Energetic footprints of irreversibility in the quantum regime

The unavoidable presence of irreversibility in classical thermodynamic processes carries two energetic footprints - the reduction of extractable work from the optimal, reversible case, and the generation of a surplus of heat that is irreversibly dissipated to the environment. Optimal thermodynamic protocols hence attempt to minimize irreversibility, quantified by the entropy production, subject to practical constraints. Recently it has been shown that in the quantum regime an additional quantum entropy production occurs, that can be linked to the fundamental irreversibility of a quantum system decohering into the energy basis. Here we employ quantum trajectories to construct distributions for classical heat and quantum heat exchanges, and show that the heat footprint of quantum irreversibility differs markedly from the classical case. We also quantify how the occurrence of quantum irreversibility reduces the amount of work that can be extracted from a state with coherences. Our results show that decoherence leads to both entropic and energetic footprints which play an important role in the optimization of controlled quantum operations at low temperature, including quantum processors.

In classical thermodynamics irreversibility occurs whenever a non-thermal system is brought into contact with a thermal environment. The ensuing relaxation of the system leads to exchanges of energy that cannot be reversed with the same thermodynamic cost. In thermodynamics this irreversibility is quantified by the positive irreversible entropy production S irr := ∆S − Q T 0, which measures the discrepancy between the system's entropy increase ∆S = S fin − S ini during any thermodynamic process and * m.hamed.mohammady@savba.sk † alexia.auffeves@neel.cnrs.fr ‡ janet@qipc.org the heat Q absorbed by the system from the environment divided by the environment's temperature T . I.e. the heat fraction Q absorbed by the system in an irreversible process will be be less than T ∆S giving a positive definite entropy production S irr . Hence when a process has entropy production there is a surplus of heat, that is irreversibly dissipated from the system to the environment [46]. Irreversibility also puts a fundamental bound on the amount of work W ext that can be extracted during isothermal processes [46,47], where ∆F = F fin −F ini is the system's free energy increase. The more irreversible a process is, the less work can be extracted and the term W lost = T S irr may be called the "lost" or non-recoverable work [48]. Eq. (1) and Eq. (2) link entropy production, S irr to a surplus in heat dissipation, Q sur diss 0, and a reduction in work extraction, W ext −∆F . These relationships are the well-known energetic footprints of irreversibility in classical thermodynamics. However, in the quantum regime, the link between entropy production and energetic footprints, such as the occurrence of a surplus of dissipated heat Q sur diss and the lost work W lost have remained elusive.
In this paper we establish the energetic footprints of irreversibility in the quantum regime, which arise whenever a system is brought in contact with a thermal environment. A quantum system can be out of equilibrium in two ways: by maintaining energetic probabilities that are nonthermal, and by maintaining coherences between energies. It has been shown that contact with the thermal environment gives rise to a classical and a quantum aspect of irreversibility [34,35]. Here we go further and identify how each of the quantum and classical aspects of irreversibility leads to non-trivial energetic exchanges, i.e. we will discuss (classical) heat and work footprints, as well as the footprint of a uniquely quantum energy exchange known as quantum heat [23,33,36,[49][50][51].
For concreteness we here discuss a specific protocol that extracts work from a qubit's quantum coherences [27]. We extend the protocol here to capture irreversible steps that are unavoidable in any experimental implementation and which will affect heat and work exchanges. Employing quantum trajectories that describe the system's evolution during the entire protocol, we first show that the two entropy productions originate from the microscopic time-reversal asymmetry of quantum trajectories. Here we identify the distributions of classical and quantum heat, and evidence that purely quantum contributions to the entropy production are not related to the average quantum heat, in stark contrast to the classical regime, cf. Eq. (1). Instead, we show that the average quantum entropy production is correlated with the variance in quantum heat and, in the special case where the system is a qubit, both grow monotonically with the coherence in the quantum state with reference to the Hamiltonian, in contrast to the classical case. Finally, we show that the classical and quantum entropy production reduce the coherence-work in equal measure, cf. Eq. (2). The results show that when experimental imperfections are unavoidable, any work-optimization strategy needs to consider the tradeoff between a system having a certain degree of classical non-thermality or quantum coherence, or both.
Besides being of fundamental importance for the development of a general quantum thermodynamics framework that includes irreversibility, these relations will also be crucial for the assessment of the energetic cost of quantum control protocols, that aim to optimize performance of computation and communication in the presence of decoherence and noise.
II. Imperfect protocol for work extraction from coherences Readers familiar with work extraction from coherences [27] may skip the details of this section by looking at Fig. 1 and proceeding to Section III.
For a system with Hamiltonian H and quantum state ρ we denote by (ρ, H) any non-equilibrium configuration of the system, and by (τ, H) T with τ := e −H/(k B T ) /Z and partition function Z := tr[e −H/(k B T ) ] its equilibrium configuration at temperature T [52]. For concreteness, we will here assume that the system is a qubit, but generalisations to larger dimensions are straightforward [27]. The qubit Hamiltonians throughout the protocol are chosen diagonal in the same basis, H j := 1 2 ω j (|e + e + | − |e − e − |) for j = 0, 1, ..., N with |e + the excited state and |e − the ground state and E (j) ± = ± ω j /2 the energy eigenvalues. This means that only the spectrum of the Hamiltonian varies during the protocol.
The qubit's general initial state, written in its diagonal basis, is FIG. 1. State evolution during work extraction protocol. For the example of a qubit, in Step (I) the initial state ρ θ is unitarily rotated (green arrow) to ρθ, which may still have coherences with respect to the energy basis |e± . The quench in Step (II) changes the splitting of the energetic levels but does not alter the state. The thermalization process (purple arrow) in Step (III) transforms the state ρθ to τ1, a thermal state diagonal in the energy basis |e± . This process has a quantum component, i.e. decoherence (blue arrow) to state ηθ and a classical thermalization (red arrow) to state τ1.
Step (IV) realises a (classical) quasistatic isothermal process (orange arrow) which transforms the state τ1 to the protocol's final state η θ . The quench in Step (V) leaves this state unaltered. This protocol realises the thermodynamic removal of coherences, i.e. transforming ρ θ to η θ , while irreversibility arises due to the mismatches between ρθ and ηθ as well as ηθ and τ1.
The protocol transfers ρ θ to the fixed final state η θ chosen to have the same energetic probabilities as the initial state ρ θ but with the energetic coherences removed [27]. I.e. η θ := k Π[e k ] ρ θ Π[e k ] with |e k the energy eigenstates and Π[ψ] = |ψ ψ| denoting projectors onto pure states |ψ . Hence the qubit's final state is with r θ := e − |ρ θ |e − quantifying the projection of ρ θ onto the ground state |e − . We also define q j := e − |τ j |e − as the ground state probabilities of the thermal states τ j of the Hamiltonians H j . Analogously to Eq. (3) and Eq. (5), states ρθ and ηθ can be defined for an angleθ. Without loss of generality, we will limit θ andθ to fall in the range [−π/2, π/2]. The optimal, reversible, implementation of the ρ θ to η θ transfer was proposed in [27] and it was shown that the "average" work extracted is where S vN is the Von Neumann entropy, defined as S vN (ρ) := −tr[ρ log ρ]. This is in agreement with equality in Eq. (2) assuming the free energy of a quantum non-equilibrium configuration is defined as [46,47,[53][54][55], and realising that the state change ρ θ to η θ carries no energy change, ∆U = 0, and hence ∆F = −k B T ∆S vN . We re-mark that no distribution of work was provided in [27] with respect to which W ext is an "average".
Generalizing first the steps of the optimal protocol [27] to include irreversibility will allow us to investigate the impact of entropy production on distributions of work and heat below.
The new protocol consists of the following five steps, and the state evolution is visualised for a qubit in Fig. 1: (I) Use a unitary V to rotate the quantum system's configuration (ρ θ , H 0 ) into configuration (ρθ, H 0 ). In the reversible protocol, V is chosen such thatθ = 0 and hence [ρθ, H 0 ] = 0 [27]. Here we allow V to be imperfect and henceθ = 0.
(II) Change the Hamiltonian rapidly resulting in a quench from (ρθ, H 0 ) to (ρθ, H 1 ). In the reversible protocol, the energetic levels of H 1 are chosen such that the configuration (ρθ, H 1 ) is thermal at temperature T [27]. Here we consider the case that the energetic levels of H 1 are adjusted imperfectly, and hence configuration (ρθ, H 1 ) is not necessarily thermal.
(III) Put the quantum system in thermal contact with a heat bath at temperature T . Assuming this step lasts longer than the system relaxation time, it brings the system from (ρθ, H 1 ) into the thermal configuration (τ 1 , H 1 ) T .
(IV) Change the system's Hamiltonian slowly from H 1 to H N , keeping the system in thermal contact with the heat bath. The evolution is chosen quasi-static (i.e. very slow), such that the thermal equilibrium at T is maintained throughout this step. The final Hamiltonian H N is chosen so that the system's thermal state is the desired final state, i.e., τ N = η θ .
(V) Decouple the system from the thermal bath and quench the Hamiltonian back to H 0 , changing the system's configuration from (η θ , H N ) T to the desired configuration (η θ , H 0 ).
Since steps (I), (II), (IV) and (V) are either unitary or quasi-static, they are reversible. The thermodynamic irreversibility of the protocol occurs when the quantum system is put in contact with the thermal bath in Step (III). The irreversible thermalization (ρθ, H 1 ) → (τ 1 , H 1 ) T leads to a reduction in free energy, i.e.
where D[ρθ τ 1 ] = tr[ρθ log ρθ − log τ 1 ] 0 is the quantum relative entropy between the state before thermalization, ρθ, and the state after thermalization, τ 1 , which vanishes if and only if ρθ = τ 1 . Observing that no work is exchanged during thermalization (W ext = 0), and based on the assumption that Eq. (2) holds in the quantum regime, the term k B D[ρθ τ 1 ] is often identified with the entropy S (III) irr that is produced during the thermalization step [35,56].
As recently discussed in [34,35], the geometric measure of irreversibility given by the relative entropy splits into a quantum and a classical part, which can be obtained as averages over the entropy produced along decoherence trajectories and classical thermalization trajectories [35], see also below. This splitting reflects the fact that the quantum configuration (ρθ, H 1 ) is out of equilibrium in two distinct ways: it can have non-Boltzmann probabilities for the energies and quantum coherences between energies. The coherence of ρθ with respect to the energy basis |e ± of H 1 can be quantified by the overlap of ρθ's eigenstates, i.e.
The classical non-thermality of ρθ compared to the thermal state τ 1 for H 1 can be quantified by the logarithm of the ratio of ground state probabilities [57], i.e.
where q 1 := e − |τ 1 |e − and rθ = e − |ρθ|e − = e − |ηθ|e − are the ground state populations of τ 1 and ρθ, respectively, see Fig. 1. When both coh(ρθ) and nonth(ρθ) are zero then ρθ is an equilibrium state. Positive (negative) nonth(ρθ) corresponds to a lower (higher) ground state population in ρθ than that of the thermal state τ 1 , corresponding to a down (up) red arrow in Fig. 1. The relaxation of a quantum state which is put in contact with a thermal bath naturally gives rise to two components of entropy production: a quantum contribution associated with the removal of coherences and a classical contribution associated with the classical thermalization of energetic state populations to become Boltzmann distributed. This splitting into decoherence (blue arrow) and classical thermalization (red arrow) is indicated in Fig. 1.

III. Stochastic quantum trajectories
Working on the level of density matrices of the qubit during the protocol, see Fig. 1, limits the discussion of thermodynamic quantities to macroscopic expectation values only. In contrast, stochastic thermodynamics associates heat Q(Γ), work W (Γ) and entropy production s irr (Γ) to individual microscopic trajectories Γ forming the set of possible system evolutions [58,59]. In this more detailed picture the macroscopic thermodynamic quantities Q , W and S arise as weighted averages over these trajectories. In the quantum regime, quantum stochastic thermodynamics captures the set of possible trajectories that, in addition to classical trajectories, are determined by quantum coherences and non-thermal sources of stochasticity [36,38,[60][61][62][63]. These trajectories consist of time-sequences of pure quantum states taken by an open system in a single run of an experiment.
One way to experimentally 'see' quantum trajectories is by observing a sequence of stochastic outcomes of a generalized measurement performed on a system [64]. Immense experimental progress in the ability to measure quantum Pure-state qubit trajectories for the work extraction protocol. Illustration of the evolution of the qubit during the work protocol on the trajectory level and on the density matrix level, cf. Fig. 1. The qubit's trajectories are deterministic during Steps (I) (unitary, green arrows), (II) (quench, black arrows), and (V) (quench, black arrows), i.e. they take one state to a unique other state. In contrast, during the decoherence part in Step (III) (blue dashed arrows) the qubit stochastically jumps from one of the states |θ± to one of the energy eigenstates |e± , thus losing any quantum coherence in an irreversible manner. During the classical thermalization part in Step (III) (red arrows) the qubit stochastically jumps from one of the energy eigenstates to another energy eigenstate, thus losing any classical non-thermality in an irreversible manner. The qubit's trajectories during the classical quasistatic isothermal change of H (Step (IV), orange arrows), are stochastic but reversible, due to infinitely small thermalizations taking place throughout.
states with high efficiency has enabled the observation of individual jumps in photon number, and more recently the tracking of single quantum trajectories of superconducting qubits [65][66][67][68]. The natural set of quantum trajectories is a function of how the system is measured, and various quantum trajectory sets have been discussed in the literature each corresponding to different measurement setups: the so-called "unravellings" [69,70]. Averaging the system's pure states over many experimental runs then gives back the density matrix describing the system's mixed state, whose evolution is governed by completely positive, trace preserving maps, also known as a quantum channel. Using the methods of quantum stochastic thermodynamics we here access a system's fluctuations in work, heat and entropy production, when quantum coherences are involved and irreversibility occurs. This allows us to expose the microscopic links between irreversibility and energetic exchanges in the quantum regime.
We here use "eigenstate trajectories" that describe a system that travels through a sequence of eigenstates of its time-local density operators. Namely, the system is measured at instances in time j = 1, 2, ... in its instantaneous eigenbases of the states ρ j that are assumed to be known e.g. from a master equation that describes the open system dynamics. We note that this is an idealized scenario as in general one does not know what the density operators ρ j are and cannot guarantee to measure in the correct eigenbases. The eigenstate trajectories are analytically tractable, and provide a first convenient analytical tool to investigate the energetic footprints of irreversibility, as we will see below.
In the following we will work with the idealized eigenstate trajectories. The ensemble of trajectories {Γ} taken by a quantum system (here a qubit) when undergoing the work extraction protocol outlined in the previous section can be broken up into trajectories for each of the Steps, see Fig. 2. We will here focus on discussing the thermalization of the qubit in Step (III), for which the initial qubit density matrix ρθ can host coherences coh and classical non-thermality nonth at the point when it is brought in contact with the thermal bath. The trajectories for the full protocol are detailed in Appendix A.
The thermalization process in Step (III) may be described by the quantum channel Λ(ρ) : the initial thermal state of the bath with Hamiltonian H B and partition function Z B , and V is a unitary operator that commutes with H 1 + H B , hence Λ is a "thermal operation" [71][72][73]. Since Λ shall here be a fully thermalizing map we require that Λ(ρ) = τ 1 for all ρ. This map exists, for example, when the bath is chosen as an infinite ensemble of identical particles, each with the same Hamiltonian as the system, and with V implementing a sequence of partial swaps between the system and each bath particle [74]. Minimal trajectories for the thermalization process can now be constructed as Γ (III) (l,n) ≡ |θ l → |e n , see Fig. 2. The probability of this transfer to occur is P Γ (III) (l,n) = θ l |ρθ|θ l e n |Λ |θ l θ l | |e n , which is obtained by first projectively measuring the system with respect to the eigenbasis |θ l of ρθ, then applying the thermalization channel Λ, and finally measuring the system with respect to the eigenbasis |e n of τ 1 . Since V commutes with the total Hamiltonian while τ B commutes with the bath Hamiltonian, it can be shown (see Theorem 1 in [75]) that e n |Λ |θ l θ l | |e n = m | e m |θ l | 2 e n |Λ (|e m e m |) |e n , where |e m are eigenstates of the system Hamiltonian H 1 .
We may therefore "augment" our trajectories by project-ing the system onto the energy basis |e m first before letting it thermalize classically. Such an augmentation was also performed in [35]. The augmented trajectories are denoted Γ (III) (l,m,n) ≡ |θ l → |e m → |e n , with probabilities P Γ (III) (l,m,n) = θ l |ρθ|θ l | e m |θ l | 2 e n |τ 1 |e n . (9) It can be shown that the minimal trajectories Γ (III) (l,n) and the augmented trajectories Γ (III) (l,m,n) are thermodynamically equivalent, as they result in the same entropy production [35]. However, the augmented trajectories have the benefit of naturally splitting into a decoherence trajectory Γ q (l,m) ≡ |θ l → |e m , followed by a classical thermalization trajectory Γ cl (m,n) ≡ |e m → |e n , as depicted in Fig. 2. Their probabilities to occur are and respectively. Here Γ q (l,m) are the trajectories the system undertakes as it undergoes the decoherence process ρθ → ηθ, while Γ cl (m,n) are the trajectories that the system undertakes as it undergoes the classical thermalization process ηθ → τ 1 .

IV. Stochastic quantum entropy production
Within quantum stochastic thermodynamics the entropy production along a quantum trajectory Γ is exposing the entropy production's microscopic origin as the imbalance between the probabilities P (Γ) and P * (Γ * ) of a forward trajectory Γ and its corresponding backward trajectory Γ * , respectively [38,63]. As shown in Appendix A, the stochastic entropy production for the thermalization Step (III) can be expressed as which is the sum of a stochastic quantum entropy production, and a stochastic classical entropy production, Since the probability of the augmented trajectories, P Γ as marginals (see Eq. (10) and Eq. (11)), the average entropy production in Step (III) can also be split into an average quantum entropy production s qu irr Γ q , and an average classical entropy production, s cl irr Γ cl . One finds, see Appendix A, that each of these averages reduces to a relative entropy between two pairs of system states, . (17) This shows that the relative entropies D[ρθ ηθ] and D[ηθ τ 1 ], which geometrically link density matrices, are physically meaningful as the average entropy productions associated with the evolution of the quantum system along ensembles of quantum trajectories. The two separate contributions to the entropy production arise because the qubit has two distinct non-equilibrium features, coh and nonth.
Each is irreversibly removed when the qubit is brought into contact with the thermal bath and undergoes decoherence trajectories followed by classical thermalization trajectories. Moreover, we show in Appendix A that the average entropy production for the full protocol reduces to s qu irr Γ q + s cl irr Γ cl in the limit where Step (IV) becomes a quasistatic process. I.e. in this limit the average entropy production for the full protocol coincides with the average entropy production for the thermalization step alone. Also in Appendix A we provide expressions for the variances of the two entropy productions.

V. Classical and quantum heat distributions
Having introduced the quantum decoherence and classical thermalization trajectories, Γ q (l,m) and Γ cl (m,n) , respectively, we now analyze the energetic fluctuations in Step (III) associated with each source of irreversibility for the thermalization with the bath. Since there is no external control in Step (III), such as a change of Hamiltonian, no work is done on the system and hence the energetic change of the system consists entirely of heat. But since we identified two contributions to irreversibility, namely quantum decoherence and classical thermalization, it stands to reason that we should obtain two types of heat [36,49].
The microscopic mechanisms associated with classical thermalization of the system with the bath are the quantum jumps from |e m to |e n with m, n ∈ {±}, which give rise to energetic fluctuations. The heat the qubit absorbs from the bath is which is the standard classical stochastic heat. Since the classical thermalization jumps are between well-defined quantum energy states, these fluctuations take one of the quantized values − ω 1 , 0, or + ω 1 . The probability of observing each value of stochastic classical heat is given by Eq. (11).
On the other hand, the microscopic mechanisms associated with decoherence are the quantum jumps from |θ l to Step (III). Histograms of classical heat Q cl (red circles) and quantum heat Qqu (blue squares) for (a) an initial state ρθ that hosts classical nonthermality: nonth(ρθ) = log(0.2/0.3) and coh(ρθ) = 0, and for (b) an initial state ρθ that hosts quantum non-thermality: coh(ρθ) = sin 2 (π/6) = 1/4 and nonth(ρθ) = 0. For comparison, grey circles and grey diamonds in both panels show the classical and quantum heat histograms, respectively, for when Step (III) is fully reversible, i.e. ρθ = τ1 and hence coh(ρθ) = 0 = nonth(ρθ). Note that, even then the system can exchange heat with the bath leading to a classical heat distribution with non-zero but symmetrical values (dashed line) that give a zero average classical heat. In (a) the only quantum heat value with non-zero probability is 0 (no quantum heat when thermalising a classical state), while in (b) four non-trivial quantum heat values occur since coh(ρθ) = 0. |e m , which give rise to energetic fluctuations of the system that are entirely quantum mechanical. The system's energy increase due to decoherence is and arises purely due to the loss of the system's initial coherences due to projective energy measurements. It has no classical counterpart and is hence referred to as quantum heat [36,49]. The energetic fluctuations for these jumps take four values ω1 2 (m − l cosθ) for l, m ∈ {±}. Contrary to the classical stochastic heat which has fixed quantized values given by the Hamiltonian H 1 alone, the stochastic quantum heat's values vary as a function of the eigenbasis of the initial state ρθ. The probability of these quantum heat values to be realised is given by Eq. (10). When the initial state has no quantum coherences (θ = 0 → coh(ρθ) = 0) the only realised value of the stochastic quantum heat is 0, i.e. in the absence of coherences, decoherence has no effect on the system's state and no energetic fluctuations result from it. Fluctuations of the quantum heat take place as soon as coh(ρθ) = 0. Histograms of the classical stochastic heat Q cl and the quantum heat Q qu are shown in Fig. 3(a) and 3(b) for initial states ρθ that have only classical nonthermality while coh = 0, and states that have only coherences while nonth = 0, respectively. Finally, Step (IV) also incurs classical heat. We do not discuss this contribution here, as the stochastic thermo-dynamic description is well established for heat exchanges during this classical quasistatic isothermal process [58,59].

VI. Heat footprints of classical and quantum irreversibility
We are now ready to discuss the energetic footprints of irreversibility in the quantum regime. The energetic footprints of classical entropy production during Step (III) are made immediately apparent from the stochastic equation (15) which, in conjunction with the classical heat value given by Eq. (18), can be re-expressed as When averaged over the classical thermalization trajectories Γ cl (m,n) , the above expression links the average absorbed heat Q cl to the average entropy production s cl irr as This thermodynamic equality, going back to Clausius, is the well-known energetic footprint of entropy production in the classical regime. It can be used to define the irreversibly dissipated heat, which is positive whenever the energetic difference between the entropy change, ∆S cl = k B (S vN (τ 1 ) − S vN (ηθ)), multiplied by bath temperature T , and the absorbed heat Q cl is non-zero. This occurs when the entropy production s cl irr is non-zero, which in turn arises when the process is irreversible, see (12). In other words, the energetic footprint of non-zero Q sur diss gives thermodynamic testament to the arrow of time.
Meanwhile, the stochastic quantum entropy production s qu irr Γ q (l,m) in Eq. (14) is given purely by a stochastic quantum entropy change and does not appear to involve any contributions from the stochastic quantum heat Q qu whatsoever. When averaged over all quantum decoherence trajectories, the quantum heat in fact vanishes, see Appendix B, while the average quantum entropy production can formally be rewritten as This quantum thermodynamic equality shows that the energetic footprint of quantum entropy production, i.e. a fixed relationship between average heat absorption and average entropy production, is mute in the quantum regime. This indicates a fundamental difference in how quantum and classical heat relate to the entropy production.

FIG. 4. Heat footprint of irreversibility for
Step (III). (a) Classical entropy production s cl irr plus the absorbed heat divided by the temperature Q cl /T gives the entropy change ∆S cl for any classical non-thermality parameter nonth(ηθ). With increasing nonth the entropy production s cl irr first decreases and then increases, while the variance of the classical heat, Var (Q cl ), increases monotonously with nonth. Positive (negative) nonthermality nonth corresponds to a lower (higher) ground state population in ηθ than that of the thermal state τ1, corresponding to a down (up) red arrow in Fig. 1. Qubit spacing vs thermal energy ( ω1/kBT ) is here set such that q1 = e−|τ1|e− = 0.85 while p ∈ [0.5, 1]. (b) Quantum entropy production s qu irr plus zero average quantum heat Qqu equals the entropy change ∆Squ for any quantum coherence parameter coh of initial states ρθ. Also shown is the quantum heat variance Var (Qqu) in natural units ( ω1) 2 . Both, Var (Qqu) and s qu irr , increase monotonously as coh tends to its maximum value of 0.5. Initial state mixing probability is here set to p = 0.95 while θ ∈ [0, π/2].
While prima faciae Eq. (14) and Eq. (24) seem to suggest that the quantum entropy production is completely dissociated from quantum heat, such a conclusion is premature. Indeed, on closer examination we discover that the average quantum entropy production s qu irr is correlated with the fluctuations in quantum heat, as quantified by its variance, Var (Q qu ). In particular, for a qubit system, we show explicitly that both s qu irr and Var (Q qu ) are monotonically increasing functions of the coherence coh of the state ρθ with respect to the Hamiltonian H 1 .
This property can be seen by straightforward evaluation of the quantum heat variance, see Appendix C, which monotonically increases as coh ranges from 0 to 1/2. As for s qu irr , note that the eigenvalues of ηθ can be arranged in the vector r coh := (rθ, 1 − rθ), where rθ = p(1 − coh(ρθ)) + (1 − p)coh(ρθ), see Eq. (5) and Eq. (7). As coh increases, coh 2 coh 1 , the eigenvalue rθ will decrease making r coh2 more mixed, i.e. r coh1 r coh2 . It follows that S vN (ηθ), and hence s qu irr , monotonically increase with coh [76,77]. Fig. 4 puts in perspective the two drastically different energetic footprints of irreversibility in the classical and quantum regime. On the well-known classical side, see Fig. 4a, the average entropy production s cl irr is equal to the difference between the fixed entropy change ∆S cl associated with the transfer ηθ → τ 1 , and an absorbed heat Q cl when this transfer is achieved by an irreversible thermalization process, divided by the temperature T . The classical heat footprint Q cl scales as the thermal energy k B T , an energy scale set by the temperature of the bath that thermalizes the qubit. The more non-thermal the initial (diagonal) qubit state ηθ is, the more irreversibility will occur during its thermalization. Hence the classical entropy production s cl irr increases as the classical non-thermality parameter nonth(ηθ) deviates from 0. Note, however, that the entropy production is not a monotonic function of nonth(ηθ), since it decreases as nonth(ηθ) approaches zero from below.
On the quantum side, see Fig. 4b, the average entropy production s qu irr equals the entropy change ∆S qu = k B (S vN (ηθ) − S vN (ρθ)) associated with the decoherence ρθ → ηθ and does not link to an absorbed quantum heat Q qu , as this is always zero. However, both s qu irr and the quantum heat fluctuations Var (Q qu ) monotonously increase with the coherence parameter coh(ρθ), showing an implicit link between entropy production and quantum heat for quantum decoherence processes. This behaviour differs from the classical counterpart, where the classical entropy production first drops and then increases again with increasing nonth, see Fig. 4a, while the classical heat variance increases monotonically with nonth, implying a nonmonotonous relationship between s cl irr and Var (Q cl ). Finally, we remark that unlike the classical case, the heat footprint does not scale with temperature but with the system energy gap, here ω 1 , an energy scale set by the quantum character of the system rather than the thermodynamics implied by the bath.
For general quantum systems (not qubits) coh as defined in Eq. (7) is not a good measure of coherence. However, one can quantify the coherence in a general state ρ, with reference to Hamiltonian H, by the relative entropy of coherence D[ρ η] [78]. Since this is simply s qu irr /k B as ρ decoheres to η, it trivially follows that the quantum entropy production will always be a monotonic function of coherence. However, proving that the quantum heat variance also increases monotonously with coherence remains an open question. What one can show for general quantum systems, is that the variance of the quantum heat is bounded, see Appendix B, as FIG. 5. Average work extraction as a function of coh and nonth. Work (grey) for the full protocol is optimal when neither quantum coherence nor classical non-thermality is present, i.e. coh = 0 = nonth, and the protocol is run reversibly [27]. Wext decreases monotonously with increasing coh(ρθ) (blue line for nonth = 0) and increasing and decreasing nonth(ρθ) (red line for coh = 0). At large deviations from the reversible protocol, Wext becomes negative (crosses yellow plane at zero) and work would need to be invested to run the protocol. Parameter choices for initial qubit state ρ θ are p = 0.8 and θ = π/3.
of the observable H in the state ρ [79][80][81]. While the energy variance ∆(H, ρ) includes quantum and classical uncertainty, the skew information I α (H, ρ) measures purely quantum contributions to the variance. Inequality (26) shows that all quantum uncertainty, and some contribution from the classical uncertainty, limit the magnitude of quantum heat variance Var (Q qu ). It has been suggested that the skew information can act as a measure for coherence [82]. Therefore, for general quantum systems, we can conclude that the average quantum entropy production s qu irr as well as the lower bounds I α (H, ρ) to the quantum heat variance Var (Q qu ) monotonically increase with coherence.
The skew information is also intimately linked to the resource theory of asymmetry [83][84][85] as it quantifies how asymmetric a quantum state ρ is with reference to unitary representations of a symmetry group G, generated by the self-adjoint operator H. It has been shown [86] that the extractable work from asymmetric states will be reduced if the work extraction process obeys symmetry constraints; in our protocol, the process in Steps (III) to (V) is symmetric with respect to unitary evolutions generated by H 1 , and so the more asymmetric ρθ is with respect to H 1 , the less work can be extracted from it.

VII. Fundamental bounds for work extraction
Finally, we check the validity of the work footprint of entropy production, Eq. (2), in the quantum regime. Here it suffices to demonstrate the link to the work average only, higher moments of work can straightforwardly be analysed using quantum trajectories. The average energy change over the entire protocol vanishes by construction, and equals the average heat absorbed by the system minus the average work extracted from the system throughout the entire protocol, Hence the work averaged over the entire protocol's trajectories is where we have assumed quasistatic isothermal trajectories ). Substituting the entropy change across the entire protocol ∆S prot = ∆S qu + ∆S cl + ∆S (IV) , Clearly, the optimum work value −∆F prot is obtained when neither classical nor quantum entropy production are present and the process is run fully reversibly, as discussed in Ref. [27]. Equation (29) further shows that the classical and quantum entropy productions limit work extraction in a similar way. In addition to the classical entropy production s cl irr , in the quantum regime the quantum entropy production s qu irr , linked to the irreversible removal of coherences, reduces work extraction in a completely symmetrical manner, see Eq. (2). The footprint of irreversibility on work extraction is shown in Fig. 5, where W ext is plotted as a function of the two parameters that give rise to irreversibility, the quantum coherence coh and classical non-thermality nonth of the state ρθ before thermal contact.
While work extraction is mathematically limited in a symmetrical manner, the physical mechanism is drastically different depending on if the irreversibility of the protocol is of classical or of quantum nature. In the classical regime the irreversibly dissipated heat Q sur diss is the physical cause of non-optimal work extraction and exactly compensates the non-recoverable work, i.e. the term T s cl irr = Q sur diss in Eq. (29). This energetic footprint of irreversibility equals the average energy change of the qubit during the irreversible thermalization step. But the quantum decoherence step does not give rise to any average energy change -the work extraction is here reduced solely because the system entropy increases, reducing the extracted work by a proportional amount T s qu irr = T ∆S qu .

VIII. Discussion
The notion of irreversibility and how it affects heat and work exchanges is a core pillar of thermodynamics. This paper brings together several strands of recent research in quantum thermodynamics, including stochastic thermodynamics and quantum work extraction protocols, to provide a comprehensive picture of when irreversibility arises in the quantum regime and details the ensuing energetic footprints of irreversibility. Specifically, we have shown that the geometric entropy production D[ρθ τ 1 ], which can be calculated using density matrices, can be understood as arising from the time-reversal asymmetry of quantum stochastic trajectories, Eq. (16) and Eq. (17), in a similar way to classical stochastic thermodynamics. Our arguments follow similar lines of reasoning as the recent work in [35]. In addition, the quantum trajectories allowed for a detailed assessment of work and heat exchanges of a quantum system that can host coherences. While reversible work extraction from quantum coherences has been found [27] to give an "average" work of W ext rev = −∆F prot , no distribution of work was provided with respect to which W ext rev is an "average". Here we showed that quantum trajectories naturally give rise to heat as well as work distributions, for which moments, such as the work "average", can be readily calculated. For the sake of brevity we have here obtained the average work from averaging over the heat distributions in (28), avoiding the need to explicitly construct the work distribution. By here including irreversible steps in the work extraction protocol, the reduction of work due to irreversibility has been quantified in Eq. (29). Understanding how imperfect experimental control, which leaves either quantum coherences, or classical non-thermality, or both present in a quantum system before thermal contact, reduces work extraction, is important for identifying experimental protocols that are optimal within realistic technical constraints.
While the first moments of heat and work coincide with the values obtained on the density matrix level, the trajectories approach allows access to higher moments of heat and work. This proved insightful for the discussion of the footprint of quantum irreversibility. We found that the average classical entropy production is linked to the surplus of dissipated heat, see Eq. (22), which is fully analogous to the classical regime, see Eq. (1). Conversely, no such link can be made in regards to quantum entropy production, see Eq. (24). Instead, we show that the quantum entropy production is correlated with the fluctuations in quantum heat. Specifically, for a general quantum system, the average quantum entropy production, and the lower bound of the quantum heat variance, monotonically increases with the initial state's coherence, coh (see Eq. (26)). In the special case of qubits, we explicitly show that the variance in quantum heat also grows monotonously with the coherence. This footprint was hidden from view before as it cannot be described on the density matrix level alone. By using the quantum trajectories approach we have here uncovered this energetic footprint for the first time. A comparable link does not exist in the classical regime where Var (Q cl ) grows monotonically with the initial state's non-thermality nonth, but the classical entropy production s cl irr does not. It would be interesting to see if the same conclusions hold true when the eigenstate trajectories are replaced by experimentally measured trajectories and their probabilities, for which the analysis presented here can be implemented in an analogous manner.
Another open problem is to establish the relationship between quantum entropy production and the fluctuations in quantum heat for arbitrarily large quantum systems with states that may host degeneracy in their eigenvalues. On the one hand, the inequality in Eq. (26), which is valid for arbitrary dimensions, does not establish a monotonicty relationship between coherence and the variance in quantum heat -only the lower bound in the variance of quantum heat increases with coherence. On the other hand, when the system's state has degeneracies, it will offer infinitely many spectral decompositions, and the corresponding quantum heat fluctuations will not be equivalent. Here a physical principle may need to be be invoked to select a particular decomposition, such as including explicit projective measurements on the system before it undergoes decoherence with respect to the Hamiltonian.
Performing coherent manipulations of quantum systems requires ω 1 k B T , so that the quantum energetic fluctuations due to quantum irreversibility largely overcome the mean energetic exchanges due to classical entropy production. Therefore this energetic footprint is expected to play an important role in the energetic assessment of quantum control operating at low temperature, including quantum processors in cryostats. is described by the channels where Γ S := (l, n 0 , ..., n N ) ≡ |θ l → |θ l → |e n0 → · · · → |e n N is the sequence of time-local eigenstates of the system during the protocol. Note that, here, we identify n 0 ≡ m and n 1 ≡ n as the eigenstate labels during Step (III). The bath indices (µ i , ν i ) merely indicate the sequence of energy measurement outcomes on the baths, and they only contribute to the probabilities of the system trajectories Γ S . The probability of the trajectory Γ is evaluated to be where we have introduced the full Kraus operator for the protocol, Averaging over all the measurement outcomes on the bath, meanwhile, yields the probabilities for the system-only trajectories Γ S , given as Note that we may recover the probability for any subtrajectory of the system by summing over all other indices of Eq. (A4). For example, summing over the indices of Steps (I) and (IV), and the classical thermalization of Step (III), the probabilities for the system's quantum decoherence trajectories Γ q (l,m) are obtained as We may also reconstruct the full density operator for the system, at any point along the trajectory, see The time-reversed trajectories can be defined by reversing the order of the protocol. Here we have Step (IV): quasistatic reversed isothermal jumps |e n N → · · · → |e n1 ; Step (III) reversed thermalization |e n1 → |e n0 followed by reversed decoherence |e n0 → |θ l ; and Step (I): reversed unitary evolution |θ l → |θ l . Moreover, we shall consider the time-reversed thermalization maps Λ Note that the only difference between Λ i and Λ * i is that we have applied the time reversal operation on the unitaries V i , transforming them to V † i . But since the sequence of measurements on the bath during the forward protocol was (µ i , ν i ), we shall take the time-reversal sequence of these outcomes, namely, (ν i , µ i ). As such, the corresponding time-reversed Kraus operators for the ther-malization channels will be where q (j) ni := e ni |τ j |e ni . Here we have used the fact that, given the energy conservation of the thermalization unitary V i , it follows that where E ni (j) := e ni |H j |e ni . Finally, the time-reversed trajectories can be denoted as ..,n0,l),(ν1,µ1),(ν2,µ2),...,(ν N ,µ N ) , (A10) which occur with the probability where we introduce the time reversed Kraus operators for the full protocol, Now we may evaluate the entropy production for the full protcol, which is given by Eq. (A2) and Eq. (A11) to be where we have used the fact that K Γ 2 = K † Γ 2 . Note that the entropy production is independent of the bath measurement results. In other words, the entropy production can be purely determined by the system trajectories Γ S .
It is trivial to show that this entropy production can be split into the three terms is the entropy production of Step (IV).
Since the average entropy production is additive, i.e s irr Γ = s qu irr Γ q + s cl irr Γ cl + s cl irr Γ (IV) , we will compute each term separately. Let us first turn to the last term, namely, the entropy production in Step (IV). We verify that averaging over the trajectory probabilities, one obtains When Step (IV) approaches the quasistatic limit, we will have D[τ i−1 τ i ] → 0, and so s irr Γ = s qu irr Γ q + s cl irr Γ cl . Now we turn to the average entropy production during Step (III). Using Eq. (10) and Eq. (14), and introducing the labels p l := θ l |ρθ|θ l and r m := e m |ηθ|e m , the average quantum entropy production can be shown to be where here q m := e m |τ 1 |e m . Finally, we wish to determine the fluctuations in the entropy production during Step (III), quantified by the variance. The variance in the quantum entropy production can therefore be shown to be Here, V [ρ σ] := tr[ρ(log ρ − log σ − D[ρ σ]1) 2 ] is the relative entropy variance [90].
Similarly, the variance in the classical entropy production can be obtained as It is straightforward to verify that the variance in entropy production across all of Step (III) is V [ρθ τ 1 ]. To see this, note that the entropy production along the full augmented trajectories Γ (III) (l,m,n) is given by combining Eq. (14) and Eq. (15), to obtain k B (log p l − log q m ). Since the index n, corresponding to the final measurement, does not appear here, these entropy productions are associated with probabilities P (Γ q (l,m) ). As such, the variance in entropy production is evaluated completely analogously as with Eq. (A19), and merely replacing ηθ and r m with τ 1 and q m , respectively. However, note that unlike the average entropy production, the variance in entropy production is generally not additive across the decoherence and thermalization trajectories: We note that, in particular, V [ηθ τ 1 ] has been identified as determining the efficiency of work extraction from an energy incoherent state ηθ, with only a finite number of copies of the system [91,92]. Namely, the work that can be -deterministically extracted per n copy of ηθ, with small values of infidently , is given as W = k B T D[ηθ τ 1 ] − ∆W , where ∆W is a positive term that is proportional to k B T V [ηθ τ 1 ]/n.

B. Quantum heat for a d-dimensional system
We here consider a finite d-dimensional system with Hamiltonian H = d m=1 E m |e m e m | and in a state ρ = d l=1 p l |ψ l ψ l |. As the system decoheres with respect to the Hamiltonian, we obtain trajectories Γ q (l,m) := |ψ l → |e m , with probabilities P (Γ q (l,m) ) = p l | ψ l |e m | 2 and quantum heat Q qu (Γ q (l,m) ) := e m |H|e m − ψ l |H|ψ l . The average quantum heat for a decoherence process is always