The role of quantum coherence in non-equilibrium entropy production
Jader P. Santos Lucas C. Céleri Gabriel T. Landi Mauro Paternostro
quant-ph





Instituto de Física da Universidade de São Paulo,  05314-970 São Paulo, Brazil

Instituto de Física, Universidade Federal de Goiás, Caixa Postal 131, 74001-970, Goiânia, Brazil

gtlandi@if.usp.br
Instituto de Física da Universidade de São Paulo,  05314-970 São Paulo, Brazil

Centre for Theoretical Atomic, Molecular and Optical Physics,
School of Mathematics and Physics, Queen's University Belfast, Belfast BT7 1NN, United Kingdom




Thermodynamic irreversibility is well characterized by the entropy production arising from  non-equilibrium quantum processes. We show that the entropy production of a quantum system undergoing open-system dynamics can be formally split into a  term that only depends on population unbalances, and one that is underpinned by quantum coherences. 
This allows us to identify a genuine quantum contribution to the entropy production in non-equilibrium quantum processes. 
We discuss how these features emerge both in Lindblad-Davies  differential maps and finite maps subject to the constraints of thermal operations. 
We also show how this separation naturally leads to two independent entropic conservation laws for the global system-environment dynamics, one referring to the redistribution of populations between system and environment  and the other describing how the coherence initially present in the system is distributed into local coherences in the environment and non-local coherences in the system-environment state.
Finally, we discuss how the processing of quantum coherences and the incompatibility of non-commuting measurements  leads to fundamental limitations in the description of quantum trajectories and fluctuation theorems. 

The role of quantum coherence in non-equilibrium entropy production
    Mauro Paternostro
    December 30, 2023
===================================================================












§ INTRODUCTION


Irreversible processes undergone by an open quantum system are associated with a production of entropy  that is fundamentally different from the possible flow of entropy resulting from the mutual coupling of the system with its  environment. 

Such inevitable contribution to the entropy change of the state of a system is termed entropy production. If entropy is labelled as S, we describe its rate of change as 

    S/ t = Π - Φ,

where Φ is the entropy flux rate from the system to the environment and Π is the entropy production rate. 
According to the second law of thermodynamics, we should have Π≥ 0, with Π = 0 if and only if the system is at equilibrium. 
The characterization of the degree of irreversibility of a process, and thus the understanding of entropy production is both fundamentally relevant and technologically desirable. On one hand, such understanding would provide the much needed foundations to the emergence of time-symmetry breaking entailed by irreversibility and epitomized, for instance, by seminal results such as Onsager's theory of irreversible currents <cit.>. On the other hand, a characterization of irreversible entropy could help us designing thermodynamically efficient quantum technologies <cit.>. 

In general, the open dynamics of a quantum system gives rise to two processes. 
The first  corresponds to the transitions between energy levels of the system, which will cause the populations to adjust to the values imposed by the bath. 
Measures of the entropy production for this kind of processes have been known for many years, particularly in the context of Fokker-Planck equations <cit.> and Pauli master equations <cit.>.

The second process undergone by an open quantum system is the loss of coherence in the energy eigenbasis. 
Coherence is an essential resource for quantum processes <cit.>, likely representing the ultimate feature setting quantum and classical worlds apart. 
Only very recently have steps towards the formulation of a unified resource theory of coherence been made <cit.>. 
While the role of quantum coherences in thermodynamics is yet to be fully understood (both qualitatively and quantitatively), it is known that coherence plays a role in the allowed transitions generated by thermal operations 
<cit.>.
Moreover, it also affects the performance of non-adiabatic work protocols <cit.>. 



Understanding the interplay between population dynamics and loss of coherence represents a pressing problem in the field of thermodynamics of quantum systems. 
In particular, one is naturally led to wonder how entropy production is underpinned by the dynamics of quantum coherences in irreversible open dynamics. 

Shedding light on such an intimate relation is the main motivation of this paper, 

where we put forward a formal description of entropy production in terms of two contributions,  one clearly related to the dynamics of populations and the other depending explicitly on the coherence within the state of the system undergoing the open process. 

We first discuss how these features emerge in the case of Davies-type master equations, for which the formulation is transparent. 
We then address more general dynamical maps satisfying the constraints of thermal operations <cit.>. 
In this case, we show how our result can be connected to recent resource theoretical developments <cit.>. 

Afterwards, we address the main physical  implications of these two contributions to the entropy production. 
First, we discuss how it enables one to construct two independent entropic conservation laws for the global system-environment dynamics. 
The first is entirely classical and relates to the redistribution of populations between system and enviroment. 
The second, on the other hand, dictates  how the coherence initially prepared in the system is distributed among local coherences in the environment and  non-local coherences in the global system-environment state. 
Lastly, we address the issue of how to access entropy production by means of quantum measurements and  quantum trajectories. 
We show that even in the presence of coherence, it is possible to construct measurement protocols which satisfy fluctuation theorems. 
However, unlike in the case of closed systems <cit.>, it is not possible to consider a single measurement protocol in which the fluctuation theorems are satisfied for both contributions of the entropy production individually. 


Several advances in  the past decade have consistently shown that it is possible to engineer  systems in which thermodynamics  coexists with quantum effects. 
It is our hope that the framework put forth in this  paper may contribute for the development of a unified theory describing both thermal and quantum resources. 











§ RESULTS




 §.§  Entropy production due to the processing of coherence


Coherence is a basis dependent concept and, in principle, no preferred basis exists. 
Here we adopt the viewpoint according to which a preferred basis only emerges when it is imposed by the environment <cit.>, a perspective that is typically referred to as einselection.
There are several scenarios in which a preferred basis may emerge. We shall focus on two of them.
The first are Davies maps <cit.>, which make use of the weak-coupling approximation, and the second are the so-called thermal operations <cit.>. 
Both scenarios lead to energy conservation so that the energy eigenbases  are naturally selected as preferred bases. 
However, while in the weak-coupling scenario, energy conservation is taken as an approximation midway through the calculations, in the case of thermal operations this is imposed from the start, as an assumption of the model. 
Consequently, thermal operations are much more versatile, while simultaneously offering a much clearer physical interpretation.
We also assume, for simplicity, that the energy eigenvalues of the system are non-degenerate, since this would cause the preferred basis to depend on additional details of the system-environment interaction, which one seldom has access to. 




  §.§.§ Lindblad-Davies maps







We begin by analyzing the so-called Lindblad-Davies maps, which generally describe the weak contact of a system with a thermal environment. They have the form  <cit.>

    ρ_S/ t = - i [H_S,ρ_S] + D(ρ_S),

where ρ_S is the density matrix of the system, H_S is the Hamiltonian  and D(ρ_S) is a Lindblad dissipator having the Gibbs state ρ^eq_S=e^-β H_S/Z_S as a fixed point; i.e. D(ρ_S^eq) = 0  (here Z_S =  e^-β H_S is the partition function and β the inverse temperature of the bath). 

Davies maps are known to lead  to a separation between the evolution of populations, p_n = ⟨ n | ρ_S | n ⟩, of the density matrix 
 and the  off-diagonal coherences p_nm = ⟨ n | ρ_S | m ⟩. Here, we have called {|n⟩} the elements of the basis imposed by the specific environment being considered. In the specific case of a Davies map, such basis is that of the energy eigenstates, whose set of eigenvalues we call {E_n}.

Following Eq. (<ref>), the populations evolve following the Pauli master equation

    p_n/ t = ∑_m[ W(n|m) p_m - W(m|n) p_n],

where W(n|m) are the transition rates from the energy level E_n to level E_m. They satisfy the detailed-balance condition 

    W(n|m)/W(m|n) = p_n^eq/p_m^eq = e^-β (E_n - E_m),

where p_n^eq=⟨ n | ρ^eq_S | n ⟩. As for the coherences, provided that the Bohr frequencies ω_nm=E_n-E_m are non-degenerate, they evolve independently of each other according to the equation

    p_nm/ t = - {i ω_nm -  1/2∑_k [ W(k|n) + W(k|m) ]} p_nm.

As the second term in the right-hand side can be seen as an effective (generally temperature-dependent) damping term, the evolution of the coherences can be interpreted as that resulting from damped oscillations.



We now define  the non-equilibrium free energy as 

    F(ρ_S) =(H_S ρ_S) +T (ρ_S lnρ_S),
 
where T is the temperature of the bath. While Eq. (<ref>) reduces to the usual expression F_eq = - T ln Z_S at equilibrium, for general non-equilibrium states, we can write

    F(ρ_S) = F_eq + T S(ρ_S||ρ_S^eq),

where  S(ρ||σ) = (ρlnρ - ρlnσ) is the quantum relative entropy. As S(ρ_S||ρ_S^eq) ≥ 0, we have that F(ρ_S) ≥ F_eq. 
This condition thus defines the equilibrium state of a system as the state that minimizes the free energy <cit.>. 
Moreover, it establishes that, during relaxation, the free energy is a monotonically decreasing function of time whose value is determined by the distance, in state space, between the instantaneous state of the system ρ_S and its equilibrium version ρ_S^eq.
Hence, one is naturally led to define the entropy production rate as 
<cit.>

    Π = - 1/T F(ρ_S)/ t,

which ensures that Π≥ 0 and  Π = 0 iff ρ_S = ρ_S^eq. 

Next we notice that  in terms of the eigenbasis of H_S, we can always split S(ρ|| ρ_eq) as 

    S(ρ_S|| ρ_S^eq) =  S(p_S||p_S^eq) +  C(ρ_S).

Here, S(p_S||p_S^eq) = ∑_n p_n ln p_n/p_n^eq is the Kullback-Leibler divergence of the classical probability distribution entailed by the populations p_S={p_n} from that of the equilibrium state p_S^eq={p_n^eq}. Moreover, we have introduced the relative entropy of coherence <cit.>

    C(ρ_S) = S(Δ_H_S(ρ_S))  - S(ρ_S),

where Δ_H_S(ρ) is the dephasing map, acting on the density matrix ρ_S, which removes all coherences from the various energy eigenspaces of H_S. 
With this at hand, Eq. (<ref>) becomes

    F(ρ_S) = F_eq + T  S(p_S||p_S^eq) + T  C(ρ_S).

This is one of the central results of this work: It shows that quantum coherence is actually a part of the non-equilibrium free energy, and thus contributes significantly to the determination of the non-equilibirum thermodynamics stemming from Eq. (<ref>). The second term in F(ρ_S) quantifies the increase in free energy due to population imbalance with respect to the equilibrium configuration and, as such, is a purely classical term. The last term, which is of a genuine quantum nature, determines the surplus in free energy that a non-equilibrium state with quantum coherences offers with respect to its diagonal (and thus classical) counterpart. 


Let us now use the formal splitting in Eq. (<ref>) to recast the entropy production rate in Eq. (<ref>) as

    Π = Π_d + Υ.

The first term is written as 

    Π_d    = - / t S(p_S|| p_S^eq)
       =
    1/2∑_n,m[ W(n|m) p_m - W(m|n) p_n ] lnW(n|m) p_m/W(m|n) p_n.

and is then exactly the classical result derived in <cit.>.
The second contribution reads 

    Υ = -  C(ρ_S)/ t,

which thus shows that the rate of loss of coherence that might ensue from the dynamics of the system enters quantitatively into the entropy production: the rate at which entropy is produced in a process where quantum coherences are destroyed as a result of the coupling with an environment surpasses the corresponding classical version. 
Clearly both Π_d and Υ are non-negative and null only for ρ_S = ρ_S^eq. 


Finally, let us address the entropy flux defined in  Eq. (<ref>). Using Eq. (<ref>) we find 





    Φ = Φ_E/T = - 1/T∑_n E_n^S  p_n/ t,

where Φ_E is the energy flux from the system to the environment. In deriving this equation we have used p_n^eq = exp[- β E_n^S]/Z_S.
The entropy flux has thus no contribution arising from quantum coherences: entropy (and energy) will only flow due to imbalances in the populations. 
Any loss of coherence contributes only to the entropy production rate and has no associated flux.
It is important to emphasize that this is a feature of the present type of master equation. 
In other situations, such as strong coupling dynamics, the coherences in the system may play a relevant role in the entropy and heat fluxes.

















  §.§.§  Thermal Operations


We now address the case of more general maps with the aim of gaining access to the environmental degrees of freedom, hence enhancing our understanding of the two contributions to the entropy production from the perspective of the joint system-environment properties.

We consider explicitly thermal operations, which have been the subject of numerous recent investigations in the context of resource theories <cit.>. 
A thermal operation is physically described as the interaction of the system with an arbitrary environment, initially prepared in equilibrium ρ_E^eq = e^-β H_E/Z_E, through a unitary U which conserves the total energy, that is an operation such that [U, H_S+H_E]=0. 
In this sense, the thermal operation hypothesis reminds us of the framework set by the weak-coupling approximation. 
However, it allows us to go significantly beyond the limitations of  weak-coupling, and thus extend our analysis to a larger set of physically meaningful cases <cit.>.
We also call attention to the fact that in thermal operations the energy conservation condition is only imposed on the global unitary U, irrespective  of how this is generated. 
One way to do so is by means of a time-dependent interaction that is turned on and off. 
Another, much simpler approach, is to simply have any potential V which satisfies [V, H_S+H_E] = 0. This will then generate a time-independent unitary U = e^- (H_S+H_E+V)t which will be energy conserving.
This type of thermal operation was recently used in Ref. <cit.> to study the heat exchange between two qubits in a magnetic resonance setup.


The state of the composite system after the evolution in a thermal operation will be 

    ρ_SE' = U (ρ_S ⊗ρ_E^eq) U^†.

We label the environmental energies and eigenstates as {E_μ^E} and {|μ⟩}, respectively. We also call
q_μ = e^-β E_μ^E/Z_E  its initial thermal populations. 
Energy conservation then implies that 

    ⟨ m,ν| U |n,μ⟩∝δ(E_n^S + E_μ^E - E_m^S - E_ν^E).

Tracing out the environment one obtains the Kraus map  for the system

    ρ_S' = _E[ U(ρ_S ⊗ρ_E^eq) U^†] = ∑_μ,ν M_μ,νρ_S M_μ,ν^†

where M_μ,ν = √(q_μ)⟨ν | U | μ⟩. 
Clearly, the Gibbs state ρ_S^eq is a fixed point of this equation. 
Moreover, the Davies maps studied in the previous Section correspond to particular Markovian limits of Eq. (<ref>). 






The energy conservation condition implies that, when the eigenvalues of H_S are non-degenerate, Eq. (<ref>) is an incoherent operation in the sense of Ref. <cit.>.
That is, defining the  energy eigenstates as the set of incoherent states,  this process always maps incoherent states into incoherent states.
This in turn allows for an independent processing of both populations and coherences. 
The diagonal entries will, in particular, evolve according to the classical Markov chain 

    p_m' = ∑_n Q(m|n) p_n,

where Q(m|n) = ∑_μ,ν |⟨ m |M_μ,ν| n ⟩ |^2 is the transition probability from state n to state m, a quantity playing the role of 
the transition rate W(m|n) in Eq. (<ref>). The processing of the coherences, on the other hand, takes place independently of the changes in populations. 
In particular, if the Bohr frequencies ω_mn are non-degenerate, this processing takes the simple form 

    p_n,m' = α_n,m p_n,m,
        α_n,m = ∑_μ,ν⟨ n | M_μ,ν | n ⟩⟨ m |M_μ,ν^† |m⟩.
 
As shown in Ref. <cit.>, the processing of coherence is not independent of the population changes, but must satisfy 
the inequality

    |α_n,m|^2 ≤ Q(n|n) Q(m|m),

which provides a bound to the maximum amount of coherence that may be lost in a thermal operation.

We now turn to the analysis of the entropy production in this scenario. 
Unlike the previous Section, as the dynamics in Eq. (<ref>) is in general non-Markovian and we only have access to the global map, it is not possible to address the rate of entropy production Π, but only the total entropy Σ produced in the process. 
In this case, using the contractive property of the relative entropy <cit.>, we have 
S(ρ_S'|| ρ_S^eq) ≤ S(ρ_S|| ρ_S^eq).
Consequently, the free energy  Eq. (<ref>) remains a non-increasing function,
thus justifying the following definition of total entropy production

    Σ = - Δ F/T  =  S(ρ_S|| ρ_S^eq) - S(ρ_S'|| ρ_S^eq) ≥ 0.

This expression may be taken as a basic postulate in our framework, motivated by physical consistency arguments that can be even reinforced by a quantum trajectory point of view, as discussed e.g. in Refs. <cit.>. Other approaches have also been used elsewhere <cit.>.

As Σ is also based on the quantum relative entropy, a splitting akin to Eq. (<ref>) into non-negative population-related and a coherence-related terms is in order, and we have 

    Σ = Σ_d + Ξ,

where 
rCl

Σ_d    =    S(p_S||p_S^eq)- S(p_S'||p_S^eq),


Ξ   =     C(ρ_S) - C(ρ_S').


The non-negativity of Σ_d follows immediately from the fact that in thermal operations diagonal elements evolve independently of coherences. 

The positivity of  Ξ, on the other hand, follows from the fact that a thermal operation is incoherent <cit.>. 
In the limit where the Davies maps are recovered,  Σ_d and Ξ become respectively the integrated versions of Π_d and Υ in Eq. (<ref>).
This demonstrates the generality, under suitable and reasonable assumptions on the form of the system-environment coupling, of the central result of our investigation. 







 §.§ Implications of the central results






We now explore which considerations  can be drawn in light of the formal splitting of the entropy production demonstrated above.






  §.§.§ Entropic conservation laws





The structure of thermal operations  and Eq. (<ref>) imply a series of conservation rules for the processing of populations and coherences. 
First, energy conservation implies that the total entropy production in Eq. (<ref>) can be written as <cit.>

    Σ = S(ρ_E'||ρ_E^eq) + ℐ(ρ_SE'),

where ρ_E'=_S(ρ'_SE)  and ℐ(ρ_AB) = S(ρ_A) + S(ρ_B) - S(ρ_AB) is the mutual information of a bipartite system. 
This gives an interesting interpretation of Σ as being related to the change in the environmental state, measured by the first term, and the total degree of correlations created by the thermal operation, measured by the mutual information.
As discussed in Ref. <cit.>, Eq. (<ref>) also provides a clear interpretation of how irreversibility emerges from a global unitary dynamics, ascribing it to two reasons. One is the creation of correlations between system and environment, which are never recovered once one traces out the environment (hence  giving rise to an irretrievable loss of information). 
The second is related to the fact that the system pushes the environment away from equilibrium. 
As shown in Ref. <cit.>, if ρ_E' - ρ_E^eq∼ϵ then S(ρ_E'|| ρ_E^eq) ∼ϵ^2, whereas S(ρ_E') - S(ρ_E^eq) ∼ϵ. 
Thus, when the reservoir is large, the first term becomes negligible and the main contribution to the entropy production will come from the total correlations created between system and environment. 


Next we note that as the map (<ref>) is unitary, it follows that S(ρ_SE') = S(ρ_SE). 
However, as [U,H_S+H_E]=0, 
the dephasing operation Δ_H_S+H_E commutes with the unitary evolution so that, in addition to the total entropy being conserved, the same is also true for the dephased entropies 
rCl

S(Δ_H_S+H_E(ρ_SE')) 

   =    S(Δ_H_S(ρ_S)) + S(ρ_E^eq).

This result reflects how the changes in population in the system and environment affect the  information content of the diagonal elements of ρ_SE'. 
Note that the left-hand side contemplates, at most, coherences in the degenerate subspaces of H_S+H_E, which are not resources from the perspective of this operation. 

From Eqs. (<ref>) and  (<ref>), it follows that a similar law must also hold for the relative entropy of coherence

    C(ρ_SE') =  C(ρ_S).

Where C(ρ_SE')=S(Δ_H_S+H_E(ρ_SE'))-S(ρ_SE').
Thus, we see that the reduction in the coherence of the system after the map is due to a redistribution of this coherence over the global system-environment state. 

Substituting Eq. (<ref>) into Eq. (<ref>) shows that the contribution of the entropy production due to quantum coherences may be written as 

    Ξ =  C(ρ_SE') -  C(ρ_S').


Thus, the entropy production due to quantum coherences can be seen as the mismatch between the global coherences in the correlated system-environment state and the local coherences in the final state.
We can also relate Ξ to the notion of correlated coherence, introduced recently in <cit.> and defined as

    C_cc(ρ_SE') =  C(ρ_SE') -  C(ρ_S') -  C(ρ_E') ≥ 0,

where C(ρ_E')=S(Δ_H_E(ρ_E'))-S(ρ_E').
This quantity therefore represents the portion of coherence that is contained within the correlations between system and environment. 
Combining the Eqs. (<ref>) and (<ref>), it is then possible to write

    Ξ =  C(ρ'_E) +   C_cc(ρ'_SE).

This has the same form as Eq. (<ref>), with the first term representing the local coherences developed in the environment and the second term representing the non-local contribution. 
Thus, we may conclude from this result that entropy production has a clearly local contribution, related to the creation of coherences in the environment, and a non-local contribution related to the creation of shared coherences in the system-environment state. 







  §.§.§ Stochastic trajectories and fluctuation theorems





Lastly, let us consider the stochastic version of the entropy production arising from quantum trajectories. 
In order to correctly treat the coherences present in the system, we adopt the following procedure.
In the forward protocol, the environment is prepared in the thermal state ρ_E = ∑_μ q_μ |μ⟩⟨μ|, where 
q_μ = e^-β E_μ^E/Z_E.
The system, on the other hand, is taken to be in an arbitrary state ρ_S = ∑_α p_α |ψ_α⟩⟨ψ_α|, which in general contain coherences, so that the basis |ψ_α⟩ is  incompatible with the energy basis |n⟩. 
As the first step in the protocol, we then perform local measurements in the bases |ψ_α⟩ and |μ⟩ of S and E, obtaining the state |ψ_α,μ⟩  with probability p_α q_μ. 
Next we evolve both with a joint unitary U.
Finally, in the third step we measure only the environment, again in the energy basis |ν⟩.
Due to the measurement backaction, the state of the system then collapses to the pure state

    |Φ_F|αμν⟩ = ⟨ν | U | ψ_α, μ⟩/√(P_F(ν|αμ)),

where P_F(ν|αμ) = ||⟨ν | U | ψ_α, μ⟩||^2. 
For a discussion on the effects of choosing different bases for the second measurement in the environment, see Ref. <cit.>.

The final state  ρ_S' of the system will then be given by an ensemble average over all possible final states (<ref>), weighted by the probability of the stochastic trajectory (α, μ, ν); viz., 

    ρ_S' = ∑_α, μ, ν P_F(ν|α,μ) p_α q_μ |Φ_F|αμν⟩⟨Φ_F|αμν|.

One may directly verify that this state is indeed equal to the unmeasured final state ρ_S' in Eq. (<ref>).
The states (<ref>), however, are not the eigenstates of ρ_S' and in fact  don't even form a basis. The diagonal structure of ρ_S' will thus be of the form ρ_S' = ∑_β p_β' |ψ_β'⟩⟨ψ_β'|, where |ψ_β'⟩ is a new basis set that is not trivially related to neither |ψ_α⟩ nor |Φ_F|αμν⟩. 
This is the key difference that appears due to the presence of coherences (if the initial state of the system were diagonal, the same would be true for the final state, since this is a thermal operation). 
The probabilities   p_β' will be given by 

    p_β' = ∑_α, μ, ν p_β|α, μ, ν  P_F(ν|α, μ)   p_α  q_μ  ,

where p_β|α, μ, ν = |⟨ψ_β' | Φ_F|αμν⟩|^2 is the conditional probability of finding the system in |ψ_β'⟩ given that it ended the forward protocol in |Φ_F|αμν⟩.


The stochastic trajectory generated by the measurement outcomes is specified by the three quantum numbers (α, μ, ν). 
However, following Ref. <cit.>, we may augment the trajectory (an idea first introduced by Dirac <cit.>) by introducing β as an additional quantum number, so that the trajectory 𝒳 = (α, μ, β, ν) is defined by the probability 

    𝒫_F[𝒳] = p_β|α, μ, ν  P_F(ν|α, μ)   p_α  q_μ .

Indeed, using the definition (<ref>),   one readily finds that

    p_β|α, μ, ν  P_F(ν|α, μ)  = |⟨ψ_β', ν | U | ψ_α, μ⟩ |^2 = P(β,ν | α, μ),

which is nothing but the transition probability of observing a transition from |ψ_α, μ⟩ to |ψ_β', ν⟩. 




Next we define the backward protocol. 
The initial state of the system is drawn from one of the possible  eigenstates |ψ_β'⟩ of ρ_S' <cit.>, whereas the environment is taken to be in equilibrium and is again measured in the energy basis |ν⟩.
This yields the state |ψ_β', ν⟩ with probability p_β' q_ν. 
We then apply the time-reversed unitary U^† and, in the end, measure E in the basis |μ⟩. 
As a consequence the system collapses to 

    |Φ_B|βνμ⟩ = ⟨μ | U^† | ψ_β', ν⟩/√(P_B(μ|β,ν)),

where P_B(μ|β,ν) = ||⟨μ | U^† | ψ_β', ν⟩||^2.



The backward trajectory is specified by the quantum numbers (β, ν, μ). 
However, as in the forward case, we can define the augmented trajectory 𝒳 = (α, μ, β, ν) by introducing the conditional probability p_α|β, ν,μ = |⟨ψ_α | Φ_B|βνμ⟩|^2. 
The probability for the augmented backward trajectory will then be given by 

    𝒫_B[𝒳] = p_α|β, ν,μ  P_B(μ|β,ν)   p_β'   q_ν .

With the path probabilities (<ref>) and (<ref>), we can now define the entropy production in the usual way, as 

    σ[𝒳] = ln𝒫_F[𝒳] /𝒫_B[𝒳] .

By construction, σ satisfies a detailed fluctuation theorem <cit.>

    ⟨ e^-σ[𝒳] ⟩ = 1.

Moreover, similarly to Eq. (<ref>), it follows that p_α|β, ν,μ  P_B(μ|β,ν) = P(β,ν|α, μ), so that 

    σ[𝒳] = lnp_β|α, μ, ν  P_F(ν|α, μ)   p_α  q_μ/p_α|β, ν,μ  P_B(μ|β,ν)   p_β'   q_ν = lnp_α q_μ/p_β' q_ν.

Thus, we see that the conditional terms cancel out. 
Physically, this means that there is no additional entropic cost in introducing the augmented trajectories, which is a consequence of the fact that the augmentation was done using the eigenstates |ψ_β'⟩ of ρ_S'. 
One may also directly verify that  ⟨σ[𝒳]  ⟩ = Σ is the average entropy production in Eq. (<ref>).































Next we address the question of how to define stochastic quantities for the two contributions to the entropy production in  Eq. (<ref>).
That is, we wish to separate 

    σ[𝒳] = σ_d[𝒳] + ξ[𝒳],

such that ⟨σ_d[𝒳]  ⟩ = Σ_d and ⟨ξ[𝒳]  ⟩ = Ξ.
This can be accomplished by augmenting the trajectory once again to include the populations of the system in the energy basis. That is, we define 𝒳̃ = {α, μ, n, β, ν, m}, 
with associated path probabilities
𝒫_F(B)[𝒳̃] = 𝒫_F(B)[𝒳] p_n|α p'_m|β, where 
we defined the  conditional probabilities p_n|α = |⟨ n |ψ_α⟩|^2 and p_m|β = |⟨ m | ψ_β'⟩|^2.





We then define the stochastic quantities
rCl

σ_d[𝒳̃]    =     ln( p_n q_μ/p_m' q_ν),


ξ[𝒳̃]    =     ln( p_αp_m'/p_β' p_n),

where p_n = ⟨ n | ρ_S | n ⟩ and p_m' = ⟨ m |ρ_S' |m ⟩ are the populations in the energy eigenbasis at the initial and final states [cf. Eq. (<ref>)]. 
Summing  these two contributions immediately yields Eq. (<ref>).
Moreover, one may also verify that ⟨σ_d[𝒳̃]  ⟩ = Σ_d and ⟨ξ[𝒳̃]  ⟩ = Ξ.
Hence, these quantities do indeed represent the stochastic counterparts of the two contributions to the entropy production. 


A second glance at Eq. (<ref>) reveals that, on the stochastic level, the entropy production ξ due to the loss of coherence,  is nothing but the change in  information between  incompatible bases, a quantity sometimes referred to as information gain <cit.>.
Hence, we arrive at the important conclusion that the incompatibility between classical and quantum entropy production can be traced back, at the stochastic level, to the basis incompatibility of the quantum rules.
Indeed, if we  rewrite the  fluctuation theorem (<ref>) as 
⟨   e^-σ_d[𝒳̃]  - ξ[𝒳̃]̃⟩ = 1,
we can clearly see that due to the presence of coherences, the classical fluctuation theorem, that one finds for diagonal initial states, is not satisfied. 
Instead, it must be corrected by the information gain. 
It is also possible to draw an alternative interpretation in terms of the entropy production due to quantum measurements, as studied for instance in Refs. <cit.>. 
When a measurement is performed in a basis which commutes with the system's density matrix, no entropy is produced. 
Non-commuting measurements, on the other hand, have an associated entropy production related to the loss of coherence. 
This is precisely the content of Eq. (<ref>).
No such additional entropy production was generated for the first augmentation that led us to Eq. (<ref>), as in this case there is no information gain since the basis |ψ_β'⟩ is the basis that diagonalizes ρ_S'.





§ DISCUSSION


We have addressed the role played by quantum coherence in determining the behaviour of the entropy production, a fundamental quantifier of thermodynamic irreversibility. By making physically reasonable assumptions on the form of the dynamics undergone by a quantum system and its environment, we have been able to single out the contribution that quantum coherences, a genuine non-classical feature of the state of a given dynamical system, play in quantifying the rate of irreversible entropy production. Such contribution appears to be fully distinct from the one arising from unbalances in the energy eigenbasis of the state of the system, which brings about a classical flavour. Moreover, it can be interpreted in a physically transparent manner as the thermodynamic cost that one has to pay for the destruction of coherences that were seeded in the state of the system itself. In turn, our results have interesting consequences for the interpretation of the process of producing entropy as a result of the dynamical generation of correlations (or equivalently coherences) between a quantum system and its environment.

We believe that the theory put forth in this paper may prove  a useful step forward towards the setting up of a self-contained framework for the interpretation of thermodynamic irreversibility at the quantum nanoscale, which is still sorely missing despite the key role that entropy production will play in the quantification of the thermodynamic fingerprint of managing quantum dynamics. 
For instance, it could serve as a starting point for the development of a theory of quantum entropy production in non-equilibrium steady-states of systems connected to multiple reservoirs. Or, as a tool for quantifying the contribution of loss of coherence in the operation of finite-time quantum heat engines.



§ DATA AVAILABILITY STATEMENT


No data sets were generated or analyzed during the current study.



§ ACKNOWLEDGMENTS
 

GTL acknowledges the funding from the São Paulo Research agency, under grant number 2016/08721-7. MP acknowledges funding from the SFI-DfE Investigator Programme, the EU H2020 Collaborative Project TEQ, the Leverhulme Trust, and the Royal Society. 
JPS and GTL acknowledge the Brazilian funding agency CAPES. 



§ ADDITIONAL INFORMATION


Competing interests: The authors declare no competing interests.



§ AUTHOR CONTRIBUTIONS


JPS developed the detailed formal calculations in collaboration with GTL; LCC, GTL, and MP conceived the original idea and shaped it with the significant help of JPS. 
All authors contributed to the research and the preparation of the manuscript. 



50

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