{"id":"f8a8fb4a-494c-4bc8-a84c-73f8bafd8aa1","arxiv_id":"2608.06056","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"General sufficient criteria now tell when P/Wigner/Q phase-space Monte Carlo SDEs exist for the GKSL equation, and when mean-field theory fails.","lead":"This paper derives sufficient conditions for phase-space stochastic Monte Carlo simulations of open bosonic quantum systems to be possible, by requiring the diffusion matrix to be positive semidefinite. It also identifies jump-operator types for which mean-field theory fails in the high-occupancy limit, making the stochastic simulation necessary.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"High-occupancy PSD claim in Sec 5.3 relies on discarding 1/N corrections; leading-order factorization is insufficient because PSD is not robust to subleading perturbations in null directions.","rationale":"The central sufficient-condition theorem (Eqs. (37)-(38) and Appendix B) appears algebraically sound: the quadratic form in Eq. (B.7)-(B.9) is a genuine sum of two absolute squares, and the derivation of the SDE from a PSD diffusion matrix in Appendix A is standard. The benchmark calculations provide credible support for the main constructive claims. However, the universal high-occupancy PSD claim is a headline result and is the least secure step. The proof factorizes only the leading-order λ and Λ, but PSD is not a property that survives arbitrary subleading perturbations when the leading matrix is rank-deficient. The explicit two-mode example L = α1α2 + α1 shows that exact finite-N corrections can produce a negative eigenvalue even when the leading-order matrix is PSD with a null direction; under the paper's own scaling this negative eigenvalue tends to zero, but it is nonzero at any finite occupancy, so the assertion that an SDE 'can always be constructed' is at best an asymptotic statement and not a finite-N feasibility criterion. This does not invalidate the sufficient conditions or the benchmarks, but it directly affects the abstract's and Sec. 5.3's strongest claims. The same issue is not present in the Sec. 4.1 closed-form B_s concern: although the derivation of Eq. (40) is called heuristic, the authors state that direct substitution verifies (33) and (35), and the Appendix B proof gives the needed PSD foundation. Therefore the reader's conditional verdict should stand, with the high-occupancy universal-PSD claim made conditional on either a subleading-order proof or an explicit limit-only qualification.","tokens_in":45425,"tokens_out":21846,"duration_ms":218644,"concrete_test":"Compute the exact Wigner diffusion matrix for \\hat L = \\hat a1\\hat a2 + \\hat a1 with H=0, s=0, α1=α2=√N, and γ=N^{-3} using Eqs. (27)-(28) without discarding Moyal-star corrections, then diagonalize A_s=0 for N=10^2, 10^4, 10^6. If the smallest eigenvalue is negative at any finite N, the Sec. 5.3 universal-PSD claim must be qualified to an asymptotic leading-order statement; if it is nonnegative for all tested N, the reader's concern is resolved and the leading-order factorization suffices.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The load-bearing weakness is exactly the reader's: Sec 5.3 proves only that the leading-order matrices (69)-(70) factor as in (37)-(38). Positive semidefiniteness is not preserved under adding O(1/N) perturbations when the leading matrix has a null direction, and no bound on the subleading correction is supplied. This is not merely formal. For a single two-mode jump operator with Wigner symbol L = α1α2 + α1 (e.g. \\hat L = \\hat a1\\hat a2 + \\hat a1), set H=0, s=0, and α1=α2=r=√N. Using the exact s=0 formulas (27)-(28) with the Moyal star, one finds λ=0 and Λ = (γ/4)[[(r+1)^2 - 1/2, r(r+1)-1/2], [r(r+1)-1/2, r^2 - 1/2]], whose determinant is -γ²/32. Hence A_s=0 has a negative eigenvalue for every finite N, although its leading-order part is the rank-one PSD matrix (γ/4) ww†. Under the paper's case-(a) scaling γ=O(N^{-3}) the negative eigenvalue is O(N^{-4}) and vanishes as N→∞, but the exact matrix is indefinite at every finite N. Thus the statement that 'a well-defined SDE can always be constructed' is not true as stated for finite-N simulations: Eq. (35) requires PSD of the exact matrix at each time step. The authors must either prove that subleading corrections are nonnegative on the leading null space, or explicitly restrict the claim to the rescaled leading-order diffusion matrix in the strict N→∞ limit.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies when the Fokker–Planck approximation to a bosonic Markovian open quantum dynamics, obtained from an s-ordered phase-space path integral, admits a Monte Carlo simulation via stochastic differential equations. The main formal result is a sufficient condition, Eqs. (37) and (38), under which the diffusion matrix A_s factorizes and is positive semidefinite, with an analytic square root B_s; Sec. 4.2 extends this to sums of such factorizations, and Sec. 4.3 gives an explicit operator condition for the Wigner function for quadratic jump operators. In the high-occupancy limit the paper classifies jump operators according to whether the classical (mean-field) equation remains valid, and it claims that for s=0 the diffusion matrix is always positive semidefinite in this limit, so that an SDE always exists. It also gives sufficient conditions for higher-order quantum fluctuations to vanish and benchmarks the resulting SDEs against exact dynamics for two- and three-site models.","tokens_in":45784,"tokens_out":17182,"duration_ms":159439,"significance":"If the claims hold, the criteria would be practical and general: they give a constructive route to analytic F/G decompositions, avoid numerical eigen-decompositions, identify models where stochastic simulation is necessary rather than optional, and delimit exact second-order descriptions. Appendix B is a clean exact proof of positive semidefiniteness under the factorization conditions, and the benchmark comparisons with exact correlation dynamics for the small models are a genuine strength. The high-occupancy universal PSD claim, however, is not proven as stated: the leading-order factorization in Sec. 5.3 controls only the N→∞ limit, not finite large N, and the exact finite-N diffusion matrix can be indefinite while its leading part is PSD. This weakens the paper's headline claim but does not undermine the Sec. 4 and Sec. 6 sufficient-condition results, which remain valid.","major_comments":[{"comment":"The argument proves positive semidefiniteness only for the leading-order matrices λ^{s=0(LO)} and Λ^{s=0(LO)}, obtained by replacing L_s with L_s^{(LO)} and ⋆_s with 1. Positive semidefiniteness is not stable under O(1/N) perturbations when the leading-order matrix has a null direction, and the paper provides no estimate of the discarded subleading contributions. This is not a purely formal concern. For H=0, s=0, and the jump-operator Wigner symbol L = α1α2 + α1 (corresponding to \\hat L = \\hat a1 \\hat a2 + \\hat a1), a direct evaluation of Eqs. (27) and (28) at α1=α2=√N gives λ=0 and Λ = (γ/4)[[(r+1)^2 - 1/2, r(r+1)-1/2],[r(r+1)-1/2, r^2 - 1/2]], whose determinant is -γ²/32. Hence the exact matrix A_{s=0} has a negative eigenvalue for every finite N, although its leading-order part is the rank-one PSD matrix (γ r²/4) ww† with w=(1,1). The statement in Sec. 5.3 that a well-defined stochastic differential equation can always be constructed is therefore false if read at finite N, and Eq. (35) requires PSD of the exact matrix at each time step. The authors should either prove that the subleading corrections are nonnegative on the leading null space, or explicitly restrict the universal claim to the rescaled leading-order diffusion matrix in the strict N→∞ limit.","section":"Sec. 5.3, Eqs. (69)–(71)"},{"comment":"The numerical benchmarks do not test the universal high-occupancy PSD claim. Every SDE used in Sec. 7 is derived from the exact sufficient conditions of Sec. 4 (Tab. 1, Eq. (53), or the exactly solvable two-site correlation hierarchy), not from the leading-order factorization of Eqs. (69)–(71). Thus the null-space issue described above is not addressed by the good agreement shown in Figs. 2–6, and the universal statement in the Abstract and Introduction remains unsupported both analytically and numerically.","section":"Sec. 5.3 and Sec. 7"}],"minor_comments":[{"comment":"The phrase 'by substituting by substituting' is duplicated; please remove one occurrence.","section":"Sec. 7.3, text before Eq. (100)"},{"comment":"The caption specifies 'U/(ℏγ)' while the main text specifies 'U11/(ℏγ) = 1'; please unify the notation.","section":"Fig. 6 caption"},{"comment":"The term 'thermodynamic limit' is used for the high-occupancy limit defined by the scaling assumptions α_m=O(N^{1/2}), η_m=O(N^{-1/2}), and the corresponding parameter scalings; using 'high-occupancy limit' consistently throughout would avoid confusion with the standard thermodynamic limit.","section":"Sec. 5.1 and Abstract"},{"comment":"The standard-error expression should state more explicitly that the limit N_initial→∞ is taken at fixed N_stoch, because the prefactor 1/√N_initial refers to initial-condition sampling only.","section":"Sec. 7.1, Eq. (75)"}],"recommendation":"major_revision","confidential_remarks":"The paper is built on the authors' previous path-integral formalism in Refs. [16,17], which are self-citations; this is not circular because the new contribution is an algebraic sufficient-condition analysis of those expressions, but it does mean the novelty is incremental. The main obstacle to publication is the Sec. 5.3 universal PSD claim; if the authors qualify it as a strict N→∞ leading-order statement, or supply the missing bound on subleading corrections, the paper would be a solid contribution within the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives a real advance: from their earlier path-integral formalism, the authors extract clean algebraic sufficient conditions (37)-(38) for positive semidefiniteness of the phase-space diffusion matrix, now covering jump operators that couple different degrees of freedom. The Appendix B proof is exact and short, and the Wigner-function condition (53) connecting PSD to the infinite-temperature steady state is a nice practical handle. The benchmarks are thorough: two-site and three-site models reproduce exact dynamics, and the demonstration that the mean-field (first-order) description breaks down for Hermitian or Hermitian-paired jump operators is convincing and well illustrated.\n\nThe main soft spot is the claim in Sec 5.3 that for the Wigner function in the high-occupancy limit, the diffusion matrix is always positive semidefinite and a well-defined SDE can always be constructed. The proof only factorizes the leading-order terms (69)-(70); subleading 1/N corrections are discarded without a bound. PSD is not stable under small perturbations when the leading matrix has a null direction. This is not a formal quibble. Take H=0, s=0, single jump operator L = a1 a2 + a1, and α1=α2=r=√N. Using the exact formulas (27)-(28) with the Moyal star, the diagonal entries of Λ acquire -1/2 corrections, so Λ = (γ/4)[[(r+1)^2 - 1/2, r(r+1)], [r(r+1), r^2 - 1/2]]. The determinant is -(γ^2/16)(r+1/2)^2, negative for every finite r. (The stress-test's off-diagonal -1/2 is not what the formulas give, but the conclusion survives.) So the exact A_s=0 is indefinite at every finite N, even though its leading-order part is rank-one PSD. Under the paper's case-(a) scaling γ=O(N^{-3}) the negative eigenvalue is O(N^{-3}) and vanishes in the limit, but for any finite-N simulation Eq. (35) needs the exact matrix PSD at each step. The authors should either bound the subleading corrections on the leading null space or explicitly restrict the universal claim to the strict N→∞ limit.\n\nThe asserted closed-form B_s in Sec 4.1 is heuristic (they admit this), but it is verified algebraically and is a minor issue. The self-citation burden is acceptable: the new results are an independent algebraic extension, not a restatement.\n\nThis paper deserves serious peer review. The core sufficient-condition theorem is solid and the benchmarks are valuable. The Sec 5.3 claim needs revision, not rejection. I would send it to a good referee and expect a conditional accept after the high-occupancy statement is sharpened.","headline":"Useful sufficient conditions for phase-space Monte Carlo, but the high-occupancy PSD claim overreaches: subleading 1/N corrections can make the exact diffusion matrix indefinite for finite N.","tokens_in":46298,"tokens_out":18468,"would_cite":true,"duration_ms":144579,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A factorized diffusion matrix makes phase-space Monte Carlo simulation of open bosonic quantum dynamics feasible, and in the high-occupancy limit the Wigner distribution is always simulation-ready.","keywords":["open quantum systems","GKSL equation","phase-space methods","stochastic differential equations","Fokker-Planck equation","truncated Wigner approximation","quasiprobability distributions","high-occupancy limit"],"falsifier":"For a model with a Hermitian jump operator at a large but finite per-mode occupation $N$, assemble the full $\\lambda_{mn}$ and $\\Lambda_{mn}$ from Eqs. (27)-(28) without discarding subleading terms and diagonalize $A_{s=0}$. The high-occupancy positive-semidefiniteness claim would be refuted if the smallest eigenvalue is negative at any such $N$; a uniform positive lower bound as $N\\to\\infty$ would confirm it.","tokens_in":45189,"feed_emoji":"🎲","tokens_out":12624,"duration_ms":113650,"temperature":0.7,"pith_summary":"Bosonic open quantum systems described by the GKSL master equation can be studied by Monte Carlo sampling of stochastic differential equations in phase space only when the Fokker–Planck diffusion matrix is positive semidefinite, i.e. the stochastic noise has no negative-variance directions. This paper provides a general sufficient condition for that property: if the blocks $\\lambda^s_{mn}$ and $\\Lambda^s_{mn}$ of the diffusion matrix can be written as symmetrized products of two arbitrary complex functions $F^s_m$ and $G^s_m$, then the diffusion matrix is positive semidefinite and an explicit stochastic differential equation follows. For the Wigner function, a concrete operator-level version of the condition is proven for quadratic jump operators, and in the high-occupancy limit the Wigner diffusion matrix is claimed to be always positive semidefinite, whatever the jump operators are. The paper also identifies the jump-operator forms for which the mean-field (first-order) description breaks down in the high-occupancy limit, making the stochastic simulation necessary, and gives sufficient conditions under which higher-order quantum-fluctuation terms vanish, making the stochastic simulation exact. The value of these results is that feasibility is reduced from a case-by-case numerical check on the diffusion matrix to checkable conditions on the Hamiltonian and jump operators.","feed_headline":"Diffusion factorization certifies open-boson Monte Carlo SDEs","feed_subtitle":"New criteria tell when the phase-space noise is valid, when mean field fails, and when simulations are exact.","key_machinery":"The central object is the $2M\\times 2M$ Hermitian diffusion matrix $$A_s = 2\\begin{pmatrix} \\Lambda^s & \\$\\lambda$^s \\\\ \\$lambda^{{s*}}$ & \\$Lambda^{{s*}}$ \\end{pmatrix},$$ whose entries $\\lambda^s_{mn}$ and $\\Lambda^s_{mn}$ are built from first and second derivatives of the s-ordered phase-space symbols of the jump operators, contracted with the strengths $\\gamma_k$ and with the $\\star_s$ product, the s-ordered multiplication that encodes operator ordering. The load-bearing identity is the factorization $$\\$\\lambda$^s_{mn}=\\tfrac12($G^{{s*}}$_mF^s_n+F^$s_mG^{{s*}}$_n),\\qquad \\Lambda^s_{mn}=\\tfrac12($G^{{s*}}$_mG^s_n+F^$s_mF^{{s*}}$_n),$$ which makes $A_s$ a sum of manifestly non-negative rank-one terms and yields the explicit noise matrix $B_s$ from $F^s$ and $G^s$. In the high-occupancy limit $N\\to\\infty$ the $\\star_s$ product reduces to the identity and only the leading-order parts of $\\lambda$ and $\\Lambda$ survive, so the factorization holds automatically for the Wigner function; the dissipative drift $K^s_m$ plays the complementary diagnostic role, with its leading and next-to-leading parts vanishing exactly when the mean-field description fails.","core_discovery":"On its own terms, the paper claims that feasibility of the stochastic Monte Carlo method for the GKSL equation is decided by a factorization of the diffusion matrix. For the s-ordered quasiprobability functions ($s=1,0,-1$ for the Glauber–Sudarshan P, Wigner, and Husimi Q functions), the second-order truncation is a Fokker–Planck equation with diffusion matrix $A_s$, and an equivalent stochastic differential equation exists only when $A_s$ is positive semidefinite. The paper proves that $A_s\\succeq 0$ holds whenever $$\\$\\lambda$^s_{mn}=\\tfrac12($G^{{s*}}$_mF^s_n+F^$s_mG^{{s*}}$_n),\\qquad \\Lambda^s_{mn}=\\tfrac12($G^{{s*}}$_mG^s_n+F^$s_mF^{{s*}}$_n),$$ and that the noise matrix can then be constructed explicitly; summing finitely many such terms generalizes the criterion to jump operators that couple different degrees of freedom. It further claims that, in the high-occupancy limit, the Wigner-function diffusion matrix is always positive semidefinite regardless of the Hamiltonian and jump operators, so a stochastic differential equation always exists there, even in the cases where the mean-field description breaks down. Finally, it derives sufficient conditions, summarized in tables, under which third- and fourth-order quantum fluctuations vanish identically, so that whenever a stochastic differential equation exists the Monte Carlo simulation reproduces the exact GKSL dynamics.","pith_inferences":["Since the factorization conditions are sufficient but not necessary, they suggest a low-cost numerical diagnostic: attempt to fit the instantaneous $\\lambda^s$ and $\\Lambda^s$ to the $F,G$ form and use the fit residual as a feasibility alarm during a simulation.","The high-occupancy proof keeps only leading-order terms in $1/N$; a natural testable extension is a perturbation analysis around null directions of the leading-order diffusion matrix to see whether subleading terms ever violate positive semidefiniteness at finite large occupation.","The mean-field breakdown criterion picks out dissipative processes with Hermitian or Hermitian-paired jump operators; such systems are natural hunting grounds for fluctuation-driven transport or ordering that survives the thermodynamic limit, a direction the paper itself leaves open.","The exactness conditions effectively certify reference models: any setup meeting them can be used as an exactly simulated benchmark for other numerical approaches to open many-body dynamics."],"forward_implications":["Whenever the diffusion entries admit a single factorization, the stochastic differential equation is explicitly known and uses only two Wiener processes rather than $2M$, avoiding numerical diagonalization or decomposition of $A_s$ at every time step.","For the Wigner function and jump operators that are at most quadratic, the operator condition $\\sum_k\\gamma_k[\\hat L_k^\\dagger,\\hat L_k]_- = \\sum_m(l_m\\hat a_m+\\bar l_m\\hat a_m^\\dagger)+\\mathrm{const.}$ guarantees $A_{s=0}\\succeq 0$, so explicit stochastic differential equations follow for models such as unidirectional incoherent hopping with an interacting Hamiltonian.","In the high-occupancy limit the Wigner-function diffusion matrix is always positive semidefinite, so stochastic simulation remains feasible exactly in the regime where it becomes necessary because the classical first-order description fails.","The mean-field description breaks down in the high-occupancy limit whenever the leading and next-to-leading parts of the dissipative drift vanish, which includes Hermitian jump operators and equal-strength Hermitian-conjugate pairs; there the second-order stochastic description is indispensable.","Under the sufficient conditions of Sec. 6, the GKSL equation reduces exactly to the Fokker–Planck equation in phase space, so the stochastic Monte Carlo simulation is exact; the three-site benchmark shows that the first-order approximation misses the steady-state current while the second-order Wigner simulation matches the exact result."],"supporting_citations":[{"why":"defines the GKSL equation whose phase-space dynamics and stochastic simulation feasibility are the paper's subject.","marker":"[7]"},{"why":"supplies the Lindblad form of the generator used in Eq. (1).","marker":"[8]"},{"why":"states the standard Fokker–Planck/SDE content that a positive-semidefinite diffusion matrix is required for the stochastic formulation.","marker":"[3]"},{"why":"derives the path-integral truncated-Wigner framework from which the diffusion-matrix entries in Eqs. (27)-(28) are taken.","marker":"[16]"},{"why":"gives the path-integral representation for the P, Wigner, and Q functions and the restricted positive-semidefiniteness conditions that this paper generalizes to coupled jump operators.","marker":"[17]"},{"why":"defines the s-ordered expansions used for the quasiprobability distributions and phase-space symbols.","marker":"[52]"},{"why":"provides the phase-space representation and star-product expansion underlying the order-by-order fluctuation analysis.","marker":"[21]"},{"why":"fixes the Ito interpretation and Wiener-process calculus used in the stochastic equations.","marker":"[65]"},{"why":"gives the infinite-temperature steady-state condition that underlies the Wigner-function criterion of Sec. 4.3.","marker":"[67]"}],"fun_headline_variants":["Phase-space noise valid when diffusion factorizes","Exact Monte Carlo for open bosons: new conditions","Diffusion matrix factorization dictates Monte Carlo use","Mean-field failure makes Monte Carlo indispensable","New criteria for feasible open-boson SDE sampling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claim that the Wigner diffusion matrix is always positive semidefinite in the high-occupancy limit rests on dropping the subleading $1/N$ corrections to $\\lambda$ and $\\Lambda$; if the surviving leading-order matrix has a flat direction, those dropped corrections could in principle make the full matrix slightly indefinite at finite occupation.","fun_headline_variants_meta":{"raw":{"variants":["Phase-space noise valid when diffusion factorizes","Exact Monte Carlo for open bosons: new conditions","Diffusion matrix factorization dictates Monte Carlo use","Mean-field failure makes Monte Carlo indispensable","New criteria for feasible open-boson SDE sampling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000431,"raw_usage":{"total_tokens":2314,"prompt_tokens":1173,"completion_tokens":1141,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":789,"completion_tokens_details":{"reasoning_tokens":1070}},"tokens_in":789,"tokens_out":1141,"duration_ms":10817,"temperature":1.0,"reasoning_tokens":1070,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T18:30:01.264983+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a model with a Hermitian jump operator at a large but finite per-mode occupation $N$, assemble the full $\\lambda_{mn}$ and $\\Lambda_{mn}$ from Eqs. (27)-(28) without discarding subleading terms and diagonalize $A_{s=0}$. The high-occupancy positive-semidefiniteness claim would be refuted if the smallest eigenvalue is negative at any such $N$; a uniform positive lower bound as $N\\to\\infty$ would confirm it.","supporting_citations":[{"cited_title":"Gorini, A","cited_arxiv_id":null,"evidence_quote":"defines the GKSL equation whose phase-space dynamics and stochastic simulation feasibility are the paper's subject."},{"cited_title":"Lindblad, On the generators of quantum dynamical semi groups, Commun","cited_arxiv_id":null,"evidence_quote":"supplies the Lindblad form of the generator used in Eq. (1)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"states the standard Fokker–Planck/SDE content that a positive-semidefinite diffusion matrix is required for the stochastic formulation."},{"cited_title":"Y oneya, K","cited_arxiv_id":null,"evidence_quote":"derives the path-integral truncated-Wigner framework from which the diffusion-matrix entries in Eqs. (27)-(28) are taken."},{"cited_title":"Y oneya, K","cited_arxiv_id":null,"evidence_quote":"gives the path-integral representation for the P, Wigner, and Q functions and the restricted positive-semidefiniteness conditions that this paper generalizes to coupled jump operators."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the s-ordered expansions used for the quasiprobability distributions and phase-space symbols."},{"cited_title":"Polkovnikov, Phase space representation of quantum dynamics, Ann","cited_arxiv_id":null,"evidence_quote":"provides the phase-space representation and star-product expansion underlying the order-by-order fluctuation analysis."},{"cited_title":"Risken, The Fokker-Planck equation, 2nd Edition, Sp ringer-V erlag, Berlin Heidelberg, 1996","cited_arxiv_id":null,"evidence_quote":"fixes the Ito interpretation and Wiener-process calculus used in the stochastic equations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the infinite-temperature steady-state condition that underlies the Wigner-function criterion of Sec. 4.3."}],"review_version":1}