Thermodynamic length in open quantum systems

1 Introduction

• Classical thermodynamic geometry links geodesics to minimally dissipative processes, while quantum Gibbs-state geometry alone omits dynamical features important in finite-time protocols.The classical construction was developed for macroscopic endoreversible thermodynamics and extended to microscopic systems.Source: §1 · p. 1
• Other approaches describe arbitrary nonequilibrium evolution using the global unitary, or dissipation near equilibrium through Kubo linear-response theory.The global-unitary requirement can be difficult to apply when the bath is accessible only through an effective dynamical description.Source: §1 · p. 1
• The framework constructs a Lindblad-system thermodynamic metric by applying the Drazin inverse, which encodes dissipative equilibration timescales, to equilibrium Gibbs-state geometry.It is used to find minimally dissipating protocols for a transverse-field Ising chain and a qubit in a bosonic bath with different spectral densities.Source: §1 · p. 1

2 Dissipation in quantum systems

• The nonequilibrium free energy is defined as F(ρ, H) = ⟨H⟩ρ − β^-1S(ρ), with ⟨A⟩ρ = Tr[Aρ] and S(ρ) the von Neumann entropy.For a Gibbs state, the notation is ρ ≡ ωβ(H) = e^−βH/Tr(e^−βH), where β is the bath's inverse temperature.Source: §2 · pp. 1–2
• Relative entropy from the thermal state is positive and measures statistical difference; it also corresponds to extractable work, motivating the name availability.Source: §2 · p. 2
• Dissipation is the path-dependent contribution to work beyond the endpoint free-energy difference, and its integral is the correction to the second law.The integrand is identified as entropy production rate and is positive under Markovian dynamics.Source: §2 · p. 2
• In slow driving, entropy production depends on the current Hamiltonian and its velocity, so minimally dissipating paths can be sought as geodesics of the induced metric.The driving timescale is assumed longer than the relaxation timescales.Source: §2 · p. 2

3 Metric structure in open quantum systems

• The approximation assumes a sufficiently smooth Hamiltonian trajectory, with bounded derivatives of Ht, and a protocol duration much longer than the equilibration timescales encoded in L+.In the slow-driving limit, it truncates the perturbative expansion at the second term, with corrections of order O(1/T^2).Source: §3 · p. 3
• The metric is built from entropy production and is symmetric, positive definite, and smoothly dependent on the Gibbs-state base point.The construction assumes a unique stationary Gibbs state and other Lindbladian eigenvalues with strictly negative real parts.Source: §3 · pp. 2–4
• In the slow-driving regime, geodesics of the introduced metric correspond to protocols that minimize dissipation to first order in O(1/T).The metric’s action is the dissipation functional for paths through the controllable parameters.Source: §3 · p. 4

3.1 Connection with classical results

• For discrete protocols with perfect equilibration at each step, the entropy-production rate coincides with that of a uniform-timescale relaxation model, and the metric reduces to the KMB inner product.In that model, all observables relax exponentially with the same timescale τ.Source: §3.1 · p. 4
• When [Ht, ˙Ht] = 0, the generalized covariance reduces to the standard covariance; non-commuting protocols are where qualitatively different quantum results are expected.Near a critical point, the order-parameter Fisher-information eigenvalues and dissipation diverge for any N.Source: §3.1 · pp. 4–5
• The metric extends equilibrium geometry by incorporating relaxation dynamics through the Lindbladian Drazin inverse, which favors parameters that thermalize faster.This inverse encodes the system’s thermalisation timescales.Source: §3 · p. 3, §3.1 · p. 5

4 Constructing minimally dissipating trajectories

• The optimal path is obtained by solving geodesic equations in the controllable parameters, although computing the metric still scales with total system dimension.For the specified relaxation dynamics, the KMB metric can be computed from the partition function when its analytic form is known.Source: §4 · p. 5

4.1 Ising chain in a transverse field

• Near g = 1, the Ising metric increases as temperature falls, and the geodesic slows near the phase transition to compensate for greater dissipation.The changing metric and Christoffel symbol track changes in the system’s physical behavior.Source: §4.1 · p. 6, Figure 2 · p. 6

4.2 Qubit in contact with a bosonic bath

• For the qubit, radial energy-spacing changes and angular coherence-generating controls have different metric eigenvalues, with the quantum-to-classical eigenvalue ratio diverging as r grows.Changing the spacing affects an exponentially small fraction of the population, whereas creating coherence requires manipulating the whole system.Source: §4.2 · p. 7
• The bath contributes a factor r^-α to the metric, so optimal radial protocols spend more time at low r, increasingly so for higher α.The factor captures the overall dependence of equilibration timescales on energy spacing.Source: §4.2 · p. 7
• For non-commuting endpoints, optimal trajectories create coherence, and bath ohmicity qualitatively changes their behavior.Along these protocols, [Ḣt, Ht] ≠ 0.Source: §4.2 · p. 7
• At large Ef, the linear-protocol first-order dissipation term diverges linearly in Ef, while the optimal geodesic result approaches π²/4.The comparison uses boundary conditions HA = 0 and HB = Efσz under trivial dynamics; the geodesic advantage grows for large gaps or low temperatures.Source: §4.2 · p. 7

5 Comparison to other approaches

• When d ≪ D, the geometric approach approximates the optimal solution with d nonlinear equations instead of D, becoming exact in the slow-driving limit.Building the metric generally requires diagonalizing the Hamiltonian, but specified relaxation dynamics allow it to be computed from the partition function.Source: §5 · p. 8
• Quasistatic protocols are independent of total duration T, so the path need not be optimized separately for each protocol time.Source: §5 · p. 8
• In the quasistatic limit, minimizing dissipation also minimizes fluctuations for classical systems and for commuting quantum protocols.Source: §5 · p. 8
• Metric eigenvalues provide a way to infer a system’s dissipative properties by inspection.Source: §5 · p. 8
• A qualitative comparison for a two-level system found good agreement between the geometric approach and exact approaches even when T is comparable to the relaxation timescale.Source: §5 · p. 8

6 Conclusions and outlook

• The framework defines a thermodynamic metric for Lindblad systems whose geodesics are minimally dissipative paths in the quasistatic regime.Source: §6 · p. 8
• The approach could extend to generalized Gibbs ensembles and nonequilibrium steady states by adapting the free energy or entropy-production reference state.Source: §6 · p. 8
• Although the framework assumes weak coupling through Lindblad dynamics, reaction-coordinate mapping may allow strong-coupling extensions with qualitatively changed optimal trajectories.Source: §6 · p. 8
• The authors identify collisional-model thermalization, thermodynamic uncertainty relations, and shortcuts to thermalisation as directions for further study.Source: §6 · p. 8

B The entropy production rate

• For recurrent dynamics, integrated relative-entropy decrease can be positive even though the quantity first increases and later returns to zero.Source: B The entropy production rate · p. 15
• When the evolution is a dynamical semigroup, relative entropy to the stationary Gibbs state is nonincreasing, making the entropy production rate positive.Source: B The entropy production rate · p. 15
• For slowly driven adiabatic master equations with L(ωβ(Ht)) = 0, the fixed-Hamiltonian reasoning applies locally around each instantaneous Hamiltonian.Source: B The entropy production rate · p. 15

C Drazin inverse of the Lindblad operator

• The appendix derives an integral expression for the Lindblad operator's Drazin inverse by checking its defining conditions.Source: C Drazin inverse of the Lindblad… · pp. 15–16

• On density matrices, the operator introduced in the derivation is the inverse of Jρ.Source: D Metric in open quantum systems · p. 16
• The calculation expands the entropy-production rate using the Lindblad equation and the quasistatic state expansion to obtain the main-text result.Source: D Metric in open quantum systems · pp. 16–17
• The derivation uses self-adjointness of J−1 under the trace inner product and the tracelessness of the Lindblad evolution term.Source: D Metric in open quantum systems · p. 17

• For a protocol of N quenches, work is defined using the Hamiltonian and state at each step, yielding a discrete dissipation split analogous to the continuous-process expression.Source: E Work for discrete processes · pp. 17–18
• If the system thermalizes at every step, the discrete-process dissipation simplifies.Source: E Work for discrete processes · p. 18
• In the quasistatic regime, the discrete dissipation becomes an integral through the Riemann-sum limit, formally matching a fictitious master-equation result.Source: E Work for discrete processes · p. 18

F Expression of the metric in coordinates

• Operator vectorization turns linear superoperators into d^2 × d^2 matrices, with d the Hilbert-space dimension.A Hilbert-Schmidt inner product is used on the real vector space of Hermitian operators.Source: F Expression of the metric in… · p. 18
• For a mixing Lindbladian, the unique zero eigenvector is the Gibbs state, while all other eigenvalues have negative real part.Trace preservation makes the corresponding nonzero-eigenvalue operators traceless.Source: F Expression of the metric in… · pp. 18–19
• The Drazin inverse of a mixing Lindbladian is the usual inverse restricted to the traceless subspace.The appendix gives an explicit construction using the Lindbladian eigenbasis.Source: F Expression of the metric in… · pp. 18–19
• With control restricted to selected observables, the metric entries reduce to a matrix product and form a matrix whose dimension matches the number of controllable parameters.The derivation uses the operator-coordinate expressions and the Drazin inverse.Source: F Expression of the metric in… · p. 19
• For an evolution of the stated form, substituting the Lindbladian’s Drazin inverse into the metric expression yields the claimed coordinate formula.The appendix derives the inverse from the Lindbladian representation before substitution.Source: F Expression of the metric in… · p. 19

G Two level system coupled to a bosonic bath

• The bath dynamics are specified by a master equation for a bosonic bath with spectral density J(r) ∝ r^α.The rate parameters are defined in the appendix, which also sets the ground-state population to one and introduces Γ_r = γ_r(2P_r + 1).Source: G Two level system coupled to a… · p. 20
• The qubit’s traceless density-matrix component is parametrized with Pauli matrices, and the Drazin inverse is obtained by changing basis when dissipation depends only on the energy spacing.This assumes the dissipative dynamics do not depend on the Hamiltonian’s unit vector.Source: G Two level system coupled to a… · p. 20
• In spherical coordinates, the metric is computed by differentiating the coordinate chart to obtain the tangent-space basis before substituting it into the metric expression.The Hamiltonian is parametrized by r, θ, and ϕ.Source: G Two level system coupled to a… · p. 20
• Short-duration protocols quench at both endpoints, whereas at long durations they become equivalent to the thermodynamic-length protocol.The figure considers protocols between ϵ(0) = −2β and ϵ(1) = 2β.Source: Figure 4 · p. 21

H Comparison with optimal control optimisation

• The geometric approach requires d differential equations, compared with 2D differential equations and d algebraic constraints for the optimal-control approach.The authors argue that the reduction can make the geometric approach considerably simpler when d ≪ D, as in the Ising-chain case with d = 1 and D → ∞.Source: H Comparison with optimal control… · p. 21
• The geometric solution is approximate at finite driving speed but becomes exact in the slowly driven limit.The comparison assumes a sufficiently smooth trajectory and a duration much larger than the equilibration timescales.Source: H Comparison with optimal control… · pp. 21–22
• In the quantum-dot comparison, exact and geodesic protocols coincide asymptotically as T → ∞, when the endpoint quenches vanish.The optimal finite-time protocols have initial and final Hamiltonian jumps whose sizes shrink with duration.Source: H Comparison with optimal control… · pp. 21–22
• At T ≈ τ, the two minimisation protocols are already reasonably similar, with a ∼10% divergence at T = τ.The comparison concerns work required to excite the quantum dot from ϵ(0) = −2β to ϵ(1) = 2β.Source: H Comparison with optimal control… · p. 22
• The authors present the approaches as complementary: exact optimisation contrasts with simpler geometric minimisation, which is particularly suited to few-control mesoscopic and many-body systems.The geometric approach remains approximate except in the slow-driving limit.Source: H Comparison with optimal control… · pp. 21–22