Thermodynamic length in open quantum systems

The dissipation generated during a quasistatic thermodynamic process can be characterised by introducing a metric on the space of Gibbs states, in such a way that minimally-dissipating protocols correspond to geodesic trajectories. Here, we show how to generalize this approach to open quantum systems by finding the thermodynamic metric associated to a given Lindblad master equation. The obtained metric can be understood as a perturbation over the background geometry of equilibrium Gibbs states, which is induced by the Kubo-Mori-Bogoliubov (KMB) inner product. We illustrate this construction on two paradigmatic examples: an Ising chain and a two-level system interacting with a bosonic bath with different spectral densities.

A central task in finite-time thermodynamics is to design protocols that maximise the extracted work while minimising the dissipation of heat in the surrounding environment. In the regime of slow processes, a powerful approach consists of equipping the thermodynamic state space with a metric, in such a way that minimally dissipative processes correspond to geodesics -whose length is known as thermodynamic length [1][2][3][4][5][6][7][8][9][10][11]. This idea was developed in the 80s for macroscopic endoreversible thermodynamics in a series of seminal papers [3][4][5][6], and more recently was extended to the microscopic regime [8,12,13], leading to several applications in, e.g., molecular motors [14] and small-scale information processing [15,16]. While this geometric approach is well established for classical systems, the quantum regime has remained less explored. As noted in [17], classical results on the minimisation of dissipation can be extended to the quantum regime by using Kubo linear response theory [18], a connection that we will here investigate further. Moreover, a thermodynamic length for arbitrary out-of-equilibrium quantum systems was proposed in [19,20]; yet, this approach assumes full knowledge of the global state of system and bath, making it difficult to apply in common situations where only partial information about the bath is available. The goal of this article is to provide a general and flexible framework to construct metrics, whose geodesics correspond to optimal thermodynamic processes, for small quantum systems and given different effective descriptions of the surrounding thermal bath.
We consider thermodynamic processes where the Hamiltonian H of a system S is modified between two fixed points, H A and H B , while being in contact with an external thermal bath B. A protocol is defined by a Hamiltonian path γ between H A and H B , and a total time T . In the limit of a quasi-static process, T → ∞, the work extracted W is given by the change of equilibrium free energy between the two endpoints, W = ∆F , and thus is independent of γ. The situation changes when the process takes finite time: A path dependent dissipation will lower the amount of work extracted from S. This is quantified by introducing W dis ≡ ∆F − W , known as dissipated, irreversible, or excess work. If the process takes a large but finite time T , one can expand W dis as The key idea of [3][4][5][6] is to reduce the task of finding the optimal path γ, i.e., the one satisfying min γ ξ(γ), to the problem of finding geodesics of a given metric. To analyse this question for quantum systems, here we consider two paradigmatic models of thermodynamic protocols, corresponding to different effective descriptions of B. First, we assume step processes, where H S is modified by quenches followed by thermalizations induced by B, as originally considered in [5]. In this case, by expanding W dis using linear response theory [17,21,22], the variance metric from [8] naturally extends to the generalized covariance of linear response theory in the quantum regime [18]. This metric has been extensively studied from a mathematical point of view in [23][24][25][26][27].
Second, we consider a framework where H S is continuously modified in time while being in contact with B, which is described by a Lindbladian master equation, an ubiquitous scenario in quantum thermodynamics [28][29][30]. The available information on the dissipative dynamics induced by B allows us to build more refined metrics, and hence better finite-time protocols, when compared to the previous approach. Indeed, we show that the linear-response metric can be recovered as a particular case of the continuous approach. Within this context, we note the interesting results of [31,32], where optimal control theory is used to minimize thermodynamic cost functionals in the presence of a Lindbladian evolution.
These two frameworks can be thought of limiting cases of a more general one, where discrete and continuous evolutions are combined, which is characterized in Appendix C. Finally, we calculate the different metrics, together with the corresponding optimal protocols, for a two-level open quantum system, where in particular we consider trajectories involving non-commuting Hamiltonians.
Discrete perfectly thermalizing processes. The protocols considered in this section can be thought as a discrete sequence of the form: H (1) → H (2) → ... → H (N ) , in which the system Hamiltonian is changed discretely while being in contact with B at inverse temperature β. After each step, the system is allowed enough time to perfectly thermalize to the corresponding equilibrium state, .
(2) arXiv:1810.05583v1 [quant-ph] 12 Oct 2018 Moreover, one imposes the two constraints, H (1) = H A and H (N ) = H B . The total time T of the protocol is given by T = N τ , where τ is the time of each equilibration step, assumed to be larger than the time scale of equilibration. The total average work of the process is the sum of work at each step, . It is related to the change of equilibrium free energy, β∆F = ln(Z B /Z A ) with Z X = Tr(e −βH X ) and X ∈ {A, B}, via the exact identity (see e.g. [33,34] and Appendix A) where S(ρ||σ) = Tr (ρ(log ρ − log σ)) is the relative entropy. Due to Klein's inequality, the sum in equation (3) is positive, so that W dis ≥ 0 and W ≤ ∆F . It is interesting to point out that W dis is connected to the entropy production Σ simply by Σ = βW dis (compare with the results in Appendix B as well). If the number of steps is large, ) in powers of 1/N , obtaining [35]: where, cov ω β (H) (A, B) is the generalized covariance in the linear-response regime [18,36], also known as Kubo-Mori-Bogoliubov (KMB) inner product [37,38], where we introduced the linear operator, At the same order of approximation, one can substitute the discrete description with a continuous one: consider a smooth one-parameter family H(τ ) (τ ∈ [0, 1]) such that H(i/N ) ≡ H (i) , and similarly for the thermal states, (4), and using the definition of Riemann sum, we obtain: We stress that to (7) holds in the linear-response regime [18], and that it is equivalent to previous expressions for W dis obtained in the context of geometric magnetism [21], shortcuts to adiabaticity [17], and quantum thermoelectrics [22]. Interpreting the KMB inner product as a metric on the space of quantum states, the integral in (7) can be thought as the energy functional, or the action, of the curve H(τ ). Using the Euler-Lagrange equations of motion [39,40], one can then construct curves H(τ ) that minimize W dis at first order in 1/N (see Appendix D for details). That is, geodesics of the KMB metric correspond to optimal thermodynamic paths.
It is interesting to compare this result with that of [8] (see also [1,5]), where the concept of thermodynamic length is applied to small (non-quantum) systems. The starting point of [8] is to decompose the Hamiltonian H as βH = i λ i (t)X i , where X i are the collective variables and λ i (t) the conjugated generalized forces [41], which are assumed to be experimentally controllable parameters. Then, plugging in βḢ = iλ i (t)X i in (7) we have which should be compared with Eq. (6) of [8]: The covariance in classical systems is replaced by the generalised covariance (5) in quantum systems. In fact, classical protocols can be recovered from the quantum formalism when the perturbation commutes with the Hamiltonian, [H(τ ),Ḣ(τ )] = 0, as the KMB inner product reduces to the standard covariance, cov(A, B) = AB − A B . At this point we can also make a connection with statistical estimation theory: in the same way that the metric for the classical case is identical to the Fisher information (FI) matrix [8,42], the KMB metric is a quantum Fisher information (QFI) matrix [23,43,44]. One crucial difference between the two cases is that, while for commuting variables the definition of FI is unique, there are uncountably many extensions in the quantum case [26], and the chosen one highly depends on the problem in consideration (for example, in quantum metrology the most common choice is given by the Bures metric [45]). The connection between dissipation and the KMB metric highlights its key role in quantum thermodynamics.
The derivation just presented (and similarly for the previous results [3-6, 8, 17, 21, 22]) relies on an important assumption: the only information available on B is its temperature 1/β. In quantum physics, often a more detailed description of the action of B on S is available; for example, the action of many physical baths (e.g. bosonic or fermionic) is encapsulated in its spectral density [46]. Furthermore, in the weak coupling limit, thermalization processes can be described by means of a Lindbladian master equation. In what follows, we construct a new class of thermodynamic metrics, whose geodesics are optimal thermodynamic processes for a given Lindbladian master equation.
Continuous processes. We consider a system with time dependent Hamiltonian H(t), with H(0) = H A , H(T ) = H B and where T is the total time of the process. The system is in contact with B, and we assume that the dissipative dynamics of S can be described by a (time-dependent) Lindbladian equation, where we set = 1. Furthermore, we assume that at each time t, the dissipative dynamics tend to bring ρ t to the corresponding equilibrium state ω β (H(t)), with ω β (H(t)) ≡ e −βH(t) /Tr(e −βH(t) ). More precisely, we assume that this zero-value (instantaneous) eigenstate is unique and that all the other eigenvalues have strictly negative real part. This type of dynamics are known as relaxing or mixing [47]. By condition (11), we are focusing our attention in the adiabatic limit, where the transformation of H(t) is much slower than the relaxation dynamics induced by B. Later we will relax this condition in order to account for corrections to the adiabatic master equation [48,49]. Given (10) and (11), we use the recent results of [50] which provide a series expansion of ρ(t) in powers of 1/T . Since the unique zero eigenvector of L t is traceful, one can define a family of invertible linear forms Λ (t) on the space of traceless operators, such that the action of the Lindbladian on states can be decomposed as: where the hat denotes the traceless component of ρ. Then, following [50], we set the parameter of the curve to the unit interval (τ ∈ [0, 1]) so that t = τ T , and define the rescaled operators ρ τ := ρ(τ T ), ω τ := ω β (H(τ T )), H τ = H(τ T ), and Λ τ := Λ (τ T ) . Then one can expand ρ τ as [50] withω τ = dω τ /dτ . Using the formula of the derivative of a matrix exponential [35] and the definition (6), we also havė Now we move to thermodynamics. For continuous evolutions, the total work is given by W = − 1 0 dτ Tr(ρ τḢτ ). Combining the definition W dis = ∆F − W with (12) and (13), we obtain This quantity is always non-negative, and the integrand is minus the entropy production rate (see Appendix B). In order to get a geometric notion of W dis , we introduce the scalar product, so that we can write W dis as At this point, we can use the same logic of the previous section (see also Appendix D) to find optimal thermodynamic paths, replacing the KMB metric by (15). Now we make an important connection with the previous discrete case. By taking Λ τ = −1, it is easy to see that the m Λτ ωt metric reduces to the KMB metric; or, put it differently, the discrete description is equivalent (at leading order in N ) to the continuous dynamics induced by the dissipative dynamics, by identifying T = 2N . This is a very important insight: it means that the KMB metric (5) is a particular case of (15). The main advantage of (15) is that it accounts for the structure of the thermal bath while keeping an efficient description of the problem (recall that the dissipator L can be expressed by a matrix of size d 2 S * d 2 S , where d S is the dimension of the the Hilbert space of S). Indeed, while the KMB metric only depends on β and H τ , the metric (15) depends also on how the system reaches equilibrium through Λ τ . As we shall see later, this extra information enables the design of better finitetime protocols.
A natural extension of this result concerns dissipative dynamics that do not satisfy (11) exactly. At the end of Appendix C, we generalize our results to Lindbladians for which is a general correction that can appear, e.g., due to corrections of the Lindbladian beyond the adiabatic limit [48]. Furthermore, we note that if a protocol combines discrete with continuous transformations, it is not difficult to adapt our results to find the corresponding metric; this is done in Appendix C. Putting everything together, the techniques developed here allow to deal with a very general class of thermodynamic protocols.
Applications. We now illustrate our results for a qubit system in contact with a bosonic bath. Optimal finite-time thermodynamic protocols for two-level systems in contact with a single thermal bath have been treated in previous works, in particular in a single-level quantum dot [51], for bosonic and fermionic baths in [31,32], and for a qubit in an optical cavity [52]. Notably, these results provide exact finite-time protocols, whereas our results are valid at leading order in 1/T . Yet, our considerations contribute to the previous results by: (i) using a geometric approach which, as we shall see, allows to infer general properties of the protocols by the form of the metric; (ii) designing optimal protocols in the presence of non-commuting Hamiltonians; (iii) characterizing the dependence of the optimal protocols on the structure of the bath (e.g. its ohmicity), and (iv) designing a simple optimal finite-time protocol for Landauer erasure.
The Hamiltonian of the system expressed in spherical coordinates takes the general form, H = r cos θ sin φ σ x + r sin θ sin φ σ y + r cos φ σ z , (18) where the parameters (r, θ, φ) are time-dependent and can be controlled at will. The bath tends to bring the state of S to ω β (H) at all times, i.e., L t (ω β (H(t))) = 0 ∀t. Details about L t are provided in Appendix E. Here it is enough to note that B has spectral density J(ω) = γω α , where α > 0 characterizes the ohmicity of the bath: for α = 1, α > 1, α < 1, B is ohmic, superohmic and subohmic, respectively.
In spherical coordinates the basis of the tangent space is given by the three operators: Then, the metric tensor in the two cases is given by . Thanks to the convenient choice of coordinates, the metrics take the particularly simple form Isolating the expression of the Euclidean metric in spherical coordinates, diag{1, r 2 sin 2 (φ), r 2 }, we can express the two metrics in the form diag{λ 1 , λ 2 r 2 sin 2 (φ), λ 2 r 2 }, separating the radial direction from the solid angle. This means that the system will dissipate differently if one changes the spacing of the two levels, or if only the eigendirection is moved. Moreover, the ratio λ 2 /λ 1 diverges for r → ∞, as an evidence of the existence of directions which are exponentially more dissipative than others. This fact has a simple physical explanation: changing r, i.e., the energy spacing between the two levels, only affects an exponentially small fraction of the population; on the other hand, in order to create coherence, the whole population has to be manipulated, causing exponentially more dissipation. It is worth noticing that this conclusion can be reached without referring to any physical intuition about the system or the process involved, simply by inspection of the metric structure. In this way, it is possible to gain insights on which parameters of H can generate more dissipation in an automatic way. The term r −α in g m Λ accounts for the dependence of the thermalization timescale τ F on the ohmicity of B, τ F ∝ r −α . In general, larger values of α (i.e., large ohmicity) imply larger τ F for low rs, and hence optimal protocols naturally spend longer times within this regime. This is illustrated in Fig. 1, where we construct optimal trajectories r(t) between two fixed endpoints, H A = r 1 σ z and H B = r 2 σ z . As expected, the optimal trajectory vary as a function of the ohmicity of B: as α becomes smaller, the m Λ trajectories become closer to the KMB one, suggesting that the KMB metric corresponds to a flat spectra. However, this is not quite true: in general, the factor tanh(βr) makes the two metrics distinct for all values of α. We also observe that all optimal trajectories lie below the naive trajectory where r is increased linearly with t. This is perfectly reasonable: as the thermal population decreases exponentially with r, it is in general favorable to move the levels slower for small values of r, which are the ones that contribute the most to the extracted work.
An important question is how relevant these effects are in practice. In Appendix F, we focus on protocols in which H A = 0 and H B = E f σ z , and consider the KMB metric as it allows for a simple analytic solution. We compare the optimal KMB protocol with a linear increase in t of r(t), and find N W lin increases linearly with E f , highlighting the relevance of a proper choice of the trajectory. This is particularly relevant in the Landauer principle (or the Szilard engine) where E f → ∞, and in this sense we note that current experimental implementations of the Landauer principle rely on a linear increase of r(t) [53][54][55][56]. Our results clearly demonstrate that a substantial improvement of the convergence to the thermodynamic bound β −1 ln 2 in these experimental setups could be achieved by an engineered r(t) (see Appendix F for details on the optimal trajectory).
Finally, we analyze a scenario involving non-commuting Hamiltonians, H = xσ x + zσ z , with the two endpoints H A = σ z and H B = σ z + xσ x . The optimal trajectory for the KMB and the m Λ metric with α = 3 are shown in Fig. 2 with x = 0.4. Both geodesics differ from the linear (Euclidian) trajectory, and it is interesting to note that the latter lies in between. In practice, and rather counterintuitively, this might imply that for some processes the KMB geodesic creates more dissipation than the linear choice. For example, given the parameters in Fig. 2, we find that W m Λ dis /W lin dis ≈ 0.96 but W KMB dis /W lin dis ≈ 1.014, when we perform a continuous variation of H in contact with a superohmic bath with α = 3 and β = 1. We stress that this does not contradict any of our considerations: Strictly speaking, the KMB metric is valid for dissipative dynamics of the form (17), and here we do the simulations with a superohmic bath with α = 3. Finally, we note that while the advantage from the geometric approach in this example is rather small, from the form of the metric it is expected it will increase as we increase the spacing of the qubit. Similarly, we have observed numerically a linear increase with β of the advantage from the geometric approach (with respect to the linear choice), as in the previous case of commuting Hamiltonians. This suggests that these advantages are most relevant at low temperatures.
Conclusions. In this article we have developed a geometric approach for constructing optimal finite-time thermodynamic protocols in the quantum regime, given different effective descriptions of the surrounding bath B. First, we considered a minimal description of B, where only its temperature is known, and showed that the covariance metric of [8] naturally extends to the generalized covariance of linear response theory in the quantum regime [18]. Second, we generalised this approach by considering that the action of B on S can be described by a Lindbladian master equation; and found the corresponding thermodynamic metric. Geodesics of such metrics correspond to optimal finite-time protocols in the limit of slow (but finite-time) processes.
One of the strengths of this geometric approach is its versatility, which we have shown here by considering different descriptions of B in the weak coupling regime. There are natural extensions of these results, which notably include: (i) strong coupling quantum thermodynamics [57][58][59][60][61][62][63][64], and (ii) thermodynamic protocols with generalised Gibbs ensembles [65][66][67][68], both of them of crucial interest for small quantum systems [69]. In fact, it is rather straightforward to formally extend the considerations of Appendix A and C to such scenarios, thus opening the possibility of designing optimal protocols for strongly coupled and, e.g., squeezed baths [70]. These are interesting questions that we shall explore in the future. systems with multiple conserved quantities," Nature communications 7, ncomms12049 (2016).
[66] Matteo Lostaglio, David Jennings, and Terry Rudolph, "Thermodynamic resource theories, non-commutativity and maximum entropy principles," New Journal of Physics 19, 043008 (2017). The derivation of formula (7) outlined in the text is presented in more details here. Starting from the definition of the work for a discrete process: , one can rewrite [64]: where in the second line we used the definition of free energy functional, the third line is just a reshuffling of the terms in the sum, and in the last line the identity S(ρ||ω β , H (i) )] has been used, together with the constraints H (1) = H A , and H (N ) = H B . The first two terms are equal to ∆F = ln(Z B /Z A )/β. Then, we can pass to study the behavior when the number of steps N is big, N 1. In this context, it is convenient to define the operator associated to a full rank density matrix ρ, J ρ and its inverse: The connection with the operator should be pointed out. Then, using the Fréchet derivative of the logarithm of an operator, it can be shown that [35]: Moreover, using the Dyson series of the exponential, one can expand a Gibbs family of states corresponding to a smoothly parametrized Hamiltonian H (t) as: Plugging this expansion in equation (A3), we obtain the result: where we used the definition of generalized covariance to clean up the notations. Going back to the study of the sum in (A1), using the expansion for N → ∞, the final result is obtained: This derivation should be compared with the one for classical systems given in [8]. In that case, one starts by dividing the Hamiltonian in collective variables X i and generalized forces λ i , as: where the λs are considered to be experimentally controllable parameters [41]. Using the first and second law of thermodynamics, one can express the work produced by the system as: where one can recognize the terms inside the parenthesis to be the change of free energy evaluated at the endpoints of the process, while the integral accounts for the loss of available work due to irreversibility, σ( λ i (t) ) being the entropy production rate. In order to give an explicit expression for the dissipation term, one models a quasi-reversible protocol as the continuous limit of a step process in which the parameters λ i (t) are varied in a discrete manner and for which, after each step, the system is allowed to thermalize until it reaches equilibrium (the time of thermalization is denoted by τ ). In this case, the expression of the change of energy and of the work produced during a single step is given by: where the first equality is simply a way of rewriting the Hamiltonian, while the second comes from an extension of the usual definition of work as force times displacement to the case of generalized variables. Hence, applying the first law, one obtains the expression of the heat absorbed by the system as: In this setting it is possible to give a closed expression for the entropy production: if one considers, for example, a cyclic process performed in two steps, the first from state A to state B, and second in the opposite direction, then the total heat absorbed during the transformation is given by: It is important to notice that, in this case, ∆Q T OT coincides with the entropy produced, because for any cyclic transformation the following identity holds: thanks to the fact that the entropy is a function of state. Now, considering a general protocol, if one assumes that the curve λ i (t) depends smoothly on t (where t, up to a rescaling, can be consider to be contained in the interval [0, 1]) then it is possible to apply a similar argument and give an explicit expression of the heat absorbed during the process run first in the forward direction and then reversed, in a total of 2N steps: In particular, since the expression obtained is quadratic in the velocities λi (t) , the heat absorbed during the forward transformation exactly equals the one for the reversed protocol; for this reason, one can rewrite equation (A7) in the form: giving an explicit expression of the dissipation which accompanies a thermodynamic transformation close to equilibrium. This formulation is equivalent to the result of equation (A6) when the Hamiltonian commutes with the perturbation, [H(τ ),Ḣ(τ )] = 0. In fact, on the one hand it is a canonical result of classical statistical mechanics that: on the other, the generalized covariance reduces to the classical covariance for commuting variables.

Appendix B: Entropy production during a thermodynamic protocol
The expression of the work given in equation (A1), together with its similarity with equation (A7), suggests the definition of the instantaneous entropy production rate [46]: Before investigating the positivity of this quantity, it is useful to show a weaker equation for ρ (t) which are induced by CPTP maps. In particular, assuming the system is initially uncorrelated with the environment (ρ (0) U N IV = ρ S ⊗ ρ E ), and that the Gibbs state is stationary under the evolution, one has that: where we used the fact that the relative entropy decreases when one traces out part of the system, together with its invariance under unitary evolution. Denoting with V t the dynamical map which bring the reduced density matrix of the system from the original state ρ to ρ (t) , we can rewrite the equation as: which ensures that on any finite time interval the quantity ∆F ρ ρ (t) is positive. Unfortunately, this result does not give any information about the monotonicity of the change of entropy: it is sufficient to consider an exactly recurrent system, with periodicity T , for which the the quantity ∆F ρ ρ (t) will first decrease, to only go back to its initial value at time T . Nonetheless, when V t is a dynamical semigroup, meaning that the identity V t+s = V t V s holds for any t, s ≥ 0, one can further prove that: This result implies that the quantity in (B1) is positive. As it was stated above, this result is true in general only if the dynamics can be described by a dynamical semigroup, that is, if the evolution is Markovian. In the continuos case, all the CPTP maps of this type can be written as V t = exp Lt, where L is a Lindblad operator [71]. Therefore, in this case equation (B1) becomes: It is interesting to show what happens in the quasi-isothermal limit, that is, when the state of the system can be described by the expansion (12). In this case, the entropy production rate can be expressed to second order in 1/T as: Comparing this expression with equation (14), we can see that the quantity inside the integral is the entropy production rate, and therefore it is non-negative.
Appendix C: Unified derivation of KMB metric and m Λ metric The connection between the thermodynamic length and the entropy production rate suggests that the KMB metric and the m Λ one should be manifestation of the same phenomenon. In this section a unified derivation will be given starting from formula (A1), in the quasi-isothermal approximation: Expanding the work as a function of the free energies in the same way as we did in (A1), one has: The term in the first parenthesis is of order O ε 2 , so it will be neglected in the rest of the derivation. We now study the sum: in the limit of N 1 and ε 1. Recalling equation (A3), we rewrite the first term inside the parenthesis as: so that we can expand it as: The second term, on the other hand, simply gives: Examining together the terms in the two expansions which are quadratic in ε, we can see that: and therefore this contribution can be neglected in the sum ( . Then, βW diss can be rewritten as: In the first line, plugging in the expansion (A4), we regain the KMB metric, in the way it was shown in appendix A. In the second line, it should be noticed that J ρ is a self-adjoint operator with respect to the Hilbert-Schmidt scalar product, meaning that for any two self-adjoint matrices A and B the following holds: This allow us to sum together the two terms, and using again equation (A4), we can obtain from the second line: Therefore, the work extracted during a quasi-isothermal protocol can be rewritten as: At this point, one can go back to applying the treatment of the main text, having in mind that the sum of two scalar products is again a scalar product. In this way, we can present a metric for discrete processes, taking into account the thermalization dynamics as well.
It is interesting to better characterize βW diss : comparing equation (C3) with (B1), it is immediate to rewrite: where σ t (ρ (t) ) indicates the entropy production rate with respect to ω (t) β . In this way, the positivity of the integrand in βW diss is equivalent to the validity of the differential version of the second law: which, as we saw in the previous appendix, holds only for Markovian dynamics. If this is the case, since the perturbation ρ is expected to only depend on the velocity at which the Hamiltonian is changed and on the thermal state, one can define an operator O, such that the perturbation to the thermal state can be written as: This simple rewriting allow us to use the symmetrization technique described in the main text to provide a scalar product on Hermitian operators: so that we can automatically generalize the thermodynamic length formalism to other perturbative expansion. For example, consider that the steady state of the Lindbladian is corrected as: where ζ is a perturbation depending only on the local features of the trajectory. Then, one can apply an analogous perturbation expansion as the one in [50]: and obtain the expression of the state as: Then, if adding ζ does not affect the validity of (C13), the technique explained above can be applied to give: where we implicitly used the bijection ω t ↔ H t to pass from the state to the corresponding Hamiltonian in the scalar product.
Plugging these vectors in g(∂ i , ∂ j ), we obtain the metric in (20). Given a metric, one can define the length of a curve γ : [0, 1] → M as: where the norm is defined by |γ (t)| := g(γ (t), γ (t)). It is straightforward to verify that this definition is invariant under reparametrization of the curve. Moreover, there always exists a smooth reparametrization such that v(t) := |γ (t)| is constant for all t ∈ [0, 1]. Such a parametrization is called by arc-length since we have that: A similar definition, which is useful in the calculus of variation, is the one of energy functional of a curve: It is worth to point out that this functional is not invariant under arbitrary reparametrization. Defining the Lagrangian: one can interpret (D7) to be twice the action of the trajectory γ. This quantity is connected with the length of the curve via the Cauchy-Schwarz inequality: where the equality is attained only for arc-length parametrization. A problem of interest is to find the shortest curve between two endpoints, called geodesics. We know from classical mechanics that the solutions of the Euler-Lagrange equations in absence of an external potential have constant velocity, which means that we have an equality in (D9). For this reason, the curve which minimize the action will even be the shortest. In this way, one can apply the formalism coming from analytical mechanics to solve the problem. The Euler-Lagrange equations read: d dt (D10) Using the definition of the Lagrangian (D8), the right hand side simply gives: The left hand side of the equation gives: where we split ∂ k g i,jẋ jẋk in two parts. Putting everything together and multiplying by the inverse of the metric g i,j , we get the geodesics equation:ẍ i + Γ i j,kẋ where we implicitly defined the Christoffel symbols: Γ i j,k = 1 2 g i,l (∂ j g l,k + ∂ k g j,l − ∂ l g j,k ) .
We consider two possible transformations: (i) the naive one of going linearly from one Hamiltonian to the other (H(t) = E f t σ z ); (ii) the one along the optimal trajectory. For (i), a direct calculation shows that For (ii), we need to solve the geodesic equation. For that, first we parametrise the Hamiltonian (and consequently of the states) as: (p,θ,φ) −→ H(p,θ,φ) = tanh −1 (1 − 2p) β cosθ sinφ σ x + β sinθ sinφ σ y + cosφ σ z In the coordinates (p, θ, φ), the Christoffel symbol Γ p p, p takes the particularly simple form: suggesting that this choice correctly mirrors the physical structure of the problem. The geodesics equation is then given bÿ The solution is analytically solvable and, for the boundary conditions p i = 1/2, p f = (1 − tanh βE f )/2, we find two solutions, and the one minimising W dis reads p = 1 2 (sin(At) + 1) Then, a direct computation shows To compare both quantities, we introduce: The ratio χ/κ is depicted in Fig. 3. The behaviour for βE f 1 can be understood by a Taylor expansion which yields κ ≈ E f β, χ ≈ π 2 /4. As it can be seen κ is approximately linear for E f β 1, while χ approaches the asymptotic value of π 2 /4. Therefore, after an initial transient, the ratio κ/χ is asymptotically linear in E f β. This means, that for a fixed protocol, the geometric approach becomes more and more relevant as either E f increases or the temperature decreases. The latter may be surprising, but is a consequence of the fact that for β 1 changing the energy spacing will not significantly affect the state of the system, which will always be close to the fully mixed density matrix ω β 1 (E f σ z ) ≈ I/2, while for lower temperatures even small changes in the Hamiltonian will reflect in the form of the state. Moreover, it should be kept in mind that ∆F as well is monotone in β, so that the intuitive insight that lowering the temperature will augment the amount of work extractable is regained in the quantum formalism.