Quantum work: reconciling quantum mechanics and thermodynamics

It has been recently claimed that no protocol for measuring quantum work can satisfy standard required physical principles, casting doubts on the compatibility between quantum mechanics, thermodynamics

In the quantum realm, the pursuit of a universal definition of work resulted in multiple definitions , allowing the thermodynamic analysis of various quantum systems [22,[45][46][47].Nevertheless, the statistical characterization of work continues to be the subject of intense debate .For instance, in scenarios without heat flux, work average is expected to equal the variation of mean energy, a feature not generally fulfilled by broadly used work measurement schemes, such as the two-point-measurement (TPM) [28,62,65,68], the Gaussian [28,72], and post-selection [53] schemes.Other significant discussions in the literature include the definition of work in scenarios encompassing the energetic effects of measurements and quantum resources [14,15,53,[73][74][75][76][77][78][79][80].Successfully overcoming these challenges can propel advancements in quantum thermodynamics and the design of efficient quantum devices [45][46][47][81][82][83][84].
To elucidate quantum work, it has been generally considered that its statistics should comply with the following criteria, provided by the basic structure of quantum mechanics and thermodynamics [68,69]: work statistics should (i) be described by a positive operator-valued measure (POVM) [85,86] independent of the system's initial state, (ii) yield an average work equal to the variation of energy over time for externally controlled systems not interacting with a heat bath, and (iii) reproduce the classical limit.Surprisingly, it was demonstrated that no protocol for raising work statistics would comply simultaneously with criteria (i) and (ii) and a condition inspired by (iii) [68].These findings have prompted the quest for new measurement protocols and a better understanding of the statistics of work [48-53, 63, 64, 69-71].Remarkably, there is still no statistical framework for describing work that aligns with criteria (i)-(iii), suggesting an incompatibility between quantum mechanics, its convergence to the classical limit, and thermodynamics.
In this Letter, we propose a solution to this challenge.We demonstrate that there is at least one protocol for raising work statistics in consonance with the basic criteria (i)-(iii).To demonstrate our results, we adopt criteria (i) and (ii) in a manner analogous to Ref. [68] while imposing necessary conditions for the classical limit as stipulated by criterion (iii).Specifically, we require that in the classical limit, the average of general quantum observables approximates their corresponding classical counterpart and that the commutation of quantum observables is sufficiently small.Guided by these conditions, we demonstrate that treating work as a twotime quantum observable [87][88][89], criteria (i)-(iii) are satisfied.Furthermore, we shed light on previous forms of imposing criterion (iii), highlighting their reliance on the twopoint-measurement (TPM) methodology [21,67,72], which we show neither guarantees the recovery of the classical limit nor represents the unique approach capable of achieving it.We expect that our study showcases the feasibility of achieving a satisfactory reconciliation between quantum and classical work statistics and contributes to the ongoing quest for a unified framework for quantum work statistics.

REQUIREMENTS FOR CONSISTENT WORK STATISTICS
We first present criteria (i)-(iii) in detail.Our analysis consists of a closed system whose initial state is described by ρ governed by an externally controlled time-dependent Hamiltonian H(t) with eigenbasis {|e j (t) }.The system's dynamics are dictated by the evolution operator U t , which satisfies the Schrödinger equation i ∂ t U t = H(t)U t .Since the system is isolated, no heat is exchanged, and work equals the system energy variation.We thus posit that there exists a general protocol for measuring work, such that for any process defined by operators H(t) and U t , this protocol can describe the probability P(w) of obtaining a value of work w.Following the quantum mechanics paradigm [85,86], we require that P(w) = Tr[M(w) ρ], where M(w) are the elements of a POVM {M(w)}, i.e., a set of non-negative Hermitian operators M(w) satisfying the relation w M(w) = 1.Considering this formalism, a work measurement scheme implemented by {M(w)} should satisfy the following criteria: (i) The operators M(w) may depend on H(t) and U t , but are independent of ρ.With this criterion, we anticipate that the measurement protocol must depend on the process described by U t and H(t) but is independent of the initial state, as one should not expect to adjust the measurement apparatus to the system's initial state [68].Importantly, an equivalent formulation of criterion (i) allows for state-dependent POVM, as long as P(w) exhibits linear dependence on the state [68,69].
(ii) For any state ρ, the work average equals the variation in the average energy over time, i.e.W = w w P(w) = Tr[H(t) U t ρU † t ] − Tr[H(0) ρ] = H(t) − H(0) .Considering the convention in which work is deemed positive when the energy H(t) increases, this statement aligns with the first law of thermodynamics, as there is no heat flux for this nonautonomous and closed quantum scenario.
(iii) In the classical limit, the statistics raised via the POVM M(w) must approach the classical results.More precisely, it is anticipated that as the system approaches the classical limit, the quantum distribution P(w) obtained through M(w) should exhibit the same statistics as the classical distribution associated with the classical limit.This criterion put forward the expectation that the classical world can be reproduced within quantum mechanics in semi-classical regimes [78,[90][91][92][93][94].
Criteria (i)-(iii) collectively provide a comprehensive framework that aligns with the foundations of quantum mechanics and thermodynamics and the convergence to classical behavior.In fact, these have served as the basis for many recent works (see [53,64,[68][69][70][71] and references therein).
We can straightforwardly examine how criteria (i) and (ii) can constrain work POVMs considering two notable examples that shall be treated frequently in this Letter.On the one hand, we consider the two-point-measurement (TPM) POVM, which is succinctly described as follows [21,67,72].At an initial time, the system is submitted to a projective measurement of energy, collapsing to an H(0) eigenstate |e n (0) with probability p n = e n (0)|ρ|e n (0) .The system then evolves unitarily until the instant τ, when a second measurement is performed and a random eigenvalue e m (τ) of H(τ) is obtained with probability p m|n = | e m (τ)| U t |e n (0) | 2 .The TPM work probability is thus computed as [68,72] P TPM (w) = mn p m|n p n δ[w − (e m (τ) − e n (0))] = Tr[M TPM (w) ρ], where M TPM (w) = mn δ[w − (e m (τ) − e n (0))]p m|n |e n (0) e n (0)| are the elements of the TPM POVM and δ is the Dirac's delta.Although TPM complies with criterion (i), it generally does not satisfy criterion (ii), as the first measurement destroys any coherence the initial state may possess [64,68,69].
On the other hand, the observable (OBS) POVM is connected with the two-time work observable [49,50,[87][88][89] where H h (t) = U † t H(t)U t represents the Hamiltonian in the Heisenberg picture at time t.Notably, W(τ, 0) is a Hermitian operator possessing a set of eigenvalues w j (τ, 0) and corresponding eigenvectors {|w j (τ, 0) }, such that W(τ, 0) = j w j (τ, 0) |w j (τ, 0) w j (τ, 0)|.Accordingly, the probability of finding a value w of work is defined as P OBS (w) = Tr[M OBS (w)ρ], where M OBS (w) = j δ(w − w j (τ, 0)) |w j (τ, 0) w j (τ, 0)|.As can be checked from the discussion in [49], there always exists a corresponding Schrödinger operator S diagonalized by the same basis as {|w j (τ, 0) } for each arbitrary interval [0, τ].Therefore, by measuring S at time 0 and collapsing the state to an eigenstate |w j (τ, 0) , one is thus "preparing" a state whose amount of work done on the system is known to be w j (τ, 0) during [0, τ].By making several measurements of S at time 0, one thus raises P OBS (w).We remark that criteria (i) and (ii) are fulfilled if and only if M(w) are state-independent operators that satisfy the relation [68,69].Comparing this equation with Eq. ( 1) and considering the explicit form of M OBS (w), we readily confirm that the OBS POVM satisfies criteria (i) and (ii).

TPM AND THE CLASSICALITY CRITERION
Unlike criteria (i) and (ii), verifying criterion (iii) for a specific POVM is challenging due to the elusive nature of the classical limit in the quantum realm-a topic that has been intensely debated since the inception of quantum mechanics [95][96][97][98][99][100]. Still, there is no unique form of characterizing the classical limit in the quantum context [101][102][103][104][105]. Notwithstanding these difficulties, Perarnau-Llobet et al. [68] boldly proposed what we refer to as the classical stochastic (CS) hypothesis, aiming to implement criterion (iii): for initial states with no quantum coherences in the energy basis, the results of classical stochastic thermodynamics should be recovered.More specifically, they suggested that for incoherent initial states where [ρ, H(0)] = 0, the work probability should be equal to the TPM probability, i.e., [ρ, H(0)] = 0 =⇒ P(w) = P TPM (w).The CS hypothesis was justified on the basis that the TPM scheme used for incoherent states enabled classical stochastic thermodynamics relations to be reproduced similarly in the quantum regime [10-13, 21, 26, 30, 106] and because of its good convergence in the classical limit [78,[90][91][92][93][94].Remarkably, the authors demonstrated that no POVM could simultaneously satisfy criteria (i) and (ii) along with the CS hypothesis.However, even though the CS hypothesis has been considered in many works aiming to recover the classical limit [53,64,69], we present throughout this section three arguments showing that demanding CS as a way of implementing criterion (iii) may not be the most general or accurate approach.
1) There is no fundamental principle dictating that quantum processes beginning with initial incoherent states should recover classical stochastic thermodynamic results [49,63,65,78].Because the Hamiltonian is time-dependent, an initial incoherent state will not always time-evolve to an incoherent state with respect to the instantaneous Hamiltonian energy bases.The emergence of coherences throughout the process may result in a different amount of work exchange relative to what is classically expected.Moreover, for incoherent states quantum corrections can still display significant ther-modynamics effects [107,108], as has been recognized ever since Wigner's seminal paper [109].
2) TPM is not the unique approach successful in reproducing classical outcomes when coherences are absent.Indeed, the agreement of TPM with the classical results in the semiclassical regime [78,[90][91][92][93][94] has served as a justification for its selection in the CS hypothesis [68,69].However, TPM is not necessarily unique in this sense.To prove this statement, consider a process described by H(t) and U t and a initial state ρ taking place within the time interval [0, τ].Consider also that the average of energy is bounded, i.e., H(t) < ∞, for any t ∈ [0, τ].Let C(ρ(t)) be the 1-norm measure of coherences [110] for the evolved state ρ(t) = U t ρU † t : where Result 1 arises from the fact that processes with a small amount of coherence require the Hamiltonians at different times to almost commute.Otherwise, a large amount of coherences will emerge.The difference between TPM and OBS, as well as the creation of coherences, stem from the lack of Hamiltonian commutation and will be reduced as the Hamiltonians approach to a perfect commutation.Therefore, the TPM protocol will approach the OBS scheme for processes close to the classical limit where coherences are all the time small.Consequently, justifying the preference for TPM over OBS based on its convergence to classical results can be misleading.We confirm this physical intuition by rigorously showing that C 1 is bounded and independent of ǫ 1 (see the Supplemental Material (SM) [111]).The case of zero coherence, ǫ 1 = 0 can be directly proved considering the results presented in Ref. [67].
3) The CS hypothesis lacks the necessary generality to encompass a broader range of processes in the classical limit.Consider, for instance, an oscillator with a time-dependent frequency described by the Hamiltonian operator [90,91] where ω 2 (t) = ω 2 0 + (ω 2 1 − ω 2 0 ) t τ , and ω 0,1 and τ are fixed parameters.When initialized in a coherent state ρ = |α α|, defined as then, for a sufficiently large |α|, the center of the wave packet exhibits classical particle-like behavior [112], following dynamics governed by a classical Hamiltonian analogous to Eq. ( 3).On the one hand, following criterion (iii), if we consider this scenario a classical limit, the quantum work statistics should converge to its classical counterpart.Under this perspective, this example extends beyond the scope of the CS hypothesis, exposing its limited coverage of the classical limit by associating it solely with incoherence in the energy basis.On the other hand, despite the oscillator's classical behavior, one might expect that a large amount of coherence could lead to work statistics diverging from the classical counterpart.This example thus highlights the need to clarify the classical limit beyond the CS hypothesis and when we expect the quantum statistics to converge to it.We address this issue in the next section and in [111], where we explicitly calculate the work statistic for a coherent state, revealing that the OBS protocol appropriately converges in the classical limit while the TPM does not.

OBS PROTOCOL AND THE CLASSICAL LIMIT
As exemplified above, formally characterizing criterion (iii) is daunting and the CS hypothesis is not able to implement it.In order to circumvent these difficulties and still obtain valuable results from criterion (iii), we propose an alternative strategy.Here, we first identify some necessary conditions for a quantum process to approach the classical limit.Then, we show that these are enough to obtain general results for quantum work statistics.
We consider again a process described by U t and H(t), but now the Hamiltonian H(t) depends on a Schrödinger coordinate observable X and its conjugate momentum P ([X, P] = i ½), and time t within the interval [0, τ].Consequently, U t and the Heisenberg version of the Hamiltonian H h (t) are continuous functions of X, P, and t.We can identify H h (t) = G(X, P, t), where G is a smooth function that can be written as powers of X, P, and t.
In order to establish the necessary conditions for the convergence to the classical limit of an initial preparation ρ and the quantum process described by U t , H h (t), it is essential to introduce the following quantities: the classical probability distribution ̺(Γ 0 ), representing the likelihood of finding a classical system at the phase points Γ 0 = (x 0 , p 0 ) at time 0; and a classical Hamiltonian functions H t CL (Γ t , t), characterizing the energy of the system at time t with respect to the phase point Γ t = (x t , p t ).Importantly, within the Hamiltonian formalism, the evolution of Γ t (Γ 0 , t) = (x t (Γ 0 , t), p t (Γ 0 , t)) can be expressed in terms of Γ 0 and t, allowing the Hamiltonian function to also be written as a function of Γ 0 and t, so that Here, H CL corresponds to the energy at time t related to the system that was at Γ 0 at time 0. With these definitions in place, the following conditions are required for the convergence to the classical limit of the process characterized by {X, P, ρ, H h (t)}: (A) There exists at least one classical distribution ̺(Γ 0 ) along with a Hamiltonian function H CL (Γ 0 , t), such that for any integers m and n, and where G(x 0 , p 0 , t) has the same functional form as G(X, P, t), previously defined, t ′ can be either 0 or τ, and ǫ A ≪ 1. Equation (5) asserts that there must be at least one classical scenario whose averages approach the quantum ones at t = 0.. Furthermore, Eq. ( 6) states that the quantum energy in the classical limit should be described in terms of the same quantities that define its classical version at t ′ = 0, τ.
(B) For any functions g l ≡ g l (X, P) and g r ≡ g r (X, P) describing product of powers of X and P, it follows that and where ǫ B ≪ 1.This requirement resembles the usual classical limit assumption → 0. In our formulation, however, we allow the non-commutation to have a non-null yet significantly small value compared to the system's inherent scales.
It is important to emphasize that conditions (A) and (B) alone are not sufficient to certify the classical limit.Specifically, they do not rule out the effects of quantum correlations, measurement's disturbance, or any other quantum phenomena.In our framework, processes influenced by these quantum effects may still adhere to conditions (A) and (B) without truly approximating the classical limit.Conversely, these conditions are necessary for the classical limit.Therefore, we expect that any quantum scenario genuinely approaching the classical limit rigorously satisfies conditions (A) and (B).In these circumstances, the set {x 0 , p 0 , ̺, H CL } describe the classical scenario to which the process characterized by {X, P, ρ, H h (t)} converges.Interestingly, we demonstrated in [111] that the system described in Eqs. ( 3) and (4) satisfies conditions (A) and (B) for |α| → ∞.Now, consider a scenario fulfilling conditions (A) and (B) converging to the classical limit.. Within this context, the classical work is described as W CL (Γ 0 , τ, 0) = H CL (Γ 0 , τ) − H CL (Γ 0 , 0), with the associated classical distribution defined as According to criterion (iii), a consistent approach to describing quantum work statistics should converge to the results provided by P CL (w) in the classical limit.Following this reasoning, we derived the central result of this Letter.
where P OBS (w) is the probability related to the OBS protocol, P CL (w) is defined in Eq. (9), ǫ max is the maximal value of ǫ A and ǫ B as defined in conditions (A) and (B), and K is a bounded real positive scalar independent of ǫ A and ǫ B .
The OBS protocol, derived from classical work using standard quantization rules, reveals differences between classical and quantum observables in scenarios with significant lack of commutation.This suggests that the difference in quantum and classical probabilities is tied to the lack of commutation, approaching zero when commutation is negligible [condition (B)], and the quantum scenario exhibits similar averages as a classical counterpart [condition (A)].This intuition supports result 2, which we rigorously proved in [111], employing the Cauchy-Hadamard formula [113,114] to demonstrate the boundedness of K.
According to result 2, the OBS protocol will converge to the description via Eq.( 9) for any given quantum state ρ and process {U t , H(t)} that satisfies condition (A) and (B).Since these conditions are necessary for the classical limit, then we can deduce that the statistics of work obtained through the OBS protocol invariably replicate the classical work statistics in the classical limit, thereby satisfying criterion (iii).Furthermore, from the review conducted in Ref. [69], the OBS protocol is the only well-known protocol that generally adheres to (i) and (ii).As a consequence of the result 2 and given our current understanding, we can also assert that OBS is the only well-known protocol capable of meeting criteria (i)-(iii) simultaneously.

CONCLUSIONS
Quantum work is a fundamental concept for extending thermodynamics to general quantum scenarios.Its statistical description, however, still presents intriguing challenges.Therefore, testing its consistency with criteria (i)-(iii), expressing general features that any quantum work protocol should possess, is a paramount goal.By adopting a novel strategy to consider the classical limit, we demonstrated that OBS is the only well-known scheme for quantum work measurement that complies with all these criteria simultaneously.
Our results are entirely general concerning the internal structure of the closed system.The latter may comprise multiple subsystems, including scenarios where heat is transferred among them.However, we did not address open quantum scenarios with external sources transferring heat to the system of interest.Although we showed that the OBS POVM is the only well-known scheme capable of satisfying simultaneously all the criteria (i)-(iii) in the closed-case scenario, it is not necessarily expected that an observable precisely as it is defined in Eq. ( 1) should be the correct observable describing thermodynamics work in the open quantum case.This leads to an intriguing question: Is there a broader, more encompassing quantum work observable that not only adheres to criteria (i)-(iii) in open quantum scenarios but also aligns with the observable in Eq. ( 1) in the specific closed case?We believe our current work lays the groundwork for addressing this question, hopefully inspiring future research to expand and refine our understanding of the concept of work.
Interestingly, our results pave the way for many research opportunities exploring superposition, entanglement, and other correlations regarding OBS statistics of work and other two-time quantities [49].Conditions (A) and (B) introduced here could be used to study the classical limit in diverse fields like quantum chaos [115], quantum optics [103][104][105]116], and the semiclassical formalism.A promising avenue in these lines involves conducting a comparative study between the results obtained in [78,[90][91][92][93][94] and result 2.Moreover, our findings support the viewpoint that conventional fluctuation theorems aimed at recovering classical expressions necessitate a departure from their conventional forms [117], revealing new connections between equilibrium properties and information resources with quantum work statistics.Furthermore, we hope our approach stimulates the analysis of the relation between work and other quantum thermodynamic facets, such as the lack of detailed balance at equilibrium (nonreciprocity) and persistent quantum currents [118,119].These lines of research, we expect, will propel the field of quantum thermodynamics to new frontiers.

Supplemental Material for "Quantum work: reconciling quantum mechanics and thermodynamics"
Thales A. B. Pinto Silva * and David Gelbwaser-Klimovsky

Schulich Faculty of Chemistry and Helen Diller Quantum Center, Technion-Israel Institute of Technology, Haifa 3200003, Israel
We provide detailed proofs of the results presented in the main text.Specifically, we derive results 1 and 2, followed by an in-depth analysis of the model described in Equations ( 4) and (5).
Throughout this supplement, we consider that a system initially in a state ρ (t = 0) undergoes a quantum process of duration τ.The processes' dynamics at time t is described by the unitary evolution operator U t , which is the solution of the Schrödinger equation i Here, H(t) = j e j (t) |e j (t) e j (t)| is a time-dependent Schrödinger Hamiltonian whose eigenstates and eigenvalues are represented by {|e j (t) } and {e j (t)}.In the Heisenberg picture H(t) is thus defined as The quantum process is completely characterized by the quadruple {H(t), U t , [0, τ], ρ}.

DERIVATION OF RESULT 1
To derive our first result, we focus on the work statistics of the quantum processes described by {H(t), U t , [0, τ], ρ}.Specifically, we analyze the observable (OBS) and two-point measurement (TPM) statistics.As discussed in the main text, the TPM work probability is computed as where M TPM (w) are the elements of the TPM POVM, satisfying ∞ −∞ dw M TPM (w) = ½, and δ is the Dirac's delta.On the other hand, the OBS probability distribution is described by where M OBS (w) are the elements of the OBS POVM ( ∞ −∞ dw M TPM (w) = ½) and {|w j (τ, 0) } and {w j (τ, 0)} are respectively the eigenvectors and eigenvalues of the work operator To make the comparisons between P TPM (w) and P OBS (w) statistics, we considered in our proofs their associated characteristic functions, defined as [1][2][3]  (τ,0)  and where, for any operator A, we denote as the dephasing map of A with respect to the energy basis at time t and as its average with respect to the initial state.T > is the time ordering operator, such that for any two arbitrary operators A(t ′ ) and The use of characteristic functions is a well-established strategy in quantum thermodynamics literature [3][4][5].Notably, characteristic functions possess important properties worth highlighting.Firstly, for any work distribution P(w), i.e. the probability distribution is the Fourier transform of its characteristic function.In fact, it was proven in [3] that whenever the initial state is initially incoherent ρ = Φ H(0) [ρ(t)], and [H h (τ), H(0)] = 0, then χ TPM (u) = χ OBS (u) and, from Eq. ( 7), P TPM (w) = P OBS (w).It follows directly from Eqs. ( 4) and ( 7) and the fact that P TPM (w) and P OBS (w) are non-negative and normalized, that It is important to highlight the fact that Eq. ( 8) is not exclusive for the TPM or OBS statistics, but hold for general pairs of consistent characteristic functions.Finally, we notice that we can rewrite the OBS characteristic function as where W n (τ, 0) can be written, for any power n, as where a ni = ±1 and the sum over i covers all the 2 n possible combinations of product of powers Here, s(i) ≤ n denotes the maximum number of powers I s(i)i and J s(i)i for each i-th combination.The form of Eq. ( 10) of writing W n (τ, 0) can be proved by induction.For instance, notice that Eq. ( 10) . Then, by induction, we can show that it is valid for n = 3, 4, • • • and so on.Similarly, the TPM characteristic function can be written as where with the same coefficients a ni as in Eq. (10).As a last requirement to establish Result 1, we assume that the average energy is always bounded.This condition has been previously explored in the context of coherences to avoid singularities related to measures of coherence and entanglement [6][7][8].Incorporating these properties into our analysis, we derive Result 1 presented in the main text: Result 1.Consider a process {H(t), U t , [0, τ], ρ} in which, for any initial incoherent state ρ = Φ H(0) [ρ(t)], the average of energy is bounded H(t) = Tr H h (t)ρ < ∞ and the 1-norm measure of coherences [6] is bounded as follows where ǫ 1 is a non-null real scalar.Then, it follows that where C 1 is a bounded positive real scalar that is independent of ǫ 1 .Therefore, Proof.Let us first consider the case in which C(ρ(t)) = 0, i.e. ǫ 1 = 0, for every initially incoherent state.It then follows that C(ρ(t)) = 0 for ρ = |e l (0) e l (0)|.From Eq. ( 14 0)] = 0.The proof for the case in which ǫ 1 → 0 is completed employing the results of reference [3], where it was shown that whenever [H h (t), H(0)] = 0, then P OBS (w) = P TPM (w).Now, consider the case in which ǫ 1 > 0. Since Eq. ( 13) holds for any incoherent initial state ρ at t = 0, τ, then it must be valid for any initial arbitrary eigenstate ρ = |e l (0) e l (0)|, for any l.Consequently, where α l j (t) = e l (0)| U † t |e j (t) and α * l j its conjugate.Taking into account Eq. ( 16), let us compare the characteristic functions χ OBS (u) and χ TPM (u) considering an arbitrary initial incoherent state ρ and Eqs. ( 9) and (11).We first notice, from Eq. ( 16), that the components of the commutator [H m (t), U τ H n (0)U † t ] for any power m and n can be written as Here, we considered that From Eq. ( 16), we defined since the upper bound of |α l j (t)α * lk (t)| decays linearly with ǫ 1 .γ jkl (t, ǫ 1 ) is a complex-valued function satisfying j k |γ jkl (t, ǫ 1 )| ≤ 1 and lim ǫ 1 →0 γ jkl (t, ǫ 1 )ǫ 1 = 0 (see Eq. ( 16) , then it follows that, for any initial incoherent state ρ, the powers in Eq. ( 10) can be written from Eq. ( 17) as where A similar procedure as the one used to obtain Eq. ( 19) can be considered to deduce where Repeating this procedure s − 1 times, we obtain where and Considering the OBS and TPM characteristic functions in Eqs.(9) and Eq. ( 11), the cyclic property of the trace, and Eqs. ( 10), (12), and ( 23), we obtain with the function As a result, where The proof of the result is finished by showing that y ′ (t, ǫ 1 ) is bounded for every ǫ 1 .This is proved in the subsection "Boundness of y ′ (t, ǫ 1 )" below so that we can define where C 1 < ∞ as a positive bounded real scalar.As a result, it follows that Consequently, lim which implies in Eq. (15).
It is important to note that the l 1 norm measure of coherences, C[ρ(t)] in Eq. ( 13), is commonly employed in finite-dimensional cases.However, in the context of infinite-dimensional systems, certain challenges may arise with this measure, as discussed in [6,7].Remarkably, our results extend naturally to infinite-dimensional scenarios when considering the relative entropy of coherence [9] where S (ρ) = − Tr[ln(ρ)ρ] represents the von Neumann entropy.This measure was shown to be consistent in both the finite-and infinite-dimensional cases [6,7,9].By employing an argument similar to that used to derive Eq. ( 16), we can establish which implies This can be proven by contradiction.Let us assume that Eq. ( 34) is not valid so that there is at least one j and k such that which holds only if α max ≥ ln 4, a contradction with i |α li (t)| 2 = 1.Consequently, Result 1 can be derived, substituting ǫ 1 by √ ǫ 1 , using the same arguments and equations as presented after Eq. ( 16) in the previous proof.
Boundness of y ′ (t, ǫ 1 ) To prove that y ′ (t, ǫ 1 ) is bounded, we first compute its limit when ǫ 1 → 0. For that, let us first consider the limit ǫ 1 → 0 for y(u, t, ǫ 1 ) as defined in Eq. ( 26): lim where the indexes , and C mi (ǫ 1 ) is defined in Eq. (24).Notice that, because N n=0 ) converges to y(u, t, ǫ 1 ) in the limit N → ∞ for any ǫ 1 (therefore pointwise in ǫ 1 ), then we could exchange the limit ǫ 1 → 0 and the sum ∞ n=0 from the second to the third equality , our focus is now to compute lim ǫ 1 →0 C mi (ǫ 1 ), so that, by considering Eq. ( 24), we aim to explicitly compute lim Since the sum N l l=1 N j j=1 N k k=1,k j converges to C mi (ǫ 1 ) for every ǫ 1 in the limit lim {N l ,N j ,N k }→{∞,∞,∞} (thus pointwise converging in terms of ǫ 1 ), then we could exchange the limit ǫ 1 with the limit lim {N l ,N j ,N k }→{∞,∞,∞} and the summations from the second to the third equality.Moreover, from now on, we order all the eigenvalues of energy in ascending order e i (0) ≤ e i+1 (0) and e i (t) ≤ e i+1 (t), where for each fixed finite value i, e i (t) is also finite, but can be unbounded when i → ∞.
Just as deduced in the beginning of the proof of result 1, we notice that as Taking these relations into account in Eq. ( 37), we obtain lim Considering the expansion of the initial incoherent state as ρ = l ′ p l |e l ′ (0) e l ′ (0)|, we deduce lim where we considered the notation lim k=1,k j and the following definition Considering the definition α lk (t) = e l (0)| U † t |e k (t) and the expansions H (0) |e q (0) e q (0)|, we can rewrite C ′ mi (ǫ 1 ) as Inserting Eq.( 41) in Eq. ( 39), we obtain lim where and In the limit ǫ 1 → 0, |α l ′ p (t)α * l ′ k (t)| → 0 for all p k and |α * qp α q j (t)| → 0 for all p j (see Eq. ( 16)).Therefore, when ǫ 1 → 0 the product α * qp α q j (t)α l ′ p (t)α * l ′ k (t) can only be non-null when k = p and p = j.However, since in the sum over k and j we are considering only cases in which k j, then it cannot be the case where k = p and p = j.Therefore, lim ǫ 1 →0 α * qp α q j (t)α l ′ p (t)α * l ′ k (t) = 0 for k j.Moreover, since p l ′ ≤ 1 and |γ jkl (t, ǫ 1 )| ≤ 1, then it follows that lim Moreover, since for every fixed natural i ∈ {l, l ′ , j, k, q, p}, e i ≤ e N i is finite, then E I i J i ll ′ jkqp is finite as well and lim To deduce the second part of Eq. ( 46), we considered the following reasoning.Using the notation N l , N l ′ , N j , N k , N q , N p → N l,l ′ , j,k,q,p , we can define the sum C l,l ′ , j,k,q,p mi which converges to C mi (ǫ 1 ) for every ǫ 1 whenever N l,l ′ , j,k,q,p → ∞.Therefore, for every ε > 0, there exist finite natural numbers N ′ l,l ′ , j,k,q,p such that sup ǫ 1 |C l,l ′ , j k,q,p mi (ǫ 1 ) − C mi (ǫ 1 )| < ε, for any N l,l ′ , j k,q,p > N ′ l,l ′ , j k,q,p .Since the limits can be exchanged whenever the sequence converges for every ǫ 1 (see footnote [16]), then for any naturals N l,l ′ , j k,q,p > N ′ l,l ′ , j k,q,p , it follows that Therefore, given that lim ǫ 1 →0 C I i J i (ǫ 1 ) = s(i)−1 m=1 lim ǫ 1 →0 C mi (ǫ 1 ) = 0, and inserting this result in the sum (36), we deduce, analogously as was done for lim ǫ 1 →0 C mi (ǫ 1 ) in Eq. ( 48), the following result lim where we considered the fact that u is bounded, since from the Riemman-Lebesgue Lemma (Lemma 7 in section 15.6 in [2]), where G possesses the same functional form as the function defining the Heisenberg Hamiltonian H h (t), with the substitution of operators X and P by x 0 and p 0 , respectively.The parameter t ′ can take on values of either 0 or τ.
(B) For any smooth functions g l ≡ g l (X, P) and g r ≡ g r (X, P) which can be written as a sum of powers of X and P, it follows that where ǫ B ≪ 1.
Our approach to establish Result 2 as presented in the main text is as follows: we demonstrate that when conditions (A) and (B) are satisfied, the difference between the characteristic functions of the OBS protocol and the classical work statistics are both bounded by ǫ A and ǫ B .Subsequently, we illustrate how this boundedness affects into probability distributions of work.
As stated in the Eq. ( 4) in the previous section, the OBS characteristic function is Taking into account that the hamiltonian operators are described in terms of sufficiently smooth functions such that H h (t) = G(X, P, t), then W(τ, 0) = G(X, P, τ) − G(X, P, 0), and its exponential can be written as a power series in the general form where a IJ (u) are u-dependent complex-valued functions, indexed by The summation IJ is done over all values of the integer powers IJ.Inserting Eq. ( 61) in Eq. ( 60), it follows that The work distribution for the classical scenario initiated in a classical distribution ̺(Γ 0 ) was introduced in the main text as and the associated classical characteristic function is Under the assumption that condition (A) is satisfied, we have H CL (Γ 0 , t) = G(x 0 , p 0 , t), where the function G maintains the same functional form as that defining H h (t).Consequently, the classical work assumes a form akin to the operator W(τ, 0), substituting the operators X and P by the variables x 0 and p 0 , respectively.Accordingly, W CL (Γ 0 , τ, 0) = G(x 0 , p 0 , τ)−G(x 0 , p 0 , 0).Furthermore, the classical characteristic function can be expressed as follows: where a IJ (u) refer to the same coefficient functions and powers IJ as in the quantum case in Eq. ( 62).Taking into account Eqs. ( 61) and ( 65), we deduce the following result: where and ξIJ and ξ′ Proof.From condition (B), we obtain Similarly, we derive where we considered condition (B) to obtain In fact, we can derive, by induction, that for 1 ≤ l ≤ J n .This can be proved by noticing first that Eq. ( 71) holds for l = 1: Also, if Eq. ( 71) holds for l, then it also holds for l + 1: Therefore, by induction from l = 1 to 2 and so on, then Eq. ( 71) should be valid for any 1 ≤ l ≤ J n .Another expression which can also be obtained by induction is the following: Indeed, from Eqs. ( 71) and ( 74), it follows that Using the induction rule (74), we deduce: Repeating this procedure J n times, we obtain ) From the previous induction rules, it follows that Then, the following new induction rules can be derived: We can use further these induction rules for I n power of X, obtaining Then From a similar procedure, it follows that Doing this iteratively we get where Therefore, it follows that where ξ IJ are complex scalars satisfying |ξ IJ | ≤ 1.Consequently, the OBS characteristic function can be written as Moreover, considering condition (A), we have Using a similar procedure to deduce Eq. ( 79), we derive via induction that Therefore, we can define complex scalars ξ ′ IJ such that |ξ ′ IJ | ≤ 1 and so that: Taking into account the definition of the classical characteristic function in Eq. ( 65), it follows that Defining ǫ max = max{ǫ A , ǫ B } and considering the explicit form of C IJ (ǫ B ) in Eq. ( 84), the result of the Proposition is obtained: where and where To establish that the above result implies the convergence of the OBS characteristic function to classical statistics as ǫ A and ǫ B tend to zero, we must demonstrate the boundedness of the function g(u, ǫ A , ǫ B ) defined in the preceding result.To accomplish this, we invoke the following outcome, inspired by the Cauchy-Hadamard formula (see [10] and Theorem 2 on page 38 in [11]): Proposition 2. Consider the series of the form where b IJ and s IJ are complex scalars indexed by where I + J = n j=1 I j + J j .Proof.First, suppose that {R IJ } exist and satisfy conditions in Eq. ( 96).Then it follows that for any real ζ, there is only a finite number of natural numbers To prove this statement, assume its negation, i.e. that there is a ζ such that for infinitely many natural numbers This is in direct contradiction with the limit in (96).Therefore, for every ζ, there is only a finite number of natural numbers As a result, one can rewrite the series (95) as where > and ≤ represents the sum over all IJ indexes in which Because the sum > is finite and |b IJ | is bounded, then there exists a real non-negative value C > such that On the other hand, if then it follows that where C ≤ and S < 1 are bounded real non-negative scalars.Notice that the infinity series of the type ≤ S I+J is always convergent for S < 1, justifying the last inequality in Eq. ( 101).Because this result holds for any ζ, then it follows that the series is convergent when |s IJ | < R I+J IJ .For the converse, we prove by contradiction.Let us thus suppose that the series in Eq. ( 95) converge but for all set {R IJ } such that |s IJ | ≤ R I+J IJ , the limit in ( 96) does not hold.Then there is a real scalar ζ > 0, such that Since this relation holds for any set {R IJ }, then we can choose so that from Eq. ( 102), it follows that for any IJ, and the series therefore cannot converge, a contradiction.The converse is thus proved.
We expect the series that define χ OBS (u) and χ OBS (u) to absolutely converge, leading to proper characteristic functions, satisfying: and lim u→∞ χ OBS (u) = lim u→∞ χ CL (u) = 0.It thus follows that a IJ (u) are bounded functions, since they are sum of polynomials of u < ∞.As consequence of the Proposition 2, this implies, after the substitutions b IJ → a IJ (u) and s IJ → x n i=1 I i 0 CL , that there exist real bounded non-negative scalars {R IJ } and {R ′ IJ } such that and Taking into account these facts, we thus deduce the following important result.
Proposition 3. The function g(u, ǫ A , ǫ B ) defined in Eq. ( 93) absolutely converges for sufficiently small ǫ max = max{ǫ A , ǫ B } ≪ 1.Therefore, if conditions (A) and (B) hold, then there is a bounded real non-negative scalar C ′ , such that 93), then it follows that where we considered the notation I + J = n j=1 I j + J j , the fact that n j=1 I j ( n m= j J m ) ≤ (I + J) 2 , and I+J in the Proposition 2, then it follows that the series will absolutely converge if there is a set of bounded real non-negative scalars {R ′′ IJ } such that Now, from the convergence of the OBS characteristic function, it follows that there exists scalars {R IJ } satisfying Eq. ( 106).Let us thus define

IJ
and the limit in Eq. ( 112) is satisfied.Therefore, from Proposition 2, the series in Eq. ( 111) absolutely converges and g(u, ǫ A , ǫ B ) is bounded.Hence, there exists a real non-negative bounded scalar A crucial consequence of the Proposition 3 is that whenever ǫ A , ǫ B → 0, then χ OBS (u) → χ CL (u).Therefore, in the classical limit, the OBS statistics converge to the classical one.In this sense, we deduced the main result of this paper.

Result 2. If conditions (A) and (B) hold, then lim
where K is a bounded non-negative real scalar that is independent of ǫ A and ǫ B and ǫ max = max{ǫ A , ǫ B } > 0.
Proof.Notice that, from Proposition 3, lim ǫ max →0 χ OBS (u) = χ CL (u).Therefore, P OBS (w) → P CL (w) (see the Continuity Theorem at section 10.3 in [1] or Theorem 15 at section 14.7 together with Proposition 3 in section 14.3 in [2]).As a result, by Scheffe's Theorem (Theorem 2.5.4 in [1]), lim To prove Eq. ( 114), we consider Eqs. ( 92) and ( 93) and the definition of the characteristic function, to deduce that where Using Proposition 2, we prove below in subsection "Boundness of g ′ (ǫ A , ǫ B )" that g ′ (ǫ A , ǫ B ) must be bounded for any ǫ max > 0, so that we can define where K < ∞ is a positive bounded real scalar.Therefore, it follows that and the result is proved.
It is important to remark that the results carried here can all be naturally extended to the scenario of many systems described by Schrödinger observables X and P satisfying [X, P] = i ½.In this sense, by suitably adapting conditions (A) and (B), similar lines of reasoning and deductions as the ones done here can be used to deduce Result 2.
To deduce that g ′ (ǫ A , ǫ B ) is bounded for any ǫ B , we first consider the definition where satisfying, respectively, Considering the definitions in Eqs. ( 84), ( 85), (89), and (94), we can explicitly write ξIJ where, for any state satisfying conditions A and B, | ξIJ |, | ξ′ IJ | ≤ 1.As a result, we have that Taken into account that P OBS (w) − P CL (w) is purely real, then it follows from Eq. ( 122) that ∞ −∞ du e −iuw g(u, ǫ A , ǫ B ) must also be a purely real number.Expanding g(u, ǫ A , ǫ B ) considering Eq. ( 124), we get where Since the right-hand side of Eq. ( 125) is real, its absolute value can only be considering the case in which it is either positive or negative, respectively.Denoting ≥(<) dw as the integral over all intervals of w in which ∞ −∞ du e iuw g(u, ǫ A , ǫ B ) is non-negative (negative), we thus obtain where Moreover, since Before proving that g ′ (ǫ A , ǫ B ) is bounded, it is important to demonstrate that a ≥ IJ 2π are bounded.For that, we first notice that, from Eq. ( 122), As we demonstrated in Eqs. ( 106) and ( 107), there exist real bounded non-negative scalars {R IJ } and {R ′ IJ } such that As a consequence, IJ , to deduce that there exist real bounded non-negative scalars {A IJ } such that Therefore, |a ≥ IJ | is bounded for every indexes IJ in which We are now in position to show that g ′ (ǫ A , ǫ B ) is bounded for any ǫ max > 0, even when ǫ max → 0. To check this, we consider ǫ A > 0 or ǫ B > 0, such that ǫ max = max{ǫ A , ǫ B } > 0, and Eq. ( 123) to rewrite Eq. (131) as If the initial quantum state ρ and classical distribution ̺(Γ 0 ) are such as to satisfy conditions A and B, then, as we deduced in Proposition 1, |ξ IJ |, | ξIJ | ≤ 1.Moreover, for ǫ A and ǫ B not tending to 0, g ′ (ǫ A , ǫ B ) is surely bounded, since, from Eq. ( 122), Because a ≥ IJ are bounded, then we can consider again Proposition 2 regarding the series in the last line of Eq. (136) to show that there exist real bounded non-negative scalars { RIJ } such that and lim sup Considering that ǫ B > 0, and the fact that |ξ IJ |, | ξIJ | ≤ 1, I + J = n j=1 I j + J j , so that n j=1 I j ( n m= j J m ) ≤ (I + J) 2 , we can deduce that and As a result, it follows that where we considered the fact that ξ′ is bounded for any indexes IJ.From Eq. ( 142) and Proposition 2, it thus follows that is bounded, since RI+J IJ > | X I 1 P J A similar result can be deduced for either ǫ B = 0 and ǫ A > 0 or ǫ A = 0 and ǫ B > 0. As a result, g ′ (ǫ A , ǫ B ) is bounded for any case in which ǫ A > 0 or ǫ B > 0.Even when ǫ max → 0, g ′ (ǫ A , ǫ B ) is bounded, since, considering Eq. ( 145), it follows that Consequently, we can consider the bounded real positive constant K such that

EXAMPLE: NON-AUTONOMOUS HARMONIC OSCILLATOR
In this section, we analyze in detail the model presented in Eqs. ( 4) and (5) in the main text.We consider the following Hamiltonian operator and function: where ω 2 (t) = ω 2 0 + (ω 2 1 − ω 2 0 ) t τ .This operators and function model the motion and energy of a particle within a time-dependent trap in the quantum and classical scenarios, respectively.Solving the Heisenberg and Hamilton equations, we obtain Ẍh (t) + ω 2 (t)X h (t) = 0 and ẍ(t) + ω 2 (t)x(t) = 0, with solutions given by X h (t) = A(t)X + B(t)P, P h (t) = C(t)X + D(t)P, and It is worth noting that (X, P) represent the quantum Schrödinger operators of position and momentum, with their Heisenberg versions (X h (t), P h (t)).Additionally, Γ 0 = (x 0 , p 0 ) denotes the classical initial phase point, evolving to the phase point Γ t (Γ 0 , t) = [x t (Γ 0 , t), p t (Γ 0 , t)] at time t.The functions A(t), B(t), C(t), and D(t) are defined as H CL .This represents the classical limit of this system.Indeed, by equating the quantum averages in the coherent scenario with the classical initial phase spaces, i.e., X = x 0 and P = p 0 , and assuming /|x 0 p 0 | → 0, it directly follows from Eqs. ( 149) and (163) that |α| → ∞, and the averages X(t) , P(t) , X 2 (t) , P 2 (t) , and X(t)P(t) exactly match their classical counterparts.Taking into account the explicit form in Eq. ( 12) for computing the statistical moments of work, for any initial state ρ, the m-th power of work using the OBS and TPM protocols are, respectively, It follows directly from Eqs. ( 161) and (164) assuming X = x 0 , P = p 0 , and /|x 0 p 0 | → 0 that the OBS average of work results in which precisely matches the classical expression in Eq. ( 159).On the other hand, from Eq. ( 164), the TPM statistics result in the expression where we have utilized the cyclic property of the trace.To compute W(t 2 , t 1 ) TPM , first notice that the incoherent part of the coherent state is given by where |e n (0) ≡ |n is the eigenvector of both H(0) and the number operator N, given by where a and a † are the annihilation and creation operators, related to position and momentum operator as follows Using the well-known algebra rules for a and a † , it follows that Consequently, for the incoherent state it follows: where we considered Eqs.( 167) and (170) and Considering Eq. ( 163), it follows that then we can rewrite Eq. (171) as Comparing TPM with the classical regime by assuming X = x 0 , P = p 0 , and /|x 0 p 0 | → 0, it follows from Eqs. ( 161), (164), and (174) that which completely differs from the classical expression for work in Eq. ( 159).Indeed, to illustrate these disparities, we have depicted in Figure 1 the quantity (W CL (Γ 0 , τ, 0) − W(τ, 0) TPM )/W CL (Γ 0 , τ, 0) as functions of x 0 and p 0 .These results were obtained using the model parameters τ −1 = ω 0 = ω 1 /3 = 1 s −1 , m = 1 kg, and /|x 0 p 0 | → 0. Notably, the difference can range between 10 − 40 %, highlighting a significant and non-negligible divergence between the two approaches in an explicitly classical limit scenario.Consequently, we can conclude that TPM fails to provide the correct statistics in this classical limit.The observed divergence in this case is primarily attributable to the pronounced impact of the first measurement's vanishing due to the coherences, a contrast that becomes evident when compared with the OBS average.Indeed, if we take into account the same parameters {ω 0 , ω

OBS versus classical statistics
In this subsection, our objective is to demonstrate that the OBS statistics can recover the classical limit for the trapped harmonic oscillator modeled in Eq. (148), whenever |α| ≫ 1. Referring to Result 2, it therefore suffices to establish that the oscillator satisfies all the classicality conditions, (A) and (B), as elaborated in the previous section and the main text.This path will guide our discussion throughout the remainder of this supplementary material.
Initially, it becomes immediately apparent from Eqs. ( 157) and (160) that the second part of condition (A) is satisfied, i.e.Eq. ( 58) holds.Consequently, our task is to establish that the first part of condition (A) and the full condition (B) are satisfied.To this end, we employ the Wigner-Weyl formalism [15].In this case, the expectation value of any momentum or position-dependent observable can be expressed as where is the so-called Wigner quasi-probability distribution and is the Weyl transform of G(X, P).In the special case in which we consider the initial coherent state ρ = |α α| described in Eq. ( 162), the Wigner function assumes a Gaussian form where refer to the Gaussian distribution centered around r with uncertainty σ r .The position and momentum uncertainties of the coherent state (see Eq. ( 163)) are With these definitions and the commutation relation [X, P] = i , we now show that condition (A) is satisfied for the coherent state.
Let us first compute the Weyl transformation of PX m .Let us call it g 1,m (q, p).From the very definition of the Weyl transformation (Eq.( 178)), we get: Similarly, for P n X m , it follows: where we used the identity dx f [14].To express this relation in a more suitable form, we first prove the following identity: Indeed, Eq. ( 184) can be derived by induction.First, if Eq. ( 184) holds for k, then for k + 1, we obtain that k! (k − j)! j! p k+1− j e ipy/ (−i ∂ y ) j (q + y/2) m + + k+1 j=1 k! (k − j + 1)!( j − 1)! p k+1− j e ipy/ (−i ∂ y ) j (q + y/2) m = p k+1 e ipy/ (−i ∂ y ) 0 (q + y/2) m + e ipy/ (−i ∂ y ) k+1 (q + y/2) m + + k j=1 k! (k − j)! j! + k! (k − j + 1)!( j − 1)! p k+1− j e ipy/ (−i ∂ y ) j (q + y/2) m . (185) Recognizing that and (k + 1)! (k + 1 − j)! j! p k+1− j e ipy/ (−i ∂ y ) j (q + y/2) m = p k+1 e ipy/ (−i ∂ y ) 0 (q + y/2) m , for j = 0, e ipy/ (−i ∂ y ) k+1 (q + y/2) m , for j = k + 1, (187) then it follows that (−i ∂ y ) k+1 e ipy/ (q + y/2) m = k+1 j=0 (k + 1)! (k + 1 − j)! j! p k+1− j e ipy/ (−i ∂ y ) j (q + y/2) m (188) Therefore, if Eq. ( 184) is valid for k, then it is also valid for k + 1.Since it can be directly checked that the expression Eq. ( 184) holds for k = 1, then, by induction, it is valid for any natural k ≥ 1.We thus have the general formula: (n)! (n − j)! j! p n− j e ipy/ (−i ∂ y ) j (q Since (−i ∂ y ) j (q + y/2) m = (−i ) j j!(1/2) j (q + y/2) m− j for m ≥ j and 0 for j > m, then (−i /2) j p n− j q m− j , for n ≤ m, m j=0 n! (n − j)! (−i /2) j p n− j q m− j , for n > m. ( It then follows that n! (n − j)! (−i /2) j p n− j q m− j , for n ≤ m, n! (n − j)! (−i /2) j p n− j q m−1 , for n > m. (191) Comparing Eq. ( 190) with (191), we obtain g n,m (q, p) = p n q m − n(i /2)g n−1,m−1 (q, p). ( Multiplying both sides by the Winger function and integrating over q and p, it follows: Using iteratively this equation, we deduce P n X m − dq d p p n q m W(q, p) = − ni 2 dq d p p n−1 q m−1 W(q, p) − (n − 1)i 2 which results in ( Considering the limit in which the quantum system approach the classical regime, then x α → x 0 , p α → p 0 , and |x 0 p 0 | → 0. As a result from Eq. ( 181), the Wigner distribution then approximates Dirac's delta distribution W α (q, p) = G q, x α , σ α x G[p, p α , σ α p ] ≈ δ(q − x 0 )δ(p − p 0 ).(196) As a result, the quantum scenario approximates a classical distribution of a particle with well-defined position and momentum.In this classical limit, we can deduce directly from Eq. (195) that P n X m − dq d p p n q m W(q, p) dq d p p n q m W(q, p) → where n min = min{n, m}.Therefore, P n X m − dq d p p n q m W(q, p) | P n X m | = P n X m − dq d p p n q m W(q, p) dq d p p n q m W(q, p) + P n X m − dq d p p n q m W(q, p) P n X m − dq d p p n q m W(q, p) for any n min .Notice that this result holds even if n min → ∞.Therefore, taking into account the fact that Eq. ( 58) holds, condition (A) is fully satisfied when x α → x 0 , p α → p 0 , and /|x 0 p 0 | → 0. Now we prove that condition (B) is satisfied for x α → x 0 , p α → p 0 , and /|x 0 p 0 | → 0. Let us first notice that, by using a similar induction procedure as used in the Results 1 and 2 above or considering the results in the reference [12], we can deduce that for any arbitrary set of powers IJ ≡ {I 1 , I 2 , • • • I n , J 1 , J 2 , • • • J n } ⊂ , the following relations can be written: where c k and c ′ k are coefficients dependent on IJ and N = max{ µ I µ , µ J µ } = N ′ + 1.To show that the condition (B) is satisfied for our example with /2|x 0 p 0 | → 0, we analyze the ratio Denoting c max = max{{c k } N 1 , {c ′ k } N ′ 1 } and taking into account Eq. ( 204), it follows that Therefore, whenever /|x 0 p 0 | → 0, then and a similar deduction can be made for As a consequence, condition (B) is satisfied.With all the conditions (A) and (B) fulfilled, it follows from Result 2 that the work statistics for both the classical and quantum scenarios will be identical in the limit where x α → x 0 , p α → p 0 , and /|x 0 p 0 | → 0. Consequently,

k 2 k
dq d p p n−k q m−k W(q, p) + n! − i 2 n X m−n , for n ≤ m, m−1 k=0 n! (n − k)! − i dq d p p n−k q m−k W(q, p) + n! (n − m)! − i 2 m P n−m , for n > m.