  76. [84]

    Francis, D

    A. Francis, D. Zhu, C. Huerta Alderete, S. Johri, X. Xiao, J. K. Freericks, C. Monroe, N. M. Linke, and A. F. Kem- per, Many-body thermodynamics on quantum comput- ers via partition function zeros, Science Advances7, 2447 (2021)

    2021

  77. [85]

    Xu and A

    Z. Xu and A. del Campo, Probing the full distribution of many-body observables by single-qubit interferometry, Phys. Rev. Lett.122, 160602 (2019)

    2019

  78. [86]

    Y. Liu, J. Tian, R. Betzholz, and J. Cai, Pulsed quantum-state reconstruction of dark systems, Phys. Rev. Lett.122, 110406 (2019)

    2019

  79. [87]

    X. Nie, X. Zhu, Y.-a. Fan, X. Long, H. Liu, K. Huang, C. Xi, L. Che, Y. Zheng, Y. Feng, X. Yang, and D. Lu, Self-consistent determination of single-impurity anderson model using hybrid quantum-classical approach on a spin quantum simulator, Phys. Rev. Lett.133, 140602 (2024)

    2024

  80. [88]

    H. Liu, T. Hur, S. Zhang, L. Che, X. Long, X. Wang, K. Huang, Y.-a. Fan, Y. Zheng, Y. Feng, Y. Zhou, J. Ng, X. Nie, D. K. Park, and D. Lu, Neural quantum embed- ding via deterministic quantum computation with one qubit, Phys. Rev. Lett.135, 080603 (2025)

    2025

  81. [89]

    A. M. Childs and N. Wiebe, Hamiltonian simulation us- ing linear combinations of unitary operations, Quantum Information and Computation12, 0901 (2012)

    2012

  82. [90]

    Bravyi and A

    S. Bravyi and A. Kitaev, Universal quantum computa- tion with ideal clifford gates and noisy ancillas, Phys. Rev. A71, 022316 (2005)

    2005

  83. [91]

    Veitch, C

    V. Veitch, C. Ferrie, D. Gross, and J. Emerson, Negative quasi-probability as a resource for quantum computa- tion, New Journal of Physics14, 113011 (2012)

    2012

  84. [92]

    Howard, J

    M. Howard, J. J. Wallman, V. Veitch, and J. Emerson, Contextuality supplies the ‘magic’ for quantum compu- tation, Nature510, 351 (2014)

    2014

  85. [93]

    Delfosse, P

    N. Delfosse, P. Allard Guerin, J. Bian, and R. Raussendorf, Wigner function negativity and contextuality in quantum computation on rebits, Phys. Rev. X5, 021003 (2015)

    2015

  86. [94]

    Campos-Uscanga, E

    A. Campos-Uscanga, E. Ben ´ ıtez Rodr ´ ıguez, E. Pi- ceno Mart ´ ınez, and M. A. Bastarrachea-Magnani, Magic states in the asymmetric quantum rabi model, Phys. Rev. A113, 012412 (2026)

    2026

  87. [95]

    Zhang, S

    P. Zhang, S. Zhou, and N. Sun, Stabilizer r´ enyi en- tropy and its transition in the coupled sachdev-ye-kitaev model, Phys. Rev. Lett.136, 080201 (2026)

    2026

  88. [96]

    Hoshino and Y

    M. Hoshino and Y. Ashida, Stabilizer r´ enyi entropy en- codes fusion rules of topological defects and boundaries, Phys. Rev. Lett.136, 080402 (2026)

    2026

  89. [97]

    Preskill, Quantum Computing in the NISQ era and beyond, Quantum2, 79 (2018) Supplemental Material CONTENTS References 5 I

    J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum2, 79 (2018) Supplemental Material CONTENTS References 5 I. An alternative derivation of Eq. (2) in the main text 8 II. Discrete phase space and single-shot positivity 9 III. Decoherence time versus the finite v...

    2018

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