Juliette Monsel

Cyril Elouard

Alexia Auff

An autonomous quantum machine to measure the thermodynamic arrow of time

Keywords: quantum thermodynamics, quantum optics, opto-mechanics

published or not. The documents may come

An autonomous quantum machine to measure the thermodynamic arrow of time

Juliette Monsel, Cyril Elouard, Alexia Auffèves

Irreversibility is a fundamental feature of our physical world. The degree of irreversibility of thermodynamic transformations is measured by the entropy production, which is always positive according to the Second Law. At the microscopic level, stochastic thermodynamics [START_REF] Sekimoto | Stochastic Energetics[END_REF][START_REF] Seifert | Stochastic thermodynamics: Principles and perspectives[END_REF] has extended this concept to characterize the evolution of small systems coupled to reservoirs and driven out of equilibrium. Such systems follow stochastic trajectories Σ and the stochastic entropy production ∆ i s[ Σ] obeys the integral fluctuation theorem (IFT) exp -∆ i s[ Σ] Σ = 1 where • Σ denotes the average over all trajectories Σ. Jarzynski's equality (JE) [START_REF] Jarzynski | Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach[END_REF] is a paradigmatic example of such IFT, that constrains the fluctuations of the entropy produced while driving some initially thermalized system out of equilibrium. Experimental demonstrations of JE especially require the ability to measure the stochastic work W [ Σ] exchanged with the external entity driving the system. In the classical regime, W [ Σ] can be completely reconstructed from the monitoring of the system's trajectory, allowing for successful experimental demonstrations [START_REF] Saira | Test of the Jarzynski and Crooks Fluctuation Relations in an Electronic System[END_REF][START_REF] Douarche | An Experimental Test of the Jarzynski Equality in a Mechanical Experiment[END_REF][START_REF] Hoang | Experimental Test of the Differential Fluctuation Theorem and a Generalized Jarzynski Equality for Arbitrary Initial States[END_REF] .

Defining and measuring the entropy production in the quantum regime is of fundamental interest in the perspective of optimizing the performances of quantum heat engines and the energetic cost of quantum information technologies [START_REF] Mancino | Geometrical bounds on irreversibility in open quantum systems[END_REF][START_REF] Mancino | The entropic cost of quantum generalized measurements[END_REF][START_REF] Santos | The role of quantum coherence in non-equilibrium entropy production[END_REF][START_REF] Francica | The role of coherence in the nonequilibrium thermodynamics of quantum systems[END_REF] . However, measuring a quantum fluctuation theorem can be problematic in the genuinely quantum situation of a coherently driven quantum system, because of the fundamental and practical issues to define and measure quantum work [START_REF] Talkner | Aspects of quantum work[END_REF][START_REF]Fluctuating work in coherent quantum systems: proposals and limitations[END_REF][START_REF] Engel | Jarzynski equation for a simple quantum system: Comparing two definitions of work[END_REF][START_REF] Campisi | Colloquium: Quantum fluctuation relations: Foundations and applications[END_REF] . So far the quantum JE has thus been extended and experimentally verified in closed quantum systems, i.e. systems that are driven but otherwise isolated. In this case work corresponds to the change in the system's internal energy, accessible by a two-points measurement protocol [START_REF] Talkner | Aspects of quantum work[END_REF] or the measurement of its characteristic function [START_REF] Mazzola | Measuring the Characteristic Function of the Work Distribution[END_REF][START_REF] Dorner | Extracting Quantum Work Statistics and Fluctuation Theorems by Single-Qubit Interferometry[END_REF][START_REF] De Chiara | Ancilla-assisted measurement of quantum work[END_REF] . Experimental demona) Electronic mail: juliette.monsel@neel.cnrs.fr b) Electronic mail: alexia.auffeves@neel.cnrs.fr strations have been realized, e.g. with trapped ions [START_REF] An | Experimental Test of the Quantum Jarzynski Equality with a Trapped-Ion System[END_REF][START_REF] Xiong | Experimental Verification of a Jarzynski-Related Information-Theoretic Equality by a Single Trapped Ion[END_REF] , ensemble of cold atoms [START_REF] Cerisola | Using a quantum work meter to test non-equilibrium fluctuation theorems[END_REF] , and spins in Nuclear Magnetic Resonance (NMR) [START_REF] Batalhão | Experimental Reconstruction of Work Distribution and Study of Fluctuation Relations in a Closed Quantum System[END_REF] where the thermodynamic arrow of time was successfully measured [START_REF] Batalhão | Irreversibility and the Arrow of Time in a Quenched Quantum System[END_REF] .

On the other hand, realistic strategies must still be developed to measure the fluctuations of entropy production for quantum open systems, i.e. that can be simultaneously driven, and coupled to reservoirs. Since work is usually assumed to be provided by a classical entity, most theoretical proposals so far have relied on the measurement of heat fluctuations, i.e. small energy changes of the reservoir. Experimentally, this requires to engineer this reservoir and to develop high efficiency detection schemes, which is very challenging [START_REF] Pekola | Calorimetric measurement of work in a quantum system[END_REF][START_REF] Horowitz | Quantum-trajectory approach to the stochastic thermodynamics of a forced harmonic oscillator[END_REF][START_REF] Elouard | Probing Quantum Fluctuation Theorems in Engineered Reservoirs[END_REF] . Experimental demonstrations have remained elusive.

In this article, we propose a new and experimentally feasible strategy to measure the thermodynamic arrow of time for a quantum open system in Jarzynski's protocol, that is based on the direct measurement of work fluctuations. We investigate a so-called hybrid opto-mechanical system [START_REF] Treutlein | Hybrid Mechanical Systems[END_REF] , that consists in a two-level system (further called a qubit) strongly coupled to a mechanical oscillator (MO) on the one hand, and to a thermal bath on the other hand. Studying single quantum trajectories of the hybrid system, we show that the MO and the qubit remain in a product state all along their joint evolution, allowing to unambiguously define their stochastic energies. We evidence that the mechanical energy fluctuations can be identified with the stochastic work received by the qubit and satisfy JE. Therefore the MO plays the role of a quantum battery, the ensemble of the qubit and the battery forming an autonomous machine [START_REF] Frenzel | Quasi-autonomous quantum thermal machines and quantum to classical energy flow[END_REF][START_REF] Tonner | Autonomous quantum thermodynamic machines[END_REF][START_REF] Holmes | Coherent fluctuation relations: From the abstract to the concrete[END_REF] . Originally, the battery behaves as an embedded quantum work meter, encoding information on the stochastic work exchanges. We show that the evolution of the complete machine is characterized by a generalized IFT, that quantitatively involves the amount of extracted information. This situation gives rise to so-called absolute irreversibility, in agreement with recent theoretical predictions and experimental results [START_REF] Murashita | Nonequilibrium equalities in absolutely irreversible processes[END_REF][START_REF] Funo | Quantum nonequilibrium equalities with absolute irreversibility[END_REF][START_REF] Murashita | Fluctuation theorems in feedback-controlled open quantum systems: Quantum coherence and abso-lute irreversibility[END_REF][START_REF] Masuyama | Information-to-work conversion by Maxwell's demon in a superconducting circuit-QED system[END_REF] . Our proposal is robust against finite measurement precision [START_REF] Lahaye | Approaching the Quantum Limit of a Nanomechanical Resonator[END_REF][START_REF] Schliesser | Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit[END_REF] The paper is divided as follows. Firstly, we introduce hybrid opto-mechanical devices as autonomous machines, and build the framework to model their evolution on average and at the single trajectory level. Focusing on Jarzynski's protocol, we define stochastic heat, work and entropy production and study the regime of validity and robustness of JE as a function of the parameters of the problem and experimental imperfections. Finally, we derive and simulate an IFT for the complete machine, evidencing the presence of absolute irreversibility. Our results demonstrate that work fluctuations can be measured directly, by monitoring the energetic fluctuations of the quantum battery. They represent an important step towards the experimental demonstration of quantum fluctuation theorem in a quantum open system.

RESULTS

Hybrid opto-mechanical systems as autonomous machines. A hybrid opto-mechanical system consists in a qubit of ground (resp. excited) state |g (resp. |e ) and transition frequency ω 0 , parametrically coupled to a mechanical oscillator of frequency Ω ω 0 (See Fig. 1a). Recently, physical implementations of such hybrid systems have been realized on various platforms, e.g. superconducting qubits embedded in oscillating membranes [START_REF] Pirkkalainen | Hybrid circuit cavity quantum electrodynamics with a micromechanical resonator[END_REF] , nanowires coupled to diamond nitrogen vacancies [START_REF] Arcizet | A Single Nitrogen-Vacancy Defect Coupled to a Nanomechanical Oscillator[END_REF] , or to semiconductor quantum dots [START_REF] Yeo | Strain-mediated coupling in a quantum dot-mechanical oscillator hybrid system[END_REF] . The complete Hamiltonian of the hybrid system reads [START_REF] Treutlein | Hybrid Mechanical Systems[END_REF] , where H q = hω 0 |e e| ⊗ 1 m and H m = 1 q ⊗ hΩb † b are the qubit and MO free Hamiltonians respectively. We have introduced the phonon annihilation operator b, and 1 m (resp. 1 q ) the identity on the MO (resp. qubit) Hilbert space. The coupling Hamiltonian is

H qm = H q + H m + V qm
V qm = hg m |e e| ⊗ (b + b † )
, where g m is the qubit-mechanical coupling strength. Of special interest for the present paper, the so-called ultra-strong coupling regime is defined as g m ≥ Ω, with ω 0 g m . It was recently demonstrated experimentally [START_REF] Yeo | Strain-mediated coupling in a quantum dot-mechanical oscillator hybrid system[END_REF] .

The Hamiltonian of the hybrid system can be fruitfully rewritten

H qm = |e e| ⊗ H e m + |g g| ⊗ H g m with H g m = hΩb † b and H e m = hΩB † B + h(ω 0 -g 2 m /Ω)1 m , with B = b + (g m /Ω)1 m .
It appears that the qubit bare energy states = e, g are stable under the dynamics and perfectly determine the evolution of the MO ruled by the Hamiltonian H m . Interestingly, H m preserves the statistics of coherent mechanical states, defined as |β = e β * b-βb † |0 , where |0 is the zerophonon state and β the complex amplitude of the field. Consequently, if the hybrid system is initially prepared in a product state | , β 0 , it remains in a similar product state | , β t at any time, with |β t = exp(-iH m t/h) |β 0 . The two possible mechanical evolutions are pictured in Fig. 1b between time t 0 = 0 and t = Ω/2π, in the phase space defined by the mean quadratures of the MO x = b + b † and p = -i b -b † . If the qubit is initially prepared in the state |e (resp. |g ), the mechanical evolution is a rotation around around the displaced origin (-g m /Ω, 0) (resp. the origin (0, 0)). Such displacement is caused by the force the qubit exerts on the MO, that is similar to the optical radiation pressure in cavity opto-mechanics. Defining δβ t = β e t -β g t , it appears that the distance between the two final mechanical states |δβ t | scales like g m /Ω. In the ultra-strong coupling regime, this distance is large such that mechanical states are distinguishable, and can be used as quantum meters to detect the qubit state.

Since the hybrid system remains in a pure product state at all times, its mean energy defined as E qm ( , β t ) = , β t | H qm | , β t naturally splits into two distinct components respectively quantifying the qubit and the mechanical energies:

E q ( , β t ) = hω(β t )δ ,e (1)
E m (β t ) = hΩ|β t | 2 , (2)
where δ ,e is the Kronecker delta and ω(β) is the effective transition frequency of the qubit defined as:

ω(β t ) = ω 0 + 2g m Re(β t ). (3)
The frequency modulation described by Eq. ( 3) manifests the back-action of the mechanics on the qubit. Note that the case g m /Ω

|β 0 | corresponds to |δβ t | |β g t |:
Then the frequency modulation is independent of the qubit state and follows ω(β t ) ∼ ω(β 0 e -iΩt ), even in the ultra-strong coupling regime. In what follows, we will be especially interested in the regime where 1 g m /Ω |β 0 |, where the mechanical evolution depends on the qubit state, while the qubit transition frequency is independent of it.

We now take into account the coupling of the qubit to a bath prepared at thermal equilibrium. The bath of temperature T consists of a spectrally broad collection of electromagnetic modes of frequencies ω , each mode containing a mean number of photons nω = (exp(hω /k B T ) -1)

-1 . The bath induces transitions between the states |e and |g , and is characterized by a typical correlation time τ c giving rise to a bare qubit spontaneous emission rate γ.

The hybrid system is initially prepared in the product state ρ qm (0) = ρ q (0) ⊗ |β 0 β 0 |. ρ q (0) is the qubit state, taken diagonal in the {e, g} basis. |β 0 β 0 | is the mechanical state, that is chosen pure and coherent. In the rest of the paper, we shall study transformations taking place on typical time scales t ∼ Ω -1 , such that the mechanical relaxation is neglected. From the properties of the interaction with the bath and the total hybrid system's Hamiltonian H qm , it clearly appears that the qubit does not develop any coherence in its bare energy basis. We show in the Supplementary 39 that as long as |β 0 | g m t, the MO imposes a well defined modulation of the qubit frequency ω(β 0 (t)) with β 0 (t) = β 0 e -iΩt . This defines the semi-classical regime, where the hybrid system evolution is ruled by the following master equation [START_REF] Monsel | Supplementary Material for "An autonomous quantum machine to measure the thermodynamic arrow of time[END_REF] :

ρqm (t) = - i h [H qm , ρ qm (t)] + γ nω(β0(t)) D[σ † ⊗ 1 m ]ρ qm (t) + γ nω(β0(t)) + 1 D[σ ⊗ 1 m ]ρ qm (t). (4)
We have defined the super-operator D[X]ρ = XρX † -1 2 {X † X, ρ} and σ = |g e|.

Product states of the form ρ qm (t) = ρ q (t) ⊗ ρ m (t) are natural solutions of Eq. ( 4), giving rise to two reduced coupled equations respectively governing the dynamics of the qubit and the mechanics:

ρq (t) = - i h [H q (t), ρ q (t)] + γ nω(β0(t)) D[σ † ]ρ q (t) + γ nω(β0(t)) + 1 D[σ]ρ q (t), (5)
ρm (t) = - i h [H m (t), ρ m (t)]. (6)
We have introduced the effective time-dependent Hamiltonians:

H q (t) = Tr m [ρ m (t)(H q + V qm )] = hω(β 0 (t)) |e e| and H m (t) = Tr q [ρ q (t)(H m + V qm )].
The physical meaning of these semi-classical equations is transparent: The force exerted by the qubit results into the effective Hamiltonian H m (t) ruling the mechanical evolution. Reciprocally, the mechanics modulates the frequency ω(β 0 (t)) of the qubit (Eq. ( 3)), which causes the coupling parameters of the qubit to the bath to be time-dependent. The semi-classical regime of hybrid opto-mechanical systems is especially appealing for quantum thermodynamical purposes, since it allows modeling the time-dependent Hamiltonian ruling the dynamics of a system (the qubit) by coupling this system to a quantum entity, i.e. a quantum battery (the MO). The Hamiltonian of the compound is time-independent, justifying to call it an "autonomous machine" [START_REF] Frenzel | Quasi-autonomous quantum thermal machines and quantum to classical energy flow[END_REF][START_REF] Tonner | Autonomous quantum thermodynamic machines[END_REF][START_REF] Holmes | Coherent fluctuation relations: From the abstract to the concrete[END_REF] . As demonstrated in a previous work 40 , this scenery suggests a new strategy to measure average work exchanges in quantum open systems. Defining the average work rate received by the qubit as Ẇ = Tr q [ρ q (t) Ḣq (t)], we have shown that this work rate exactly compensates the mechanical energy variation rate: Ėm = Tr m [ ρm (t)H m ] = -Ẇ . Remarkably, this relation demonstrates the possibility of measuring work "in situ", directly inside the battery. This strategy offers undeniable practical advantages, since it solely requires to measure the mechanical energy at the beginning and at the end of the transformation. The corresponding mechanical energy change is potentially measurable in the ultra-strong coupling regime g m /Ω

1 40 , which is fully compatible with the semiclassical regime g m t |β 0 |. Our goal is now to extend this strategy to work fluctuations. A key point is to demonstrate that the qubit and the mechanical state remain in a pure product state along single realizations of the protocol, allowing to unambiguously define stochastic energies for each entity. This calls for an advanced theoretical treatment based on the quantum trajectories picture.

Quantum trajectories. We shall now describe the evolution of the machine between the time t 0 and t N by stochastic quantum trajectories of pure states

Σ := {|Ψ Σ (t n ) } N n=0
, where |Ψ Σ (t n ) is a vector in the Hilbert space of the machine and t n = t 0 + n∆t with ∆t the time increment. To introduce our approach we first consider the semi-classical regime where the master equation ( 4) is valid: The initial state of the machine |Ψ Σ (t 0 ) is drawn from the product state ρ q (0) ⊗ |β 0 β 0 | where ρ q (0) is diagonal in the {e, g} basis, and the evolution is studied over a typical duration (t N -t 0 )

|β 0 |g -1 m . Eq. ( 4) is unraveled in the quantum jump picture [START_REF] Plenio | The quantum-jump approach to dissipative dynamics in quantum optics[END_REF][START_REF] Gardiner | Quantum noise : a handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics[END_REF][START_REF] Carmichael | Statistical Methods in Quantum Optics 2: Non-Classical Fields[END_REF][START_REF] Wiseman | Quantum measurement and control[END_REF][START_REF] Haroche | Exploring the Quantum: Atoms, Cavities, and Photons[END_REF] , giving rise to the following set of Kraus operators {J -1 (t n ); J +1 (t n ); J 0 (t n )}:

J -1 (t n ) = γ∆t(n ω(β0(tn)) + 1) σ ⊗ 1 m , (7)
J +1 (t n ) = γ∆tn ω(β0(tn)) σ † ⊗ 1 m , (8)
J 0 (t n ) = 1 qm - i∆t h H eff (t n ). (9)
We have introduced 1 qm = 1 m ⊗ 1 q the identity operator in the Hilbert space of the machine. J -1 and J +1 are the so-called jump operators. Experimentally, they are signaled by the emission or absorption of a photon in the bath, that corresponds to the transition of the qubit in the ground or excited state respectively. The mechanical state remains unchanged. Reciprocally, the absence of detection event in the bath corresponds the no-jump operator J 0 , i.e. a continuous, non Hermitian evolution governed by the effective Hamiltonian

H eff (t n ) = H qm + H nh (t n ). Here H nh (t) = -(ih/2)(J † +1 (t)J +1 (t) + J † -1 (t)J -1 (t)) is the non-hermitian part of H eff .
Let us suppose that the machine is initially prepared in a pure state |Ψ(t 0 ) = | 0 , β 0 . The quantum trajectory Σ is then perfectly defined by the sequence of stochastic jumps/no-jump

{K Σ (t n )} N n=1 where K = 0, -1, +1. Namely, |Ψ Σ (t N ) = N n=1 J KΣ(tn) |Ψ(t 0 ) / P [ Σ|Ψ(t 0 )] where we have introduced P [ Σ|Ψ(t 0 )] = N n=1 P [Ψ Σ (t n )|Ψ Σ (t n-1 )] the probability of the trajectory Σ conditioned to the initial state |Ψ(t 0 ) . P [Ψ Σ (t n )|Ψ Σ (t n-1 )] = Ψ Σ (t n-1 )|J † KΣ(tn) J KΣ(tn) |Ψ Σ (t n-1
) denotes the probability of the transition from |Ψ Σ (t n-1 ) to |Ψ Σ (t n ) at time t n . At any time t N , the density matrix of the machine, i.e. the solution of Eq. ( 4), can be recovered by averaging over the trajectories:

ρ qm (t N ) = Σ P [ Σ] |Ψ Σ (t N ) Ψ Σ (t N )| . ( 10
)
We have introduced the probability of the trajectory

P [ Σ] = p[Ψ(t 0 )]P [ Σ|Ψ(t 0 )],
where p[Ψ(t 0 )] the probability that the machine is initially prepared in |Ψ(t 0 ) . Interestingly from the expression of the Kraus operators, it appears that starting from the product state | 0 , β 0 , the machine remains in a product state

|Ψ Σ (t n ) = | Σ (t n ), β Σ (t n )
at any time t n , which is the first result of this paper. The demonstration is as follows: At each time step t n , either the machine undergoes a quantum jump J ±1 , or it evolves under the no-jump operator J 0 . In the former case, the qubit jumps from | Σ (t n ) into | Σ (t n+1 ) and the mechanical state remains unchanged, such as |β Σ (t n+1 ) = |β Σ (t n ) . In the latter case, the evolution of the machine state is governed by the effective Hamiltonian H eff , whose non-hermitian part can be rewritten H nh = (-ih/2)1 m ⊗ H q nh with H q nh diagonal in the bare qubit energy eigenbasis. It naturally derives from the evolution rules that H nh has no effect on a machine state of the form | Σ (t n ), β Σ (t n ) , such that the nojump evolution reduces to its unitary component defined by H qm . As studied above, the qubit energy state is stable under such evolution, such that | Σ (t n ) = | Σ (t n+1 ) . Reciprocally, the coherent nature of the mechanical field is preserved by

H Σ (tn) m . Thus the mechanics evolves into |β Σ (t n+1 ) = exp -i∆tH Σ (tn) m |β Σ (t n ) , completing the demonstration.
This result invites to recast the machine trajectory as a set of two reduced trajectories

Σ = { , β[ ]} where = {| Σ (t n ) } N n=0
is the stochastic qubit trajectory with Σ (t n ) = e, g. In the semi-classical regime considered here, the jump probabilities solely depend on ω(β 0 (t)), such that the qubit reduced evolution is Markovian. Conversely,

β = {|β Σ (t n ) } N n=0 is the continuous MO trajectory verify- ing |β Σ (t n ) = n-1 k=0 exp -i∆tH Σ (t k ) m |β 0 .
At any time t N , the mechanical state depends on the complete qubit trajectory .

Examples of numerically generated mechanical trajectories β[ ] (See Methods) are plotted in Fig. 1c. As it appears in the figure, at the final time the mechanical states |β Σ (t N ) are restricted within an area of typical dimension g m /Ω. Splitting the mechanical amplitude as

β Σ (t N ) = β 0 (t N ) + δβ Σ (t N ), the semi-classical regime is characterized by |δβ Σ (t n )| |β 0 (t N )| while in the ultra-strong coupling regime |δβ Σ (t N )|
1. These two regimes are compatible, which is the key of our proposal as we show in the next Section.

Interestingly, the modeling of the machine stochastic evolution can be extended over timescales t ≥ |β 0 |g -1 m , beyond the semi-classical regime. The key point is that the trajectory picture allows keeping track of the mechanical state at each time step |β Σ (t n ) . Therefore at each time t n , a master equation of the form of Eq. ( 4) can thus be derived and unraveled into a set of trajectory-dependent Kraus operators similar to Eq. ( 7), taking now ω(β Σ (t n )) as the qubit effective frequency. In this general situation, the machine stochastic evolution still consists in trajectories of pure product states

|Ψ Σ (t n ) = | Σ (t n ), β Σ (t n ) , but the mechanical fluctuations |δβ Σ (t n )|
cannot be neglected anymore with respect to the mean amplitude |β 0 (t n )|. Consequently, Eq.( 10) can not be written as an average product state of the qubit and the MO, resulting in the emergence of classical correlations between the qubit and the MO average states. Moreover, the jump probabilities at time t n now depend on nω(βΣ(tn)) , such that the reduced qubit trajectory is not Markovian anymore. As we show below, this property conditions the validity of our proposal, which is restricted to the Markovian regime.

Stochastic thermodynamics. From now on we focus on the following protocol: At the initial time t 0 the machine is prepared in a product state ρ qm (t 0 ) = ρ ∞ q (β 0 ) ⊗ |β 0 β 0 | where ρ ∞ q (β 0 ) is the qubit thermal distribution defined by the effective frequency ω(β 0 ). Note that ρ qm (t 0 ) is not an equilibrium state of the whole machine. One performs an energy measurement of the qubit, preparing the state

|Ψ(t 0 ) = | (t 0 ), β 0 with probability p ∞ β0 [ ] = exp(-hω(β 0 )δ ,e /k B T )/Z(β 0 ). Z(β 0 ) = 1 + exp(-hω(β 0 )/k B T ) is the partition function.
The machine is then coupled to the bath and its evolution is studied between t 0 = 0 and t N = π/2Ω. Depending on the choice of thermodynamical system, this physical situation can be studied from two different perspectives, defining two different transformations. If the considered thermodynamical system is the machine, then the studied evolution corresponds to a relaxation towards thermal equilibrium. Since the machine Hamiltonian H qm is time-independent, energy exchanges reduce to heat exchanges between the machine and the bath. On the other hand, if the considered thermodynamical system is the qubit, then the studied transformation consists in driving the qubit out of equilibrium through the time-dependent Hamiltonian H q (t), the driving work being provided by the mechanics. In the semi-classical regime, the qubit evolution is Markovian, such that this last situation simply corresponds to Jarzynski's protocol with H q (t) = hω(β 0 (t)) |e e|.

We now define and study the stochastic thermodynamical quantities characterizing the transformation experienced by the system (qubit or machine) for the protocol intro-duced above. As shown previously, starting from a product state |Ψ(t 0 ) = | 0 , β 0 the machine remains in a product state at any time

|Ψ Σ (t n ) = | Σ (t n ), β Σ (t n ) .
Defining as E qm (Ψ Σ(tn) ) = Ψ Σ(tn) H qm Ψ Σ(tn) the machine internal energy, it thus naturally splits into a sum of the qubit energy E q ( Σ (t n ), β Σ (t n )) (See Eq. ( 1)) and mechanical energy E m (β Σ (t n )) (See Eq. ( 2)). Along the trajectory, the set of internal energies can change in two distinct ways. A quantum jump taking place at time t n stochastically changes the qubit and the machine energies by the same amount δE q [Σ, t n ] = δE qm [Σ, t n ], leaving the MO energy unchanged. Following standard definitions in stochastic thermodynamics [START_REF] Horowitz | Quantum-trajectory approach to the stochastic thermodynamics of a forced harmonic oscillator[END_REF][START_REF] Alicki | The quantum open system as a model of the heat engine[END_REF][START_REF] Elouard | The Role of Quantum Measurement in Stochastic Thermodynamics[END_REF] , the corresponding energy change is identified with heat q[Σ, t n ] provided by the bath. Conversely in the absence of jump, the qubit remains in the same state between t n and t n+1 while its energy eigenvalues evolve in time due to the qubitmechanical coupling. Such energy change is identified with work denoted w[Σ, t n ] and verifies δE q [Σ,

t n ] = w[Σ, t n ].
During this time interval, the machine is energetically isolated such that δE qm [Σ, t n ] = 0. Therefore the work increment exactly compensates the mechanical energy change δE m [Σ, t n ] = -w[Σ, t n ]. Finally, the total work (resp. heat) received by the qubit is defined as

W [ Σ] = N -1 n=0 w[Σ, t n ] (resp. Q[ Σ] = N -1 n=0 q[Σ, t n ])
. By construction, their sum equals the qubit total energy change between t 0 and t N ,

∆E q [ Σ] = W [ Σ] + Q[ Σ].
From the analysis conducted above, it appears that the heat exchange corresponds to the energy change of the machine,

∆E qm [ Σ] = Q[ Σ].
Reciprocally, the work received by the qubit is entirely provided by the mechanics and verifies:

W [ Σ] = -∆E m [ Σ], (11)
which is the second result of this article. Eq. ( 11) extends the results obtained for the average work in a previous work 40 , and explicitly demonstrates the one-by-one correspondence between the stochastic work received by the qubit and the mechanical energy change between the start and the end of the trajectory. The MO thus behaves as an ideal embedded quantum work meter at the single trajectory level.

We finally derive the expression of the stochastic entropy production ∆ i s[ Σ]. It is defined by comparing the probability of the forward trajectory in the direct protocol P [ Σ] to the probability of the backward trajectory in the time-reversed protocol P [ Σ] 48 :

∆ i s[ Σ] = log P [ Σ] P [ Σ] . ( 12
)
The probability of the direct trajectory reads:

P [ Σ] = p ∞ β0 [ Σ (t 0 )] N n=1 P [Ψ Σ (t n )|Ψ Σ (t n-1 )], (13)
The state of the hybrid system averaged over the forward trajectories at time t N is described by Eq. (10). At the end of the protocol, the reduced mechanical average state defined as ρ m (t N ) = Tr q [ρ qm (t N )] thus consists in a discrete distribution of the final mechanical states {|β Σ (t N ) }. Introducing the probability p m [β f ] for the mechanical amplitude to end up in a state of amplitude β f , we shall denote it in the following

ρ m (t N ) = Σ βf p m [β f ] |β f β f | where Σ βf p m [β f ] = 1.
Reciprocally, the time-reversed protocol is defined between t N and t 0 . It consists in time-reversing the unitary evolution governing the dynamics of the machine, keeping the same stochastic map at each time t n . This leads to the expression of the time dependent reversed Kraus operators [START_REF] Elouard | The Role of Quantum Measurement in Stochastic Thermodynamics[END_REF][START_REF] Crooks | Quantum Operation Time Reversal[END_REF][START_REF] Manzano | Quantum Fluctuation Theorems for Arbitrary Environments: Adiabatic and Nonadiabatic Entropy Production[END_REF][START_REF] Manikandan | Time reversal symmetry of generalized quantum measurements with past and future boundary conditions[END_REF] :

J0 (t n ) = 1 qm + i∆t h H † eff (t n ), (14)
J-1 (t n ) = J +1 (t n ), (15)
J+1 (t n ) = J -1 (t n ), (16)
The initial state of the backward trajectory is defined as follows: The mechanical state |β Σ (t N ) is drawn from the final distribution of states {|β f } generated by the direct protocol with probability p m [β f ], while the qubit state is drawn from the thermal equilibrium defined by β Σ (t N ) with probability p ∞ βΣ(t N ) . The probability of the backward trajectory reads

P [ Σ] = p m [β Σ (t N )]p ∞ βΣ(t N ) [ Σ (t N )] × 1 n=N P [Ψ Σ (t n-1 )|Ψ Σ (t n )]. (17)
We have introduced the reversed jump probability at time

t n P [Ψ Σ (t n-1 )|Ψ Σ (t n )] = Ψ Σ (t n )| J † KΣ(tn) JKΣ(tn) |Ψ Σ (t n ) . Based on Eqs.
(11), ( 12), ( 13), (17), we derive in the Supplementary [START_REF] Monsel | Supplementary Material for "An autonomous quantum machine to measure the thermodynamic arrow of time[END_REF] the following expression for the stochastic entropy produced along Σ:

∆ i s[ Σ] = σ[ Σ] + I Sh [ Σ], (18)
where σ[ Σ] and I Sh [ Σ] are defined as

σ[ Σ] = - ∆E m [ Σ] + ∆F [ Σ] k B T , (19)
I Sh [ Σ] = -log(p m [β Σ (t N )]). (20)
We have introduced the quantity

∆F [ Σ] = k B T log(Z(β 0 )/Z(β Σ (t N ))
) that extends the notion of the qubit free energy change to cases where the reduced qubit trajectory is non-Markovian. In the Markovian regime, we simply recover Z(t N ) = 1 + exp(-hω(β 0 (t N ))/k B T ) and ∆F [ Σ] = ∆F . As we show below, in this case σ[ Σ] can be interpreted as the entropy produced along the reduced trajectory of the qubit, that gives rise to a reduced JE. Conversely, I Sh [ Σ] measures the stochastic entropy increase of the MO and is involved in a generalized IFT characterizing the evolution of the whole machine. We now study in detail these two fluctuation theorems. Reduced Jarzynski's equality. We first focus on the transformation experienced by the qubit. As mentioned above, in the Markovian regime the applied protocol corresponds to Jarzynski's protocol: The qubit is driven out of thermal equilibrium while it experiences the frequency modulation ω(β 0 (t)). Since the stochastic work W [ Σ] is provided by the mechanics, one expects the mechanical energy fluctuations to obey a reduced Jarzynski's equality. We derive in the Supplementary [START_REF] Monsel | Supplementary Material for "An autonomous quantum machine to measure the thermodynamic arrow of time[END_REF] the following IFT:

exp ∆E m [ Σ] k B T Σ = exp - ∆F k B T . (21)
Eq. ( 21) corresponds to the usual Jarzynski's equality, with the remarkable difference that the stochastic work involved in σ[ Σ] is now replaced by the mechanical energy change ∆E m [ Σ]. This is the third and most important result of this paper, which now suggests a new strategy to measure work fluctuations. Instead of reconstructing the stochastic work by monitoring the complete qubit trajectory, one can simply measure the mechanical stochastic energy at the beginning and at the end of the protocol. This can be done, e.g. in time-resolved measurements of the mechanical complex amplitude through to optical deflection techniques [START_REF] Sanii | High Sensitivity Deflection Detection of Nanowires[END_REF][START_REF] De Lépinay | A universal and ultrasensitive vectorial nanomechanical sensor for imaging 2D force fields[END_REF] . To do so, the final mechanical states |β Σ (t N ) should be distinguishable, which requires to reach the ultra-strong coupling regime. As mentioned above, this regime has been experimentally evidenced [START_REF] Yeo | Strain-mediated coupling in a quantum dot-mechanical oscillator hybrid system[END_REF] with typical values Ω ∼ g m ∼ 400kHz. The strategy we suggest here is drastically different from former proposals aiming at measuring JE in a quantum open system, that involved challenging reservoir engineering techniques [START_REF] Horowitz | Quantum-trajectory approach to the stochastic thermodynamics of a forced harmonic oscillator[END_REF][START_REF] Elouard | Probing Quantum Fluctuation Theorems in Engineered Reservoirs[END_REF] or fine thermometry [START_REF] Pekola | Calorimetric measurement of work in a quantum system[END_REF] in order to measure heat exchanges. We have simulated the reduced JE (See Fig. 2a). As expected, JE is verified in the Markovian limit where we have checked that the action of the MO is similar to a classical external operator imposing the qubit frequency modulation ω(β 0 (t)) (Fig. 2b). On the contrary, the Markovian approximation and JE break down in the regime

(g m /Ω)/|β 0 | ≥ 10 -2 .
In what follows, we restrict the study to the range of parameters (g m /Ω)/|β 0 | < 10 -2 .

The results presented in Fig. 2 presuppose the experimental ability to measure the mechanical states with an infinite precision. To take into account both quantum uncertainty and experimental limitations, we now assume that the measured complex amplitude β M corresponds to the mechanical amplitude β f in the end of the protocol with a finite precision δβ.

For our simulations we have chosen δβ = 2 which corresponds to achievable experimental value [START_REF] Sanii | High Sensitivity Deflection Detection of Nanowires[END_REF][START_REF] De Lépinay | A universal and ultrasensitive vectorial nanomechanical sensor for imaging 2D force fields[END_REF] . To quantify this finite precision, we introduce the mutual information between the final distribution of mechanical states p m [β f ] introduced above, and the measured distribution p m [β M ], defined as:

I[β f , β M ] = βf,β M p(β f , β M ) log p(β f , β M ) p m [β f ]p m [β M ] . (22)
p(β f , β M ) denotes the joint probability of measuring β M while the mechanical amplitude equals β f . If the measurement pre-cision is infinite, the mutual information I[β f , β M ] exactly matches the Shannon entropy characterizing the final distribution of mechanical states

S Sh [β f ] = -βf p m [β f ] log(p m [β f ]).
On the opposite, it vanishes in the absence of correlations between the two distributions. The simulation of the measured JE and the mutual information I[β f , β M ] are plotted in Fig. 3 for the measurement precision δβ = 2, as a function of the parameter g m /Ω (See Methods). We have introduced the measured reduced entropy production σ

M [ Σ] = (W M [ Σ] -∆F )/k B T where W M [ Σ] is the measured work distribution W M [ Σ] = -∆E M m [ Σ] = hΩ(|β M 0 | 2 -|β M Σ (t N )| 2 ).
As expected, small values of g m /Ω correspond to a poor ability to distinguish between the different final mechanical states, hence to measure work, which is characterized by a non-optimal mutual information. In this limit, the measured work fluctuations W M [ Σ] do not verify JE. Increasing the ratio g m /Ω allows to increase the information extracted on the work distribution during the readout. Thus the mutual information converges towards S Sh [β f ] despite the finite precision readout. JE is recovered for g m /Ω ∼ 50. Such high rates are within experimental reach, by engineering modes of lower mechanical frequency [START_REF] Yeo | Supplementary information for "Strain-mediated coupling in a quantum dot-mechanical oscillator hybrid system[END_REF] . PSfrag replacements Generalized integral fluctuation theorem. We finally consider the complete machine as the thermodynamical system under study. Based on Eq. ( 12) and ( 18), we show in the Supplementary [START_REF] Monsel | Supplementary Material for "An autonomous quantum machine to measure the thermodynamic arrow of time[END_REF] that the entropy produced along the stochastic evolution of the hybrid system obeys a modified IFT of the form:

hω 0 /k B T e -∆ i s[ Σ] Σ ∆ i s[ Σ] Σ e -(σ[ Σ]+I Sh [ Σ]) Σ Σ P [ Σ] = 1 -λ (b)
hω 0 /k B T e -∆ i s[ Σ] Σ ∆ i s[ Σ] Σ log P [ Σ]/ P [ Σ] Σ σ[ Σ] + ISh[ Σ]
exp -∆ i s[ Σ] Σ = 1 -λ. (23)
Following [START_REF] Murashita | Nonequilibrium equalities in absolutely irreversible processes[END_REF][START_REF] Funo | Quantum nonequilibrium equalities with absolute irreversibility[END_REF][START_REF] Murashita | Fluctuation theorems in feedback-controlled open quantum systems: Quantum coherence and abso-lute irreversibility[END_REF][START_REF] Masuyama | Information-to-work conversion by Maxwell's demon in a superconducting circuit-QED system[END_REF] , we have defined the parameter λ as

Σ P [ Σ] = 1 -λ.
The case λ = 0 signals the existence of backward trajectories Σ without any forward counterpart, i.e. P [ Σ] = 0, a phenomenon that has been dubbed absolute irreversibility (See Supplementary [START_REF] Monsel | Supplementary Material for "An autonomous quantum machine to measure the thermodynamic arrow of time[END_REF] ). From Eq. ( 23) and the convexity of the exponential, it is clear that absolute irreversibility characterizes transformations associated to a strictly positive entropy production. This is the case in the present situation, which describes the relaxation of the machine towards a thermal equilibrium state: Such transformation is never reversible, unless for T = 0.

The IFT (Eq. ( 23)) and the mean entropy production

∆ i s[ Σ]
Σ are plotted in Fig. 4a and Fig. 4b respectively, as a function of the bath temperature T (See Methods). The limit hω 0 k B T corresponds to the trivial case of a single reversible trajectory characterized by a null entropy production and λ → 0. In the opposite regime defined by k B T hω 0 , a mean entropy is produced while λ → 1: In this situation, most backward trajectories have no forward counterpart. As we show in the Supplementary [START_REF] Monsel | Supplementary Material for "An autonomous quantum machine to measure the thermodynamic arrow of time[END_REF] , such effect arises since a given β f of the final distribution of mechanical states can only be reached by a single forward trajectory, while it provides a starting point for a large number of backward trajectories.

As noticed in [START_REF] Murashita | Nonequilibrium equalities in absolutely irreversible processes[END_REF][START_REF] Masuyama | Information-to-work conversion by Maxwell's demon in a superconducting circuit-QED system[END_REF][START_REF] Manikandan | Time reversal symmetry of generalized quantum measurements with past and future boundary conditions[END_REF] , absolute irreversibility can also appear in IFTs characterizing the entropy produced by a measurement process. In particular, λ = 0 can signal a perfect information extraction: This typically corresponds to the present situation which describes the creation of classical correlations between the qubit reduced trajectory and the distributions of final mechanical states β[ ]. Interestingly, the two FTs ( 21) and ( 23) are thus deeply related. To be experimentally checked, Eq. ( 21) requires the MO to behave as a perfect quantum work meter, which is signaled by absolute irreversibility Eq. (23). Therefore absolute irreversibility is constitutive of the protocol, and a witness of its success.

DISCUSSION

We have evidenced a new protocol to measure stochastic entropy production and thermodynamic time arrow in a quantum open system. Based on the direct readout of stochastic work exchanges within an autonomous machine, this protocol is experimentally feasible in state-of-the-art opto-mechanical devices and robust against finite precision measurements. It offers a promising alternative to former proposals relying on the readout of stochastic heat exchanges within engineered reservoirs, which require high efficiency measurements. Originally, our proposal sheds new light on absolute irreversibility, which quantifies information extraction within the quantum work meter and therefore signals the success of the protocol.

In the near future, direct work measurement may become extremely useful to investigate genuinely quantum situations where a battery coherently drives a quantum open system into coherent superpositions. Such situations are especially appealing for quantum thermodynamics since they lead to entropy production and energetic fluctuations of quantum nature [START_REF] Elouard | The Role of Quantum Measurement in Stochastic Thermodynamics[END_REF][START_REF] Elouard | Extracting Work from Quantum Measurement in Maxwell's Demon Engines[END_REF] , related to the erasure of quantum coherences [START_REF] Santos | The role of quantum coherence in non-equilibrium entropy production[END_REF][START_REF] Francica | The role of coherence in the nonequilibrium thermodynamics of quantum systems[END_REF] . Recently, small amounts of average work have been directly measured, by monitoring the resonant field coherently driving a superconducting qubit [START_REF] Cottet | Observing a quantum maxwell demon at work[END_REF] . Generalizing our formalism to this experimental situation would relate measurable work fluctuations to quantum entropy production, opening a new chapter in the study of quantum fluctuation theorems.

METHODS

The numerical results presented in this article were obtained using the jump and no-jump probabilities to sample the ensemble of possible direct trajectories [START_REF] Haroche | Exploring the Quantum: Atoms, Cavities, and Photons[END_REF] . The average value of a quantity A[ Σ] is then approximated by

A[ Σ] Σ 1 Ntraj Ntraj i=1 A[ Σ i ]
where N traj = 5 × 10 6 is the number of numerically generated trajectories and Σ i denotes the i-th trajectory.

The reduced entropy production σ[ Σ] used in Fig. 2 and4 was calculated with the expression (19), using the numerically generated values of β 0 and β Σ (t N ) in the trajectory Σ. One value of β Σ (t N ) can be generated by a single direct trajectory Σ: Below we use the equality p m [β Σ (t N )] = P [ Σ]. Using the expression (17) of the probability of the reversed trajectory, the average entropy production becomes:

∆ i s[ Σ] Σ = log P [ Σ] P [ Σ] Σ = -log p ∞ βΣ(t N ) [ Σ (t N )] × N n=1 P [Ψ Σ (t n-1 )|Ψ Σ (t n )] Σ -1 N traj Ntraj i=1 log p ∞ β i Σ (t N ) [ i (t N )] × N n=1 P [Ψ i Σ (t n-1 )|Ψ i Σ (t n )] ,
and,

Σ P [ Σ] = Σ p ∞ βΣ(t N ) [ Σ (t N )]p m [β Σ (t N )] × N n=1 P [Ψ Σ (t n-1 )|Ψ Σ (t n )] = p ∞ βΣ(t N ) [ Σ (t N )] N n=1 P [Ψ Σ (t n-1 )|Ψ Σ (t n )] Σ 1 N traj Ntraj i=1 p ∞ β i Σ (t N ) [ i (t N )] × N n=1 P [Ψ i Σ (t n-1 )|Ψ i Σ (t n )].
The plotted error bars represent the statistical error σ/ N traj , where σ is the standard deviation.

To obtain Fig. 3, we considered that the preparation of the initial MO state was not perfect. So instead of starting from exactly |β 0 , the MO trajectories start from |β Σ (t 0 ) with the β Σ (t 0 ) uniformly distributed in a square of width 2δβ, centered on β 0 . Similarly, the measuring apparatus has a finite precision, modeled by a grid of cell width 2δβ in the phase plane (Re β f , Im β f ). Instead of obtaining the exact value of β Σ (t N ), we get β M Σ (t N ), the center of the grid cell in which β Σ (t N ) is. The value used to compute the thermodynamical quantities are not the exact β Σ (t 0 ) and β Σ (t N ) but β M 0 = β 0 and β M Σ (t N ). Here we describe the coupling of a hybrid opto-mechanical system to a thermal bath of temperature T . The total Hamiltonian reads H = H qm + H b + V qb , where H b = k hω k a † k a k is the free Hamiltonian of the bath, a k is the annihilation operator of the k-th electromagnetic mode of frequency ω k . The coupling Hamiltonian between the qubit and the bath in the Rotating Wave Approximation equals

V qb = k hg k (a k σ † + a † k σ)
, where σ = |g e| and g k is the coupling strength between the qubit and the k-th mode. We denote γ = k g 2 k δ(ω 0 -ω k ) the spontaneous emission rate of the bare qubit in the bath. The typical correlation time of the bath verifies τ c γ -1 , g -1 m , Ω -1 . The hybrid system is initially prepared in a factorized state ρ qm (0) = ρ q (0) ⊗ |β 0 β 0 | where ρ q (0) is diagonal in the bare qubit energy basis and |β 0 is a pure coherent state. We can define a coarse grained time step ∆t, fulfilling τ c ∆t γ -1 such that under these assumptions, the hybrid system and the bath are always in a factorized state (Born-Markov approximation). Moreover, the coupling to the bath solely induces transitions between the qubit bare energy states, such that the hybrid system naturally evolves into a classically correlated state of the form ρ qm (t

(t) = ρ I qm (t + ∆t) -ρ I qm (t) = - 1 h2 t+∆t t dt t t dt Tr b V I qb (t ), V I qb (t ), ρ I qm (t) ⊗ ρ b ,
where we have defined the interaction representation with respect to the free Hamiltonians of the hybrid system and the bath ρ I qm (t) = e it(Hqm+Hb)/h ρ qm (t)e -it(Hqm+Hb)/h , V I qb (t) = e it(Hqm+Hb)/h V qb (t)e -it(Hqm+Hb)/h . Tr b is the trace over the bath's Hilbert space and ρ b is the bath's density matrix. We have used that the term of first order in V I qb vanishes. Because of the presence of V qm = hg m |e e| (b + b † ) in H 0 , V I qb also acts on the MO. It can be split in the following way:

V I qb (u) = R † b (u) ⊗ S(u) + R b (u) ⊗ S † (u), with R b (u) = h k g k a k e -iω k
u and S(u) = e iuHqm/h (σ ⊗ 1 m )e -iuHqm/h . Then, expanding the commutators, the trace over the bath's degrees of freedom can be computed. For any two times u and v, the correlation functions of the bath read: u-v) . nω k is the average number of photon of frequency ω k in the bath. As a result, only terms containing one S and one S † remains in ∆ρ I qm . The integral t t dt can then be changed into an integral over τ = t -t : t -t 0 dτ . Since g s (u, v) = g s (u -v), with s ∈ {+, -}, is non zero only for |u -v| < ∼ τ c ∆t, the upper bound can be set to infinity. In addition, the coarse-graining time can be chosen such that ∆t γ -1 , g -1 m , Ω -1 and the MO does not evolve during the integration. As a consequence, the operator S(u) becomes S(u) = σe -iω0u e -igm(b+b † )u .

Tr b [ρ b R b (u)R b (v)] = Tr b [ρ b R † b (u)R † b (v)] = 0, g -(u, v) = Tr b [ρ b R b (u)R † b (v)] = h2 k g 2 k (n ω k + 1)e -iω k (u-v) and g + (u, v) = Tr b [ρ b R † b (u)R b (v)] = h2 k g 2 k nω k e iω k (
As long as u

|β 0 |g -1 m , ω(β e (u)) ω(β g (u)) ω(β 0 (u))
where ω(β) = ω 0 + g m (β + β * ) and β 0 (t) = β 0 e -iΩt . Moreover, the state of the system ρ qm (t) can be approximated by the factorized state ρ q (t)

⊗ |β 0 (t) β 0 (t)|. Denoting |E(u) = |e e| ⊗ |β 0 (u) β 0 (u)| (resp. |G(u) = |g g| ⊗ |β 0 β 0 | (u)), S(u) thus verifies E(u)|S(u)|E(u) = G(u)|S(u)|G(u) = E(u)|S(u)|G(u) = 0 and G(u)|S(u)|E(u)
e -iω(β0(u))u . ∆ρ I qm (t) can then be decomposed over the states |E(t) , |G(t) and the integral over τ make the system interacts only with bath photons of frequency ω(β 0 (t)).

As announced in the main text, the master equation describing the relaxation of the hybrid system in the bath can finally be written as

ρqm (t) = - i h [H qm , ρ qm (t)] + γ nω(β0(t)) D[σ † ⊗ 1 m ]ρ qm (t) + γ nω(β0(t)) + 1 D[σ ⊗ 1 m ]ρ qm (t), (24)
where D[X]ρ = XρX † -1 2 {X † X, ρ}, and nω = (exp(hω/k B T ) -1) -1 .

B. Entropy production for the autonomous machine.

Starting from the definition ∆ i s[ Σ] = log P [ Σ]/ P [ Σ] and using Eqs. ( 13) and ( 17) from the main text, the entropy production can be written:

∆ i s[ Σ] = log p ∞ β0 [ Σ (t 0 )] p m [β Σ (t N )]p ∞ βΣ(t N ) [ Σ (t N )] N n=1 P [Ψ Σ (t n )|Ψ Σ (t n-1 )] N n=1 P [Ψ Σ (t n-1 )|Ψ Σ (t n )] . (25)
From the expressions of the jump and no-jump operators, we obtain

P [Ψ Σ (t n )|Ψ Σ (t n-1 )] P [Ψ Σ (t n-1 )|Ψ Σ (t n )] = Ψ Σ (t n-1 )|J † KΣ(tn) J KΣ(tn) |Ψ Σ (t n-1 ) Ψ Σ (t n )| J † KΣ(tn) JKΣ(tn) |Ψ Σ (t n ) = exp(-q[Σ, t n-1 ]/k B T ), (26)
and, using the expression of the thermal distribution p

∞ β [ ] = exp(-hω(β)δ ,e /k B T )/Z(β), we get p ∞ β0 [ Σ (t 0 )] p ∞ βΣ(t N ) [ Σ (t N )] = exp (∆E q [ Σ] -∆F [ Σ])/k B T . (27)
The initial and final thermal distributions respectively depend on β 0 and β Σ (t N ), which leads to a trajectory-dependent free energy variation ∆F [ Σ] = k B T log(Z(β 0 )/Z(β Σ (t N ))). Finally,

∆ i s[ Σ] = -log(p m [β Σ (t N )]) + ∆E q [ Σ] -∆F [ Σ] -Q[ Σ] k B T = I Sh [ Σ] - ∆E m + ∆F [ Σ] k B T = I Sh [ Σ] + σ[ Σ], (28)

C. Reduced Jarzynski Equality

We show that, in the Markovian limit, the reduced entropy production σ[ Σ ] obeys the Jarzynski like equality:

exp -σ[ Σ ] Σ = 1. (29)
The derivation starts from the sum over all reversed trajectories of the complete machine: 1 = Σ P [ Σ]. In the limit |β 0 | g m /Ω, the action of the MO on the qubit is similar to an external operator imposing the evolution of the qubit frequency ω(β 0 (t)). As a consequence, the reversed jump probability at time t n does not depend on the exact MO state β Σ (t n ), but only on β 0 (t n ) = β 0 e -iΩtn , which corresponds to the free MO dynamics. The IFT for the complete autonomous machine [Eq. ( 23) from the main text] can be derived starting from the sum over all reversed trajectories, making appear the ratio P [ Σ]/P [ Σ]. To do so, we need to ensure that P [ Σ] = 0. This requires to separate the set Σ d = { P [ Σ]|P [ Σ] = 0} of reversed trajectories with a direct counterpart from the set without:

Only the reversed trajectories Σ = {|˜ Σ (t n ), βΣ (t n ) } 0 n=N such that βΣ (t 0 ) = β 0 verify P [ Σ] = 0. Fig. 5 gives examples of both kinds of trajectories. Denoting λ = Σ / ∈Σd P [ Σ] and using Eqs. ( 13) and ( 17) from the main text we obtain:

1 = Σ P [ Σ]p m [β Σ (t N )] p ∞ βΣ(t N ) [ Σ (t N )] p ∞ β0 [ Σ (t 0 )]

Fig. 1 .

 1 Fig. 1. (a) Situation under study: a qubit exchanging work W with a mechanical resonator and heat Q with a thermal bath at temperature T . The ensemble of the qubit and mechanics constitutes an autonomous machine. (b) Evolution of the complex mechanical amplitude β if the qubit is in the |e (resp. |g ) and the MO is initially prepared in the state i|β0| . The mechanics can be used as a meter to detect the qubit state if gm/Ω 1 (ultra-strong coupling regime). The mechanical fluctuations induced by the qubit state are small w.r.t. the free evolution if |β0| gm/Ω (semi-classical regime). These two regimes are compatible (See text) (c) Stochastic mechanical trajectories β[ ] in the phase space defined by (x, p) (See text). The MO is initially prepared in the coherent state i|β0| , and the qubit state is drawn from thermal equilibrium. Inset: Distribution of final states |βΣ(tN ) within an area of typical width gm/Ω. Parameters: T = 80 K, hω0 = 1.2kBT , Ω/2π = 100 kHz, γ/Ω = 5, gm/Ω = 100, |β0| = 1000.

- 1 Fig. 2 .- 1 .

 121 Fig. 2. Jarzynski's equality for the qubit. Parameters: T = 80 K, hω0 = 1.2kBT , Ω/2π = 100 kHz, γ/Ω = 5. (a) Deviation from JE as a function of gm Ω /|β0| (|β0| = 5000). The points were computed by increasing the opto-mechanical coupling strength gm/2π from 1 MHz to 20 MHz, keeping the other parameters constant. (b) Deviation from JE as a function of |β0| with gm/Ω = 10. Red squares: Case of a classical external drive imposing the qubit frequency modulation ω(β0(t)) (See text). Blue dots: Eq. (21). Green diamonds: exp -σ[ Σ]

1 Fig. 3 .

 13 Fig. 3. Impact of finite precision readout of the mechanical amplitude. Parameters: δβ = 2, T = 80 K, hω0 = 1.2kBT , Ω/2π = 1 kHz, γ/Ω = 5. 2gm|β0| was kept constant (2gm|β0|/2π = 600 GHz) while increasing gm, such that each point corresponds to the same mean reduced entropy production σ[ Σ] Σ . Left axis, blue dots: Deviation from measured JE. Right axis, orange squares: Mutual information I[βf, β M ]. Orange dashed line: Shannon's entropy of the final distribution of mechanical states SSh[βf] (See text).

ΣFig. 4 .

 4 Fig. 4. Deviation from the integral fluctuation theorem (a) and mean entropy production (b) for the complete autonomous machine. Parameters: ω0/2π = 2 THz (amounts to hω0/kBT = 1.2 for T = 80 K used in Fig. 2), Ω/2π = 100 kHz, γ/Ω = 5, gm/Ω = 10 and |β0| = 5000. In both cases, two different expressions were used.The blue dots are computed using the final distribution of mechanical states {|βΣ(tN ) } and mimic an experiment. The red squares involve the probability of the reversed trajectory, which can only be the result of a theoretical treatment. (See Methods for more details.)

  ACKNOWLEDGMENT J.M. acknowledges J-P Aguilar Ph.D. grant from the CFM foundation. C.E. acknowledges the US Department of Energy grant No. de-5sc0017890. This work was supported by the ANR project QDOT (ANR-16-CE09-0010-01). Part of this work was discussed at the Kavli Institute for Theoretical Physics during the program Thermodynamics of quantum systems: Measurement, engines, and control. The authors acknowledge the National Science Foundation under Grant No. NSF PHY-1748958.

  where we used∆E q [ Σ] = W [ Σ] + Q[ Σ] and W [ Σ] = -∆E m [ Σ] [Eq.(11) from the main text].

N n=1 PFig. 5 .

 n=15 Fig. 5. Example of trajectories for the qubit (a) the MO (b). The solid (resp. dashed) arrows correspond to the direct (resp. reversed) protocol. For the sake of simplicity, only the trajectories without any jump are represented. βg (resp. βe) is the final state of the MO after the direct protocol when the qubit is in state |g (resp. |e ). The expressions of the MO evolution operators are: U (t) = exp(-itH m ) and Ũ (t) = U † (t), with = e, g. The reversed trajectories that do not have a direct counterpart are plotted in red and the corresponding qubit states with dashed lines. The final MO states for these trajectories are |β = Ũg(tN ) |βe and |β = Ũe(tN ) |βg , where β = β0 and β = β0. ρ ∞ q (t) (resp. ρm(t)) is the qubit thermal state (resp. the MO average state) at time t.

N

  n=1 P [Ψ Σ (t n-1 )|Ψ Σ (t n )] N n=1 P [Ψ Σ (t n )|Ψ Σ (t n-1 )] + λ = Σ P [ Σ] exp -I Sh [ Σ] -∆E q [ Σ] -∆F [ Σ] -Q[ Σ] k B T + λ = exp -(σ[ Σ] + I Sh [ Σ]) Σ + λ.(31)Thus, exp -∆ i s[ Σ] Σ = 1 -λ.

  and can be probed with state-of-the-art

	experimental devices.
	arXiv:1804.02296v2 [quant-ph] 17 Sep 2018

  ) = P e (t) |e e|⊗|β e (t) β e (t)|+P g (t) |g g|⊗|β g (t) β g (t)|. {|β (t) } =e,g are coherent states of the MO verifying β = β 0 e -iΩt + δβ (t). The mechanical fluctuations after a typical time t verify |δβ (t)| ∼ g m t. They become potentially detectable as soon as |δβ (t)| ≥ 1, i.e. t ≥ g -1 m . Conversely, the mechanical fluctuations have no influence on the qubit frequency as long as |δβ (t)| |β 0 |, i.e. t |β 0 |g -1 m (See main text).

	The precursor of the master equation reads
	∆ρ I qm

  Therefore, we can get rid of the state dependencies in the MO state:P [Ψ Σ (t n-1 )|Ψ Σ (t n )] = P [ Σ (t n )| Σ (t n+1 )] and p ∞ βΣ(t N ) [ Σ (t N )] = p ∞ β0(t N ) [ Σ (t N )]. Therefore,

				N
	1 =		p m [β Σ (t N )] 	p ∞ β0(t N ) [ Σ (t N )]	P [ Σ (t n-1 )| Σ (t n )]
		βΣ(t N )		n=1
			N
	=	p ∞ β0(t N ) [ Σ (t N )]	P [ Σ (t n-1 )| Σ (t n )]
			n=1
	=	P [ ]	p ∞ β0(t

N ) [ Σ (t N )] N n=1 P [ Σ (t n-1 )| Σ (t n )] p ∞ β0 [ Σ (t 0 )]

SUPPLEMENTARY INFORMATION FOR "AN AUTONOMOUS QUANTUM MACHINE TO MEASURE THE THERMODYNAMIC ARROW OF TIME"

A. Master equation.


MD Ondrej Derner

PhD Anneke Kramer
email: a.kramer@amsterdamumc.nl

MD Zdenka Hruskova

MD Mustafa Arici

MD Frederic Collart

MD, PhD Patrik Finne

Laura Fuentes Sanchez

Jerome Harambat

PhD Marc H Hemmelder

MD Kristine Hommel

PhD Laura Fuentes Sánchez

MD Jérôme Harambat

MD, MSc, MPH Julia Kerschbaum

MD, MSc, MPH Johan De Meester

PhD Runolfur Palsson

MD Mårten Segelmark

MD Rannveig Skrunes

MD Jamie P Traynor

MD, PhD Oscar Zurriaga

MD, PhD Ziad A Massy

Prof Kitty J Jager

PhD Vianda S Stel

Prof Vladimir Tesar

MD, PhD Jérôme Harambat

Incidence of Kidney Replacement Therapy and Subsequent Outcomes Among Patients With Systemic Lupus Erythematosus: Findings From the ERA Registry

Keywords: Systemic lupus erythematosus, lupus nephritis, kidney failure, kidney replacement therapy, kidney transplantation, incidence, prevalence, survival

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ABSTRACT Rationale and objective:

There is a dearth of data characterizing patients requiring kidney replacement therapy (KRT) for kidney failure due to systemic lupus erythematosus (SLE) and their clinical outcomes. The aim of this study was to describe trends in incidence and prevalence of KRT among these patients as well as to compare their outcomes to patients treated with KRT for diseases other than SLE.

Study design: Retrospective cohort study based on kidney registry data.

Setting & participants: Patients recorded in 14 registries of patients receiving kidney replacement therapy that provided data to the European Renal Association (ERA) Registry between 1992 and 2016.

Predictor: SLE as cause of kidney failure.

Outcomes: Incidence and prevalence of KRT, patient survival while receiving KRT, patient and graft survival after kidney transplantation, and specific causes of death.

Analytical Approach: Kaplan-Meier methods and Cox regression models were fit to compare patient survival between the SLE and non-SLE groups, overall KRT, dialysis and patient and graft survival after kidney transplantation.

Results: In total, 1826 patients commenced KRT for kidney failure due to SLE, representing an incidence of 0.80 per million population (pmp) per year. The incidence remained stable during the study period (annual percent change=0.1 [95%CI: -0.6 ; 0.8]). Patient survival among patients with SLE receiving KRT was similar to survival within the comparator group (HR=1.11 [95%CI: 0.99-1.23]). After kidney transplantation, the risk of death was greater among patients with SLE than among patients within the comparator group (HR=1.25 [95%CI: 1.02-1.53]), while the risk of all-cause graft failure was similar (HR=1.09 [95%CI: 0.95-1.27]). Ten-year J o u r n a l P r e -p r o o f Introduction Systemic lupus erythematosus (SLE) is an autoimmune disease involving different organ systems. It is typically a disease of young women in reproductive age with a women-to-men ratio averaging about 9:1 [START_REF] Feldman | Epidemiology and sociodemographics of systemic lupus erythematosus and lupus nephritis among US adults with Medicaid coverage, 2000-2004[END_REF] . Lupus nephritis (LN) is one of the most frequent and important types of organ involvement in SLE with a negative impact on patient outcome, partly because of the morbidity and mortality associated with chronic kidney disease (CKD) [START_REF] Coresh | Evidence for increased cardiovascular disease risk in patients with chronic kidney disease[END_REF] . The risk of developing LN varies significantly between regions and ethnicities, ranging from 10% to 70% of the SLE population [START_REF] Cervera | Morbidity and mortality in systemic lupus erythematosus during a 10-year period: a comparison of early and late manifestations in a cohort of 1,000 patients[END_REF][4][START_REF] Ingvarsson | Variations in the epidemiology of systemic lupus erythematosus in southern Sweden[END_REF][START_REF] Bastian | Systemic lupus erythematosus in three ethnic groups. XII. Risk factors for lupus nephritis after diagnosis[END_REF][START_REF] Hanly | The frequency and outcome of lupus nephritis: results from an international inception cohort study[END_REF] . Kidney failure has been reported in 10% to 30% of patients with LN [START_REF] Hanly | The frequency and outcome of lupus nephritis: results from an international inception cohort study[END_REF][START_REF] Tektonidou | Risk of End-Stage Renal Disease in Patients With Lupus Nephritis, 1971-2015: A Systematic Review and Bayesian Meta-Analysis[END_REF] . About 1% of patients undergoing kidney replacement therapy (KRT) in US [START_REF] Nee | Survival Disparity of African American Versus Non-African American Patients With ESRD Due to SLE[END_REF] do so because of LN.

In recent decades, registry-based data on the outcomes of SLE patients on KRT have been presented by the US Renal Data System (USRDS) [START_REF] Nee | Survival Disparity of African American Versus Non-African American Patients With ESRD Due to SLE[END_REF][START_REF] Wallace | Renal Transplantation and Survival Among Patients With Lupus Nephritis: A Cohort Study[END_REF][START_REF] Wallace | All-Cause and Cause-Specific Mortality Trends of End-Stage Renal Disease Due to Lupus Nephritis From 1995 to 2014[END_REF][START_REF] Sexton | ESRD from lupus nephritis in the United States, 1995-2010[END_REF][START_REF] Costenbader | Trends in the incidence, demographics, and outcomes of end-stage renal disease due to lupus nephritis in the US from 1995 to 2006[END_REF][START_REF] Sule | Increased risk of death in pediatric and adult patients with ESRD secondary to lupus[END_REF][START_REF] Sule | Increased risk of death in African American patients with end-stage renal disease secondary to lupus[END_REF][START_REF] Chelamcharla | The outcome of renal transplantation among systemic lupus erythematosus patients[END_REF] , the Taiwanese National Registry (NHIRD) [START_REF] Chang | Survival analysis in systemic lupus erythematosus patients on maintenance dialysis: a nationwide population-based study in Taiwan[END_REF][START_REF] Chen | Predictors of longterm mortality in patients with and without systemic lupus erythematosus on maintenance dialysis: a comparative study[END_REF] , the French Registry of Renal Epidemiology and Information Network (REIN) [START_REF] Levy | Outcome of patients with systemic lupus erythematosus on chronic dialysis: an observational study of incident patients of J o u r n a l P r e -p r o o f the French National Registry 2002-2012[END_REF] and the Australia and New Zealand Dialysis and Transplant Registry (ANZDATA) [START_REF] Zhang | Long-term outcomes of end-stage kidney disease for patients with lupus nephritis[END_REF] . These studies have shown mixed results, also for patient survival on dialysis and patient and allograft survival after kidney transplantation. To our knowledge, although SLE patients differ substantially from patients on KRT for kidney failure due to other causes, only a few of these studies used advanced analytical methods to compare patients with SLE patients receiving KRT to patients without SLE receiving KRT. Furthermore, no registry-based study has examined time trends in the incidence and mortality of SLE patients on KRT outside the US.

Using data from the European Renal Association (ERA) Registry, we aimed to compare patient characteristics, overall survival on KRT, survival after kidney transplantation, and causes of death for SLE patients starting KRT with their age-, sex-and time period-matched comparator group of patients without SLE. In addition, we aimed to explore the trends in incidence and J o u r n a l P r e -p r o o f prevalence of KRT for kidney failure due to SLE and in survival of SLE patients over the past 25 years.

Methods

Data collection

The ERA Registry collects data annually on patients who initiate KRT from national and regional kidney registries in Europe. The details of data collection and data processing methods are described elsewhere [START_REF]ERA-EDTA Registry Annual Report 2016[END_REF] . For this study, data from kidney registries providing individual patient data to the ERA Registry for patients receiving KRT between 1992 and 2016 were used, including Andalusia (Spain), Austria, Basque country (Spain), Belgium (French-speaking), Catalonia (Spain), Denmark, Finland, Greece, Iceland, The Netherlands, Norway, Scotland (UK), Sweden and the Valencian region (Spain). Together, the Spanish regions covered 49.2% of the general population of Spain. Data on children were not available for French-speaking Belgium. All participating national and regional renal registries provided full coverage of the population within their corresponding region, accounting for 85 million Europeans in 1992 and 96 million in 2016. The national and regional registries complied with national legislation with regard to ethics committee approval and with European and national data protection regulations.

Informed consent was not obtained separately for the present study, as data collection was part of the routine work of the participating registries for which -according to each country's rules and regulationsinformed consent was or was not required. Cause of death was defined and categorized according to the ERA-EDTA coding system [START_REF]ERA-EDTA Registry Annual Report 2016[END_REF] .

SLE patients and the matched comparison group

J o u r n a l P r e -p r o o f

The analyses included data on patients requiring KRT (dialysis and kidney transplantation) for kidney failure due to SLE (ERA-EDTA primary renal disease code 84) [START_REF]ERA-EDTA Registry Annual Report 2016[END_REF] (Figure S1). These patients will hereafter be referred to as SLE patients, and all other patients will be indicated as non-SLE patients. The median age at the onset of KRT differed substantially between SLE patients (42.9 years) and non-SLE patients (66.4 years). Therefore we matched the entire cohort of SLE patients who started KRT between 1992 and 2016 to non-SLE patients starting KRT within the same period, by age at KRT initiation (per five years), by sex and year of KRT initiation (per five years) at a ratio of 1 to 5. In addition, we separately matched SLE patients who received a kidney transplant between 1992 and 2016 to non-SLE transplant recipients, by age at the time of kidney transplantation (per five years), sex and year of transplantation (per five years) at a ratio of 1 to 5. The same strategy was applied separately for recipients of living and deceased donor kidneys. SLE patients who started KRT between 1992 and 2016 and received a kidney transplant in the same period were included in both the KRT cohort and the kidney transplant cohort, however, the comparator group was assembled separately as the matching variables for the KRT cohort (e.g. age at KRT onset) and the kidney transplant cohort (e.g. age at kidney transplantation) differed.

Statistical analysis

The incidence of KRT per million population (pmp) was studied per country/region over the entire study period. For the time trend analysis the incidence rate was studied by year of KRT onset, and prevalence was assessed on 31 December of each year for all participating European countries/regions combined. To enable comparisons over time and between countries the adjusted incidence and prevalence were calculated using the age and sex distribution of the EU J o u r n a l P r e -p r o o f Population in 2005. Time trends were examined using Joinpoint regression, with the observed rate as the outcome and year as the explanatory variable. The average annual percent change was computed using Poisson regression as provided by the Joinpoint Regression Program (National Cancer Institute; version 4.6.0). [START_REF]Joinpoint Regression Program. Version 4.6.0. Statistical Research and Applications Branch[END_REF] To compare the characteristics of the SLE patients to the comparator group, the Mann-Whitney test was used for continuous variables and the chi-square test for categorical variables. A two-tailed p-value of less than 0.05 was considered as statistically significant.

The survival analyses were performed using the Kaplan-Meier method and Cox regression analysis, and were performed unadjusted and adjusted for age, sex, time period and country. In order to allow all patients to complete the follow-up period of ten years for all survival analyses, a subset of the initial 1992-2016 cohort was used, only including patients until 2006 (Figure S1).

For patient survival on KRT, individuals who initiated treatment between 1992 and 2006 were included. The first day on KRT was defined as the starting point, and patient death was the event studied. Follow-up time included treatment changes from dialysis to kidney transplantation and vice versa, and was censored at recovery of kidney function (defined as interruption of dialysisexcluding kidney transplantation -for more than 30 days), at loss to follow-up, at the end of the follow-up period on 31 December 2016, or at ten years of follow-up, whichever occurred first.

Patient survival on KRT was compared between SLE patients and the comparison group, between men and women with SLE, and among the periods of 1992-1996, 1997-2001 and 2002-2006. Furthermore, we examined the impact of time spent on dialysis versus with a functioning kidney transplant during follow-up on the risk of death in a sensitivity analysis in which treatment modality for kidney failure was incorporated as a time-dependent covariate in a Cox regression model that compared patients with SLE to the comparator group.

J o u r n a l P r e -p r o o f

For patient survival on dialysis (hemodialysis or peritoneal dialysis), individuals who initiated dialysis between 1992 and 2006 were included. The starting point was defined as the first day on dialysis, and the event studied was patient death. Follow-up time was censored at recovery of kidney function, loss to follow-up, on the day of kidney transplantation (unless a patient restarted dialysis within 7 days after transplantation), at the end of the follow-up period on 31 December 2016, or at ten years of follow-up, whichever occurred first. Patient survival on dialysis was compared between SLE patients and the comparator group, between patients on hemodialysis and peritoneal dialysis at day 90, and among the periods of 1992-1996, 1997-2001 and 2002-2006. Patient and graft survival after kidney transplantation was investigated for patients who received their first kidney transplant between 1992 and 2006. The date of the first kidney transplantation was defined as the first day of follow-up. For patient survival after kidney transplantation, the event studied was death, and for graft survival the event was all-cause graft failure (defined as return to dialysis, retransplantation or death). Patient and graft survival after kidney transplantation were compared between SLE patients and the comparator group, between recipients of living and deceased donor kidneys, and among the periods of 1992-1996, 1997-2001 and 2002-2006. Hazard ratios for patient and graft survival after kidney transplantation were additionally adjusted for dialysis vintage.

Competing risk analyses were carried out to calculate the cumulative incidence of kidney transplantation from the onset of KRT in SLE patients and the comparator group, taking into account the competing risk due to death. To examine the trend in the probability of transplantation over time, this analysis was done separately for patients starting KRT in the periods 1992-1996, 1997-2001 and 2002-2006.

J o u r n a l P r e -p r o o f

All analyses were performed using SAS 9.4.

Results

Incidence and prevalence

Out of 280,892 patients who started KRT between 1992 and 2016, 1,826 commenced this treatment due to SLE-related kidney failure (0.65%, ranging from 0.46% to 1.00% in kidney registries across Europe) (Table 1). The age-and sex-standardized incidence of KRT for kidney failure due to SLE ranged from 0.46 pmp in Finland to 1.24 pmp in the Valencian region (Spain).

Characteristics of SLE patients and the non-SLE comparator group

The 1826 SLE patients starting KRT between 1992 and 2016 were matched to 9130 patients without SLE by age (median 42.9 years), sex (79% women) and year of KRT initiation (Table 2). SLE patients more frequently started KRT with hemodialysis (73.8%) than matched non-SLE patients (70.4%; p=0.003), but after 90 days this difference became non-significant. By contrast, SLE patients underwent pre-emptive kidney transplantation less often (5.0%) than matched non-SLE patients (7.8%; p<0.001), and during the first ten years of follow-up after J o u r n a l P r e -p r o o f KRT initiation fewer SLE patients (46.9%) received a first kidney transplant than matched non-SLE patients (51.9%, p<0.001).

In addition, 999 patients with SLE who received a first kidney transplant (SLE transplant recipients) between 1992 and 2016 were matched with 4995 without SLE (non-SLE transplant recipients) who received a first transplant (Table 2). In the SLE transplant recipients, the median age at the time of kidney transplantation was 39.1 years (interquartile range [IQR]: 31.3-48.9), and 82.2% were women. SLE transplant recipients spent more time on dialysis before transplantation (median: 2.0 years [IQR: 0.9 -3.9]), than matched non-SLE transplant recipients (1.5 years [IQR: 0.5-3.0], p<0.001).

Patient survival after KRT initiation

Within 90 days after commencing KRT, a larger proportion of SLE patients than matched non-SLE patients died (3.1% vs. 2.2%, p=0.03) (Table 2). The unadjusted patient survival on KRT in SLE patients and in the comparator group is shown in Figure 2A and Table 3. SLE patients showed lower survival probabilities during the first 10 years after KRT initiation than did the comparator group without SLE, and this was also the case for the patient and graft survival after a first kidney transplantation. Although increased, the difference in overall adjusted risk of death on KRT between SLE patients and matched non-SLE patients was not statistically significant (adjusted hazard ratio [HR]=1.11 [95%CI: 0.99-1.23], p=0.06). In sensitivity analyses additionally adjusting for treatment modality as time-dependent covariate, the HR was 1.04 (95%CI: 0.93-1.16). Within the group of SLE patients, survival on KRT tended to be worse in men, with an adjusted HR for women of 0.81 (95%CI: 0.64-1.02, p=0.07) (Table S1). As shown in Table S2, patient survival was better in both SLE patients and the comparator group who 3. SLE patients had lower patient and graft survival probabilities during the first 10 years after a first kidney transplantation than the matched comparator group without SLE, but only the risk of death was statistically significantly higher for SLE patients (HR=1.25 [95%CI: 1.02-1.53], p=0.03). This increase in the risk of death for SLE kidney transplant recipients was also observed after deceased donor kidney transplantation (Table S3), while this difference was statistically insignificant after living donor kidney transplantation. The risk of all-cause graft failure was similar for SLE patients and matched non-SLE patients (HR=1.09 [95%CI: 0.95-1.27], p=0.2, Table 3), and this was also the case for both living and deceased donor kidney transplantation (Table S3). Within the group of SLE patients, patient mortality after a first kidney transplantation was similar for recipients of living and deceased donor grafts (HR for living versus deceased donor grafts =: 0.64 [95%CI: 0.38-1.08], p=0.09), while the risk of all-cause graft failure was lower for recipients of a living donor grafts (HR for living versus deceased donor grafts = 0.60 [0.41-0.88], p=0.009, Table S1).

Patient and graft survival of patients receiving their first kidney transplant improved over time in both SLE patients and the non-SLE comparator group (Table S2). This finding was accompanied by an increase in the median age of SLE patients commencing KRT and undergoing transplantation (from 37.6 and 30.7 years in period 1992-1996 to 43.6 and 35.1 years in the most recent period, respectively).

J o u r n a l P r e -p r o o f

Figure S2 shows a longer median time to a first kidney transplantation for SLE patients than within the non-SLE comparator group. However, as the median time to first kidney transplantation in the non-SLE comparator group increased over time, the difference in access to kidney transplantation between SLE patients and the comparator group became similar in the most recent period.

Causes of death

As shown in Table 4, the percentage of deaths due to cardiovascular diseases among patients on KRT was similar in the SLE group and the comparator group (31.1% vs. 34.3%, p=0.2). However, the percentage of deaths due to infection was significantly higher in SLE patients (24.8% vs. 16.9%, p<0.001). On the other hand, a higher percentage of patients in the non-SLE comparator group died from malignancy (8.7% vs. 5.4%, p=0.03). There was no statistically significant difference in the causes of death between the SLE group and the comparator group after kidney transplantation.

Discussion

In this study of a European cohort with SLE, we found a stable incidence of KRT for kidney failure. The results suggest similar patient survival of SLE patients on KRT when compared with an age-and sex-matched group of patients on KRT due to other causes of kidney failure. Patient survival after a first kidney transplantation appeared to be worse in SLE patients, although graft survival was similar. Cardiovascular disease was the most common cause of death J o u r n a l P r e -p r o o f in both groups. A greater probability of death due to infection was observed in the SLE patients.

Finally, we found significant improvement in patient survival on KRT over the studied period.

Our results show that though the incidence of KRT for kidney failure due to SLE varied between countries (0.46 to 1.24 pmp), it has remained stable over the last 25 years. This finding corresponds with the most recently reported data from the USRDS by Ward [START_REF] Ward | Changes in the incidence of endstage renal disease due to lupus nephritis in the United States, 1996-2004[END_REF] and Sexton et al. [START_REF] Sexton | ESRD from lupus nephritis in the United States, 1995-2010[END_REF] , showing a stable or declining incidence of KRT among patients with SLE. However, it remains unclear whether the risk of kidney failure in patients with SLE and LN has changed over time.

Lack of national data on the annual incidence and prevalence of LN precludes the opportunity to define the number of SLE patients at risk of kidney failure. Data from the first decade of the 21 st century in a cohort from the UK [START_REF] Rees | The incidence and prevalence of systemic lupus erythematosus in the UK, 1999-2012[END_REF] suggested a gradual decline in SLE incidence, which might result in a decreased incidence of LN and eventually also of SLE-related kidney failure, but this still needs to be determined. Nevertheless, a potential benefit of new therapies for LN introduced at the turn of century, such as mycophenolate mofetil and rituximab, could not be demonstrated, since the coverage of this era by the study period was too short. However, we observed an increase in the median age of SLE patients at the start of KRT over time by six years, implying that some improvement in the outcome of the disease has been achieved, although not yet reflected in the incidence of patients with LN-related kidney failure.

Our data show that the survival of SLE patients on KRT is similar when compared with patients with kidney failure due to other causes. Comparable mortality of SLE and non-SLE patients on KRT has also been published by Sexton and co-workers [START_REF] Sexton | ESRD from lupus nephritis in the United States, 1995-2010[END_REF] , even though the studied cohort included a high percentage of African-Americans who are expected to have a less favorable prognosis when diagnosed with SLE. The median follow-up of patients in this study was 4.4 years, which may not allow for the development of specific complications that are J o u r n a l P r e -p r o o f unique to patients with SLE. A variety of factors may contribute to adverse outcomes in SLE patients such as history of aggressive immunosuppressive treatment and presence of antiphospholipid antibodies potentiating the risk of vascular access failure and thrombotic complications, including allograft thrombosis [START_REF] Ames | Antiphospholipid antibodies and renal transplant: A systematic review and meta-analysis[END_REF][START_REF] Fuentes | Survival of lupus nephritis patients after renal transplantation in Malaga[END_REF] . Delayed wait-listing for kidney transplantation, causing the prolongation of time to transplant as shown in our study, likely due to persistent disease activity or complications arising from the disease itself [START_REF] Cheigh | End-stage renal disease in systemic lupus erythematosus[END_REF] results in overall lower transplantation rate, including pre-emptive kidney transplantation.

While the survival of SLE patients after kidney transplantation was inferior when compared with the matched non-SLE comparator group in our study, there was no difference in graft survival, suggesting that SLE patients are more prone to fatal outcomes of complications, especially infections. Similar patient survival disadvantage after kidney transplantation and tendency to comparable graft survival was noted in two other registry studies [START_REF] Chelamcharla | The outcome of renal transplantation among systemic lupus erythematosus patients[END_REF][START_REF] Zhang | Long-term outcomes of end-stage kidney disease for patients with lupus nephritis[END_REF] . Recurrence of LN does not appear to have significant clinical relevance [START_REF] Contreras | Recurrence of lupus nephritis after kidney transplantation[END_REF] . Only a few registry studies have analyzed the prognosis of patients with SLE after kidney transplantation to date. Marked improvement in patient outcomes has recently been demonstrated by Jorge et al. [START_REF] Wallace | Renal Transplantation and Survival Among Patients With Lupus Nephritis: A Cohort Study[END_REF] The investigators compared outcomes of SLE patients on KRT who were wait-listed for kidney transplantation and those who were eventually transplanted. Patients who received a kidney transplant had a 70% reduction in all-cause mortality, when compared with patients remaining on hemodialysis (adjusted HR, 0.30 [95% CI, 0.27 -0.33]).

Although SLE has been shown to be among important non-traditional risk factors of CV disease, we found no difference in CV mortality in patients with SLE on KRT compared with matched non-SLE patients. Comparable CV morbidity measured by hospitalization rate in SLE patients on KRT was observed by Ward [START_REF] Ward | Cardiovascular and Cerebrovascular Morbidity and Mortality Among Women With End-Stage Renal Disease Attributable to Lupus Nephritis[END_REF] , suggesting that SLE might not increase the risk of CV J o u r n a l P r e -p r o o f complications beyond the CV risk of patients on KRT in general. This notion could in part be a explained by burnt-out SLE in patients on dialysis [START_REF] Mojcik | End-stage renal disease and systemic lupus erythematosus[END_REF][START_REF] Mattos | Disease activity in systemic lupus erythematosus patients with endstage renal disease: systematic review of the literature[END_REF] , therefore not contributing to an inflammatory environment that could enhance the progression of atherosclerosis. Similar proportions of CV and infection-related deaths were observed by Zhang et al. [START_REF] Zhang | Long-term outcomes of end-stage kidney disease for patients with lupus nephritis[END_REF] , demonstrating that SLE patients were more prone to die from infections than the comparator group, but less likely to die due to CV complications. The higher risk of infectious diseases associated with immunosuppressive therapy might partly explain this increased mortality in SLE patients on KRT.

To our knowledge, this is the largest European study thus far on the outcomes of SLE patients on KRT, with almost 100% coverage in selected regions. An important strength of the study is matching of the SLE cohort to the comparator group to reduce possible confounding factors at the time of KRT initiation. Only a few registry studies have attempted to match comparator groups [START_REF] Sexton | ESRD from lupus nephritis in the United States, 1995-2010[END_REF][START_REF] Contreras | Comparison of mortality of ESRD patients with lupus by initial dialysis modality[END_REF] , in some cases with certain shortcomings [START_REF] Levy | Outcome of patients with systemic lupus erythematosus on chronic dialysis: an observational study of incident patients of J o u r n a l P r e -p r o o f the French National Registry 2002-2012[END_REF] . Other studies used adjustment methods to eliminate the effect of known confounding factors at the onset of KRT.

This study has certain limitations that are related to the registry-based nature of the data.

The unavailability of information regarding the disease course, including duration before KRT, kidney biopsy results, severity of extrarenal involvement and treatment choices, as well as the lack of data on comorbidities, vascular access, race and socioeconomic status precludes the possibility of adjusting for these probable confounding factors. In addition, data on transplantrelated characteristics such as time on the waiting list, presence of panel-reactive antibodies, cold ischemia time and immunosuppressive regimen as possible confounders were not available.

In conclusion, data from the ERA Registry show plateauing of the incidence of KRT for kidney failure due to SLE in this large European cohort. Our findings also suggest that the J o u r n a l P r e -p r o o f prognosis of SLE patients on KRT has improved over recent decades. Finally, we show that SLE as an underlying cause of kidney failure has minimal effect on the overall patient survival after starting KRT, although it does have an unfavorable effect on overall patient survival after kidney transplantation. This may be caused by disease-specific complications but may also in part be due to complications arising from long-term immunosuppressive burden. Therapeutic advances resulting from better understanding of these complications have the potential to improve survival after kidney transplantation in patients with SLE.

Supplementary Material

Table S1 -Patient survival of systemic lupus erythematosus (SLE) patients after kidney replacement therapy (KRT) or dialysis initiation in 1992-2006, and patient and graft survival after a first kidney transplantation (KTx) in 1992-2006, analyzed for different subgroups.

Table S2 -Patient survival after kidney replacement therapy (KRT) or dialysis initiation, and patient and graft survival after a first kidney transplantation (KTx) in systemic lupus erythematosus (SLE) patients and in a matched non-SLE comparator group, compared for three time periods (1992-1996, 1997-2001, 2002-2006) according to the year of KRT initiation or the year of a first KTx.

Table S3 -

Article information

Authors' contributions

Research idea and study design: OD, ZH, VT, KJJ, VSS, AK; data acquisition: MA, FC, PF, LFS, JH, MHH, KH, JK, JDM, RP, MS, RS, JPT, OZ, ZAM; data analysis/interpretation: OD, AK, KJJ, VSS, VT, ZH; supervision or mentorship: VT, KJJ. Each author contributed important intellectual content during manuscript drafting or revision and agrees to be personally accountable for the individual's own contributions and to ensure that questions pertaining to the accuracy or integrity of any portion of the work, even one in which the author was not directly involved, are appropriately investigated and resolved, including with documentation in the literature if appropriate. J o u r n a l P r e -p r o o f J o u r n a l P r e -p r o o f J o u r n a l P r e -p r o o f

J o u r n a l P r e -p r o o f

Figure 1

 1 Figure1shows that the age-standardized incidence of KRT due to SLE was stable between 1992

  J o u r n a l P r e -p r o o f started KRT between 2002 and 2006, than for those who initiated treatment between 1992 and 1996. Patient and graft survival after kidney transplantation and access to kidney transplantation Patient and graft survival after a first kidney transplantation in SLE patients and in the comparator group is shown in Figures 2B and 2C and in Table

  Patient and graft survival after a first kidney transplantation (KTx) from a living or deceased donor in 1992-2006 among systemic lupus erythematosus (SLE) patients versus patients in the matched non-SLE comparator group.

Figure S1 -

 S1 Figure S1 -Flow diagram illustrating the number of patients selected for the analysis of the

Figure S2 -

 S2 Figure S2 -Competing risk analysis comparing the time to first kidney transplantation (KTx)

J o u rFiguresFigure 1 .

 1 Figures

Figure

  Figure Kaplan-Meier survival analysis, comparing (A) patient survival from day 1 after starting kidney replacement therapy (KRT) in 1992-2006, and (B) patient and (C) graft survival after first kidney transplantation in 1992-2006 in SLE patients and the matched comparator group. J o u r n a l P r e -p r o o f

Table 1 .

 1 Incidence of kidney replacement therapy (KRT) for kidney failure due to systemic lupus erythematosus (SLE), by country/region in 1992-2016. Adjusted for age and sex distribution using the European Standard Population of 2005 as reference population.

		All KRT patients			SLE patients on KRT
				Total group		Male patients	Female patients
	Country/region	N total	N	%	Crude PMP	Adjusted PMP a	Adjusted PMP a	Adjusted PMP a
	Andalusia (Spain)	21662	216	1.00	1.12	1.14	0.40	1.84
	Austria	27644	166	0.60	0.81	0.81	0.32	1.27
	Basque country (Spain)	5779	50	0.87	0.95	0.93	0.46	1.38
	Catalonia (Spain)	24135	127	0.53	0.74	0.74	0.36	1.10
	Denmark	16265	109	0.67	0.80	0.83	0.28	1.36
	Finland	11342	59	0.52	0.45	0.46	0.19	0.72
	French-speaking Belgium	17990	86	0.48	0.77	0.79	0.49	1.08
	Greece	45724	209	0.46	0.77	0.76	0.34	1.16
	Iceland	514	4	0.78	0.54	0.53	0	1.04
	Norway	11214	106	0.95	0.91	0.95	0.52	1.37
	Scotland (UK)	12839	77	0.60	0.60	0.61	0.20	0.99
	Sweden	27613	198	0.72	0.87	0.91	0.34	1.45
	the Netherlands	41787	278	0.67	0.69	0.70	0.35	1.03
	Valencian region (Spain)	16384	141	0.86	1.25	1.24	0.35	2.09
	All countries	280892	1826	0.65	0.80	0.82	0.35	1.26
	Abbreviations: KRT, kidney replacement therapy; PMP, per million population.

a

Table 2 .

 2 Characteristics of systemic lupus erythematosus (SLE) patients and non-SLE patients, including patients in the matched comparator group, initiating kidney replacement therapy (KRT) by either dialysis or kidney transplantation, or receiving a first kidney transplant (KTx) in 1992-2016.

			Non-SLE
	SLE	Total	Matched comparator group a	p-value*

a For SLE patients starting KRT the comparator group comprised of a subgroup of non-SLE patients starting KRT (matched by age group and sex). For SLE patients receiving a first KTx the comparator group comprised of a subgroup of non-SLE patients receiving a first KTx (matched by age group and sex).

* P-values are based on the comparison between SLE patients and the matched non-SLE comparator group.

Table 3 .

 3 Patient survival after kidney replacement therapy (KRT) overall or dialysis initiation in 1992-2006, and patient and graft survival after a first kidney transplantation in 1992-2006, in systemic lupus erythematosus (SLE) patients versus age-, sex-and time period-matched non-SLE the group. Adjusted for age, sex, time period, dialysis vintage (for patient and graft survival after kidney transplantation only) and country.

	Crude survival probabilities (95% CI)	Hazard ratios (95% CI)

*

Table 4 .

 4 Causes of death distribution among systemic lupus erythematosus (SLE) patients and patients in the matched non-SLE comparator group who died within ten years after starting kidney replacement therapy, or after receiving a first kidney transplant in 1992-2006.Note: Percentages may not add up to 100% because of rounding. For SLE patients starting kidney replacement therapy (KRT) the comparator comprised of non-SLE patients starting KRT, matched by age group and sex. For SLE patients receiving a first kidney transplant (KTx) the comparator group comprised of non-SLE patients receiving a first KTx, matched by age group and sex.

	SLE patients	Matched comparator group a
	N	%	N	%	p-value

a

Acknowledgements

We would like to thank the patients and the staff of the dialysis and transplant units for contributing the data via their national and regional kidney registries. Furthermore, we gratefully acknowledge the following registries and persons for their contribution of the data: Austrian Dialysis and Transplant Registry [OEDTR] (R. Kramar); French speaking Belgian Society of Nephrology [GNFB] (JM. des Grottes); Danish Nephrology Registry [DNS] (J.G. Heaf); Finnish Registry for Kidney Diseases (J. Helve and P.H. Groop); Hellenic Renal Registry (G. Moustakas); Icelandic End-Stage Renal Disease Registry; Norwegian Renal Registry (A.V. Reisaeter, and A. Åsberg); Swedish Renal Registry [SRR] (K.G. Prütz, M. Stendahl, M. Evans, H. Rydell and T. Lundgren); Dutch Renal Registry [RENINE]; Scottish Renal Registry [SRR] (All of the Scottish renal units); and the regional registries of Andalusia [SICATA] (P. Castro de la Nuez (on behalf of all users of SICATA)), Basque country [UNIPAR] (Á. Magaz, J. Aranzabal, M. Rodrigo, and I. Moina), Catalonia [RMRC] (J. Comas), Valencian region [REMRENAL] (N. Fuster Camarena and J. Pérez Penadés); and the other ERA Registry committee members not mentioned above for their advice in the analysis and the drafting of this paper: C. Zoccali, L. Mercadal, S.S. Sørensen, and E. Vidal; and the AMC Registry office for data collection and management. The ERA Registry is funded by the European Renal Association (ERA). This article was written by Ondrej Derner, Anneke Kramer, Zdenka J o u r n a l P r e -p r o o f

Support

OD received financial support from Charles University´s PROGRES Q25/LF1 program to cover travel expenses.

Financial disclosure

The authors declare that they have no relevant financial interests.