Single-shot energetic-based estimator for entanglement genesis

Producing and certifying entanglement between distant qubits is a highly desirable skill for quantum information technologies. Here we propose a new strategy to monitor and characterize entanglement genesis in a half parity measurement setup, that relies on the continuous readout of an energetic observable which is the half-parity observable itself. Based on a quantum-trajectory approach, we theoretically analyze the statistics of energetic fluctuations for a pair of continuously monitored qubits. We quantitatively relate these energetic fluctuations to the rate of entanglement produced between the qubits, and build an energetic-based estimator to assess the presence of entanglement in the circuit. Remarkably, this estimator is valid at the single-trajectory level. Our work paves the road towards a fundamental understanding of the stochastic energetic processes associated with entanglement genesis, and opens new perspectives for witnessing non-local quantum correlations thanks to thermodynamic quantities.


I. INTRODUCTION
In the past years, research towards quantum thermodynamics has led to tremendous experimental and theoretical achievements, using concepts and tools from three fondamental theories in physics, thermodynamics, quantum mechanics and quantum information theory, see reviews [1][2][3] and references therein. Similarly to classical thermodynamics, these advances concern as well fundamental issues, e.g. the definitions of thermodynamical quantities, heat, work, entropy, for quantum systems [4,5], and the derivation of fluctuation theorems [6][7][8][9][10][11], as more applied issues, e.g. the development of quantum thermal machines that can outperform their classical counterparts [12]. Emblematic examples are quantum thermoelectric engines [13][14][15] and quantum thermal machines able to generate quantum correlations between a reduced number of small systems [16][17][18][19].
Of particular interest for this work is the investigation of energy exchanges associated with quantum information processing (QIP) tasks, e.g.
to erase information, to exploit coherence properties of distinct quantum systems, to generate quantum correlations and to perform quantum measurements. To pursue this goal, one can distinguish between two approaches, an operational or resource-theory approach and a quantum-trajectory approach. The first one, see [3], formalizes the quantum information tasks as a set of allowed unitary operations onto a (multi-partite) quantum system and then determine the energetic cost * geraldine.haack@unige.ch of these mathematical operations, in terms of the free energy for instance. This resource-theory approach uses concepts and tools from quantum information theory and has led to groundbreaking results towards understanding thermodynamics at the quantum scale, e.g. the demonstration of a work gain during an information's erasure procedure in presence of entanglement [20], the derivation of a minimal amount of free energy necessary to generate classical and non-classical correlations [21] and the characterization of quantum thermal machines with lower bounds for their efficiency [22][23][24][25].
In contrast, the quantum-trajectory approach focuses on the stochastic dynamics of a (multi-partite) quantum system in weak interaction with an environment, e.g. a thermal bath or a measurement apparatus, and identifies trajectory-dependent thermodynamics quantities such as heat, work and entropy production [11,[26][27][28]. This identification is done within the framework of stochastic thermodynamics, initially developed to derive the thermodynamic laws from a microscopic description of small systems, for which fluctuations play an important role [29,30]. In particular, this approach was used to investigate the thermodynamics properties of a forced harmonic oscillator [31], and of a weakly monitored single quantum two-level system (qubit) [27,32], demonstrating for instance the ability to convert efficiently a quantum measurement into useful work.
Extending this quantum-trajectory approach to investigate thermodynamic quantities of quantum correlated systems is a non-trivial issue. It requires a physical model that allows for tracing back the quantum trajectories of two quantum systems while being entangled. In this work, we achieve this exciting and challenging task by exploiting a parity measurement setup, known  to induce entanglement between two qubits initially in a separable state. In the last decade, several theoretical works investigated this parity measurement within various physical systems, especially superconducting qubits jointly measured by a cavity mode [33][34][35] and semiconductor quantum dots jointly measured by a quantum point contact [36,37] or by an electronic Mach-Zehnder interferometer in quantum transport experiments [38,39]. Since then, measurement-induced entanglement genesis has eventually been implemented within circuit QED experiments [40,41]. These platforms don't only provide the technological know-how to access the quantum trajectories of individual quantum systems [42][43][44], but also of several quantum systems in the context of the parity measurement [41]. Hence, this parity-measurement setup presents all features required to investigate the thermodynamics quantities associated with the generation of entanglement within a quantum-trajectory approach in the framework provided by stochastic thermodynamics.
In this work, we conduct an analytical and numerical analysis demonstrating that quantum heat fluctuations along single trajectories constitute the energy fingerprints of the entanglement genesis process. To this end, we derive upper and lower energetic bounds for the entanglement genesis rate, based on heat fluctuations. This allows us to introduce energetic witnesses that certify the presence of entanglement, based solely on the measurements of heat flow and heat fluctuations. Importantly, these witnesses have a zero error rate (they never associate a positive answer when a trajectory does not lead to entangled qubits). Their success probability depends on the time window over which heat fluctuations are averaged and reaches unity for large enough measurement time. Our results are promising towards deriving single-shot energetic witnesses to certify the presence of quantum correlations, a novel alternative to present tomography-type measurements.
The paper is organized as follows. In Section II, we briefly present the half-parity measurement and then derive in Section III the associated quantum trajectories, which are at the basis of our thermodynamic analysis. In Section IV, we show that the average quantum heat exchange indicates whether our measurement scheme is a quantum non-demolition measurement or not, but does not contain any information about the generation of quantum correlations between the qubits. In Section V, we derive the fluctuations of the infinitesimal heat exchange along single trajectories. This allows us to derive energetic bounds for the entanglement genesis rate in Section VI that depen on these infinitesimal heat exchange fluctuations. Finally, in Section VII, we derive energetic witnesses based on the lower bound for entanglement genesis rate and characterize them in terms of their success and error rates.

II. HALF-PARITY MEASUREMENT-INDUCED ENTANGLEMENT
The physical setup we investigate is sketched up in Fig.1. It consists of two distant qubits initially in a separable state, described within the charge basis spanned by the two-qubit quantum states {|↑↑ , |↑↓ , |↓↑ , |↓↓ }.
Here |↑ and |↓ denote respectively the ground and first excited states of each qubit. The principle of a parity measurement is to distinguish the even two-qubit states {|↑↑ , |↓↓ } from the odd two-qubit states {|↑↓ , |↓↑ }, but not the quantum states within each parity subspace. This measurement is formally described by the observ- that has two eigenvalues ±1 corresponding to the measurement of a quantum coherent superposition of states within the even and odd subspaces respectively. Here σ (i) z stays for the z-Pauli matrix for qubit i.
In this work, we consider more precisely a half-parity measurement that corresponds to the joint observablê Φ = |↑↑ ↑↑| − |↓↓ ↓↓| that has three eigenvalues ±1 and 0. The first two eigenvalues correspond respectively to the even two-qubit states |↑↑ and |↓↓ , meaning that this half-parity measurement is able to distinguish the qubits when being in an even quantum state and does not induce entanglement within the even parity subspace. In contrast, the measurement outcome 0 is obtained whenever the qubits are projected into the odd parity subspace spanned by {|↑↓ , |↓↑ }. This outcome does not allow one to distinguish between these two odd states, and consequently, the two qubits are driven into a coherent superposition of those states and become entangled. Hence, this half-parity measurement presents the specificity of producing in a probabilistic way entangled and product states. This allows us to distinguish quantum trajectories associated with entanglement genesis or not using post-selection, and to compare their thermodynamical properties.
Experimentally, this half-parity measurement has been successfully implemented within a circuit QED setup [40,41]. It consists of two distant superconducting transmon qubits placed in two different single-mode cavities. Parameters of the experiments, e.g. qubits and cavities frequencies and couplings strengths between the qubits and their own cavities are adjusted such that the amplitude shift of the output mode reflects the three possible outcomes of the half-parity observableΦ. The concurrence, a monotone measure to characterize two-qubit entanglement [45], was then used to demonstrate halfparity measurement-induced entanglement. This measure is computed from the density matrix elements of the two qubits and ranges between 0 (qubits in a separable state) and 1 (maximally entangled qubits, or qubits in a Bell state). Owing to state tomography, it was demonstrated that the concurrence is finite when the obtained measurement outcome is the odd eigenvalue 0, whereas it remains null when outcomes ±1 corresponding to the even states is obtained.

III. QUANTUM TRAJECTORIES
We assume two qubits of identical energy splitting described by their HamiltonianĤ S in the two-qubit Hilbert spaceĤ In this project, we consider a weak continuous measurement of the joint observableΦ of the two qubits. The corresponding time-dependent measurement outcome denoted I(t) is stochastic and due to the inevitable backaction of theΦ-measurement onto the qubits, each realization of the measurement procedure is associated to a quantum stochastic trajectory followed by the qubits and labelled γ in the rest of the manuscript [46,47]. We assume the initial state of the qubits is a known pure state, such that the trajectory γ is made of a sequence of pure states {|ψ γ (t) }. To each trajectory γ, we associate a stochastic measurement record I γ (t).
Here, we have introduced the expectation value Φ (t) γ = ψ γ (t)|Φ|ψ γ (t) and set = 1 for clarity. The infinitesimal Wiener increment dW γ (t) is a stochastic variable characterized by a zero average and variance dt, i.e. dW γ = 0 and dW 2 γ = dt, where · denotes the average over the realizations of the measurement. Note that throughout this article, we use Ito's convention for stochastic differential calculus [48,49]. This infinitesimal Wiener increment encodes Gaussian fluctuations of the measurement record I γ (t) around its expectation value Φ (t) γ and therefore captures the detector's shot noise in the weak coupling limit [47,50]. Finally, the detector measurement rate Γ is defined as the rate at which one is able to distinguish the measurement outcomes from the detector's shot noise [38,39,50,51]. Based upon the knowledge of I γ , the conditional dynamics of the two qubits subject to the weak measurement of the half-parity observableΦ is captured by the stochastic Schrödinger equation When manipulating quantum trajectories, it is important to distinguish between the measurement record I γ (t) and the experimental measurement outcome, denoted here J γ (t). The latter corresponds to the measurement record averaged over a finite time interval t that is set by the single-realization measurement duration: with W γ (t) = t 0 dW γ (t ) a Gaussian random variable with mean zero and variance t. The probability distribution of J γ (t) is a sum of three Gaussian functions of variance 1/4Γt, each peaked around one of the eigenvalues of the measured observableΦ, i.e. 0, ±1 [47].
(1), one can solve analytically Eq.(3) at time t as a function of the stochastic measurement outcome J γ (t) [47]. For any time t, we find |ψ γ (t) = |ψ(J γ , t) with: with N γ (t) the time-dependent normalization factor N γ (t) = (1 + e −2Γt cosh(2 √ ΓW γ ))/2. The initial separable state of the two qubits is denoted |Ψ(0) . In the rest of the manuscript, we will assume that each qubit is at t = 0 in a maximal superposition state, Indeed, this state belongs to the set of optimal states that lead to maximally entangled final states (i.e. Bell states) when the qubits are subject to a (half-) parity measurement [38]. This choice will therefore allow us to compare thermodynamic properties of maximally entangled states and of product states. We quantify the presence of entanglement at final time t using the concurrence, a monotone entanglement measure for two-qubit states [45]. The qubits being in a pure state at each instant of time along their trajectory γ, their concurrence takes the simple form C γ (t) = max {0, 2|ad − bc|} where a, b, c, d are the amplitudes of |ψ γ (t) with respect to the computational states |↑↑ , |↑↓ , |↓↑ , |↓↓ respectively. Inserting Eq.(5), we get: In the long time limit t (4Γ) −1 , J γ (t) can only take one of three possible outcomes which are the eigenvalues of the half-parity measurement operator: J o = 0, J ↑↑ = 1, J ↓↓ = −1. Let us remark that the measurement becomes then projective. As a consequence we can label the trajectories with the long-time value of J γ and post-select the subsets of trajectories ending with an odd qubit's state (J γ → J o ), or with even states |↑↑ (J γ → J ↑↑ ) or |↓↓ (J γ → J ↓↓ ). We start our analysis of thermodynamic properties of entanglement genesis by investigating the heat and work exchanged along these three kinds of post-selected trajectories, depending on the final value of J γ .
In Fig. 2 a, we plot the evolution of the concurrence post-selected according to the three final outcomes J γ → J o , J ↑↑ and J ↓↓ respectively. The concurrence reaches the maximum value 1 in the first case, and zero in the two other cases, as expected for maximally entangled states and for product states. At long times, t > (6Γ) −1 , the value of the concurrence can be directly obtained by replacing in Eq.(7) J γ by the eigenvalues 0, ±1 of the observableΦ.

IV. AVERAGE THERMODYNAMIC QUANTITIES AFTER TRAJECTORY POST-SELECTION
To quantify the energetics of the entanglement genesis process, we introduce the internal energy U γ (t) of the two qubits along a given trajectory γ [28]: Considering the initial state |ψ(0) (see Eq. (6)), the initial internal energy U (0) is zero and constitutes the energy reference. In our setup, we face the simple situation whereĤ S commutes with the measurement observablê Φ. Although this is not the most general situation, it does not limit the scope of our results, and allows us to highlight the most important features towards the characterization of measurement-induced entanglement from thermodynamic properties. In the absence of driving,Ĥ S is time-independent.Hence no work is performed onto the qubits, and a change in the internal energy of the qubits can only arise from the measurement process itself. Moreover, it has been recently shown that the change in internal energy of a quantum system subject to a (quantum) measurement takes the form of a heat flow, here denoted Q [27,28,[52][53][54][55]. This form of energy exchange has no classical counter-part. As a consequence, in our model, the change of internal energy (and therefore the net heat received by the qubits) takes three different values depending on the measurement outcome at long times, t > (6Γ) −1 : At intermediate times, the heat exchanged up to time t (the only origin of change in internal energy of the qubits) depends on the exact outcome J γ and is given by This heat exchange Q γ (t), after post-selection, is plotted in Fig.2b as a function of time. The post-selection of the trajectories is made according to their long-time measurement outcome J γ (t) ∈ {0, ±1} for t (4Γ) −1 . Within each subset, an average is made. Numerically, the total number of quantum trajectories is 800 and the qubits are initially in a maximal superposition state.
The behavior of the total heat exchange for the three types of trajectories leads us to formulate interesting conclusions with respect to the half-parity measurement process. First, the presence of stochastic heat exchange hallmarks the presence of measurement-induced dephasing. Indeed, previous theoretical studies have shown that the measurement ofΦ does not induce any dephasing within the odd subspace spanned by the states |↑↓ and |↓↑ , whereas it affects the even states |↑↑ and |↓↓ with a dephasing rate given by 4Γ [38][39][40][41]. It corresponds to the situation observed in Fig.2 b. We find that the total heat exchange is zero for trajectories of qubits ending in the odd parity subspace, whereas Q γ (t) converges towards its final value ± at a rate 4Γ for the trajectories ending in an even state.
Second, the heat exchange averaged over all trajectories gives 0 at any time t. This mean value reflects that the specific measurement we are considered in our model is a quantum non-demolition measurement (QNDmeasurement) because [Ĥ S ,Φ] = 0 [56]. From a quantum information point of view, these QND-measurements do not induce any loss of information during the measurement process. In other words, the dephasing rate induced by the measurement onto the qubits corresponds exactly to the information acquisition rate of the detector, here 4Γ. From a thermodynamic point of view, this absence of information loss can be understood as an absence of any form of incoherent energy exchange: on average, Q γ = 0 ∀ t during the whole measurement process. These results show that the analysis of the total heat exchange provides us information about the characteristics of the quantum measurement and the associated measurement-induced dephasing processes and are in full agreement with previous works, see [27,28]. However, it does not provide any information about the presence or not of non-classical correlations. This is clear when comparing the curves for the concurrence and the stochastic heat exchange. No energetic hallmark of entanglement genesis can be gained directly from the value of the stochastic heat exchange. Below, we demonstrate that energetic hallmarks of quantum correlations are present in the variance of the stochastic heat exchange.

V. STOCHASTIC THERMODYNAMICS OF ENTANGLEMENT
We now focus on the fluctuations of the heat increment at time t, denoted δQ γ (t). This analysis is performed within the framework of quantum stochastic thermodynamics presented in [11,28]. The increment δQ γ (t) corresponds to the stochastic infinitesimal variation of the internal energy U γ (t) between times t and t + dt along q trajectory γ, and is related to the total heat exchange up to time t via: Using Ito's rule of stochastic calculus, this infinitesimal heat exchange is defined as Inserting Eqs. (1) and (5) into Eq. (13) and expanding the last term up to the first order in dt, we obtain For simplicity, we have introduced the populations P ij for the 4 two-qubit states defined as and we denote P e = P ↑↑ + P ↓↓ and P o = P ↑↓ + P ↓↑ the populations within the even and odd parity subspaces respectively. The infinitesimal stochastic heat increment δQ γ (t) corresponds to the infinitesimal heat flow in the time interval [t, t + dt] and depends on the Wiener increment dW that follows a Gaussian distribution. The first term in Eq. (14) is proportional to the modulus squared of the coherence between the two even states |↑↑ and |↓↓ . Here coherences refer to the off-diagonal terms in the density matrix of the two qubits ρ γ (t) = |ψ γ (t) ψ γ (t)| expressed in the basis {|↑↑ , |↑↓ , |↓↑ , |↓↓ }. Hence, the infinitesimal stochastic heat exchange term proportional to P ↑↑ P ↓↓ contains information about the dephasing processes present within the even subspace. Integrating this contribution following Eq.(11), one recovers the heat exchange proper to trajectories ending up in the even states |↑↑ and |↓↓ , compare Eq. (10) with J γ = ±1 and Eq. (5) projected onto states |↓↓ and |↑↑ .
In contrast, the second term in Eq. (14) is proportional to the coherences between even and odd states. Let us recall that a (half-) parity measurement generates entanglement by distinguishing the even states form the odd ones. It is therefore the loss of coherence between the two parity subspaces that brings the two qubits in a coherent superposition of odd states when outcome J γ = 0 is obtained. Hence, the value of the product P o P e in Eq.(14) is expected to contain information about entanglement genesis, and so does the heat increment δQ γ (t).
We therefore separate the infinitesimal stochastic heat increment into two contributions, with In order to quantify the fluctuations of heat at time t and highlight their connection with the concurrence increase, we define the ensemble average · Iγ (t) over the trajectories (or equivalently over the measurement record I γ (t)), during the time interval [t, t + dt], keeping the past records {I γ (t )} t <t fixed. This ensemble average allows us to get rid of the Wiener increment as dW 2 γ (t) Iγ (t) = dt and we now define the standard deviation of the quantum heat increment at time t as From the theoretical understanding of a half-parity measurement, the dephasing between the even and odd subspaces is responsible for the entanglement genesis, such that one can expect a connection between the concurrence increase and the standard deviation of δQ (eo) γ (t). This assumption can be validated by introducing σ eo (t) = δQ (eo) that is plotted in Fig.3 as a function of the measurement outcome J γ and time t. The contour plot superimposed on the density plot of σ eo (t) corresponds to constant values of the concurrence. Figure 3 highlights that indeed the concurrence increase is associated with non-zero fluctuations of δQ (eo) γ (t). In contrast, the concurrence plateaus correspond to the areas where σ eo vanishes. An equality linking the concurrence increase to this standard deviation can even be found in the long time limit, see App. A. However, the practical interest of such formula can be questioned as it would be impossible in a heat-sensitive measurement to distinguish the two contributions δQ Fortunately, it is possible to prove quantitative link between the concurrence increase and the total quantum heat standard deviation σ γ , as we expose in the remainder of this section.

VI. ENERGETIC BOUNDS ON ENTANGLEMENT GENESIS RATE
Remarkably, we can quantify the entanglement generation rate for by looking at the time-derivative of the concurrence which takes the form (see derivation in Appendix B): To compare this concurrence rate to the standard deviation σ γ (t), we perform the same average · Iγ (t) over dC γ /dt. As shown in Appendix B, this average can be related to the thermodynamical quantities Q γ and σ γ via the following inequality: These upper and lower bounds are exclusively expressed by thermodynamical quantities, and depend implicitly on the measurement outcome J γ . This inequality constitutes one of the main results of this work, together with the witnesses derived in the following section. Of interest towards non-equilibrium quantum thermodynamics, these bounds are valid at the level of a unique stochastic (quantum) trajectory. This validity is illustrated in Fig.4 for two examples of trajectories, leading qubits either into an entangled state or into a separable state. For the sake of clarity, we show the thermodynamical-based bounds with their fluctuations, whereas the concurrence rate has been averaged over a time interval of duration 0.4/Γ. This average is reflected in the error bars. In contrast to the lower bound for trajectories ending in a separable state that stays negative at long times, the lower bound for trajectories ending up in an entangled state rapidly decreases to 0 and remains positive and close to 0 at long times t 1/Γ. Indeed, in the long-time limit, the inequality (21) can be handled analytically: the lower bound for trajectories corresponding to qubits in an entangled state is trivially 0, whereas it reaches −4Γ for qubits ending up in a separable state, see Appendix B. These two limits are clearly distinguishable in Fig.4.
In the remainder of this article, we emphasize the particular role of the lower bound, showing that non-zero heat fluctuations are crucial to entanglement genesis and their amplitude compared to the heat flow's amplitude is crucial when time increases. Below we exploit this lower bound to build witnesses for certifying the presence of entanglement, that are solely based on thermodynamic quantities.

VII. ENERGETIC WITNESSES OF THE PRESENCE OF ENTANGLEMENT
To certify the presence of entanglement at time t, we require that This condition, together with inequality (24), allows us to derive a sufficient condition for certifying the presence of entanglement along a single trajectory in terms of thermodynamical quantities only. From the l.h.s of (21), we have at time t: In the spirit of quantum information theory where witnesses take negative value iff the state under consideration is entangled, we define a novel witness based solely on thermodynamical-based measurements: where we recall that σ γ (t) and Q γ (t) are the thermodynamical quantities averaged over several realizations in the time interval dt (through · Iγ (t) ). The usefulness of this witness is as follows. Given a trajectory γ defined by the set of measurement records {I γ (t )} t <t , the witness W will take negative values at each instant of time t iff the ensemble averaged quantities σ γ (t) and Q γ (t) during the time interval dt satisfy Eq.(23). The integration over the time interval [t i , t f ] is motivated to exploit information contained in the heat exchanged during the whole interval to predict entanglement at time t f > 1/Γ. We analyze the performance of our witness W as a function of this integration window as illustrated in Fig.5 (t i is set to 0, and t f is on the x-axis). Importantly, the witness W exhibits a zero error rate that does not depend on the integration duration, i.e. it never associates a negative answer when a trajectory does not lead to entangled qubits. In contrast, the success rate (corresponding to the ratio of detected vs. total number of trajectories leading to entangled states) does depend on the integration interval, but always stays above 60%, even for an infinitesimal integration time t f ∼ t i ∼ 0. The success rate reaches 100% for . Given a single trajectory γ, the performance of W depends on the time-integration window. Whereas the error rate remains 0 for all integration windows, its success rate can be tuned from 60% to 100% by adjusting the integration interval t f − ti. Continuous lines stay for t f = 10, whereas crosses stay for t f = 6. b) Analysis of the single-shot energetic witness Wss. Left: Error and success rates of the single-shot energetic witness Wss as a function of ti. Again, continuous lines stay for t f = 10, whereas crosses stay for t f = 6.. In contrast to W, this witness does not require any ensemble average, see Eq. (25). As expected, this implies a lower success rate when averaged over a short time interval. Remarkably, for longer time intervals, (t f − ti)/Γ > 5, the success rate is greater than 80% and rapidly increases up to 100%. The two right panels show explicit values for Wss around 0, highlighting the exact correspondence between a negative witness Wss and qubits in a maximally entangled state, concurrence C = 1.
Single-shot thermodynamic witness -Based on the promising performance of the above witness W, we propose and analyze numerically a single-shot thermodynamic witness W ss (with the label ss referring to singleshot). In contrast to W, this new witness does not depend on all realizations of stochastic trajectories during the time interval dt, but solely on the thermodynamic quantities Q γ (t) and δQ γ (t) at time t. We do not perform the average · Iγ (t) introduced above. However, as for W, we perform a time integration over the interval [t i , t f ]. This integration implements a time average that is expected to mimick the previously introduced ensemble average. Hence, this novel single-shot thermodynamic witness is defined as: Its performance, shown in Fig.5, is analyzed numerically over 800 stochastic trajectories, similarly to W. Because no ensemble-average is performed over all realizations during the time interval dt, its success rate is expected to be much smaller than the one for W when integrated over a small time interval. Indeed, the stochasticity of the trajectory hinders the witness to faithfully detect a trajectory leading to entangled qubits. However, and this is promising for developing thermodynamic-based witnesses, the success rate rapidly increases up to 60% for integration intervals over 4/Γ, and eventually reaches an efficiency above 90% for integration times over 8/Γ. Importantly, this high success rate is then associated with a zero error rate, demonstrating the reliability of the singleshot witness W ss based exclusively on the measurements of thermodynamic quantities.

VIII. CONCLUSION
In this work, we investigated the thermodynamics associated with entanglement genesis considering the paradigmatic half-parity measurement procedure. Indeed, when the parameters are appropriately chosen, this measurement applied onto two qubits initially in a separable state generates two types of extremal quantum trajectories according to quantum correlations. Either trajectories that leave the qubits in a separable state (no generation of quantum correlations), or trajectories that lead the qubits in a maximally entangled state, i.e. in the singlet state that is non-local under Bell-type inequalities. Based on a quantum-trajectory approach, and making use of the framework provided by stochastic thermodynamics, we have been able to conduct a stochastic thermodynamics analysis associated to the generation of entanglement. Whereas a heat flow is typically associated to a loss of information (also described as dephasing), we demonstrate for the first time that the generation of quantum correlations is associated with fluctuations of the heat flow increment. Analytically, we could bound the generation of entanglement, characterized by a positive value of the concurrence's derivative, by the standard deviations of the infinitesimal heat flow during a time interval dt. Remarkably, we then exploited this theoretical bound to define witnesses based exclusively on the measurements of thermodynamical quantities. These witnesses present a zero-error rate in the sense that they never detect a trajectory as leading to entangled qubits if not, and their success rates can be optimized with the integration time over which they are averaged. Of interest experimentally, the largest the integration window is, the higher the success probability will be, eventually reaching unity. Whereas the derivation of the witnesses is based on analytical results obtained within stochastic thermodynamics, their performance is analyzed numerically over 800 trajectories, providing reliable statistics. Finally, we defined a single-shot thermodynamical witness that exhibits high performance to certify the presence or not of quantum correlations at the level of a unique realization of the half-parity measurement.
Exact in the context of the half-parity measurement process, this work fixes theoretical tools for thermodynamic analysis of the generation of quantum correlations and opens the way to develop witnesses based exclusively on the measurements of thermodynamical quantities for quantum information purposes. This theoretical reserach is additionally motivated by recent experimental achievements in the emergent field of quantum caloritronics, aiming at controlling and measuring energetic observables like the heat current in various quantum circuits [57][58][59][60][61][62].
Appendix A: Long-time limit formula between the concurrence derivative and infinitesimal heat fluctuations The fluctuations of the heat increment associated to the lose of coherences between the two parity subspaces are defined in a similar was as σ γ : We can compare them to the derivative of the concurrence average over realizations occurring during the time interval dt: When t 1/Γ, the probability distribution of the measurement outcome J γ is narrowly peaked around the three values corresponding to the eigenvalues of the halfparity measurement operatorΦ [47], i.e. ±1, 0. The average value ofΦ tends also to one of the three eigenvalues. Hence, for trajectories corresponding to qubits in a maximally entangled state at long times, J γ = Φ (t) γ = 0 and the heat flow Q γ = 0. Consequently, Eqs. (A2) and (A3) simplify to When time exceeds the measurement time, the following equality holds: Although only valid and meaningful at times longer than the measurement time, this relation exemplifies the underlying fundamental role of infinitesimal heat fluctuations for the generation of entanglement. As stated in the main text, this relation is only exact in the context of the half-parity measurement considered in this work and can not be exploited experimentally. Indeed, one could not distinguish in an experiment infinitesimal heat fluctuations originating in the loss of phase coherence between the two parity subspaces (σ (eo) γ ) from total infinitesimal heat fluctuations (σ γ ).
σ γ = e −4Γt + e −2Γt cosh(4ΓtJ γ ) (1 + e −2Γt cosh(4ΓtJ γ )) 2 . (B5) We therefore have: 2e −2Γt cosh(4ΓtJ γ ) (1 + e −2Γt cosh(4ΓtJ γ )) 2 ≤ 2σ γ , and this allows us to derive upper and lower bounds for the concurrence derivative as a function of the heat flow Q γ , the std. deviation of heat fluctuations σ γ and the measurement record I γ defined in the main text, Eq. (2): These bounds depend on the fluctuations of the outcome at time t through dW γ (t). We can take the average over these fluctuations (i.e. the average over the outcome I γ (t) obtained at time t), keeping the past outcomes {I γ (t )} t <t constant. This average was defined in the main text, together with a physical motivation. Given that our goal is to derive bounds that only depend on thermodynamic quantities and the measurement outcome, we get rid of the dependance on the concurrence C γ by noting that C γ ∈ [0, 1]. Finally, using dW γ (t) = 0 and Φ (t) γ = Q γ / in the situation under study, we find: To certify the presence of entanglement, we require that This condition, together with inequality (21), allows us to derive a sufficient condition for certifying the presence of entanglement in terms of thermodynamical quantities only. Indeed, we have: If this inequality is satisfied by σ γ and Q γ , entanglement will be present. This result, exact in the context of the half-parity measurement process considered in this work, is therefore promising to further explore the possibility of certifying the presence of quantum correlations based on thermodynamical quantities in different setups and opens the way to develop thermodynamical witnesses for quantum information purposes.