Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

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

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The classical limit of double-bracket dissipation is a deformed Hamiltonian flow in phase space.

desk verdict Good semiclassical machinery with an honest but unresolved scaling caveat: the energy-dephasing limit is clean and citable, the double-anticommutator limit is only conditional. read the letter →

arxiv 2601.20925 v2 pith:YD726SEK submitted 2026-01-28 quant-ph cond-mat.stat-mech

Ankit W. Shrestha , Budhaditya Bhattacharjee , Adolfo del Campo This is my paper · ORCID
classification quant-phcond-mat.stat-mech MSC 81Q2081S30 PACS 03.65.Sq03.65.Yz
keywords double-bracketmasterequationsclassicallimitWignerfunctionMoyalbracketenergydephasingdoubleanticommutatorgradientflowspectralfiltering
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

This paper asks what happens to two standard dissipative quantum master equations when $\hbar$ is sent to zero. It shows that energy dephasing—decoherence in the Hamiltonian eigenbasis—becomes a deformed classical Liouvillian flow, $\partial_t W = L W + \gamma L^2 W$, whose solution is a heat-kernel average over Hamiltonian trajectories. It further shows that the nonlinear double-anticommutator master equation from stochastic non-Hermitian systems becomes $\partial_t W = \{H,W\}_P - 4\Gamma(H^2 - \langle H^2\rangle)W$, a balanced-gain-and-loss evolution that cools the state toward low energy. These classical limits matter because they give tractable semiclassical models for decoherence, connect quantum dephasing to classical diffusion on phase space, and identify when anti-Hermitian noise can delay the emergence of classicality.

What carries the argument

The machinery is the Wigner–Weyl phase-space formulation of quantum mechanics: operators become functions $A(x,p)$ via the Weyl transform, and the density operator becomes the Wigner function $W(x,p)$. Commutators become Moyal brackets $\{\{A,B\}\} = (A\star B - B\star A)/i\hbar$, with the star product expanding in powers of $\hbar$; taking the leading order recovers Poisson brackets and classical Liouville flow. The paper uses this expansion order-by-order for the master equations, identifies the leading nonvanishing dissipative terms, and solves the resulting semiclassical equations with a heat-kernel formula for the $L^2$ deformation and characteristic-curve integration for the gain/loss term. A gradient-flow representation of both evolutions supplies a geometric interpretation.

What would settle it

Supply a harmonic oscillator with a Gaussian Wigner function and evolve it under the exact double-anticommutator master equation (34) with the physical noise scaling $\Gamma=(\lambda/\hbar)^2$. As $\hbar\to0$, the term $-4\Gamma(H^2-\langle H^2\rangle)W$ grows without bound, so either the state is pinned to a surface of constant $H^2$ or extra counterterms from higher orders save the limit; the paper's classical equation (44), derived for fixed $\Gamma$, predicts unconstrained cooling. A numerical comparison of $\langle H^2\rangle(t)$ under these two scalings settles which behaviour is the true classical limit.

Watch

Extended reading notes

Core claim

The paper claims that energy-dephasing master equations and double-anticommutator master equations have well-defined classical limits captured by explicit phase-space equations. For double-commutator (energy dephasing) dissipation, the leading-order $\hbar\to0$ equation is $\partial_t W = L W + \gamma L^2 W$ with $L = \{H,\cdot\}_P$, meaning the classical Liouvillian generator is deformed by its own square and the solution is a heat-kernel average over Hamiltonian trajectories. For the trace-preserving double-anticommutator equation, assuming the dissipation strength $\Gamma$ is independent of $\hbar$, the leading-order equation is $\partial_t W = \{H,W\}_P - 4\Gamma(H^2 - \langle H^2\rangle)W$, whose nonlinear gain/loss term drives probability toward low energy and which admits a closed solution along characteristic Hamiltonian curves. Both equations are shown to be gradient flows, and the same phase-space technique produces classical limits for higher-order nested-bracket master equations used in spectral filtering.

Load-bearing premise

The double-anticommutator classical limit assumes $\Gamma$ is independent of $\hbar$; if $\Gamma$ scales as $1/\hbar^2$, as it does in the noise-averaged derivation of that master equation, the leading nonlinear term diverges and equation (44) is not the classical limit.

Editorial extensions

If this is right

  • Energy dephasing's classical limit is not pure Hamiltonian flow: the generator becomes $L+\gamma L^2$, so each Fourier mode in action-angle variables decays as $e^{-\gamma\omega^2 k^2 t}$, driving the Wigner function to an angle-uniform ring while preserving the energy distribution.
  • The double-anticommutator equation cools: in the energy eigenbasis populations obey $\rho_n(t)\propto \rho_n(0)e^{-4\Gamma E_n^2 t}$, and in the classical limit the energy marginal concentrates on the lowest available energy. Higher moments $\mu_2(t)$ and $\mu_4(t)$ decrease monotonically.
  • In a driven anharmonic (chaotic) oscillator, the balanced gain/loss term cannot itself create Wigner negativity, but it can amplify and sustain the negative regions produced by quantum corrections. This delays the quantum-to-classical crossover time $t_c$ for both Gaussian and Schrödinger-cat initial states.
  • The phase-space equations extend to higher-order nested brackets: frequency filters map to nested Poisson brackets $\{H,W\}_P^{2n}$, and eigenvalue filters map to multiplicative terms $2^{2n}(H^{2n}-\langle H^{2n}\rangle)W$, giving classical counterparts of spectral-filtering master equations.
  • Both double-bracket evolutions admit gradient-flow representations, with a projected-gradient structure enforcing trace preservation, so steady states are critical points of the associated potentials.

Reading between the lines

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

  • The divergence noted by the paper for $\Gamma\sim 1/\hbar^2$ suggests that the physically correct classical limit of the double-anticommutator equation is a constrained evolution on the surface $H(x,p)^2=\langle H^2\rangle$ (the energy shell), rather than the unconstrained equation (44); deriving that constrained dynamics explicitly would be a natural next step.
  • Because the $L^2$ deformation acts as a Gaussian time average over Hamiltonian trajectories, one could test whether energy dephasing is equivalent to classical dynamics with a random time offset; such an equivalence would give a simple classical explanation of the smoothing seen in numerical simulations.
  • The result that anti-Hermitian noise prolongs Wigner negativity suggests that engineering the noise statistics of a non-Hermitian drive could be used as a control knob for preserving nonclassical resources in continuous-variable and oscillator-based quantum information processing.
  • Extending the nested-bracket classical limits to many-body systems could provide a semiclassical route to spectral form factors and spectral statistics, connecting the paper's phase-space equations to the spectral-filtering techniques already used in numerical many-body physics.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies the classical (ℏ→0) limit of two classes of double-bracket master equations: energy-dephasing dynamics with a double-commutator dissipator, and a nonlinear trace-preserving evolution with a double-anticommutator dissipator motivated by fluctuating non-Hermitian Hamiltonians. Using the Wigner-Weyl transform and Moyal brackets, the authors derive ℏ-expansions for both master equations and identify order-ℏ⁰ classical equations: ∂tW = LW + γL²W for energy dephasing (Eq. 20) and ∂tW = {H,W}_P − 4Γ(H² − ⟨H²⟩)W for the double-anticommutator case (Eq. 44). They also provide a gradient-flow formulation, closed-form solutions for the harmonic oscillator, numerical studies of a driven anharmonic oscillator using Wigner logarithmic negativity, and an extension to higher-order nested commutator/anticommutator master equations motivated by spectral filtering.

Significance. If the central claims are correct, the paper provides a systematic semiclassical framework for two physically relevant classes of dissipative dynamics, including explicit closed-form solutions for the harmonic oscillator and a clear gradient-flow interpretation. The derivations are self-contained and the leading-order classical equations are obtained from the quantum master equations rather than assumed. The numerical studies of Wigner negativity and classical emergence in a chaotic driven oscillator are concrete and falsifiable. However, the significance of the double-anticommutator result is conditional on the assumed scaling of Γ with ℏ, which is not the scaling obtained from the noise-averaging derivation in the same paper.

major comments (4)
  1. [Sec. IV, Eq. (40) to Eq. (44)] The classical limit for the double-anticommutator master equation is derived under the explicit assumption that Γ is independent of ℏ (stated after Eq. (40)), but the physically motivated noise-averaged scaling from Eq. (4) is Γ = (λ/ℏ)². Under that scaling, the leading-order term −4Γ(H² − ⟨H²⟩)W diverges as ℏ→0, and the terms written as O(ℏ²) in Eq. (40) become O(1), so Eq. (44) is not a valid classical limit in the physical scaling. The manuscript acknowledges this divergence but does not provide the constrained energy-shell dynamics that would constitute the correct limit, nor does it identify any regime in which Eq. (44) governs the original quantum dynamics. Since the abstract and conclusions present the classical limit of the double-anticommutator class without this caveat, the central claim for this class is not supported for the physically relevant scaling.
  2. [Sec. III, Eq. (19)] The coefficient of the {H,{H,W}_P}_P term in Eq. (19) is written as λ/2, but the preceding derivation leading to Eq. (17) and the identification Γ = (λ/ℏ)² in Eq. (4) require the coefficient to be λ²/2. The definition γ = λ/2 in Eq. (20) then propagates the error into the deformation L → L + γL² and into all subsequent time scales such as the decay rate e^{−γω²t} in Eq. (30). If λ is intended to be dimensionless as in Eq. (3), then γ must be λ²/2; otherwise the strength of the classical dissipation is incorrect by a factor λ.
  3. [Sec. IV, Eq. (36)] The expansion of the symmetric Moyal bracket in Eq. (36) is truncated as O(ℏ⁵), but the immediately following Eq. (37) retains terms of order ℏ² and quotes O(ℏ⁴). Since the cosine series contains only even powers of ℏ, the correct remainder is O(ℏ⁴), and the O(ℏ⁵) label obscures the consistency of the expansion used for Eq. (40). This is not merely cosmetic: Eq. (40) is the basis for the classical limit and for the subsequent comparison of leading and subleading orders, so the ordering must be stated precisely.
  4. [Sec. V, Eq. (66)] The claimed gradient-flow representation of the double-anticommutator term relies on the existence of an operator K′_m such that ||{K_m,σ}||²_HS = ||[K′_m,σ]||²_HS. No construction of K′_m is provided, and the manuscript only notes that one possible route involves the inverse of the symmetric logarithmic derivative, with K′_m generally depending on σ. Without an explicit or existence proof of this equality, the gradient-flow statement for the anticommutator class (Eqs. (66)–(69)) is incomplete and should be qualified as conditional or deferred to future work.
minor comments (5)
  1. [Sec. IV, Eq. (57)] Equation (57) uses the symbol γ in the exponential amplitude, but the section consistently uses Γ for the double-anticommutator coupling; this is a notation inconsistency that may confuse readers comparing with Eq. (44).
  2. [Sec. V, text after Eq. (62)] The text reads "action of the Linbladian" — presumably a typo for "Lindbladian".
  3. [Sec. VI.A, Eq. (32)] The expression for ⟨x²⟩ in Eq. (32) is not symmetric between x(0) and p(0) in the way one would expect from the Hamiltonian flow; please verify the algebra or add a brief derivation, as this is the main closed-form result used in the harmonic oscillator example.
  4. [Sec. VI, Fig. 2 caption] The caption of Fig. 2 does not specify the parameters γ, Γ, and ℏ used for the double-commutator and double-anticommutator panels; adding these values would improve reproducibility.
  5. [Data Availability] The data availability statement references a GitHub repository without a URL or identifier, so the code is not currently accessible to the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the classical limits are obtained by explicit ℏ-expansions of the stated master equations, with the double-anticommutator case explicitly conditional on Γ being independent of ℏ.

full rationale

The derivations are self-contained. Equation (20) follows from applying the Moyal expansion to the energy-dephasing master equation (5) with the stated scaling Γ=(λ/ℏ)^2 from Eq. (4); the heat-kernel solution (24), Ehrenfest equations (25)–(26), and oscillator moments are derived from Eq. (20) rather than assumed. Equation (44) is the explicit ℏ^0 truncation of the exact phase-space equation (40) for the double-anticommutator master equation (34), and the text states the Γ-independent-of-ℏ condition under which this truncation is valid, together with the divergent behavior under the alternative scaling (4). This is a conditionality caveat, not a circular step. The only input taken from previous work is the form of the master equations themselves (e.g., Eq. (34) from Ref. [31]), which are used as starting models rather than as evidence for the classical limits; no parameter is fitted to the quantities being 'predicted', and no uniqueness theorem or ansatz is imported to force a conclusion. The gradient-flow potentials in Sec. V are explicitly constructed from the generators (e.g., Eq. (64)) and are not load-bearing for the classical-limit claims.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The main derivations use standard phase-space methods. The only ad hoc input is the ℏ-scaling of Γ for the double-anticommutator case. No new physical entities are introduced; the K'_m operator is an unfinished mathematical construction.

free parameters (2)
  • Γ = not fitted; chosen independent of ℏ (e.g., Γ=0.3 in figures)
    Decoherence strength in the double-anticommutator master equation; the classical limit derivation assumes Γ is ℏ-independent. In numerical examples Γ is set to specific values.
  • γ = set via γ = λ/2 (likely λ²/2) in Eq. 20; numerical values in figures
    Coefficient of L² in the classical limit of energy dephasing; controls the diffusion strength in the deformed Liouvillian.
assumptions (4)
  • standard math Wigner-Weyl transform and Moyal brackets provide a faithful phase-space representation of operator commutators, with the ℏ-expansion given by Poisson brackets at leading order.
    Used throughout Sec. III and IV; equations (10), (12), (14).
  • domain assumption The noise-averaged dynamics of a system with fluctuating Hamiltonian H + Σ ξ_m λ_m L_m is governed by the double-commutator master equation with Γ_m = (λ_m/ℏ)².
    Eq. (4), cited to Refs. [5,36,45,46]; this scaling is needed for a nonvanishing classical dissipator.
  • ad hoc to paper For the double-anticommutator classical limit, Γ is assumed independent of ℏ.
    Sec. IV after Eq. (40). This assumption is necessary for the expansion (40) to be the classical limit; the physical 1/ℏ² scaling leads to a divergent nonlinear term.
  • domain assumption The Wigner function is sufficiently smooth for the ℏ-expansion to be valid term-by-term.
    Gradient terms in the expansion are kept to fixed order; this breaks down for highly oscillatory Wigner functions, though the paper argues decoherence suppresses small-scale structure (Sec. VI.B).
invented entities (1)
  • Operator K'_m
    purpose: To rewrite the anticommutator {K_m, σ} as a commutator [K'_m, σ] so that the anticommutator potential Φ+ can be cast as a gradient flow on the Lie-group tangent space.
    Introduced in Sec. V (around Eq. 66) with the note that it could be constructed via the inverse symmetric logarithmic derivative; no explicit construction or uniqueness proof is given. It is a mathematical device, not a physical entity.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Double-Bracket Master Equations: Phase-Space Representation and Classical Limit." pith.science (2026). https://pith.science/paper/YD726SEK

@misc{pith2026260120925,
  author       = {Pith},
  title        = {Pith review of: Double-Bracket Master Equations: Phase-Space Representation and Classical Limit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YD726SEK}},
  note         = {Machine review of arXiv:2601.20925}
}
abstract

We systematically investigate master equations involving double-bracket dissipators in the classical limit. Dissipators defined by double commutators arise naturally in dephasing dynamics, while those defined by double anticommutators are typically associated with noisy Hamiltonians. The classical limit of such master equations is derived by reformulating the dissipative open dynamics in phase space using the Wigner-Weyl transform and Moyal bracket formalism, and by identifying the leading-order terms in a systematic $\hbar$-expansion. We first analyze a double-commutator master equation associated with energy diffusion, and then turn to master equations containing a double anticommutator with the system Hamiltonian, recently derived in the context of noisy non-Hermitian systems. For both classes of double-bracket equations, we establish a gradient-flow representation of the dynamics within both the quantum and semiclassical formalisms. We further study the semiclassical evolution generated by double-bracket master equations for the harmonic (integrable) and driven anharmonic (chaotic) oscillators, considering both classical and quantum initial states. The dynamics are characterized through several observables, including mean position, momentum, and energy. The quantumness of the time-evolved state is assessed via the Wigner logarithmic negativity, and the interplay between double-bracket dissipation and chaotic dynamics is examined in detail. Finally, we extend our analysis to generalized master equations involving higher-order nested brackets, which arise naturally as a time-continuous formulation of spectral filtering techniques widely used in the numerical simulation of quantum systems.

Figures

Figures reproduced from arXiv: 2601.20925 by the authors.

Figure 1
Figure 1. FIG. 1. Gradient flow description of the time evolution, show [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Evolution of the Wigner function of an initial Gaussian wavepacket with mean at ( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Average position, momentum, and energy of the Wigner function evolved using the semiclassical limit of the master [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Wigner logarithmic negativity [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Wigner logarithmic negativity [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Evolution of the Wigner function of an initial cat state [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Wigner Logarithmic Negativity [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

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

    quant-ph 2025-07 conditional novelty 5.0 of 10

    A single round of an N-level coherent-state POVM removes all quasiprobability negativity, and the equivalent depolarizing dynamics loses negativity abruptly at a critical time that can be shorter than the conventional...

Reference graph

Works this paper leans on

109 extracted references · 60 canonical work pages · cited by 1 Pith paper

  1. [1]

    H. P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, 2002)

    2002

  2. [2]

    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

  3. [3]

    I. L. Egusquiza, L. J. Garay, and J. M. Raya, Quantum evolution according to real clocks, Phys. Rev. A59, 3236 (1999)

    1999

  4. [4]

    I. L. Egusquiza and L. J. Garay, Real clocks and the Zeno effect, Phys. Rev. A68, 022104 (2003)

    2003

  5. [5]

    Chenu, M

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

    2017

  6. [6]

    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

  7. [7]

    J. K. Korbicz, E. A. Aguilar, P. ´Cwikli´ nski, and P. Horodecki, Generic appearance of objective results in quantum measurements, Phys. Rev. A96, 032124 (2017)

    2017

  8. [8]

    A. S. Matsoukas-Roubeas, M. Beau, L. F. Santos, and A. del Campo, Unitarity breaking in self-averaging spec- tral form factors, Phys. Rev. A108, 062201 (2023)

    2023

Show all 109 references
  1. [9]

    Z. Xu, A. Chenu, T. Prosen, and A. del Campo, Ther- mofield dynamics: Quantum chaos versus decoherence, Phys. Rev. B103, 064309 (2021)

    2021

  2. [10]

    A. S. Matsoukas-Roubeas, T. Prosen, and A. del Campo, Quantum Chaos and Coherence: Random Parametric Quantum Channels, Quantum8, 1446 (2024)

    2024

  3. [11]

    del Campo and T

    A. del Campo and T. Takayanagi, Decoherence in Con- formal Field Theory, JHEP2020(2), 170

  4. [12]

    Gisin, Quantum measurements and stochastic pro- cesses, Phys

    N. Gisin, Quantum measurements and stochastic pro- cesses, Phys. Rev. Lett.52, 1657 (1984)

    1984

  5. [13]

    I. C. Percival, Primary state diffusion, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences447, 189 (1994)

    1994

  6. [14]

    Schneider and G

    S. Schneider and G. J. Milburn, Decoherence in ion traps due to laser intensity and phase fluctuations, Phys. Rev. A57, 3748 (1998)

    1998

  7. [15]

    S. L. Adler, D. C. Brody, T. A. Brun, and L. P. Hugh- ston, Martingale models for quantum state reduction, Journal of Physics A: Mathematical and General34, 8795 (2001)

    2001

  8. [16]

    D. C. Brody and L. P. Hughston, Efficient simulation of quantum state reduction, Journal of Mathematical Physics43, 5254 (2002)

    2002

  9. [17]

    S. L. Adler, Weisskopf-Wigner decay theory for the energy-driven stochastic Schr¨ odinger equation, Phys. Rev. D67, 025007 (2003)

    2003

  10. [18]

    Bassi and G

    A. Bassi and G. Ghirardi, Dynamical reduction models, Physics Reports379, 257 (2003)

    2003

  11. [19]

    D. C. Brody and L. P. Hughston, Quantum noise and stochastic reduction, Journal of Physics A: Mathemati- cal and General39, 833 (2006)

    2006

  12. [20]

    Bassi, K

    A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ul- bricht, Models of wave-function collapse, underlying theories, and experimental tests, Rev. Mod. Phys.85, 471 (2013)

    2013

  13. [21]

    A. S. Matsoukas-Roubeas, F. Roccati, J. Cornelius, Z. Xu, A. Chenu, and A. del Campo, Non-Hermitian 16 Hamiltonian deformations in quantum mechanics, JHEP2023(1), 60

  14. [22]

    R. W. Brockett, Dynamical systems that sort lists, di- agonalize matrices, and solve linear programming prob- lems, Linear Algebra and its Applications146, 79 (1991)

    1991

  15. [23]

    Wegner, Flow-equations for Hamiltonians, Annalen der Physik506, 77 (1994)

    F. Wegner, Flow-equations for Hamiltonians, Annalen der Physik506, 77 (1994)

    1994

  16. [24]

    F. J. Wegner, Flow equations for Hamiltonians, Physics Reports348, 77 (2001)

    2001

  17. [25]

    S. D. G lazek and K. G. Wilson, Renormalization of Hamiltonians, Phys. Rev. D48, 5863 (1993)

    1993

  18. [26]

    S. D. Glazek and K. G. Wilson, Perturbative renormal- ization group for Hamiltonians, Phys. Rev. D49, 4214 (1994)

    1994

  19. [27]

    Kehrein,The Flow Equation Approach to Many- Particle Systems, Springer Tracts in Modern Physics (Springer Berlin Heidelberg, 2007)

    S. Kehrein,The Flow Equation Approach to Many- Particle Systems, Springer Tracts in Modern Physics (Springer Berlin Heidelberg, 2007)

    2007

  20. [28]

    H¨ ornedal, N

    N. H¨ ornedal, N. Carabba, K. Takahashi, and A. del Campo, Geometric Operator Quantum Speed Limit, Wegner Hamiltonian Flow and Operator Growth, Quan- tum7, 1055 (2023)

    2023

  21. [30]

    Gluza, J

    M. Gluza, J. Son, B. H. Tiang, R. Zander, R. Seidel, Y. Suzuki, Z. Holmes, and N. H. Y. Ng, Double-bracket quantum algorithms for quantum imaginary-time evo- lution, Phys. Rev. Lett.136, 020601 (2026)

    2026

  22. [31]

    Martinez-Azcona, A

    P. Martinez-Azcona, A. Kundu, A. Saxena, A. del Campo, and A. Chenu, Quantum dynamics with stochastic non-Hermitian Hamiltonians, Phys. Rev. Lett.135, 010402 (2025)

    2025

  23. [32]

    Gorini, A

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

    1976

  24. [33]

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

    2003

  25. [34]

    M. A. Nielsen and I. L. Chuang,Quantum Computa- tion and Quantum Information, 10th ed. (Cambridge University Press, Cambridge, UK, 2010)

    2010

  26. [35]

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

    2025

  27. [36]

    N. G. van Kampen,Stochastic processes in physics and chemistry(Elsevier Science B. V., Amsterdam, The Netherlands, 1992)

    1992

  28. [37]

    Ruschhaupt, X

    A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, Optimally robust shortcuts to population inversion in two-level quantum systems, New Journal of Physics14, 093040 (2012)

    2012

  29. [38]

    Kiely, J

    A. Kiely, J. G. Muga, and A. Ruschhaupt, Effect of Poisson noise on adiabatic quantum control, Phys. Rev. A95, 012115 (2017)

    2017

  30. [39]

    del Campo and K

    A. del Campo and K. Kim, Focus on shortcuts to adia- baticity, New Journal of Physics21, 050201 (2019)

    2019

  31. [40]

    Kiely, Exact classical noise master equations: Ap- plications and connections, Europhysics Letters134, 10001 (2021)

    A. Kiely, Exact classical noise master equations: Ap- plications and connections, Europhysics Letters134, 10001 (2021)

    2021

  32. [41]

    Dutta, A

    A. Dutta, A. Rahmani, and A. del Campo, Anti-Kibble- Zurek behavior in crossing the quantum critical point of a thermally isolated system driven by a noisy control field, Phys. Rev. Lett.117, 080402 (2016)

    2016

  33. [42]

    Ai, J.-M

    M.-Z. Ai, J.-M. Cui, R. He, Z.-H. Qian, X.-X. Gao, Y.-F. Huang, C.-F. Li, and G.-C. Guo, Experimental verification of anti–Kibble-Zurek behavior in a quantum system under a noisy control field, Phys. Rev. A103, 012608 (2021)

    2021

  34. [43]

    Iwamura and T

    K. Iwamura and T. Suzuki, Analytical derivation and extension of the anti-Kibble-Zurek scaling in the trans- verse field Ising model, Phys. Rev. B110, 144102 (2024)

    2024

  35. [44]

    Smith, Y

    A. Smith, Y. Lu, S. An, X. Zhang, J.-N. Zhang, Z. Gong, H. T. Quan, C. Jarzynski, and K. Kim, Verification of the quantum nonequilibrium work relation in the pres- ence of decoherence, New Journal of Physics20, 013008 (2018)

    2018

  36. [45]

    A. A. Budini, Non-Markovian Gaussian dissipative stochastic wave vector, Phys. Rev. A63, 012106 (2000)

    2000

  37. [46]

    A. A. Budini, Quantum systems subject to the action of classical stochastic fields, Phys. Rev. A64, 052110 (2001)

    2001

  38. [47]

    Jacobs and D

    K. Jacobs and D. A. Steck, A straightforward introduc- tion to continuous quantum measurement, Contemp. Phys.47, 279–303 (2006)

    2006

  39. [48]

    Jacobs,Quantum Measurement Theory and its Ap- plications(Cambridge University Press, 2014)

    K. Jacobs,Quantum Measurement Theory and its Ap- plications(Cambridge University Press, 2014)

    2014

  40. [49]

    Wigner, On the quantum correction for thermody- namic equilibrium, Phys

    E. Wigner, On the quantum correction for thermody- namic equilibrium, Phys. Rev.40, 749 (1932)

    1932

  41. [50]

    J. E. Moyal, Quantum mechanics as a statistical theory, Mathematical Proceedings of the Cambridge Philosoph- ical Society45, 99–124 (1949)

    1949

  42. [51]

    Hillery, R

    M. Hillery, R. O’Connell, M. Scully, and E. Wigner, Dis- tribution functions in physics: Fundamentals, Physics Reports106, 121 (1984)

    1984

  43. [52]

    C. K. Zachos, D. B. Fairlie, and T. L. Curtright, eds., Quantum Mechanics in Phase Space: An Overview with Selected Papers(World Scientific, Singapore, 2005)

    2005

  44. [53]

    G. S. Agarwal, Master equations in phase-space formu- lation of quantum optics, Phys. Rev.178, 2025 (1969)

    1969

  45. [54]

    Dekker, Quantization of the linearly damped har- monic oscillator, Phys

    H. Dekker, Quantization of the linearly damped har- monic oscillator, Phys. Rev. A16, 2126 (1977)

    1977

  46. [55]

    Caldeira and A

    A. Caldeira and A. Leggett, Path integral approach to quantum brownian motion, Physica A: Statistical Me- chanics and its Applications121, 587 (1983)

    1983

  47. [56]

    W. G. Unruh and W. H. Zurek, Reduction of a wave packet in quantum brownian motion, Phys. Rev. D40, 1071 (1989)

    1989

  48. [57]

    B. L. Hu, J. P. Paz, and Y. Zhang, Quantum brownian motion in a general environment: Exact master equa- tion with nonlocal dissipation and colored noise, Phys. Rev. D45, 2843 (1992)

    1992

  49. [58]

    A. Isar, A. Sandulescu, and W. Scheid, Phase space rep- resentation for open quantum systems within the Lind- blad theory, International Journal of Modern Physics B 10, 2767 (1996)

    1996

  50. [59]

    Habib, K

    S. Habib, K. Shizume, and W. H. Zurek, Decoherence, chaos, and the correspondence principle, Phys. Rev. Lett.80, 4361 (1998)

    1998

  51. [60]

    Cabrera, D

    R. Cabrera, D. I. Bondar, K. Jacobs, and H. A. Rab- itz, Efficient method to generate time evolution of the Wigner function for open quantum systems, Phys. Rev. A92, 042122 (2015)

    2015

  52. [61]

    D. I. Bondar, R. Cabrera, A. Campos, S. Mukamel, and H. A. Rabitz, Wigner–Lindblad equations for quantum friction, The Journal of Physical Chemistry Letters7, 1632 (2016). 17

    2016

  53. [62]

    W. F. Braasch, O. D. Friedman, A. J. Rimberg, and M. P. Blencowe, Wigner current for open quantum sys- tems, Phys. Rev. A100, 012124 (2019)

    2019

  54. [63]

    Steuernagel and R.-K

    O. Steuernagel and R.-K. Lee, Lindblad superoperators from Wigner’s phase space continuity equation, AIP Ad- vances15, 015009 (2025)

    2025

  55. [64]

    Pearle, Combining stochastic dynamical state-vector reduction with spontaneous localization, Phys

    P. Pearle, Combining stochastic dynamical state-vector reduction with spontaneous localization, Phys. Rev. A 39, 2277 (1989)

    1989

  56. [65]

    P. Wang, Random non-Hermitian action theory for stochastic quantum dynamics: from canonical to path integral quantization, Journal of Physics A: Mathemat- ical and Theoretical58, 135303 (2025)

    2025

  57. [66]

    Wang, Random non-Hermitian Hamiltonian frame- work for symmetry-breaking dynamics, Phys

    P. Wang, Random non-Hermitian Hamiltonian frame- work for symmetry-breaking dynamics, Phys. Rev. A 112, 012209 (2025)

    2025

  58. [67]

    Rossmann, inLie Groups: An Introduction Through Linear Groups(Oxford University Press, 2002)

    W. Rossmann, inLie Groups: An Introduction Through Linear Groups(Oxford University Press, 2002)

    2002

  59. [68]

    Helmke and J

    U. Helmke and J. B. Moore,Optimization and dynami- cal systems(Springer, 2014)

    2014

  60. [69]

    A. M. Bloch,Hamiltonian and gradient flows, algo- rithms and control(American Mathematical Society, 1994)

    1994

  61. [70]

    Schulte-Herbr¨ uggen, S

    T. Schulte-Herbr¨ uggen, S. J. Glaser, G. Dirr, and U. Helmke, Gradient flows for optimization in quantum information and quantum dynamics: foundations and applications, Reviews in Mathematical Physics22, 597 (2010)

    2010

  62. [71]

    Wiersema and N

    R. Wiersema and N. Killoran, Optimizing quantum cir- cuits with Riemannian gradient flow, Phys. Rev. A107, 062421 (2023)

    2023

  63. [72]

    Wiersema, D

    R. Wiersema, D. Lewis, D. Wierichs, J. Carrasquilla, and N. Killoran, Here comes the SU(N): multivariate quantum gates and gradients, Quantum8, 1275 (2024)

    2024

  64. [73]

    Xiaoyue, M

    L. Xiaoyue, M. Robbiati, A. Pasquale, E. Pedicillo, A. Wright, S. Carrazza, and M. Gluza, Strategies for optimizing double-bracket quantum algorithms (2024), arXiv:2408.07431 [quant-ph]

    2024 arXiv

  65. [74]

    Gluza, Double-bracket quantum algorithms for di- agonalization, Quantum8, 1316 (2024)

    M. Gluza, Double-bracket quantum algorithms for di- agonalization, Quantum8, 1316 (2024)

    2024

  66. [75]

    N. A. McMahon, M. Pervez, and C. Arenz, Equat- ing quantum imaginary time evolution, Riemannian gradient flows, and stochastic implementations (2025), arXiv:2504.06123 [quant-ph]

    2025 arXiv

  67. [76]

    Villanueva and L

    A. Villanueva and L. P. Garc´ ıa-Pintos, Hamiltonian and double-bracket flow formulations of quantum measure- ments (2025), arXiv:2512.15412 [quant-ph]

    2025 arXiv

  68. [77]

    A. M. Bloch, R. W. Brockett, and P. E. Crouch, Dou- ble bracket equations and geodesic flows on symmetric spaces, Communications in Mathematical Physics187, 357 (1997)

    1997

  69. [78]

    Mittnenzweig and A

    M. Mittnenzweig and A. Mielke, An entropic gradient structure for Lindblad equations and couplings of quan- tum systems to macroscopic models, Journal of Statis- tical Physics167, 205 (2017)

    2017

  70. [79]

    E. A. Carlen and J. Maas, Gradient flow and entropy inequalities for quantum Markov semigroups with de- tailed balance, Journal of Functional Analysis273, 1810 (2017)

    2017

  71. [80]

    Y. Cao, J. Lu, and Y. Lu, Gradient flow structure and exponential decay of the sandwiched r´ enyi diver- gence for primitive Lindblad equations with gns-detailed balance, Journal of Mathematical Physics60, 052202 (2019)

    2019

  72. [81]

    E. A. Carlen and J. Maas, Non-commutative calculus, optimal transport and functional inequalities in dissi- pative quantum systems, Journal of Statistical Physics 178, 319 (2020)

    2020

  73. [82]

    Kaplanek, A

    G. Kaplanek, A. Maloney, J. Pollack, and D. VanAllen, Lindblad evolution as gradient flow, Phys. Rev. A112, 042220 (2025)

    2025

  74. [83]

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

    1998

  75. [84]

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

    2017

  76. [85]

    W. H. Zurek, Decoherence and the transition from quan- tum to classical, Physics Today44, 36 (1991)

    1991

  77. [86]

    Beau and A

    M. Beau and A. del Campo, Nonlinear quantum metrol- ogy of many-body open systems, Phys. Rev. Lett.119, 010403 (2017)

    2017

  78. [87]

    Y. Yang, Z. Xu, and A. del Campo, Decoherence rate in random Lindblad dynamics, Phys. Rev. Res.6, 023229 (2024)

    2024

  79. [88]

    S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994)

    1994

  80. [89]

    quantum-ness

    using the Husimi distribution function, where the authors find that the driving term enhances chaos and increases the rate of quantum tunneling between stabil- ity tubes present in the classically inaccessible regions of phase space. Another canonical chaotic model studied in ...

  81. [90]

    W. A. Lin and L. E. Ballentine, Quantum tunneling and chaos in a driven anharmonic oscillator, Phys. Rev. Lett.65, 2927 (1990)

    1990

  82. [91]

    J. H. Jensen and Q. Niu, Wigner symbols, quantum dynamics, and the kicked rotator, Phys. Rev. A42, 2513 (1990)

    1990

  83. [92]

    W. H. Zurek and J. P. Paz, Decoherence, chaos, and the second law, Phys. Rev. Lett.72, 2508 (1994)

    1994

  84. [93]

    A. R. Kolovsky, Quantum coherence, evolution of the Wigner function, and transition from quantum to clas- sical dynamics for a chaotic system, Chaos: An Inter- disciplinary Journal of Nonlinear Science6, 534 (1996)

    1996

  85. [94]

    L¨ utkenhaus and S

    N. L¨ utkenhaus and S. M. Barnett, Nonclassical effects in phase space, Physical Review A51, 3340 (1995)

    1995

  86. [95]

    Kenfack and K

    A. Kenfack and K. ˙Zyczkowski, Negativity of the Wigner function as an indicator of non-classicality, Jour- nal of Optics B: Quantum and Semiclassical Optics6, 396 (2004)

    2004

  87. [96]

    Strandberg, Y

    I. Strandberg, Y. Lu, F. Quijandr´ ıa, and G. Johans- son, Numerical study of Wigner negativity in one- dimensional steady-state resonance fluorescence, Phys. Rev. A100, 063808 (2019)

    2019

  88. [97]

    K. Goto, T. Nosaka, and M. Nozaki, Probing chaos by magic monotones, Phys. Rev. D106, 126009 (2022)

    2022

  89. [98]

    Xu, Quantum-to-classical transition via single-shot generalized measurements (2025), arXiv:2507.13174 [quant-ph]

    Z. Xu, Quantum-to-classical transition via single-shot generalized measurements (2025), arXiv:2507.13174 [quant-ph]

    2025 arXiv

  90. [99]

    Pashayan, J

    H. Pashayan, J. J. Wallman, and S. D. Bartlett, Esti- mating outcome probabilities of quantum circuits using quasiprobabilities, Phys. Rev. Lett.115, 070501 (2015)

    2015

  91. [100]

    Albarelli, M

    F. Albarelli, M. G. Genoni, M. G. A. Paris, and A. Fer- raro, Resource theory of quantum non-Gaussianity and Wigner negativity, Phys. Rev. A98, 052350 (2018)

    2018

  92. [101]

    D. T. Smithey, M. Beck, M. G. Raymer, and A. Fari- dani, Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum, Phys. Rev. Lett.70, 1244 (1993). 18

    1993

  93. [102]

    Nogues, A

    G. Nogues, A. Rauschenbeutel, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, S. Haroche, L. G. Lutter- bach, and L. Davidovich, Measurement of a negative value for the Wigner function of radiation, Phys. Rev. A62, 054101 (2000)

    2000

  94. [103]

    Gerry and P

    C. Gerry and P. Knight, Quantum superpositions and schr¨ odinger cat states in quantum optics, American Journal of Physics65, 964 (1997)

    1997

  95. [104]

    M. Bild, M. Fadel, Y. Yang, U. von L¨ upke, P. Mar- tin, A. Bruno, and Y. Chu, Schr¨ odinger cat states of a 16-microgram mechanical oscillator, Science380, 274 (2023)

    2023

  96. [105]

    Vlastakis, G

    B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. De- voret, and R. J. Schoelkopf, Deterministically encoding quantum information using 100-photon Schr¨ odinger cat states, Science342, 607 (2013)

    2013

  97. [106]

    X. He, Y. Lu, D. Bao, H. Xue, W. Jiang, Z. Wang, A. Roudsari, P. Delsing, J. Tsai, and Z. Lin, Fast gen- eration of Schr¨ odinger cat states using a Kerr-tunable superconducting resonator, Nature communications14, 6358 (2023)

    2023

  98. [107]

    C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. Sliwa,et al., A Schr¨ odinger cat living in two boxes, Science352, 1087 (2016)

    2016

  99. [108]

    Leghtas, S

    Z. Leghtas, S. Touzard, I. M. Pop, A. Kou, B. Vlastakis, A. Petrenko, K. M. Sliwa, A. Narla, S. Shankar, M. J. Hatridge,et al., Confining the state of light to a quan- tum manifold by engineered two-photon loss, Science 347, 853 (2015)

    2015

  100. [109]

    X. Yu, B. Wilhelm, D. Holmes, A. Vaartjes, D. Schwien- bacher, M. Nurizzo, A. Kringhøj, M. R. v. Blankenstein, A. M. Jakob, P. Gupta,et al., Schr¨ odinger cat states of a nuclear spin qudit in silicon, Nature Physics21, 362 (2025)

    2025

  101. [110]

    Cornelius, Z

    J. Cornelius, Z. Xu, A. Saxena, A. Chenu, and A. del Campo, Spectral filtering induced by non-Hermitian evolution with balanced gain and loss: Enhancing quan- tum chaos, Phys. Rev. Lett.128, 190402 (2022)

    2022

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.