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Quantal-classical fluctuation relation and the second law of thermodynamics: The quantum linear oscillator

In this work, we study the fluctuation relation and the second law of thermodynamics within a quantum linear oscillator externally driven over the period of time t = tau. To go beyond the standard approach (the two-point projective measurement one) to this subject and also render it discussed in both quantum and classical domains on the single footing, we recast this standard approach in terms of the Wigner function and its propagator in the phase space (x,p). With the help of the canonical transformation from (x,p) to the angle-action coordinates (\phi,I), we can then derive a measurement-free (classical-like) form of the Crooks fluctuation relation in the Wigner representation. This enables us to introduce the work W_{I_0,I_{tau}} associated with a single run from (I_0) to (I_{tau}) over the period tau, which is a quantum generalization of the thermodynamic work with its roots in the classical thermodynamics. This quantum work differs from the energy difference e_{I_0,I_{tau}} = e(I_{tau}) - e(I_0) unless beta, hbar -->0. Consequently, we will obtain the quantum second-law inequality Delta F_{beta} \leq_{P} \leq_{P} = Delta U, where P, Delta F_{beta}, and_P denote the work (quasi)-probability distribution, the free energy difference, and the average work distinguished from the internal energy difference Delta U, respectively, while_P -->Delta U in the limit of beta, hbar -->0 only. Therefore, we can also introduce the quantum heat Q_q = Delta U - W even for a thermally isolated system, resulting from the quantum fluctuation therein. This is a more fine-grained result than_P = Delta U obtained from the standard approach. Owing to the measurement-free nature of the thermodynamic work W_{I_0,I_{tau}}, our result can also apply to the (non-thermal) initial states rho_0 = (1-gamma) rho_{beta} + gamma sigma with sigma \ne rho_{beta}.


I. INTRODUCTION
Fluctuation relations such as Jarzynski's equality and Crooks' theorem have attracted a great deal of interest owing to their nature of the link between non-equilibrium fluctuations and thermal equilibrium properties of small systems (either classical or quantal) [1,2]. Here, the stochastic nature of the thermodynamic work performed on a given system emerges through (infinitely) many runs of the external driving of the system; in the classical case, this nature is associated solely with a random sampling of individual microstates from the initially prepared (canonically thermal) state of the system, because after such a sampling, the system becomes thermally isolated and evolves deterministically under Hamilton's equations. In the quantum case, on the other hand, such a stochastic nature is associated not only with the random sampling from the thermal initial state (as a source of the thermal fluctuation) but also with the quantum fluctuation existing even during the external driving. In fact, an appropriate determination of the work and its probability distribution associated with both thermal and quantum fluctuations, required for a legitimate form of the quantum fluctuation relation, has been one of the central issues in the field of quantum thermodynamics.
The standard approach to the quantum fluctuation relation has been made in the so-called two-point projective measurement (TPM) framework [3][4][5][6][7][8][9][10][11][12]: An isolated quantum system is initially prepared in the thermal stateρ 0 =ρ β (with β = 1/k B T ) and then undergoes an external driving (denoted by a time-dependent Hamiltonian parameter λ t ). The probability distribution of the single-run work (w) for the system in a forward process of the external driving is then given by where the energy-eigenvalue difference ∆e nm = e m(τ ) − e n(0) , between the two outcomes e n(0) and e m(τ ) found from the initial (t = 0) and final (t = τ ) measurements, and its probability P (f) nm = P [m(τ )|n(0)] P [n(0)] consisting of both initial probability P [n(0)] = e −βe n(0) /Z β (λ 0 ) of the nth energy eigenstate, with the partition function Z β (λ 0 ) for the initial stateρ β (λ 0 ), and the conditional probability P [m(τ )|n(0)] = | m(τ )|Û|n(0) | 2 for the run |n(0) → |m(τ ) with the unitary operatorÛ . Here, the probabilistic nature of finding those two measurement outcomes gives rise to the stochastic nature of the work. As such, the single-run work is given by w nm ≡ ∆e nm .
Likewise, the work probability distribution for the system in a backward process starting fromρ β (λ τ ) is given by where P in which the free energy difference is ∆F . With the help of Eqs. (1) and (2), this will result in the quantum Jarzynski equality in its known form Therefore, the free energy difference and the non-equilibrium fluctuating work can be exactly linked. With the help of the Jensen inequality, this Jarzynski equality gives rise to as an expression of the second law of thermodynamics in the quantum domain [cf. (46)].
In spite of its great usefulness, the TPM framework has a conceptual issue when it comes to its generalization: It is nonlegitimate to apply the same form of the work probability to the processes starting from the non-thermal statesρ 0 =ρ β with coherence in the energy basis; because the initial projective measurement then destroys the initial coherence and so produces extra entropy, thus leading to disturbing the original time evolution of the system. As such, this standard approach is not fully quantum-mechanical. Moreover, we also note that an individual external driving (λ t ) itself, described by unitary dynamics, produces no entropy at all, regardless of the initial outcomes e n(0) ; but an appearance of the entropy production, achieved through the (classical) mixture over many runs, is due to the (non-unitary) final projective measurement (thus viewed as an extra non-equilibrium work).
Consequently, it still remains an open question to introduce a generalized form of quantum work legitimate for the non-thermal initial condition and the external driving only.
To go beyond such a limitation, we intend to introduce in this paper an alternative definition of the quantum work ( = w) and its distribution formulated without the projective measurements. For this purpose, we will resort to the classical phase space (x, p), in which both quantum and classical fluctuation relations can be discussed on the single footing, for example, by making use of the Wigner function and its propagator [cf. Eqs. (6)- (7)]. In fact, the Wigner representation is known to be the most classical-like in propagation among different phase-space representations [13]. Then we will formulate a Crooks fluctuation theorem in its measurement-free (classical-like) form by recasting the characteristic function of the TPM framework in the Wigner representation (cf. e.g., [14] for a different phasespace approach without such a link with the TPM approach). This will finally give a new definition of the work W as the direct quantum counterpart to the thermodynamic work which has its original roots in the classical thermodynamics. For this formulation, we will also employ the angle-action coordinates (φ, I) [cf. Eq. (21)]; as is well-known, this pair is well-defined for the separable systems (e.g., the generic one-dimensional ones) and useful for the semiclassical analysis [15][16][17][18][19]. To our best knowledge, these coordinates have not been applied extensively for the study of quantum fluctuation relations. For the sake of an explicit treatment with analytical rigor, we will restrict our analysis here to a linear oscillator with its time-dependent frequency λ t = ω(t) (cf. [8] for an analysis of this system in the TPM framework); our methodology will also apply to a more generic class of systems.
Further, as is well-known, the Wigner function can be negative-valued, which reflects the quantum fluctuation. This will result in the work distribution P(W) with its negativity.
Therefore, our concern lies in the average values only (over many runs) that have the physical meaning (cf. the unavoidable negativity of the work distribution has already been studied in the extended TPM framework where the initial state is non-diagonal in the energy basis, e.g., [20,21]). Then, it will be shown that the average work W P is distinguished from the internal energy difference ∆U which remains unaffected under this transformation of the representation. For comparison, we also point out that it is impossible to consider the same scenario (free from the projective measurements) for such an alternative definition of the classical work, because in the classical setup, no projective measurements are required anyway. Therefore, our alternative approach to the quantum work ( W P ) will produce, in the classical limit, no difference from the TPM approach (∆U). This will also enable us to introduce the quantum heat Q q = ∆U − W P ≥ 0 (with no classical counterpart) for a thermally isolated system such that Q d + Q q = ∆U − ∆F β ; here, the thermal heat Q d = W P −∆F β ≥ 0 corresponds to the dissipative heat which will go out to the environment until the system-environment equilibrium will be achieved, if additional heat exchange between system and environment is carried out after completing the external driving. Consequently, we will acquire, as one of our main findings, the quantum second-law inequality (46) in a more fine-grained form than the inequality (5). Owing to the measurement-free nature of W and P(W), our result for the thermodynamic work in the quantum regime will be further generalized to the initial states being partially thermal in the form ofρ The general layout of this paper is as follows: In Sec. II we provide the phase-space formulation needed for an introduction of our quantum work and its distribution. In Sec.
III we derive the quantal-classical Crooks fluctuation theorem in our framework and then discuss the second law of thermodynamics and its implications. In Sec. IV our framework is generalized to the partially thermal initial states and then several examples of those initial states (with coherence in the energy basis) are explicitly considered. Finally, we provide concluding remarks in Sec. V.
And it is the propagator T (f) W (x, p; τ |x ′ , p ′ ; 0) that generates the trajectory running from the position (x ′ , p ′ ) at t = 0 to (x, p) at t = τ . In the limit of → 0, those Wigner trajectories exactly reduce to the classical trajectories. It is also easy to verify that and whereK(p; τ |p ′ ; 0) = p|Û(τ )|p ′ . The time-evolutionρ τ =Û(τ )ρ 0Û † (τ ) is then rewritten in the Wigner representation as As such, a given final position (x, p) is associated with all possible initial positions (x ′ , p ′ ) through infinitely many trajectories. Likewise, the Wigner propagator for the backward process [cf. Eq. (2)] is given by T For a quantum linear oscillator with a time-dependent frequency ω(t), we have the propagator in the Gaussian form [31] K(x; t|x where both quantities X = X(t) and Y = Y  (7) will be evaluated explicitly.

B. Quantum Work Distribution and Angle-Action Coordinates
In the classical scenario, on the other hand, the work distribution for a thermally isolated system in a forward process starting from the thermal state can be expressed as [12] in which the energy difference W 0τ = E τ − E 0 as a single-run work, and its probability density (P c ) ; here, we have the instantaneous energy E t = H(z t ; λ t ) with its trajectory z t = (x t , p t ) evolving from z 0 under Hamilton's dynamics, and the initial probability density P c (E 0 ) = {e −βE 0 /Z β,c (λ 0 )} g(E 0 ) with the classical partition function Z β,c (λ 0 ) = e −βF β,c (λ 0 ) and the density of states g(E 0 ), and the conditional probability density into its phase-space counterpart: By using Eqs. (6)-(7) with (12) and (15), we can acquire where the factor With the help of the relation m W m (x, p) = (2π ) −1 , it is easy to observe here that Now we restrict our discussion to a driven linear oscillator with λ t = ω t . To obtain the quantum work distribution in the Wigner representation taking the form of its classical counterpart in Eq. (17), we employ the change of coordinates from (x ′ , p ′ ; x, p) to the angleaction pairs (φ 0 , I 0 ; φ τ , I τ ) associated with the initial position (φ 0 , I 0 ) and the final position where the symbol denotes the integral which runs over a single period in the phase space.
For a linear oscillator, it follows that x = {2I t /(mω t )} 1/2 sin φ t and p = (2mω This enables us to rewrite Eq. (18b) as where the two-state quantity [cf. (19)] and the conditional distribution are Here we adopted the Wigner function of the nth energy eigenstate [25,26] in which the nth Laguerre Polynomial L n (A) and , giving rise to the relation n=0 W n (x, p) e −βen = Z β W β (x, p) indeed; the thermal Wigner function is given by the Gaussian form [33] where the partition function is given by Z β (ω) = 2 −1 csch(β ω/2).
Here we take into consideration three particular cases for Eq. (22); first, the case of Then we can introduce, with the help of Eq. (20), the joint (quasi)probability distribution associated with a single motion from (I 0 ) at t = 0 to (I τ ) at Likewise, the joint (quasi)probability distribution for the backward process can also be acquired where ; note the discussion after Eq. (15). As a result, we observe that the (quasi)probability distributions P (f) (I 0 , I τ ) and P (b) (I τ , I 0 ) are the counterparts to (1) and (2), respectively. The second case is given by , which equals the internal energy difference ∆U(τ ). This quantum-mechanical average value can now be expressed as in which the (single-motion) energy difference between the initial position I 0 and the final position I τ is given by To explicitly evaluate Eq. (28), we obtain the first moment (cf. Appendix A) where I 0 P (f) = ( /2) coth(β ω 0 /2) and the (classical) dimensionless quantity with K 0 = 1. By using the inequality of arithmetic and geometric means, it is easy to see that K t ≥ 1; e.g., for the sudden switch in Eq. (8), we find that K t,s = (ω 0 /ω t + ω t /ω 0 )/2. If the process is carried out adiabatically, it turns out that K t = 1 and so I τ P (f) is invariant.
Eqs. (29) and (30) finally give the internal energy difference, which is, in fact, identical to as required [cf. Eqs. (10) and (A15)-(A16)]. This reduces to The third case is given by nm . This is shown to differ from (ω τ I τ −ω 0 I 0 ) 2 P (f) , though; in fact, with the help of the second-moment relations (29)]. In fact, the quantum-mechanical expectation value (in the form of the first moment), such as ∆U(τ ), is identically evaluated in both TPM and Wigner frameworks; however, it can be shown that such a framework-independent behavior is not available any longer for all higher-order moments (I τ ) n (I 0 ) m P (f) . At this point, we also remind that differing from the energy operator (Ĥ), the quantum work (w) performed by an external agent is not a quantum-mechanical observable [3].

A. Crooks Theorem in the Wigner Representation
We are ready to consider a quantum Crooks fluctuation theorem in the classical phase space: By combining Eqs. (26) and (27), leading to we can easily obtain this fluctuation theorem given by Here, the free energy difference ∆F (f) β for the forward process is explicitly given by , and the single-motion work associated with the transformation from the initial position I 0 to the final position I τ is identified as By construction, this form of the thermodynamic work in the quantum regime, linked to ∆F (f) β , was derived from requiring the Crooks theorem in the Wigner representation. Taking Eqs. (11a)-(11b) into consideration, we find that this quantum work, expressed in terms of the action coordinates and formulated without resorting to any projective measurements, is evidently not a quantum-mechanical observable. Here, we also observe that this work and ∆e (I 0 ,Iτ ) in Eq. (29) differ from each other, while both become identical (ω τ I τ − ω 0 I 0 ) in the limit of β, → 0 where the quantum Crooks theorem in Eq. (36) reduces to its classical counterpart in its known form.
Then we can introduce the quantum work distribution for the forward process in the Wigner representation [cf. Eq. (1)], which is valid in the entire quantum regime and may also be viewed as the quantum generalization of the classical work distribution in Eq. (17). Likewise, the work distribution for the backward process turns out to be Employing Eqs. (37) and (38) with (30), we can also evaluate the average work such that Here, the dissipative heat is explicitly given by where d is achieved if the entire process is carried out adiabatically (K τ = 1). With the help of the inequality given by y ≥ ln(y) + 1 (with y = 1/ K τ ), it is then easy to verify that this minimum value is non-negative, and so the second law of thermodynamics is met in Eq. (41); cf. Eq. (53) for the same discussion in other phase-space representations.
, the two first moments become identical. Therefore, it is legitimate to say that this difference should be ascribed to the (non-thermal) quantum fluctuation. In fact, Figs. 1 and 2 show that It is also tempting to examine more rigorously this inequality for its validity. To do so, we restrict ourselves to the periodic external drivings (ω τ = ω 0 ) for arbitrary pairs of (ω t , τ ).
Then, it follows from Eqs. (32) and (41) that the net external work on the system and the net internal energy difference are respectively, where y = β ω 0 /2. Because the factor y coth(y) monotonically increases for y ≥ 0, it is easy to see that Now we are in a position to discuss the quantum second law associated with W P (f) : The inequality (43), together with the Jensen inequality resulting from Eq. (40), yields the quantum-thermodynamic inequality as one of our main findings. This represents a more fine-grained result than the inequality (5) obtained from the TPM framework; the first inequality reduces to the equality if the thermodynamic quasi-static process with Q (f) d = 0 (obtained from both K τ = 1 and K τ = 1) is carried out (differing from the adiabatic process with K τ = 1). This can be implemented if additional heat exchange between system and environment is undergone infinitely slowly over the entire process. On the other hand, the second inequality reduces to the equality in the limit of β, → 0 only; therefore, we can introduce the quantum heat Q (f) q := ∆U(τ ) − W P (f) ≥ 0 (even for a thermally isolated system) which vanishes in the classical limit. This d , is accordingly interpreted as a "builtin" quantity induced by the (non-thermal) quantum fluctuation. In Fig. 3, the behaviors of It is also interesting to discuss the difference between two second moments W 2 P (f) and w 2 P (f) as a next step to the first-moment inequality (43): With the help of Eqs. (33a)-(33c), we can acquire the relative variance which is less than its counterpart ( ∆w) 2 Again, it is the (non-thermal) quantum fluctuation contribution to ( ∆w) 2 P (f) that gives this difference. On the other hand, in the high-temperature regime (β → 0), these two quantities become identical. Fig. 4 demonstrates this variance difference.
Finally, we point out that the above discussion signifies that unlike the internal energy difference, the average quantum work (not viewed as a quantal expectation value) may be contingent upon the representation in consideration; in fact, there has thus far been no broadly agreed-upon "textbook" definition of quantum work [12]. Therefore, the choice of an appropriate representation for the average thermodynamic work in the (entire) quantum regime with its direct classical counterpart on the same footing will also be a significant issue.

B. Why only the Wigner Representation for the work distribution?
In fact, in addition to the Wigner function W ρ (x, p), one has several other quasiprobability distributions such as the Husimi function Q ρ (x, p) ≥ 0, the Glauber-Sudarshan function P ρ (x, p), the Kirwood function K ρ (x, p), and the standard-ordered function F (s) ρ (x, p) [24]. Therefore, it is also tempting to discuss the quantum work (quasi)probability distribution in these additional representations. First, we point out that the two functions K ρ (x, p) and F (s) ρ (x, p) are not always real-valued and therefore will not be under our consideration here. Therefore, we focus on the two real-valued functions Q ρ (x, p) and P ρ (x, p) only, which do not fulfill the marginal-distribution condition, like in Eq. (9) for W ρ (x, p), though. In fact, the Husimi function can simply be understood as the convolution of the Wigner function with a Gaussian filter such that [24] Likewise, the relation between the Husimi and Glauber-Sudarshan functions is As an example, the thermal state for a linear oscillator is given by the Gaussian form with a Υ = 1, and P β (x, p; ω t ) obtained from Eq. (50) but with The propagators for all other phase-space representations have been discussed in [13]; in fact, they can be expressed in terms of the Wigner propagator T W (x, p; τ |x ′ , p ′ ; 0) and the evolution kernels G's such that in which the symbols Υ = W, Q, P, K, F (s) denote the respective phase-space representations; the kernels are explicitly given in [13], e.g., G W→W (x, p) = δ(x) δ(p). This means that these propagators are equivalent to transforming first into the Wigner representation (G Υ→W ) and then propagating with T W , followed by transforming into the original representation (G W→Υ ). Further, it has been shown that free propagation in the Wigner representation is in all other representations such a simple one-to-one correspondence between the initial and final phase-space points is unavailable even for free propagation. This implies that propagations in all other representations will in general possess the non-classical features in more complicated form than the Wigner propagation. Further, unlike T W , both propagators T Q and T P under our consideration have been shown to involve divergent evolution kernels G's in Eq. (51). This divergence for Υ = Q, P will make it obscure to have the symmetry given by T Υ (x ′ , p ′ ; τ |x, p; 0) between the forward and backward processes, which, for Υ = W , led to the result like in Eq. (35) and then the quantal-classical Crooks theorem in its compact form.
Nevertheless, it is still instructive to consider Eq. (35) but now expressed in terms of either Q β 's or P β 's in place of W β 's. Then, it will be straightforward to introduce the "work" in the Husimi representation given by ( = W (I 0 ,Iτ ) in (37)) and likewise the "work" W P,(I 0 ,Iτ ) in the Glauber-Sudarshan representation.
By applying the same techniques as for Eqs. (41)-(42), we can finally arrive at the "average work" (41)) expressed in terms of the "dissipative heat" Then it follows that Q As a result, we see that although the internal energy and its difference can be uniquely determined regardless of the phase-space representations under consideration, it is the Wigner representation only that propagates in the most classical way among them. Therefore, it is legitimate to say that the Wigner representation is the most appropriate choice for the study of the average thermodynamic work in the quantum regime equipped with the canonical transition to its classical counterpart W = τ 0 dtλ ∂ λ H(z t ; λ t ) in the simplest way. In fact, thermodynamics originally appeared from the classical domain. Therefore, defining quantum work in the Wigner representation is a consistent step toward a generalization of the classical work.

C. Comments on our results
Several additional comments are deserved here. First, it is instructive to compare our exact analysis with the semiclassical analysis, carried out in [12], in the context of the quantum-classical correspondence principle. In their analysis, the classical work distribution [cf. Eq. (17)], built from the classical trajectories that connect the initial and final energies, has shown an excellent approximation to its TPM counterpart in the semiclassical regime. On the other hand, our work distribution in the quantum regime, built from the Wigner trajectories that connect the initial and final action values, renders the average quantum work distinguished from the internal energy difference; however, in the classical limit, as discussed above, these trajectories exactly reduce to the classical ones and this distinguishment between the average work and internal energy difference goes away. Consequently, we may say that our quantal-classical analysis consistently accommodates this semiclassical analysis of the TPM framework.
Second, we add remarks upon the adiabatic process (K τ = 1): If an external driving acts infinitely slowly, there are no transitions between different eigenstates (cf. [16,34] for the quantum adiabatic theorem) such that the conditional probability P [m(τ )|n(0)] = δ nm , and thus no additional quantum fluctuation is produced over the driving. Therefore, while in the non-adiabatic process the quantum work w nm of the TPM framework has no independent physical reality until completion of the final measurement [12], in the adiabatic process this work has the physical reality over the driving indeed; therefore, in this case, the stochastic nature of the quantum work becomes associated solely with the random nature of the initial state, like in the classical setup. Then, the quantum heat Q (f) q in the Wigner representation also appears solely from the quantum fluctuation in the initial state.
Third, it is also instructive to emphasize that we resorted to the action variable (I) in our discussion, not to the EBK quantization rule given by I n = (n + α/4) [15,16]: For a linear oscillator with the Maslov index α = 2, this semiclassical quantization is exactly valid over the entire quantum regime such that the energy eigenvalue E n = ω I n . On the other hand, our approach simply made use of the continuous nature of the action variable, which underlies the continuous nature of the thermodynamic work W (I 0 ,Iτ ) in the quantum regime.
Finally, we remind that the work distribution in the Wigner representation is positive valued for a linear oscillator because of the Gaussian nature of the initial thermal state and its time evolution [cf. Eqs. (26)- (27)]. However, this is not the case any longer for generic systems such as a single particle confined by a one-dimensional infinite potential.
Further, in our framework of quantum thermodynamics formulated without resorting to any projective measurements, the measurability of single-motion values ∆e (I 0 ,Iτ ) and W (I 0 ,Iτ ) is inherently abandoned even for the process starting from the thermal state because of the quasi-probabilistic nature of the Wigner function; instead, our concern lies in the average values ∆U(τ ) and W P only that reveal the more fine-grained form of the quantum second law in (46) with the canonical transition to its classical counterpart. In fact, the TPM framework is viewed as the special case only that the measurability of single-run values ∆e nm is available. However, it is such a measurement-free nature of our framework that enables us to be free from a determination of the energy eigenvalues (required for the TPM framework) and straightforwardly generalize our findings to the processes starting from the non-thermal initial states with coherence in the energy basis. This subject will explicitly be covered by the following section.

TIONS
Now we generalize the Crooks fluctuation theorem in Eq. (36) by considering the partially thermal states (as a particular class of non-equilibrium initial states) such thatρ 0 = (1 − γ)ρ β (ω 0 )+γσ(ω 0 ) for the forward process andρ τ = (1−γ)ρ β (ω τ )+γσ(ω τ ) for the backward process. Here, the symbolsσ and γ (with 0 ≤ γ < 1/2) denote a non-thermal state and an imperfection in preparing the thermal stateρ β , respectively. To do so, we first modify Eq. (18b), simply by replacing W β with W ρ 0 , into being unity. Then it is straightforward to rewrite this as A , in which the joint distribution for the forward process is given by [cf. Eq. (26)] Here, the second term on the right-hand side will be determined explicitly for a given statê σ. Likewise, we have for the backward process [cf. Eq. (27)] These two distributions can, in general, be negative valued and will be used for the gener- 1 in notation); here, the generalized work is identified as for a given value of γ with γ/(1 − γ) < 1 [cf. Eq. (37)]. As such, the second term is not linear in I 0 and I τ any longer. We also find that the initial state W ρ 0,1 (x, p; ω 0 ) and so the distribution P (f) 1 (I 0 , I τ ) can be negative valued indeed if n = 0 and γ th,1 < γ < 1/2, where the threshold value γ th,1 should be determined for the respective initial state (cf. Fig. 6).
for n ≤ m, where L (k) n (· · · ) denotes the associated Laguerre polynomial, and W |n m| (x, p) = {W |m n| (x, p)} * ; e.g., We see that the angle coordinate φ will also be needed for the Moyal functions W |m n| (φ, I).
Then, we can find from Eq. (56) that the joint distribution for the forward process is where the second conditional distribution is given by In fact, we can show that C (f) (I τ |I 0 ) ≡ 0 (cf. Appendix B). Consequently, we see that the off-diagonal terms of the initial stateρ 0,2 do not contribute to P (f) 2 (I 0 , I τ ), meaning that the diagonal formσ 2 = (|n n| + |n + 1 n + 1|)/2, in place ofσ 2 , will give the same result for P (f) 2 (I 0 , I τ ). Similarly, we can acquire the joint distribution for the backward process Combining Eqs. (68) and (70), it is straightforward to acquire the Crooks fluctuation theorem in the form of Eq. (59) and the Jarzynski equality in the form of (61) as well as the second-law inequality in the form of (63), where the pertinent work W 2,(I 0 ,Iτ ) is accordingly given by Eq. (60) but with W n (I t ) → {W n (I t ) + W n+1 (I t )}/2, and its distribution is then Also, the internal energy difference ∆U 1 (τ ) → ∆U 2 (τ ) such that C. Mixture of thermal and energy-superposed states: Case 2 Letσ 3 = (|n + |n + 1 + |n + 2 )( n| + n + 1| + n + 2|)/3. Then, we have We can obtain from Eq. (56) the joint distribution for the forward process where G (f) Here, the diagonal-term contribution and the third conditional distribution are respectively (cf. Appendix B). We see that the joint distribution P 3 (without coherence), obtained from the diagonal formσ 3 = (|n n| + |n + 1 n + 1| + |n + 2 n + 2|)/3 in place ofσ 3 . Likewise, the joint distribution for the backward process is given by in which G , as well as its distribution is Also, the internal energy difference ∆U 1 (τ ) → ∆U 3 (τ ). In Fig. 8 and ∆U 3 (τ ) without coherence, respectively, where for n = 0 [cf. (A14a)-(A14b)], and (80b)

D. Comments on our results
Now we give the interpretation of our findings in the present section. Differing from the thermal initial state (with γ = 0), the partially thermal initial state (with γ > 0) results in the fact that the internal energy difference is still given by the first moments, but the average work, obtained from the generalized work in non-linear form, necessarily contains the higher-order moment contributions. Remarkably enough, such a generalized work can also be linked to the fully thermodynamic quantity ∆F β operationally through our generalized Crooks theorem (and the resulting Jarzynski equality), finally giving rise to the second-law inequality associated with the average work. If the parameter γ continues to increase such that it becomes greater than its threshold value γ th (like in Fig. 6), then a dominance of the quantum fluctuation over the thermal fluctuation will be found for the initial state in non-Gaussian form (e.g.,σ 1 = |n n| with n = 0) so that the initial Wigner function and the work distribution can be negative valued. In the classical scenario, on the other hand, this exact link between the generalized work and the free energy difference is well-defined (i.e., with no negativity of the work probability distribution) also for the single-motion values through our generalized Crooks theorem in the classical limit. In fact, a perfect preparation of the thermal state (with r = 0) could be a formidable task in reality.
It is also instructive to point out that our result obtained from the phase-space framework, free from the projective measurement, is consistent with the result obtained from the histories framework (as a different generalization of the TPM framework) in [21], which employed the time-reversal symmetrized work distributions for non-thermal initial states, concluding that thermodynamic work in the quantum regime cannot be determined by the projective measurements.

V. CONCLUSION
We studied the quantum fluctuation relations in the Wigner representation. To make our analysis as exact as possible, we restricted our discussion here to a driven quantum linear oscillator. Then we obtained the single-motion work (in closed form) in the quantum regime and its distribution expressed in terms of the action coordinates only, without resorting to any projective measurements, such that the quantum and classical setups can be analyzed on the single footing. This enabled us to derive the quantum Crooks fluctuation theorem, the quantum Jarzynski equality and the second-law inequality in more fine-grained form than their counterparts in the two-point projective measurement (TPM) framework, in that the resulting average work W in our framework notably differs from the internal energy difference ∆U(τ ) between the initial and final states, thus rendering the quantum heat Q q = ∆U(τ ) − W ≥ 0 introduced. Such a discrepancy between ∆U(τ ) and W was shown to disappear gradually in the semiclassical regime. This result contrasts with W ≡ ∆U(τ ) in the standard TPM framework. We also provided a justification for the choice of the Wigner representation for our analysis rather than any other phase-space representations.
We showed that it is the Wigner representation that behaves in the most classical-like way and is most appropriate for the quantum work with the canonical transition to its classical counterpart, in that thermodynamics originally took its shape from the classical scenario.
Our findings were straightforwardly generalized to the processes starting from a particular class of non-thermal initial states including the states with quantum coherence in the eigen-energy basis. Here, we introduced the generalized quantum work with its non-linear nature and the work distribution with its negativity. As a matter of fact, the unavoidable negativity of the work distribution has been well-known also in the extended TPM framework where the initial state is non-diagonal in the energy basis. In this paper, on the other hand, we used such a negativity resulting from the quasi-probabilistic nature of the Wigner function as our stating point for a new framework free from the projective measurements, which led to achieving the aforementioned fine-grained results in the quantum thermodynamics. As a result, it is legitimate to claim that our results, covering the genuine quantum to (semi)classical regimes, can provide a more sophisticated discussion of the second law of thermodynamics associated with the average work within an isolated quantum system. Finally, as long as the angle-action coordinates are well-defined, our methodology will continue to apply to different quantum systems including the generic one-dimensional systems.