Thermodynamic length in open quantum systems
Matteo Scandi and Martí Perarnau-Llobet
Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany
October 15, 2019
The dissipation generated during a qua-
sistatic thermodynamic process can be
characterised by introducing a metric on
the space of Gibbs states, in such a way
that minimally-dissipating protocols corre-
spond to geodesic trajectories. Here, we
show how to generalize this approach to
open quantum systems by finding the ther-
modynamic metric associated to a given
Lindblad master equation. The obtained
metric can be understood as a perturba-
tion over the background geometry of equi-
librium 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.
1 Introduction
A central task in finite-time thermodynamics is
to design protocols that maximise the extracted
work while minimising the dissipation during the
process. In the slow driving regime, a powerful
approach consists in equipping the space of ther-
modynamic states with a metric whose geodesics
correspond to minimally dissipative processes.
This geometrical construction was first developed
in the 80s for macroscopic endoreversible thermo-
dynamics in a series of seminal papers [1–8], and
more recently it was extended to the microscopic
regime [9–12], leading to several applications in,
e.g., molecular motors [13] and small-scale infor-
mation processing [14, 15].
While this approach is well established for clas-
sical systems, the quantum regime has remained
less explored. The geometry of quantum equilib-
rium Gibbs states has been characterised in [16–
20]; however the resulting metric does not take
into account dynamical features of the dissipa-
tion, which are of crucial importance in finite-
time protocols. For arbitrary out-of equilibrium
evolutions, a notion of thermodynamic length has
been put forward in [21, 22]; yet, this approach
requires full knowledge of the global unitary evo-
lution, which makes it difficult to apply in com-
mon situations where the size of the bath allows
only for an effective description of the dynamics.
Finally, Kubo linear-response theory also allows
for describing dissipation near equilibrium [23–
25], which in turn allows for defining a notion of
thermodynamic metric [24, 26].
The goal of this article is to provide a general
framework to construct a thermodynamic met-
ric whenever the evolution of the system can be
described by a Lindblad master equation (see
also [27]). The obtained metric can be expressed
as the background equilibrium geometry of [16–
19] acted upon by the Drazin inverse of the Lind-
blad operator, which encodes the different equi-
libration timescales of the dissipative dynamics.
This geometrical approach is used to find min-
imally dissipating protocols of a slowly driven
Ising chain in a transverse field, and of a qubit in
contact with a bosonic bath with different spec-
tral densities.
2 Dissipation in quantum systems
Before presenting the construction of the met-
ric, we here define the quantum correspondent
of some important quantities of classical thermo-
dynamics. A quantum system is described by
its density matrix ρ and its Hamiltonian H; if
the state is given by the Gibbs ensemble we will
use the notation ρ ≡ ω
β
(H), where ω
β
(H) =
e
−βH
/Tr(e
−βH
) and β is the inverse tempera-
ture of the surrounding bath. We will sometimes
also use the shorthand notation ω := ω
β
(H). A
functional of key importance is the non equilib-
rium free energy, which is defined by the formula
F (ρ, H) = hHi
ρ
− β
−1
S(ρ), where we define the
1
arXiv:1810.05583v5 [quant-ph] 14 Oct 2019
average as hAi
ρ
= Tr [Aρ], and we denote by S(ρ)
the von Neumann entropy of the state.
Direct calculations show that the non equilib-
rium free energy is connected to the equilibrium
one by the equality
F (ρ, H) = F (ω
β
(H), H) + β
−1
S(ρ||ω
β
(H)) (1)
where S(ρ||ω
β
(H)) is the relative entropy. This
quantity is positive definite and it can be under-
stood as a measure of how statistically different
the current state is from the thermal one. More-
over, it corresponds to the extractable work from
ρ and, for this reason, it is sometimes referred to
as availability.
We consider thermodynamic processes in which
the Hamiltonian of a system in contact with a
thermal bath is experimentally varied between
two fixed endpoints, say H
A
and H
B
. Each pro-
tocol is then defined by a curve γ in the space of
controllable parameters, and by its duration T .
The work extracted is defined as
W = −
Z
γ
dt Tr
h
ρ
t
˙
H
t
i
. (2)
It is useful for what follows to rewrite the work in
terms of the non equilibrium free energy. Using
the insight coming from (1), we integrate equa-
tion (2) by parts and obtain (see Appendix A):
W = −∆F
n.eq.
− β
−1
Z
γ
dt [−∂
ρ
t
S(ρ
t
||ω
β
(H
t
))],
(3)
where ∆F
n.eq.
is the difference in the non
equilibrium free energy at the endpoints, and
∂
ρ
t
S(ρ
t
||ω(H
t
)) is given by the limit
lim
ε→0
S(ρ
t+ε
||ω
β
(H
t
)) − S(ρ
t
||ω
β
(H
t
))
ε
. (4)
The importance of equation (3) is that it makes
manifest that we can isolate a term which only
depends on the endpoints of the protocol, and a
contribution, W
diss
(γ) = −(W + ∆F
n.eq.
), which
accounts for the dissipation during the process
and that explicitly depends on the path γ. The
quantity inside the integral can be identified with
the entropy production rate, which is guaranteed
to be positive definite whenever one has Marko-
vian dynamics (see Appendix B). This identifi-
cation is also justified by noting that the heat
supplied to the system can be rewritten as:
β∆Q = ∆S −
Z
γ
dt [−∂
ρ
t
S(ρ
t
||ω
β
(H
t
))], (5)
so that the integral accounts for the correction
to the second law in the presence of dissipation
(Appendix A).
When the driving of H
t
is slow (i.e. the charac-
teristic timescale of the driving is longer than the
relaxation timescales), it is sensible to assume
that the entropy production rate will only depend
on the base point H
t
and on the velocity
˙
H
t
. The
key idea of the classical papers [4–7] is then to
reduce the task of finding the minimally dissipat-
ing path γ
∗
to the problem of finding the geodesic
of the metric induced by the entropy production
rate on the thermodynamic space. We will now
show how to apply this idea to open quantum
systems.
3 Metric structure in open quantum
systems
We assume that the dissipative dynamics is de-
scribed by a time-dependent Lindblad equation,
˙ρ
t
= L
t
[ρ
t
] (6)
Since a Lindbladian master equation always leads
to Markovian dynamics [28], this assumption
guarantees the positivity of the entropy produc-
tion rate. Furthermore, we assume that the
Lindbladian operator has a unique zero eigen-
state, given by the instantaneous Gibbs state
(L
t
[ω
β
(H
t
)] = 0), and that all the other eigen-
values have strictly negative real part. These two
assumptions are sufficient to ensure that given
any initial conditions the system will thermalise
to the instantenous Gibbs state of H
t
,
lim
τ→∞
e
τL
t
ρ = ω
β
(H
t
). (7)
As a final remark, note that by assuming
L
t
(ω
β
(H
t
)) = 0 we are neglecting non-adiabatic
contributions to L
t
, which is justified whenever
the bath dynamics are fast compared to the driv-
ing rate of the system Hamiltonian [29–31].
For slow driving of H
t
, we can assume that
the state is always close to the equilibrium one:
ρ
t
= ω
β
(H
t
) + δω
t
. Plugging this expansion
in equation (6), we obtain an equation for δω
t
as [32–34]:
L
t
−
d
dt
[δω
t
] =
d
dt
ω
β
(H
t
). (8)
2
It is useful to express the derivative on the right
hand side using the derivative for the exponential
of operators [35]:
de
−βH
t
dt
= −β
Z
1
0
ds e
−βsH
˙
H
t
e
−β(1−s)H
, (9)
obtaining ˙ω
β
(H
t
) = −βJ
ω
β
(H
t
)
[
˙
H
t
], where we de-
fined the operator:
J
ρ
[A] :=
Z
1
0
ds ρ
1−s
(A − Tr[ρA] ) ρ
s
. (10)
In order to solve equation (8) we need to intro-
duce the Drazin inverse of the Lindbladian oper-
ator L
+
t
L
+
t
[A] :=
Z
∞
0
dν e
νL
t
ω
β
(H
t
)Tr [A] − A
. (11)
which is the unique operator satisfying the three
conditions (see [36] and Appendix C for de-
tails): (i) commutation with the Lindbladian
(L
t
L
+
t
[A] = L
+
t
L
t
[A] = A − ω
β
(H
t
)Tr [A]), (ii)
invariance of the thermal state (L
+
t
[ω
β
(H
t
)] = 0)
and (iii) tracelessness (Tr
h
L
+
t
[A]
i
= 0).
An alternative expression of L
+
t
can be con-
structed in finite dimensions as follows [32]: Con-
sider the space of traceless states, i.e., for any
state ρ
t
we consider the traceless component
given by δω
t
:= ρ
t
− ω
β
(H
t
). Thanks to the sta-
tionarity of ω
β
(H
t
), equation (6) can be rewrit-
ten in the instantaneous basis as: δ ˙ω
t
= Λ
t
[δω
t
],
where Λ
t
is the projection of L
t
on the space of
traceless states. Since the only zero eigenstate of
the Lindbladian is traceful, Λ
t
is invertible on this
subspace and we can identify L
+
t
≡ Λ
−1
t
, thanks
to the uniqueness of the Drazin inverse. From
this expression it is clear that L
+
t
encodes the
thermalisation timescales of the system.
We can proceed to solve equation (8):
multiplying both sides by L
+
t
and inverting
− L
+
t
d
dt
we obtain
ρ
t
=
∞
X
n=0
L
+
t
d
dt
n
ω
β
(H
t
) =
= ω
β
(H
t
) − βL
+
t
h
J
ω
β
(H
t
)
[
˙
H
t
]
i
+ . . . (12)
In the slow driving limit each
˙
H
t
is of order
O(1/T ) (recall that T is the total time of the pro-
cess), so that we can truncate the perturbative
expansion above at the second term, with cor-
rections of order O
1/T
2
. More precisely, this
Figure 1: A thermodynamic protocol in the slow driving
regime naturally corresponds to a trajectory in the space
of Gibbs states ω
β
(H). The metric introduced lives in
the tangent space of this manifold parametrized by ω
t
:=
ω
β
(H
t
).
approximation implicitly assumes that the trajec-
tory is sufficiently smooth so that derivatives of
H
t
are always bounded, and that the duration of
the protocol is much larger than the equilibration
timescales encoded in L
+
t
.
Plugging expression (12) into ∂
ρ
t
S(ρ
t
||ω(H
t
))
gives the following formula for the dissipation (see
Appendix D):
W
diss
= −β
Z
γ
dt Tr
h
˙
H
t
L
+
t
h
J
ω
β
(H
t
)
[
˙
H
t
]
ii
. (13)
It should be noticed that this expression could
be obtained directly by plugging the expansion
of the state in the definition of the work (2), but
we preferred to keep the discussion independent
of the particular expansion (12), so to empha-
sise the connection between W
diss
and the entropy
production rate.
At this point the introduction of a met-
ric structure on the thermodynamic space is
quite straightforward. Without loss of general-
ity, we decompose the system Hamiltonian as
H =
P
λ
i
t
X
i
, where
λ
i
t
are the time-dependent
externally controllable parameters, and {X
i
} are
the corresponding observables. Then, equa-
tion (13) can be rewritten as:
W
diss
= β
Z
γ
dt
˙
λ
i
t
m
L
ω
β
(H
t
)
(X
i
, X
j
)
˙
λ
j
t
(14)
where we introduced the bilinear form:
m
L
ω
(A, B) = −
1
2
Tr
h
A L
+
ω
[J
ω
[B]] + B L
+
ω
[J
ω
[A]]
i
(15)
which is symmetric by construction, positive def-
inite, thanks to the positivity of the entropy pro-
duction rate, and it depends smoothly on the
3
base point ω. These are the defining properties
of a metric. Note that, while we assumed Lind-
blad master equation to ensure the positivity of
(15), the positivity also holds for more general
maps [37], for which this approach can be natu-
rally generalised.
Equation (14) can be interpreted as the energy
functional, or the action, of the curve γ({λ
t
}).
This denomination can be understood thinking
of W
diss
as the action of a system of point parti-
cles moving freely along γ({λ
t
}), with coordinate
dependent mass encoded in m
L
ω
. Then, one can
use Euler-Lagrange equations of motion [38, 39]
to construct curves in the space of parameters
which minimize the dissipation at first order in
O(1/T ). This means that the geodesics induced
by the metric (15) correspond to minimally dissi-
pative protocols in the regime of slow processes.
3.1 Connection with classical results
The connection between the entropy produc-
tion rate in the slowly driven regime and the ge-
ometry of Gibbs states has been studied for clas-
sical systems since the 70s [1–7]. Historically, the
formalism of thermodynamic length has been ini-
tially developed in the context of discrete per-
fectly thermalising protocols, in which case the
thermodynamic metric can be obtained as the
second derivative of the partition function with
respect to the control parameters [6, 9]. Let us
now show how to recover this metric within our
framework.
Let us assume that the parameters of the sys-
tem are changed in discrete steps from
λ
i
0
to
λ
i
N
. After each step we wait enough time for
the system to perfectly equilibrate. In the qua-
sistatic limit (N 1) we can use the approxima-
tion
H
i+1
= H
i
+ τ
˙
H
i
+ O
τ
2
, (16)
where τ is the time spent at each step, so that
the total time of the protocol is given by T = Nτ.
Using the derivative of the exponential (9) we also
have up to order O
τ
2
that:
ω
β
(H
i+1
) = ω
β
(H
i
) − τβJ
ω
β
(H
i
)
[
˙
H
i
]. (17)
Comparing the expression just obtained with
the expansion (12), it is straightforward to see
that we could obtain the same result starting
from the master equation,
˙ρ
t
= τ
−1
(ω
β
(H
t
) − ρ
t
). (18)
Moreover, the entropy production rate for dis-
crete perfectly thermalising protocols coincides
with the one coming from this fictitious evolu-
tion (see Appendix E). This connection is a con-
sequence of the fact that for discrete protocols
details about the dynamics are disregarded, since
at each step the state completely reaches equilib-
rium. This is translated into a master equation
which has the least possible structure: in fact, the
evolution (18) for a fixed t leads to an exponen-
tial relaxation for all observables with the same
timescale τ.
In this context, we see that the metric (15) re-
duces to the Kubo-Mori-Boguliobov (KMB) inner
product
m
KMB
ω
(A, B)
τ
= cov
ω
(A, B) = Tr [A J
ω
[B]] , (19)
also known as generalised covariance in the con-
text of linear response theory [40, 41]. This quan-
tity is directly related to the partition function by
differentiation:
β
2
m
KMB
i,j
=
∂
2
∂λ
i
∂λ
j
log Z
H
, (20)
where we indicate m
KMB
i,j
≡ m
KMB
ω
(X
i
, X
j
),
and Z
H
is the partition function defined as:
Z
H
= Tr[e
−βH
].
In the classical case the generalised covari-
ance (19) is substituted by the standard covari-
ance hABi
ω
− hAi
ω
hBi
ω
[9]. Indeed, when the
protocol does not generate coherence between dif-
ferent energy levels, that is when [H
t
,
˙
H
t
] = 0,
the former reduces to the latter. Hence, it is only
in the case of non-commuting protocols, meaning
[H
t
,
˙
H
t
] 6= 0, that we expect qualitatively differ-
ent results with respect to classical systems.
Similarly to the classical case, the metric (19)
can be interpreted as a quantum Fisher infor-
mation matrix. The corresponding Riemannian
structure has been studied both from the math-
ematical and the physical point of view [16, 17,
20, 42]. In particular, (20) makes clear the re-
lation between phase transitions, statistical dis-
tinguishability and dissipation: in fact, close to a
phase transition the eigenvalues of the Fisher in-
formation matrix corresponding to the order pa-
rameters diverge. Thanks to the extreme sensi-
bility of the system, it is indeed easier to estimate
4
the values of the external parameters in this re-
gion. In the same way, the abrupt changes in the
system makes it more difficult to stay within the
quasistatic limit, so that for any N the dissipa-
tion will diverge close to the critical point.
In conclusion, the metric (15) extends the
standard thermodynamic metric (20) by includ-
ing non-trivial relaxation dynamics through the
Drazin inverse L
+
t
. This modifies the underly-
ing equilibrium geometry to account for the spe-
cific nature of the dissipative dynamics: since
observables closer to their equilibrium value dis-
sipate less when they are manipulated, the in-
troduction of L
+
t
can be interpreted as a way
to favour parameters which thermalise faster. A
similar extension to non-trivial dynamics of the
geometric framework for classical systems was ob-
tained in [12, 13] and the construction presented
here constitutes the natural generalisation to the
quantum regime. In fact, if we consider the par-
ticular case in which the controllable observables
satisfy the relaxation dynamics:
D
˙
X
i
E
ρ
t
= τ
−1
i
hX
i
i
ω
t
− hX
i
i
ρ
t
, (21)
then the metric takes the simple form (see Ap-
pendix F):
m
L
ω
(X
i
, X
j
) =
τ
i
+ τ
j
2
m
KMB
i,j
, (22)
in complete analogy with the classical result [12],
which gives the thermodynamic length as the
Hadamard product between a matrix encoding
the equilibration timescales of the system and the
Fisher information matrix.
4 Constructing minimally dissipating
trajectories
We now discuss how to find minimially dissipa-
tive processes from the metric (15), which is then
applied to an Ising chain in a transverse field,
which equilibrates through (18), and for a qubit
in contact with a bosonic bath.
Before starting with the calculations some gen-
eral remarks need to be made. First, it should be
noticed that, in the case of partial control over the
Hamiltonian H =
P
λ
i
t
X
i
, the metric m
i,j
will
be of the same dimension as the number of con-
trollable parameters {λ
i
t
} (see Appendix F for an
explicit expression). The optimal trajectory can
then be obtained by solving the geodesic equa-
tion:
¨
λ
i
t
+ Γ
i
j,k
λ
t
˙
λ
j
t
˙
λ
k
t
= 0, (23)
where Γ denotes the Christoffel symbols, which
can be computed from the formula:
Γ
i
j,k
|
λ
t
=
1
2
m
i,l
(∂
j
m
l,k
+ ∂
k
m
j,l
− ∂
l
m
j,k
) |
λ
t
,
(24)
where m
i,l
is the inverse of the metric, and
we use the shorthand notation ∂
i
m
j,k
|
λ
t
≡
(∂m
j,k
/∂λ
i
)|
λ=λ
t
.
In this way, the machinery presented here pro-
vides an automatic method to obtain differential
equations for the optimal trajectory directly in
the space of parameters, which is usually of a di-
mension much smaller than the one of the full
Hilbert space. Nonetheless, it is important to
keep in mind that the computational cost still
scales with the total dimension, where the most
expensive operation is the computation of m
i,j
itself.
In the case of the master equation (18), there
are some important simplifications coming from
the fact that the KMB metric is directly related
to the partition function. In fact, one can obtain
simply by differentiation:
β
3
Γ
i
j,k
|
λ
t
=
1
2
m
i,l
∂
3
∂λ
j
∂λ
k
∂λ
l
log Z
H
!
. (25)
This makes this approach particularly simple
when an analytical expression for Z
H
is known,
as we will illustrate in the following example.
4.1 Ising chain in a transverse field
We now consider the system to be given by an
Ising chain in a transverse field, which equili-
brates through (18). The Hamiltonian of the sys-
tem is given by:
H
I
= −J
n
X
i=1
σ
z
i
σ
z
i+1
+ gσ
x
i
(26)
where J > 0 set a measure for the energy scale
and g is a dimensionless coupling parameter,
which can be interpreted as an external mag-
netic field. For simplicity, in the following we
will measure times in units of τ and energies
in units of J. The spectrum of the system can
5
be computed analytically for periodic boundary
conditions through a Jordan-Wigner transforma-
tion, and in the thermodynamic limit the parti-
tion function is given by the integral expression
(see e.g. [43])
lim
n→∞
1
N
log Z =
Z
2π
0
dk log
2 cosh
β
ε
k
2
,
(27)
where ε
k
is the eigenvalue corresponding to the
momentum k, which reads:
ε
k
= 2J
q
1 + g
2
− 2g cos k. (28)
This system presents a phase transition at zero
temperature and g = 1 from an ordered ferromag-
netic phase to a quantum paramagnetic phase.
Moreover, at finite but low enough temperatures,
there is a rich variety of different physical regimes
(summarised in Fig. 2), characterised by different
scaling for the correlation length and the equili-
bration timescale of the system (see e.g. [43] for
details).
We study here the case in which one has control
only over g, so that the metric and the Christoffel
symbols become a scalar, which we name m and
Γ, respectively. In particular, we see that it is
sufficient to differentiate (27) with respect to g
twice, to obtain:
m(g) =
Z
2π
0
dk
¨ε
k
2
tanh
β
ε
k
2
+
˙ε
k
2
sech
β
ε
k
2
2
!
,
(29)
Similarly, we can compute Γ using (25). Both
m and Γ are shown in Fig. 2 for different tem-
peratures. Finally, we also compute numerically
optimal thermodynamic processes through the
geodesic equations (23). The results are also
shown in Fig. 2.
Since the metric is connected with the free
energy of the system, it is interesting to notice
how the Riemannian structure is affected by the
presence of a phase transition at zero tempera-
ture. Comparing the metric and, in particular,
the Christoffel symbol with the phase diagram of
the system, we can see that the change in be-
haviour of these geometric quantities retrace a
change in the underlying physical properties. Ad-
ditionally, note that m increases close to g = 1 as
Figure 2: The top two figures show the metric and the
Christoffel symbol for the Ising chain at different temper-
atures as a function of the coupling g. On the bottom,
on the left we present the behaviour of minimally dis-
sipating trajectories defined by the boundary conditions
g(0) = 0 to g(1) = 5; on the right, the phase diagram
of the system.
the temperature is decreased, illustrating how the
dissipation increases in the presence of a phase
transition. This behaviour reflects in the shape of
the geodesics, as
˙
H
t
decreases close to the phase
transition in order to compensate for the larger
dissipation.
4.2 Qubit in contact with a bosonic bath
We now move to treat an example in which L
+
t
is
non-trivial, a two level system in contact with a
bosonic bath with spectral density J(ω) = γ
0
ω
α
.
The Lindbladian we use can be obtained through
a microscopic derivation [44]; more details and
the particular form of the dynamics are given in
appendix G. The parameter α characterises the
ohmicity of the environment, and we have that
for α = 1, α > 1, α < 1, the bath is ohmic,
superohmic and subohmic, respectively. We as-
sume full control on the Hamiltonian of the two-
level system, which is parametrised by spherical
coordinates
H = r cos ϕ sin θ ˆσ
x
+ r sin ϕ sin θ ˆσ
y
+ r cos θ ˆσ
z
,
(30)
where (r, θ, ϕ) are the control parameters.
Thanks to the convenient choice of coordinates,
the metric takes the particularly simple form (see
Appendix G)
m
L
=
1
r
α
diag
n
λ
d
, λ
q
r
2
, λ
q
r
2
sin
2
θ
o
, (31)
6
Figure 3: From left to right, from top to bottom: de-
pendence of the classical and quantum component of
the metric λ
d,q
on the energy spacing r; geodesics in
the radial direction (r(0) = 0.1, r(1) = 5) for con-
stant θ and φ; geodesics in Cartesian coordinates for
non commuting Hamiltonians ({x(0), z(0)} = {0, 1},
{x(1), z(1)} = {1, 1}) and y ≡ 0; comparison of the
first order contribution to the dissipation during a linear
protocol with respect to the optimal one as a function of
the energy spacing. The energies are measured in units
of β.
where we can identify the expression of
the Euclidean metric in spherical coordinates
(diag{1, r
2
, r
2
sin
2
(θ)}), and the two eigenvalues
are given by:
λ
d
=
tanh(r)
cosh
2
(r)
, λ
q
=
2 tanh
2
(r)
r
. (32)
The existence of different eigenvalues for the ra-
dial direction and the solid angle reflects the
physical fact that the system dissipates differently
if only the energy spacing is moved, or if coher-
ence between the two levels is created. Moreover,
it is straightforward to verify that the ratio λ
q
/λ
d
diverges for r → ∞. This illustrates a general be-
haviour: the existence of exponentially more dis-
sipative parameters in thermodynamic systems.
In this case, there is a simple physical explana-
tion: changing the energy spacing only affects
an exponentially small fraction of the population,
whereas the whole system has to be manipulated
in order to create coherence. This example how-
ever illustrates the fact that inspecting the metric
eigenvalues provides us with a powerful descrip-
tion of the physics of dissipation, which is crucial
in more complex systems where a simple intuitive
understanding is out of reach.
The ohmicity of the bath contributes to the
metric with a factor r
−α
, which is the overall de-
pendence of the equilibration timescales on the
energy spacing r. Owing to this dependence, op-
timal protocols will tend to spend more time in
the region with low r. This effect is illustrated in
the right top panel of Fig. 3, where we construct
optimal trajectories in the radial direction. As ex-
pected, this effect is more pronounced for higher
αs.
Our results also allow for constructing optimal
trajectories between non-commuting endpoints.
Then we necessarily have that [
˙
H
t
, H
t
] 6= 0,
so that quantum coherence between energy lev-
els is created along the protocol. Optimal non-
commuting trajectories are shown in the bottom
left Figure of 3. Interestingly, we see that the
ohmicity of the bath qualitatively changes the be-
haviour of the optimal trajectories.
In order to illustrate the improvement obtained
by using geodesic trajectories, we compare them
to a naive choice where the parameters are lin-
early modified in time. We consider boundary
conditions H
A
= 0 and H
B
= E
f
σ
z
(energy is
measured in units of β). Restricting to trivial
dynamics (18), which allows for a simple analytic
solution, we find at first order in O(1/T ):
W
lin
dis
= E
f
tanh E
f
, (33)
W
KMB
dis
=
1
4
π − 2 tan
−1
(csch(E
f
))
2
. (34)
In the limit of E
f
1 the first term diverges
linearly in E
f
, while W
KMB
dis
≈ π
2
/4. This dif-
ference makes clear how for big energy gaps or,
equivalently, at low temperature, using a geodesic
trajectory gives an increasingly bigger advantage,
making the use of optimal trajectories particu-
larly relevant in the quantum regime. These con-
siderations are also relevant for experiments, as
current demonstrations of the Landauer principle
rely on a linear increase of E(t) [45–48].
5 Comparison to other approaches
Whereas the presented results are valid in the
slow driving limit, exact solutions for minimis-
ing dissipation also exist in the literature [49–
52]. In particular, optimal finite-time protocols
for two-level systems have been treated in a va-
riety of settings [50–54]. Besides specific solvable
systems, a general approach for minimising dissi-
pation based on optimal control theory has been
7
put forward in [51, 52, 54], which requires solving
a system of D non-linear differential equations,
where D is the Hilbert space dimension (see [51]
and Appendix H).
Conversely, the geometric approach provides
an approximation of the optimal solution (which
becomes exact in the slow driving limit) by a
set of d non-linear differential equations, where
d is the number of controllable parameters in the
Hamiltonian (see Eq. (23)). This makes this ap-
proach particularly useful in more complex sys-
tems, where d D, as we have illustrated in the
Ising chain example where d = 1 and D → ∞.
This gives the geometric approach a wide range
of applicability, which has also been illustrated in
classical systems of different nature and complex-
ity [4–7, 13–15, 55–58]. It is however important
to keep in mind that in general building the met-
ric (15) requires diagonalising the Hamiltonian of
the system of interest, as discussed in Appendix
F. A class of dynamics where the computation of
the metric notably simplifies is when the expected
value of each externally controlled observable (i.e.
hX
i
i
ρ
t
)) relaxes to its thermal expectation value
with a well defined time scale τ
i
as in (21); as
in this case the corresponding metric (22) can be
directly computed from the partition function.
Besides providing an efficient way of finding
minimally dissipative paths, there are other ad-
vantages of working in the quasistatic regime:
1. By construction, the protocols developed are
independent of the total time T . That is, one
does not need to optimise the path for every
duration of the protocol.
2. When dealing with small systems, it is not
only important to minimise the dissipated
work, but also to minimise fluctuations. In
the quasistatic limit, minimising dissipation
guarantees the minimisation of fluctuations
for classical systems [9], and for commuting
protocols in the quantum regime [59].
3. The introduction of a Riemannian structure
in the space of parameters provides an auto-
matic way to infer the dissipative properties
of the system, just by inspecting the eigen-
values of the metric.
Finally, in Appendix H we also qualitatively
compare the geometric approach with exact ap-
proaches for a two-level system finding good
agreement between them even when the total
time T of the protocol is similar to the time scale
of relaxation, see also [26].
6 Conclusions and outlook
We developed a general framework to define a
thermodynamic metric for systems that evolve
under a Lindblad master equation, in such a
way that geodesics correspond to minimally-
dissipative paths in the quasistatic regime. The
connection with the non-equilibrium free energy
makes it extremely versatile and an extension
to more general settings seems to be foreseeable
with only minor modifications. For example, it
should be possible to extend our approach to gen-
eralised Gibbs ensembles [60–63] simply by re-
defining the non equilibrium free energy (1) so
to take into account the extra conserved quan-
tities. We can even consider a dynamics which
equilibrates to a non equilibrium steady states
by simply substituting the thermal state in the
definition (4) of the entropy production rate with
the appropriate steady state π(H) [33]. For non-
equilibrium steady states, an interesting future
direction is to consider relations between this ge-
ometric approach and quantum thermodynamic
uncertainty relations [42, 64]. Moreover, despite
the fact that we assume a Lindblad master equa-
tion, and therefore weak coupling, thanks to the
reaction-coordinate mapping, an extension to the
strong coupling regime is possible [65–72]. In this
context, since the spectral density of the bath is
modified under the mapping, we expect that in
the strong coupling regime there will be qualita-
tively changes in the behaviour of optimal tra-
jectories, similarly to what we showed in Fig. 3.
Similarly, the same framework can also be applied
to other types of thermalisation such as in colli-
sional models [73]. Another interesting research
direction is to investigate the connection between
optimal thermodynamic processes and shortcuts
to thermalisation [74]. Overall, it seems that the
introduction of a metric structure on the space of
thermodynamic states is not only natural, but it
is worth of more investigation in the future.
Acknowledgements. We thank G. Crooks, S.
Deffner, M. Ueda, and V. Cavina for insightful
8
comments. M.P.-L. acknowledges support from
the Alexander von Humboldt Foundation. This
project has received funding from the European
Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie
grant agreement No 713729, and from Span-
ish MINECO (QIBEQI FIS2016-80773-P, Severo
Ochoa SEV-2015-0522), Fundacio Cellex, Gener-
alitat de Catalunya (SGR 1381 and CERCA Pro-
gramme). This research was also supported in
part by the National Science Foundation under
Grant No. NSF PHY-1748958.
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13
A Non equilibrium free energy and work extraction
In this section we give the details on how to derive equation (1) and (3) of the main text. First, it is
straightforward to show, using the defining formula F (ρ, H) = hHi
ρ
− β
−1
S(ρ), that the free energy
of a thermal state ω
β
(H) is given by
F (ω, H) = hHi
ρ
− β
−1
(β hHi
ρ
+ log Z
H
) = −β
−1
log Z
H
(35)
in complete analogy with classical thermodynamics. Then, equation (1) is obtained by direct compu-
tation, rewriting the relative entropy as:
β
−1
S(ρ||ω
β
(H)) = β
−1
Tr [ρ(log ρ − log ω
β
(H))] =
=
−β
−1
S(ρ) + hHi
ρ
+ β
−1
log Z
H
=
= F (ρ, H) − F (ω
β
(H), H), (36)
which gives the result after rearranging the terms. For what regards equation (3), it is useful to first
give the expression of the total derivative of the non equilibrium free energy along a trajectory (ρ
t
, H
t
).
This can be decomposed in the two terms:
d
dt
F (ρ
t
, H
t
) = (Tr [ ˙ρ
t
H
t
] + β
−1
d
dt
Tr [ρ
t
log ρ
t
]) + Tr[ρ
t
˙
H
t
]. (37)
The last trace can be rewritten as:
Tr[ρ
t
˙
H
t
] = lim
ε→0
F (ρ
t
, H
t+ε
) − F (ρ
t
, H
t
)
ε
=: ∂
H
t
F (ρ
t
, H
t
). (38)
For what regards the term inside the parenthesis, it should be first noticed that we can rewrite:
Tr [ ˙ρ
t
H
t
] = −β
−1
(Tr
h
˙ρ
t
log e
−βH
t
i
− log Z
H
Tr [ ˙ρ
t
]) =
= −β
−1
Tr [ ˙ρ
t
log ω
β
(H
t
)] , (39)
where in the first line we use the fact that the derivative of a state is traceless. Then, we obtain for
the whole expression:
d
dt
Tr [ρ
t
log ρ
t
] − Tr [ ˙ρ
t
log ω
β
(H
t
)] = lim
ε→0
S(ρ
t+ε
||ω
β
(H
t
)) − S(ρ
t
||ω
β
(H
t
))
ε
=: ∂
ρ
t
S(ρ
t
||ω
β
(H
t
)), (40)
where we recognise the quantity defined