# Work Sum Rule for Open Quantum Systems

Parth Kumar<EMAIL_ADDRESS>Caleb M. Webb<EMAIL_ADDRESS>Charles
A. Stafford<EMAIL_ADDRESS>Department of Physics, University of
Arizona, 1118 East Fourth Street, Tucson, Arizona 85721, USA

###### Abstract

A key question in the thermodynamics of open quantum systems is how to
partition thermodynamic quantities such as entropy, work, and internal energy
between the system and its environment. We show that the only partition under
which entropy is non-singular is based on a partition of Hilbert-space, which
assigns half the system-environment coupling to the system and half to the
environment. However, quantum work partitions non-trivially under Hilbert-
space partition, and we derive a Work Sum Rule that accounts for quantum work
at a distance. All state functions of the system are shown to be path
independent once this nonlocal quantum work is properly accounted for. The
thermodynamics of two classes of quasi-statically driven open quantum systems
is analyzed: systems with a finite environment in the grand canonical
ensemble, and systems with an unbounded environment. Our results are
illustrated with applications to a time-dependent two-level system and the
driven resonant-level model.

###### Contents

  1. I Introduction
  2. II Driven Quantum System
    1. II.1 Quasi-static driving
  3. III Hilbert-space Partition
  4. IV Closed Quantum System in the Grand Canonical Ensemble
    1. IV.1 Setup and Definitions
    2. IV.2 Sum Rule for Quantum Work
    3. IV.3 Example: Time-dependent two-level system
  5. V Open Quantum System
    1. V.1 Work Sum Rule for Open Systems
    2. V.2 First Law of Thermodynamics for an Open Quantum System
    3. V.3 NEGF formulas
    4. V.4 Illustration: Driven Resonant-Level Model
  6. VI $\alpha$-Partition of the Thermodynamics
  7. VII Conclusions
  8. A Derivation of Eq. (2)
  9. B Nonequilibrium Green’s functions in the quasi-static limit
    1. B.1 Key definitions and relations
    2. B.2 Relation of thermodynamic quantities to density of states [Eqs. (6), (7), and (9)]
  10. C Global Quasi-static dynamics and thermodynamics
    1. C.1 Derivation of the Global Fundamental Thermodynamic Identity [Eq. (12)]
    2. C.2 Derivation of Eq. (13)
  11. D Motivation for the operator Hilbert-space partition [Eq. (20)]
  12. E Derivation of the expression for partitioned power [Eq. (32)]
  13. F Quasi-static dynamics and thermodynamics of an open quantum system
  14. G $\alpha$-Partition of the Entropy

##  I Introduction

The program of scaling the laws of thermodynamics down to the nanoscale and
beyond has proven to be exceptionally challenging. While substantial progress
has been made [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21], there exists little consensus on some of the most foundational
questions, especially in the extreme quantum limit. One key question is the
validity and formulation of the First Law of Thermodynamics for open quantum
systems [22, 23, 24, 25, 26, 27, 28, 29]. In this Letter, we address this
question for quasi-statically driven open quantum systems, wherein the
interface between the system and its environment plays a central role.

Applying the laws of thermodynamics to open quantum systems begs the question
of how to partition quantum observables between the system and its
environment. Various schemes have been proposed in the literature [22, 23, 24,
25, 26, 27, 28, 29], many of which involve assigning part of the interfacial
energy to the system and the remainder to the environment. We show that
entropy is only well-defined under a Hilbert-space partition, which divides
the system-environment coupling equally between the system and the
environment.

We show that internal energy, chemical work, and entropy are partitioned
straightforwardly under a partition of Hilbert space. However, the partition
of quantum work is non-trivial in this framework. We therefore derive a Work
Sum Rule that accounts for the thermodynamic effect of quantum work at a
distance. All state functions of the system are shown to be path independent
once this nonlocal quantum work is properly accounted for.

We emphasize that our thermodynamic partition of the entropy is different than
the usual information-theoretic partition based on the reduced state of a
quantum subsystem [30]; due to the nonlocality of quantum information, the
reduced-state description measures entanglement entropy that is not associated
with any thermodynamic process [31]. While the thermodynamic states analyzed
in this Letter are generically highly entangled, this entanglement is not
associated with thermodynamic entropy.

This Letter is structured as follows: We start by defining the global
thermodynamic quantities in the quasi-static limit in Sec. II.1. The Hilbert-
space partition of the thermodynamics is introduced in Sec. III, leading to
the First Law for a quantum subsystem. The partition of a finite quantum
system in the grand canonical ensemble is analyzed in Sec. IV, wherein the
Work Sum Rule is derived. The results are illustrated with an application to a
time-dependent two-level system. The partition of a quantum system with an
unbounded environment is analyzed in Sec. V. Closed-form expressions for all
terms appearing in the First Law are derived in terms of Nonequilibrium
Green’s functions (NEGF). The results are illustrated with an application to
the driven resonant-level model. Finally, our partition of the thermodynamics
is compared with other schemes proposed in the literature in Sec. VI. Our
conclusions are presented in Sec. VII.

##  II Driven Quantum System

The Internal Energy $U(t)$ of a quantum system with Hamiltonian $H(t)$ is

$U(t):=\langle H(t)\rangle\,,$ (1)

where $\langle\,\rangle$ denotes the quantum statistical average $\langle
H(t)\rangle=\mathrm{Tr}\\{H(t)\rho(t)\\}$, where $\rho(t)$ is the density
matrix of the universe at time $t$ and $\mathrm{Tr}\\{\\}$ denotes the trace
over the full Fock space. The (inclusive) rate of Work done by external forces
on the quantum system is [32, 5, 26]

$\dot{W}_{ext}(t):=\frac{d}{dt}\langle
H(t)\rangle=\langle\dot{H}(t)\rangle\,,$ (2)

where the equality follows from the von Neumann equation for the density
matrix (see Appendix A for details).

In this Letter, we consider a time-dependent quantum system of fermions
without inter-particle interactions. In that case, the internal energy of and
the power delivered to the system can be expressed in terms of the single-
particle Green’s function $G^{<}(t,t^{\prime})$ as

$U(t)=-i\mathbb{Tr}\\{h(t)G^{<}(t,t)\\}\,,$ (3)

and

$\dot{W}_{ext}(t)=-i\mathbb{Tr}\\{\dot{h}(t)G^{<}(t,t)\\}\,,$ (4)

respectively, where $h(t)$ is the matrix representation of $H(t)$ in the
1-body Hilbert-space, and $\mathbb{Tr}$ denotes a trace over this Hilbert-
space (see Appendix B for the definition and evaluation of $G^{<}$).

###  II.1 Quasi-static driving

We consider a quantum system that can exchange energy and particles with a
reservoir in equilibrium at temperature $T$ and chemical potential $\mu$. The
system is driven quasi-statically so that it remains in equilibrium with the
reservoir throughout the driving protocol. Under these conditions, the equal-
time Green’s function takes the quasi-equilibrium form

$G^{<(0)}(t,t)=\int d\epsilon f(\epsilon)A^{(0)}(t,\epsilon)\,,$ (5)

where $A^{(0)}(t,\epsilon)=\delta(\epsilon-h(t))$ is the spectral function in
the quasi-static limit and $f(\epsilon)=(1+e^{\beta(\epsilon-\mu)})^{-1}$ is
the Fermi-Dirac distribution. Here the superscript $A^{(n)}$ denotes the order
in time derivatives of the driving Hamiltonian (see Appendix B for details.)

The relevant thermodynamic quantities for such a system are evaluated as
follows (see Appendix B.2 for details). The internal energy is given by

$U^{(0)}(t)=\int d\epsilon\,g(t,\epsilon)f(\epsilon)\epsilon\,,$ (6)

where $g(t,\epsilon)=\mathbb{Tr}\\{A^{(0)}(t,\epsilon)\\}$ is the quasi-static
density of states. Similarly, the quasi-static power delivered may be computed
using Eqs. (4) and (5) as

$\dot{W}_{ext}^{(1)}(t)=\int
d\epsilon\,f(\epsilon)\mathbb{Tr}\\{\dot{h}(t)A^{(0)}(t,\epsilon)\\}\,.$ (7)

The quasi-static entropy is given by

$\langle S(t)\rangle=S^{(0)}(t)=\int d\epsilon\,g(t,\epsilon)s(\epsilon)\,,$
(8)

where the entropy operator $S(t)=-\ln\rho(t)$,
$s(\epsilon)=\beta(\epsilon-\mu)f(\epsilon)+\ln(1+e^{-\beta(\epsilon-\mu)})$,
and we set $k_{B}=1$. The mean number of particles in the system is given by

$N^{(0)}(t)=\int d\epsilon\,g(t,\epsilon)f(\epsilon)\,.$ (9)

The grand canonical potential of the system is

$\Omega^{(0)}(t)=U^{(0)}(t)-TS^{(0)}(t)-\mu N^{(0)}(t)\,,$ (10)

which can be expressed as

$\Omega^{(0)}(t)=\int d\epsilon\,g(t,\epsilon)\omega(\epsilon)\,,$ (11)

where $\omega(\epsilon)=-\frac{1}{\beta}\ln(1+e^{-\beta(\epsilon-\mu)})$.

For quasi-static driving of the system, with the reservoir held at fixed $T$
and $\mu$, the first variations of the thermodynamic quantities satisfy

$\dot{\Omega}^{(1)}(t)=\dot{U}^{(1)}(t)-T\dot{S}^{(1)}(t)-\mu\dot{N}^{(1)}(t)\,.$
(12)

Importantly, the first variation of $\Omega$ is equal to the external work

$\dot{W}_{ext}^{(1)}(t)=\dot{\Omega}^{(1)}(t)$ (13)

(see Appendix C.1 and Appendix C.2 for derivations of Eqs. (12) and (13),
respectively).

##  III Hilbert-space Partition

A central quantity in scanning probe microscopy is the local density of states
(LDOS) [33, 34, 35, 36, 37, 38, 8, 39, 40, 41, 42, 9, 10, 12, 13, 43, 44, 16],
which provides a firm experimental basis to construct local thermodynamic
state functions in real quantum systems. The LDOS of subsystem $\gamma$ is
defined as [45, 46]

$g_{\gamma}(t,\epsilon):=\mathbb{Tr}\\{\Pi_{\gamma}A^{(0)}(t,\epsilon)\\}=\mathbb{Tr}\\{\Pi_{\gamma}\delta(\epsilon-h(t))\\},$
(14)

where $\Pi_{\gamma}=\int_{x\in\gamma}\,dx|x\rangle\langle x|$ is the
projection operator onto subspace $\gamma$ of the single-particle Hilbert-
space. $g_{\gamma}(t,\epsilon)$ gives the local spectrum of the nonlocal
operator $h(t)$.

In analogy with the global quantities defined by Eqs. (6), (8), (9), and (11),
the partitioned thermodynamic quantities of subsystem $\gamma$ are defined as
[47, 48, 49]

$U_{\gamma}^{(0)}(t)=\int
d\epsilon\,g_{\gamma}(t,\epsilon)f(\epsilon)\epsilon\,,$ (15)
$S_{\gamma}^{(0)}(t)=\int d\epsilon\,g_{\gamma}(t,\epsilon)s(\epsilon)\,,$
(16) $N_{\gamma}^{(0)}(t)=\int
d\epsilon\,g_{\gamma}(t,\epsilon)f(\epsilon)\,,$ (17)

and

$\Omega_{\gamma}^{(0)}(t)=\int
d\epsilon\,g_{\gamma}(t,\epsilon)\omega(\epsilon)\,.$ (18)

It follows trivially that

$\dot{\Omega}_{\gamma}^{(1)}(t)=\dot{U}_{\gamma}^{(1)}(t)-T\dot{S}_{\gamma}^{(1)}(t)-\mu\dot{N}_{\gamma}^{(1)}(t)\,,$
(19)

where a summation over $\gamma$ of the above identity recovers the global
thermodynamic identity [Eq. (12)].

The thermodynamic quantities partitioned on subsystem $\gamma$ as defined in
Eqs. (15)–(17) are the quantum statistical averages of partitioned quantum
observables $H|_{\gamma}$, $S|_{\gamma}$, and $N|_{\gamma}$, respectively,
where $O|_{\gamma}$ is the Fock-space operator corresponding to the following
operator defined on the single-particle Hilbert-space [31] (see Appendix D for
details)

$o|_{\gamma}=\frac{1}{2}\\{\Pi_{\gamma},o\\}\,,$ (20)

where $o$ is the single-particle Hilbert-space operator corresponding to the
global Fock-space operator $O=\sum_{\gamma}O|_{\gamma}$, and the
anticommutator (defined as $\\{a,b\\}=ab+ba$) ensures the hermiticity of
$O|_{\gamma}$.

Rearranging the terms in Eq. (19), and identifying
$\dot{\Omega}_{\gamma}^{(1)}(t):=\dot{W}_{\gamma}^{(1)}(t)$ as the rate of
thermodynamic work on subsystem $\gamma$, we obtain the First Law of
Thermodynamics for subsystem $\gamma$

$\dot{U}_{\gamma}^{(1)}(t)=T\dot{S}_{\gamma}^{(1)}(t)+\mu\dot{N}_{\gamma}^{(1)}(t)+\dot{W}_{\gamma}^{(1)}(t).$
(21)

However, in general,
$\dot{W}^{(1)}_{\gamma}(t)\neq\langle\dot{H}|_{\gamma}(t)\rangle$, so that the
rate of thermodynamic work done on a given subsystem is not equal to the
expectation value of the power operator partitioned on that subsystem.
Instead,

$\dot{W}^{(1)}_{\gamma}(t):=\dot{\Omega}_{\gamma}^{(1)}(t)=\langle\dot{H}|_{\gamma}(t)\rangle+I^{W}_{\gamma}(t)\,,$
(22)

where $I^{W}_{\gamma}(t)$ represents the instantaneous quantum flow of free
energy into subsystem $\gamma$ induced by the external drive $\dot{H}(t)$.
$I^{W}_{\gamma}(t)$ can be thought of as (the rate of) quantum work at a
distance, as discussed further below.

Figure 1: The integrated terms (a) entropy $\Delta S_{S}$, (b) occupancy
$\Delta N_{S}$, and (c) work $W_{S}$ appearing in the First Law, Eq. (21), for
subsystem $\gamma=S$ in the resonant-level model, where both the level
$\varepsilon_{s}(t)$ and the coupling $V(t)$ to the reservoir are driven (see
Sec. V.4 for details of the model). (d) Verification of the equality of the
LHS and RHS of Eq. (21). Here the reservoir $T=0.02$ and $\mu=0$, while the
Hamiltonian parameters are varied along the two paths shown in the inset with
$\varepsilon_{s}$: $0\rightarrow 1$ and $V$: $0.6\rightarrow 0.4$.

The integrated terms appearing in the First Law for subsystem $\gamma=S$ are
plotted in Fig. 1 for the resonant-level model, where both the level and the
coupling to the reservoir are driven (see Sec. V.4 for details of the model
and driving protocol). Verification of the equality of the LHS and RHS of Eq.
(21) is shown in Fig. 1(d).

##  IV Closed Quantum System in the Grand Canonical Ensemble

###  IV.1 Setup and Definitions

We first consider a (quantum mechanically) closed system of independent
fermions whose coupling to the reservoir with which it exchanges particles and
energy is infinitesimal. The Hamiltonian is

$H(t)=H_{S}(t)+H_{\bar{S}}+H_{S\bar{S}}(t)\,,$ (23)

where

$H_{S}(t)=\sum_{n,m}[h_{S}(t)]_{nm}d^{\dagger}_{n}d_{m}\,,$ (24)
$H_{\bar{S}}=\sum_{i,j}[h_{\bar{S}}]_{ij}c^{\dagger}_{i}c_{j}\,,$ (25)

and

$H_{S\bar{S}}(t)=\sum_{i,n}[V_{in}(t)c_{i}^{\dagger}d_{n}+\mbox{h.c.}]\,.$
(26)

Here $d^{\dagger}_{n}$ ($d_{n}$) and $c^{\dagger}_{i}$ ($c_{i}$) are fermionic
creation (annihilation) operators satisfying
$\\{d_{n},d^{\dagger}_{m}\\}=\delta_{nm}$,
$\\{c_{i},c^{\dagger}_{j}\\}=\delta_{ij}$, and
$\\{d_{n},d_{m}\\}=\\{c_{i},c_{j}\\}=\\{d_{n},c_{i}\\}=0$. For this system,
definition (20) implies

$H|_{S}(t)=H_{S}(t)+\frac{1}{2}H_{S\bar{S}}(t),$ (27)

so that the coupling Hamiltonian is partitioned equally between the two
subsystems.

For a finite closed system, the density of states takes the simple form
$g(t,\epsilon)=\sum_{\nu}\delta(\epsilon-\epsilon_{\nu}(t))$, where
$\epsilon_{\nu}(t)$ are the eigenvalues of $h(t)$, and the LDOS is

$g_{\gamma}(t,\epsilon)=\sum_{\nu}P_{\nu}(\gamma,t)\delta(\epsilon-\epsilon_{\nu}(t))\,,$
(28)

where $P_{\nu}(\gamma,t)=\int_{x\in\gamma}\,dx\,|\psi_{\nu}(x,t)|^{2}$, with
$\psi_{\nu}(x,t)$ an instantaneous eigenfunction of $h(t)$.

###  IV.2 Sum Rule for Quantum Work

It follows immediately from Eq. (13) and the Hilbert-space partition
introduced in the previous section that one can write the following sum rule
for quantum work

$\dot{W}_{ext}^{(1)}(t)=\langle\dot{H}|_{S}(t)\rangle+\langle\dot{H}|_{\bar{S}}(t)\rangle=\dot{\Omega}_{S}^{(1)}(t)+\dot{\Omega}_{\bar{S}}^{(1)}(t)\,.$
(29)

Even if only the subsystem Hamiltonian is driven, the instantaneous rate of
quantum work in the complementary subsystem is generally nonzero, so that

$\langle\dot{H}_{S}(t)\rangle\stackrel{{\scriptstyle\dot{H}_{S\bar{S}}=0}}{{=}}\dot{\Omega}_{S}^{(1)}(t)+\dot{\Omega}_{\bar{S}}^{(1)}(t)\,.$
(30)

Inserting Eq. (28) into Eq. (18), and taking the time derivative, one obtains
the terms on the RHS of Eq. (29) as

$\displaystyle\dot{W}^{(1)}_{\gamma}(t)$ $\displaystyle:=$
$\displaystyle\dot{\Omega}_{\gamma}^{(1)}(t)=\sum_{\nu}[\dot{P}_{\nu}(\gamma,t)\omega_{\nu}+P_{\nu}(\gamma,t)\dot{\omega}_{\nu}]$
(31) $\displaystyle=$
$\displaystyle\sum_{\nu}[\dot{P}_{\nu}(\gamma,t)\omega_{\nu}+P_{\nu}(\gamma,t)f_{\nu}\dot{\epsilon}_{\nu}]\,.$

The expectation value of the partitioned power can be computed as (see
Appendix E for details)

$\displaystyle\langle\dot{H}|_{\gamma}(t)\rangle$ $\displaystyle=$
$\displaystyle\sum_{\nu}P_{\nu}(\gamma,t)f_{\nu}\dot{\epsilon}_{\nu}$ (32)
$\displaystyle+$
$\displaystyle\sum_{\mu\neq\nu}\frac{f_{\nu}+f_{\mu}}{2}\langle\nu|\dot{h}|\mu\rangle\langle\mu|\Pi_{\gamma}|\nu\rangle\,.$

Thus the quantity $I^{W}_{\gamma}(t)$ implicity defined in Eq. (22) can be
expressed as

$\displaystyle I^{W}_{\gamma}(t)$ $\displaystyle=$
$\displaystyle\sum_{\nu}\dot{P}_{\nu}(\gamma,t)\omega_{\nu}$ (33)
$\displaystyle-\sum_{\mu\neq\nu}\frac{f_{\nu}+f_{\mu}}{2}\langle\nu|\dot{h}|\mu\rangle\langle\mu|\Pi_{\gamma}|\nu\rangle\,.$

This term is subject to a conservation condition over the subsystems:
$\sum_{\gamma}I^{W}_{\gamma}=0$, to be interpreted as a conserved _flow of
free energy_ between subsystems. Equivalently, $I^{W}_{\gamma}$ may be thought
of as the _instantaneous rate of nonlocal quantum work_ on subsystem $\gamma$.

###  IV.3 Example: Time-dependent two-level system

Figure 2: Verification of the Work Sum Rule [Eq. (29), (30)] for a quasi-
statically driven two-level system in the grand canonical ensemble ($T=0.2$,
$\mu=0$). Only $\varepsilon_{1}(t)$ is driven (from -0.5 to 0.5) and $w=1$.
(a) Changes of the partitioned grand potential, (b) integrals of the
partitioned power operator, and (c) comparison of the total external work and
change in the global grand potential.  Figure 3: Analysis of nonlocal quantum
work in a quasi-statically driven two-level system in the grand canonical
ensemble ($T=0.2$, $\mu=0$). Both the level $\varepsilon_{1}(t)$ and the
coupling $w(t)$ are driven along two different paths (see inset:
$\varepsilon_{1}$: $-1\rightarrow 1$, $w$: $0.4\rightarrow 0.04$). (a)
Comparison of the LHS and RHS of Eq. (22). (b) Integral of the partitioned
power operator on site 1, computed from Eq. (32). (c) Nonlocal quantum work on
site 1, computed from Eq. (33).

The Hamiltonian for this model is given by Eq. (23) with
$H_{S}(t)=\varepsilon_{1}(t)d^{\dagger}d$,
$H_{\bar{S}}=\varepsilon_{2}c^{\dagger}c$, and
$H_{S\bar{S}}(t)=w(t)d^{\dagger}c+\mbox{h.c.}$ The system is driven slowly
enough that it remains in equilibrium with the (weakly coupled) reservoir at
temperature $T$ and chemical potential $\mu$ throughout the driving protocol.
We consider two protocols, which consist of quasi-static variations of the
parameters of the two-level system: In protocol 1, only the level
$\varepsilon_{1}(t)$ is driven, while in protocol 2, both the level
$\varepsilon_{1}(t)$ and the coupling $w(t)$ are driven.

The non-local nature of quantum work is most clearly manifested in protocol 1,
where the power operator has no off-diagonal contribution. Figures 2(a) and
2(b) show the changes in the grand canonical potential partitioned on
subsystems 1 and 2, and the integrated expectation values of the partitioned
power operator, respectively, under protocol 1, wherein $\varepsilon_{1}$ is
varied from -0.5 to 0.5. The net changes of all quantities are plotted as
functions of the fixed level $\varepsilon_{2}$, showing the effects of
hybridization when $\varepsilon_{2}\approx\varepsilon_{1}$. While the power
operator [Fig. 2(b)] has no contribution in subsystem 2, the instantaneous
rate of quantum work on subsystem 2 is clearly nonzero [Fig. 2(a)]. Adding up
the values for the two subsystems, the Work Sum rule, Eq. (29), is verified in
Fig. 2(c).

The changes of the relevant quantities of subsystem 1 under protocol 2, and
their path dependence or lack thereof, are illustrated in Fig. 3.
$\Delta\Omega_{1}=W_{1}$ is independent of which path is taken, as is the sum
$\int(\langle\dot{H}|_{1}\rangle+I^{W}_{1})dt$, as shown in Fig. 3(a).
However, the individual terms $\int(\langle\dot{H}|_{1}\rangle dt$ and $\int
I^{W}_{1}dt$ are path dependent, as shown in Figs. 3(b) and (c), respectively.
The rate of nonlocal quantum work $I^{W}_{1}$ is appreciable when there is
substantial hybridization of the two levels.

##  V Open Quantum System

The Hamiltonian of the open quantum system is

$H(t)=H_{S}(t)+H_{R}+H_{SR}(t)\,,$ (34)

where

$H_{S}(t)=\sum_{n,m}[h_{S}(t)]_{nm}d^{\dagger}_{n}d_{m}$ (35)

is the system Hamiltonian and

$H_{R}=\sum_{k}\epsilon_{k}c_{k}^{\dagger}c_{k}$ (36)

is the reservoir Hamiltonian. The reservoir is held at fixed temperature $T$
and chemical potential $\mu$. Finally, the coupling Hamiltonian $H_{SR}(t)$,
describing the interface between the system and the reservoir, is

$H_{SR}(t)=\sum_{k,n}[V_{kn}(t)c_{k}^{\dagger}d_{n}+\mbox{h.c.}]\,.$ (37)

As in the closed system, definition (20) implies

$H|_{S}(t)=H_{S}(t)+\frac{1}{2}H_{SR}(t),$ (38)

so that the coupling Hamiltonian is partitioned equally between the system and
the reservoir.

###  V.1 Work Sum Rule for Open Systems

The sum rule for external work derived for closed systems [Eq. (29)] can be
generalized to open systems as follows (see Appendix F for a derivation)

$\dot{W}_{ext}^{(1)}(t)=\langle\dot{H}|_{S}(t)\rangle+\langle\dot{H}|_{R}(t)\rangle=\dot{\Omega}_{S}^{(1)}(t)+\delta\dot{\Omega}_{R}^{(1)}(t)\,,$
(39)

where $\delta\Omega_{R}=\Omega_{R}-\Omega_{R,0}$ is the change in the
reservoir grand potential induced due to the change in its spectrum as a
result of its coupling to the system. A similar construction was introduced by
Friedel [50] to describe the cloud of screening charge induced in a metal due
to the presence of an impurity.

An interesting and important case to consider is that where only the system is
time-dependent so that $\dot{H}|_{S}(t)=\dot{H}_{S}(t)$ and
$\dot{H}|_{R}(t)=0$. Nonetheless, generically $\delta\dot{\Omega}_{R}(t)\neq
0$, so the work sum rule becomes

$\langle\dot{H}_{S}(t)\rangle\stackrel{{\scriptstyle\dot{H}_{SR}=0}}{{=}}\dot{\Omega}_{S}^{(1)}(t)+\delta\dot{\Omega}_{R}^{(1)}(t)\,.$
(40)

Thus, even if external forces act only inside the system, the instantaneous
thermodynamic work done on the reservoir is nonzero. For this case, Eqs. (22)
and (40) imply that the rate of nonlocal quantum work done on the system is

$I^{W}_{S}\stackrel{{\scriptstyle\dot{H}_{SR}=0}}{{=}}-\delta\dot{\Omega}^{(1)}_{R}\,,$
(41)

or minus the rate of nonlocal quantum work done on the reservoir.

###  V.2 First Law of Thermodynamics for an Open Quantum System

Using Eq. (21) with the internal energy $U_{S}(t)=\langle H|_{S}(t)\rangle$,
the 1st Law for the open system becomes

$\frac{d}{dt}\langle
H_{S}(t)+\frac{1}{2}H_{SR}(t)\rangle=T\dot{S}_{S}^{(1)}+\mu\dot{N}_{S}^{(1)}+\dot{W}_{S}^{(1)},$
(42)

where $S_{S}$ is the entropy partitioned on the system Hilbert-space, given by
Eq. (16), and

$\dot{W}_{S}^{(1)}=\dot{W}_{ext}^{(1)}-\delta\dot{\Omega}_{R}^{(1)}=\langle\dot{H}|_{S}\rangle+I^{W}_{S}$
(43)

is the rate of thermodynamic work done on the system.

###  V.3 NEGF formulas

Using the NEGF formalism [51, 52, 53], each term in the First Law [Eq. (42)]
can be expressed in terms of the quasi-static system Green’s function (see
Appendix F for derivations). The terms contributing to the (partitioned)
quasi-static power [Eq. (7)] delivered by external forces are

$\langle\dot{H}_{S}(t)\rangle=\int\frac{d\epsilon}{\pi}\,f(\epsilon)\,\mathbb{ImTr}\Bigg{\\{}\dot{h}_{S}(t)\mathcal{G}^{A(0)}(t,\epsilon)\Bigg{\\}}\,,$
(44)
$\langle\dot{H}_{SR}(t)\rangle=\int\frac{d\epsilon}{\pi}\,f(\epsilon)\,\mathbb{ImTr}\Bigg{\\{}\dot{\Sigma}^{A}(t,\epsilon)\mathcal{G}^{A(0)}(t,\epsilon)\Bigg{\\}}\,,$
(45)

where $\mathcal{G}^{A(0)}$ and $\Sigma^{A}$ are the quasi-static advanced
system Green’s function and self-energy, respectively, defined in Appendix B.

The partitioned quasi-static rates of change of grand potential may be
computed from Eq. (18) using
$g_{S}(t,\epsilon)=(1/\pi)\mathbb{ImTr}\mathcal{G}^{A(0)}(t,\epsilon)$ and
$\delta
g_{R}(t,\epsilon)=-(1/\pi)\mathbb{ImTr}[\mathcal{G}^{A(0)}(t,\epsilon)\partial_{\epsilon}\Sigma^{A}(t,\epsilon)]$,
yielding

$\dot{\Omega}_{S}^{(1)}=\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\left\\{\frac{\partial\mathcal{G}^{A(0)}(t,\epsilon)}{\partial
t}\right\\}\omega(\epsilon)\,,$ (46)
$\displaystyle\delta\dot{\Omega}_{R}^{(1)}=$ $\displaystyle-$
$\displaystyle\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\frac{\partial\Sigma^{A}(t,\epsilon)}{\partial\epsilon}\frac{\partial\mathcal{G}^{A(0)}(t,\epsilon)}{\partial
t}\Bigg{\\}}\omega(\epsilon)$ $\displaystyle-$
$\displaystyle\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\frac{\partial^{2}\Sigma^{A}(t,\epsilon)}{\partial
t\partial\epsilon}\mathcal{G}^{A(0)}(t,\epsilon)\Bigg{\\}}\omega(\epsilon)\,.$

If only the system is driven, the second line in the above equation is zero.
We note that in the broad-band limit
[$\partial_{\epsilon}\Sigma^{A}(t,\epsilon)=0$] the rate of nonlocal quantum
work vanishes, in contrast to the notion proposed in Ref. [25] and discussed
in Ref. [29]. Analogous expressions for other partitioned quantities entering
the First Law—internal energy, heat, and particle current—are given in
Appendix F.

###  V.4 Illustration: Driven Resonant-Level Model

Figure 4: Verification of the Work Sum rule [Eqs. (39), (40)] for an open
quantum system: the resonant-level model. Only the level $\varepsilon_{s}(t)$
is driven (from 1 to 1.5). The reservoir is maintained at temperature $T=0.02$
and chemical potential $\mu=0$, with $t_{0}=1.25$.

The Hamiltonian is given by Eq. (34), where

$H_{S}(t)=\varepsilon_{s}(t)d^{\dagger}d\,,$ (48)
$H_{R}=t_{0}\sum_{j=1}^{\infty}(c_{j}^{\dagger}c_{j+1}+\mbox{h.c.})$ (49)

models the reservoir as a semi-infinite tight-binding chain with hopping
integral $t_{0}$, and the interface between system and reservoir is modelled
by

$H_{SR}(t)=V(t)d^{\dagger}c_{1}+\mbox{h.c.}\,.$ (50)

Two driving protocols are again investigated: In protocol 1, the level
$\varepsilon_{s}(t)$ is varied from 1 to 1.5, while $V$ is held fixed. In
protocol 2, both $\varepsilon_{s}(t)$ and $V(t)$ are varied.

A verification of the Work Sum Rule for protocol 1 [Eq. (40)] is shown in Fig.
4, where the nonlocal quantum work on the reservoir is clearly visible as the
difference between $\Delta\Omega_{S}$ (dashed blue curve) and the total work
done $W_{\rm ext}$ (green dots). The nonlocal quantum work done on the
reservoir increases as the coupling $V$ between the system and reservoir
increases.

##  VI $\alpha$-Partition of the Thermodynamics

Figure 5: Results for the $\alpha$-partition of the entropy [Eq. (VI)] in the
resonant-level model with $\mu=0$, $V=1$, and $t_{0}=1.25$. (a) Comparison of
$\alpha\mbox{-}S_{S}$ to the entropy under Hilbert-space partition
($\alpha=1/2$) at $T=0.02$. (b) Coefficient of ($1-2\alpha$) in Eq. (VI) (2nd
term on the RHS) versus inverse temperature for various values of the level
energy $\varepsilon_{s}=0,\pm 1,\pm 2$.  Figure 6: $\alpha$-Partition of the
thermodynamics in the resonant-level model, where both the level
$\varepsilon_{s}(t)$: $0\rightarrow 1$ and coupling $V(t)$: $0.6\rightarrow
0.4$ are driven along two different paths [see protocols in inset (a)]. Here
$T=0.02$, $\mu=0$, and $t_{0}=1.25$. (a) Two definitions of work, $W_{S}$
[defined in Eq. (43)] and $\alpha\mbox{-}W_{S}$ [defined in Eq. (52)] versus
$\alpha$. (b) The change in the statistical mechanical partition of the
entropy $\Delta(\alpha\mbox{-}S_{S})$ [Eq. (VI)] and $\Delta S_{S}^{\rm EOG}$
[Eq. (55)], proposed in Ref. [23], versus $\alpha$ for the same processes. The
bound $|\Delta S_{S}|\leq\ln 2$ under Hilbert-space partition [Eq. (16)] is
shown as red dashed lines.

Finally, let us compare our analysis of the thermodynamics of a quasi-
statically driven open quantum system with some previous frameworks found in
the literature [22, 23, 24, 25, 27]. These frameworks can be described as
$\alpha$-partitions of the internal energy, where $\alpha\in[0,1]$ describes
the fraction of the coupling Hamiltonian $H_{SR}$ included in the internal
energy of the open system. In the $\alpha$-partition, the internal energy of
the open quantum system is 111Ref. [23] uses the convention $\alpha\rightarrow
1-\alpha$.

$\alpha\mbox{-}U_{S}(t)=\langle H_{S}(t)+\alpha H_{SR}(t)\rangle\,,$ (51)

and the rate of external work done on the system is identified by some authors
as [23]

$\alpha\mbox{-}\dot{W}_{S}(t)=\langle\dot{H}_{S}(t)+\alpha\dot{H}_{SR}(t)\rangle\,.$
(52)

The Hilbert-space partition proposed in this Letter corresponds to setting
$\alpha=1/2$ in Eq. (51). However, as discussed above, the rate of external
work done on the system is given by Eq. (43), and cannot in general be
expressed as in Eq. (52) for any value of $\alpha$.

For a system of independent quantum particles, the entropy operator
$-\ln\hat{\rho}^{(0)}$ for a quasi-statically driven system is also a one-body
observable, and can be partitioned in the same way (see Appendix G for
details) 222The particle number $N_{S}$ that appears in the chemical work term
in the First Law [Eq. (42)] is typically independent of $\alpha$.. For a
closed system, one finds for the $\alpha$-partition of the entropy of
subsystem $\gamma$ [31]

$\displaystyle\alpha\mbox{-}S^{(0)}_{\gamma}(t)$ $\displaystyle=$
$\displaystyle 2\alpha\sum_{\nu}\langle\nu|\Pi_{\gamma}|\nu\rangle
s(\varepsilon_{\nu})$ (53) $\displaystyle+$
$\displaystyle(1-2\alpha)\sum_{\mu,\nu}|\langle\mu|\Pi_{\gamma}|\nu\rangle|^{2}$
$\displaystyle\times$ $\displaystyle[-f_{\mu}\ln
f_{\nu}-(1-f_{\mu})\ln(1-f_{\nu})],$

where $|\nu(t)\rangle$ and $\varepsilon_{\nu}(t)$ are the instantaneous
eigenstates and eigenvalues of $h(t)$, respectively, and
$f_{\nu}=f(\varepsilon_{\nu})$. For open quantum systems, Eq. (53) can be
generalized to a continuous spectrum as

$\displaystyle\alpha\mbox{-}S^{(0)}_{\gamma}(t)$ $\displaystyle=$
$\displaystyle 2\alpha\int d\epsilon\,g_{\gamma}(t,\epsilon)s(\epsilon)$
$\displaystyle+$ $\displaystyle(1-2\alpha)\int d\epsilon\int
d\epsilon^{\prime}\,\mathbb{Tr}\\{\tilde{A}_{\gamma}(t,\epsilon)\tilde{A}_{\gamma}(t,\epsilon^{\prime})\\}$
$\displaystyle\times$
$\displaystyle[-f(\epsilon)\ln(f(\epsilon^{\prime}))-(1-f(\epsilon))\ln(1-f(\epsilon^{\prime}))]\,,$

where
$\tilde{A}_{\gamma}(t,\epsilon)=\Pi_{\gamma}\delta(\epsilon-h(t))\Pi_{\gamma}$.

Although it might appear that one could construct the partitioned
thermodynamics for arbitrary values of $\alpha$, a severe problem arises in
$\alpha$-$S_{\gamma}$ if $\alpha\neq 1/2$. For $\alpha>1/2$, the term
beginning on the second lines of Eqs. (53) and (VI) is negative and unbounded,
while for $\alpha<1/2$, the partitioned entropy of the subspace complementary
to $\gamma$ is negative and unbounded. Since entropy is a non-negative
quantity, any partition that yields a negative subsystem entropy should be
ruled out on principle as unphysical. Moreover, in fermionic systems there are
generically tightly-bound core states with occupancy $f\rightarrow 1$ and
high-lying scattering states with occupancy $f\rightarrow 0$, for both of
which Eqs. (53) and (VI) are undefined for $\alpha\neq 1/2$. Thus we are
forced to conclude that the only physically allowable partitioning of the
entropy is $\alpha=1/2$, namely, the Hilbert-space partition given by Eq. (16)
and previously proposed in Ref. [48].

Figure 5(a) shows a comparison of the small physical value of the system
entropy for $\alpha=1/2$ and the large (positive or negative) unphysical
entropy for $\alpha\neq 1/2$ for the resonant-level model with
$\varepsilon_{s}=-1$, $V=1$, $\mu=0$, and $T=0.02$. Under Hilbert-space
partition, the system entropy given by Eq. (16) is bounded by $0\leq
S_{S}\leq\ln 2$. Note that the slope of the green curve representing
$\alpha\mbox{-}S_{S}$ tends to $-\infty$ as $T\rightarrow 0$, indicating a
severe contradiction of the Third Law of Thermodynamics for $\alpha\neq 1/2$,
as pointed out previously in Refs. [23, 48]. Figure 5(b) plots the coefficient
of $(1-2\alpha)$ in Eq. (VI) (second term on the RHS) as a function of inverse
temperature at $\mu=0$ for several values of the resonant level
$\varepsilon_{s}$, indicating that this unphysical contribution to the entropy
partition diverges $\propto 1/T$ as $T\rightarrow 0$, as can be readily
understood from Eq. (VI) and the functional form of the Fermi-Dirac
distribution.

Figure 6(a) plots $W_{S}$ [defined in Eq. (43)] and $\alpha\mbox{-}W_{S}$
[defined in Eq. (52)] for two different paths in protocol 2 (see inset) as a
function of $\alpha$. The quasi-static work for a system in the grand
canonical ensemble at constant $T$ and $\mu$ should be independent of path. It
is clear that $\alpha\mbox{-}W_{S}$, the definition of the work done on the
system proposed in Ref. [23], is only path independent in the limit
$\alpha\rightarrow 1$, wherein $\lim_{\alpha\rightarrow
1}\alpha\mbox{-}W_{S}=W_{ext}$, which unsurprisingly is indeed path
independent. However, our definition [Eq. (43)] of the work $W_{S}$ done on
the Hilbert space of the system, including nonlocal quantum work, is path
independent and generally not equal to the total external work $W_{ext}$ for
nonzero system-reservoir coupling.

Using their definitions of internal energy [Eq. (51)] and of work done on the
system [Eq. (52)], Esposito, Ochoa, and Galperin [23] use the thermodynamic
identity

$T\Delta S_{S}^{\rm EOG}=\Delta(\alpha\mbox{-}U_{S})-\mu\Delta
N_{S}-\alpha\mbox{-}W_{S}$ (55)

to define the change in system entropy $\Delta S_{S}^{\rm EOG}$ for a process.
$\Delta S_{S}^{\rm EOG}$ is plotted in Fig. 6(b) as a function of $\alpha$ for
two different paths in protocol 2. Also plotted is the change in
$\alpha\mbox{-}S_{S}$, the $\alpha$-partition of statistical mechanical
entropy [Eq. (VI)]. Although our model is slightly different than that used in
Ref. [23] (we utilize a semi-infinite 1D tight-binding model of the reservoir
while Ref. [23] uses a phenomenological self-energy), the quantitative results
for the two models are comparable (see Appendix B).

As shown in Fig. 6(b) and in Ref. [23], $\Delta S_{S}^{\rm EOG}$ is path
independent only for $\alpha=1$. Based on this fact, Esposito, Ochoa, and
Galperin suggest that $\alpha=1$ should be chosen in the partition of the
internal energy [Eq. (51)], despite the acknowledged violation of the Third
Law of Thermodynamics for $\alpha=1$. It is noteworthy that the magnitude of
$\Delta S_{S}^{\rm EOG}$ greatly exceeds $\ln 2$ [see red dashed lines in Fig.
6(b)], the maximum entropy of the system under Hilbert-space partition. In
fact, $\Delta S_{S}^{\rm EOG}|_{\alpha=1}=5.57$ is opposite in sign and 80.9
times larger in magnitude than the actual entropy change $\Delta S_{S}=-0.068$
for the process shown in Fig. 6. We would argue that there is no room in the
Hilbert space of the system for so much entropy, no matter how the level is
broadened and shifted due to a finite coupling to the reservoir.

The problem with the entropy [23] defined thermodynamically via Eq. (55),
which is not equal to the statistical mechanical partition
$\alpha\mbox{-}S_{S}$ for any value of $\alpha$, stems from the incorrect
definition [Eq. (52)] of the work done on the system, which does not take into
account the nonlocal quantum work $\int I_{S}^{W}dt$. Once the correct
definition of work [Eq. (43)] is used, the Hilbert-space partition
($\alpha=1/2$) of the statistical mechanical entropy [Eq. (VI)] satisfies the
First Law [Eq. (42)], and is path independent, as shown in Fig. 6(b).

##  VII Conclusions

In this Letter, we have derived a Work Sum Rule describing the thermodynamic
effects of nonlocal quantum work. An open quantum system is analogous to a
quantum impurity problem, and nonlocal quantum work is a thermodynamic effect
analogous to the screening charge that must be included in the Friedel sum
rule [50].

We have investigated various partitions of the thermodynamics proposed in the
literature [22, 23, 24, 25, 27] and have shown that the only non-singular
partition of the entropy is based on a partition of Hilbert-space, which
corresponds to a symmetric (half-and-half) partition of off-diagonal terms
between the system and its environment, consistent with the analysis of Ref.
[22]. Alternative attempts [23, 56, 29] to partition the thermodynamics of
open quantum systems have proven problematic because they did not take into
account the nonlocal character of quantum work.

We emphasize that our thermodynamic partition uses the local spectrum of the
nonlocal Hamiltonian operator to generate the local thermodynamics.
Furthermore, our thermodynamic partition of the entropy must not be confused
with an information-theoretic partition based on the reduced density matrix.
The entanglement entropy quantifying the quantum entanglement of the system
and its environment, encoded in the von Neumann entropy calculated with the
reduced density matrix, does not arise in our entropy partition.

These results have been derived for quasi-static processes. It will be
important to investigate how to generalize the thermodynamic partition of an
open quantum system to driving beyond the quasi-static limit.

###### Acknowledgements.

We thank Carter Eckel, Ferdinand Evers, and Yiheng Xu for insights developed
during the preliminary stages of this work, and acknowledge useful discussions
on the thermodynamics of the resonant-level model with Michael Galperin.

## References

  * Michel  _et al._ [2009] M. Michel, J. Gemmer, and G. Mahler,  _Quantum Thermodynamics: Emergence of Thermodynamic Behavior Within Composite Quantum Systems_, 2nd ed., Vol. 784 (Berlin, Heidelberg: Springer Berlin / Heidelberg, Berlin, Heidelberg, 2009).
  * Binder  _et al._ [2018] F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, eds.,  _Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions_, Fundamental Theories of Physics, Vol. 195 (Springer International Publishing, Cham, 2018).
  * Deffner and Campbell [2019] S. Deffner and S. Campbell,  _Quantum Thermodynamics_, 2053-2571 (Morgan & Claypool Publishers, 2019).
  * Strasberg [2022] P. Strasberg,  _Quantum Stochastic Thermodynamics: Foundations and Selected Applications_ , Oxford Graduate Texts (Oxford University Press, Oxford, New York, 2022).
  * Campisi  _et al._ [2011] M. Campisi, P. Hänggi, and P. Talkner,  _Colloquium_ : Quantum fluctuation relations: Foundations and applications, Reviews of Modern Physics 83, 771 (2011).
  * Jarzynski [2011] C. Jarzynski, Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale, Annual Review of Condensed Matter Physics 2, 329 (2011).
  * Seifert [2012] U. Seifert, Stochastic thermodynamics, fluctuation theorems, and molecular machines, Reports on Progress in Physics 75, 126001 (2012), arxiv:1205.4176 .
  * Shi  _et al._ [2009] L. Shi, J. Zhou, P. Kim, A. Bachtold, A. Majumdar, and P. L. McEuen, Thermal probing of energy dissipation in current-carrying carbon nanotubes, Journal of Applied Physics 105, 104306 (2009).
  * Kim  _et al._ [2012a] K. Kim, W. Jeong, W. Lee, and P. Reddy, Ultra-high vacuum scanning thermal microscopy for nanometer resolution quantitative thermometry, ACS Nano 6, 4248 (2012a).
  * Lee  _et al._ [2013] W. Lee, K. Kim, W. Jeong, L. A. Zotti, F. Pauly, J. C. Cuevas, and P. Reddy, Heat dissipation in atomic-scale junctions, Nature 498, 209 (2013).
  * Neumann  _et al._ [2013] P. Neumann, I. Jakobi, F. Dolde, C. Burk, R. Reuter, G. Waldherr, J. Honert, T. Wolf, A. Brunner, J. H. Shim, D. Suter, H. Sumiya, J. Isoya, and J. Wrachtrup, High-precision nanoscale temperature sensing using single defects in diamond, Nano Letters 13, 2738 (2013).
  * Mecklenburg  _et al._ [2015] M. Mecklenburg, W. A. Hubbard, E. R. White, R. Dhall, S. B. Cronin, S. Aloni, and B. C. Regan, Nanoscale temperature mapping in operating microelectronic devices, Science (New York, N.Y.) 347, 629 (2015).
  * Menges  _et al._ [2016] F. Menges, P. Mensch, H. Schmid, H. Riel, A. Stemmer, and B. Gotsmann, Temperature mapping of operating nanoscale devices by scanning probe thermometry, Nature Communications 7, 10874 EP (2016).
  * Cui  _et al._ [2017] L. Cui, W. Jeong, S. Hur, M. Matt, J. C. Klöckner, F. Pauly, P. Nielaba, J. C. Cuevas, E. Meyhofer, and P. Reddy, Quantized thermal transport in single-atom junctions, Science (New York, N.Y.) 355, 1192 (2017).
  * Mosso  _et al._ [2017] N. Mosso, U. Drechsler, F. Menges, P. Nirmalraj, S. Karg, H. Riel, and B. Gotsmann, Heat transport through atomic contacts, Nat Nano 12, 430 (2017).
  * Shastry  _et al._ [2020] A. Shastry, S. Inui, and C. A. Stafford, Scanning tunneling thermometry, Physical Review Applied 13, 024065 (2020).
  * Toyabe  _et al._ [2010] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality, Nature Physics 6, 988 (2010), arxiv:1009.5287 [cond-mat.stat-mech] .
  * Bérut  _et al._ [2012] A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Experimental verification of Landauer’s principle linking information and thermodynamics, Nature 483, 187 (2012).
  * Koski  _et al._ [2014] J. V. Koski, V. F. Maisi, J. P. Pekola, and Dmitri V. Averin, Experimental realization of a Szilard engine with a single electron, Proceedings of the National Academy of Sciences 111, 13786 (2014).
  * Devoret  _et al._ [2014] M. H. Devoret, B. Huard, R. Schoelkopf, and L. F. Cugliandolo, eds.,  _Quantum Machines: Measurement and Control of Engineered Quantum Systems_, first edition ed. (Oxford University Press, Oxford, United Kingdom, 2014).
  * Liu  _et al._ [2021] J. Liu, K. A. Jung, and D. Segal, Periodically Driven Quantum Thermal Machines from Warming up to Limit Cycle, Physical Review Letters 127, 200602 (2021).
  * Ludovico  _et al._ [2014] M. F. Ludovico, J. S. Lim, M. Moskalets, L. Arrachea, and D. Sánchez, Dynamical energy transfer in ac-driven quantum systems, Physical Review B 89, 161306 (2014).
  * Esposito  _et al._ [2015] M. Esposito, M. A. Ochoa, and M. Galperin, Nature of heat in strongly coupled open quantum systems, Physical Review B 92, 235440 (2015).
  * Bruch  _et al._ [2016] A. Bruch, M. Thomas, S. Viola Kusminskiy, F. von Oppen, and A. Nitzan, Quantum thermodynamics of the driven resonant level model, Physical Review B 93, 115318 (2016).
  * Haughian  _et al._ [2018] P. Haughian, M. Esposito, and T. L. Schmidt, Quantum thermodynamics of the resonant-level model with driven system-bath coupling, Physical Review B 97, 085435 (2018).
  * Talkner and Hänggi [2020] P. Talkner and P. Hänggi,  _Colloquium_ : Statistical mechanics and thermodynamics at strong coupling: Quantum and classical, Reviews of Modern Physics 92, 041002 (2020).
  * Strasberg and Winter [2021] P. Strasberg and A. Winter, First and Second Law of Quantum Thermodynamics: A Consistent Derivation Based on a Microscopic Definition of Entropy, PRX Quantum 2, 030202 (2021).
  * Lacerda  _et al._ [2022] A. M. Lacerda, A. Purkayastha, M. Kewming, G. T. Landi, and J. Goold, Quantum thermodynamics with fast driving and strong coupling via the mesoscopic leads approach (2022), arxiv:2206.01090 .
  * Bergmann and Galperin [2021] N. Bergmann and M. Galperin, A Green’s function perspective on the nonequilibrium thermodynamics of open quantum systems strongly coupled to baths: Nonequilibrium quantum thermodynamics, The European Physical Journal Special Topics 230, 859 (2021).
  * Breuer and Petruccione [2007] H.-P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_ (Oxford University Press, 2007).
  * Webb and Stafford [2024] C. M. Webb and C. A. Stafford, How to Partition a Quantum Observable (2024), arxiv:2402.17908 [quant-ph] .
  * Jarzynski [2007] C. Jarzynski, Comparison of far-from-equilibrium work relations, Comptes Rendus Physique 8, 495 (2007).
  * Binnig and Rohrer [1983] G. Binnig and H. Rohrer, Scanning tunneling microscopy, Surface Science 126, 236 (1983).
  * Binnig  _et al._ [1986] G. Binnig, C. F. Quate, and Ch. Gerber, Atomic Force Microscope, Physical Review Letters 56, 930 (1986).
  * Kalinin and Gruverman [2007] S. Kalinin and A. Gruverman, eds.,  _Scanning Probe Microscopy: Electrical and Electromechanical Phenomena at the Nanoscale_ (Springer, New York, NY, 2007).
  * Chen [2021] C. J. Chen,  _Introduction to Scanning Tunneling Microscopy Third Edition_ , third edition, third edition ed., Monographs on the Physics and Chemistry of Materials (Oxford University Press, Oxford, New York, 2021).
  * Muralt and Pohl [1986] P. Muralt and D. W. Pohl, Scanning tunneling potentiometry, Applied Physics Letters 48, 514 (1986).
  * Williams and Wickramasinghe [1986] C. C. Williams and H. K. Wickramasinghe, Scanning thermal profiler, Applied Physics Letters 49, 1587 (1986).
  * Kim  _et al._ [2011] K. Kim, J. Chung, G. Hwang, O. Kwon, and J. S. Lee, Quantitative Measurement with Scanning Thermal Microscope by Preventing the Distortion Due to the Heat Transfer through the Air, ACS Nano 5, 8700 (2011).
  * Yu  _et al._ [2011] Y.-J. Yu, M. Y. Han, S. Berciaud, A. B. Georgescu, T. F. Heinz, L. E. Brus, K. S. Kim, and P. Kim, High-resolution spatial mapping of the temperature distribution of a Joule self-heated graphene nanoribbon, Applied Physics Letters 99, 183105 (2011).
  * Kim  _et al._ [2012b] K. Kim, W. Jeong, W. Lee, and P. Reddy, Ultra-High Vacuum Scanning Thermal Microscopy for Nanometer Resolution Quantitative Thermometry, ACS Nano 6, 4248 (2012b).
  * Menges  _et al._ [2012] F. Menges, H. Riel, A. Stemmer, and B. Gotsmann, Quantitative Thermometry of Nanoscale Hot Spots, Nano Letters 12, 596 (2012).
  * Stafford [2016] C. A. Stafford, Local temperature of an interacting quantum system far from equilibrium, Physical Review B 93, 245403 (2016).
  * Shastry and Stafford [2016] A. Shastry and C. A. Stafford, Temperature and voltage measurement in quantum systems far from equilibrium, Physical Review B 94, 155433 (2016).
  * Datta [2005a] S. Datta,  _Quantum Transport: Atom to Transistor_ (Cambridge University Press, Cambridge, 2005).
  * Moskalets [2011] M. V. Moskalets,  _Scattering Matrix Approach to Non-Stationary Quantum Transport_ (Imperial College Press, 2011).
  * Stafford and Shastry [2017] C. A. Stafford and A. Shastry, Local entropy of a nonequilibrium fermion system, The Journal of Chemical Physics 146, 092324 (2017).
  * Shastry  _et al._ [2019] A. Shastry, Y. Xu, and C. A. Stafford, The third law of thermodynamics in open quantum systems, The Journal of Chemical Physics 151, 064115 (2019).
  * Shastry [2019] A. Shastry,  _Theory of Thermodynamic Measurements of Quantum Systems Far from Equilibrium_, Springer Theses (Springer International Publishing, Cham, 2019).
  * Friedel [1958] J. Friedel, Metallic alloys, Il Nuovo Cimento (1955-1965) 7, 287 (1958).
  * Haug and Jauho [2007] H. Haug and A.-P. Jauho,  _Quantum Kinetics in Transport and Optics of Semiconductors_ , 2nd ed. (Springer, Berlin ; New York, 2007).
  * Rammer [2007] J. Rammer,  _Quantum Field Theory of Non-equilibrium States_ (Cambridge: Cambridge University Press, Cambridge, 2007).
  * Stefanucci and van Leeuwen [2013] G. Stefanucci and R. van Leeuwen,  _Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction_ (Cambridge University Press, Cambridge, 2013).
  * Note [1] Ref. [23] uses the convention $\alpha\rightarrow 1-\alpha$.
  * Note [2] The particle number $N_{S}$ that appears in the chemical work term in the First Law [Eq. (42)] is typically independent of $\alpha$.
  * Ochoa  _et al._ [2016] M. A. Ochoa, A. Bruch, and A. Nitzan, Energy distribution and local fluctuations in strongly coupled open quantum systems: The extended resonant level model, Physical Review B 94, 035420 (2016).
  * Datta [2005b] S. Datta, Chapter 8 - Level broadening, in  _Quantum Transport: Atom to Transistor_ (Cambridge University Press, Cambridge, 2005).
  * Sakurai and Napolitano [2011] J. J. Sakurai and J. Napolitano, Modern quantum mechanics; 2nd ed. (Addison-Wesley, San Francisco, CA, 2011) pp. 346–353.

##  Appendix A Derivation of Eq. (2)

The quantum statistical average rate of external work done on the system [Eq.
(2)] is given by

$\dot{W}_{ext}(t)=\langle\dot{H}(t)\rangle\,.$ (56)

This can be shown by first noting that since $\langle
H(t)\rangle=\mathrm{Tr}\\{\rho(t)H(t)\\}$, it follows from the product rule
that

$\frac{d}{dt}\langle
H(t)\rangle=\mathrm{Tr}\\{\dot{\rho}(t)H(t)+\rho(t)\dot{H}(t)\\}\,.$ (57)

Eq. (56) then follows from inserting the von Neumann equation of motion for
the density matrix

$i\hbar\dot{\rho}(t)=[H(t),\rho(t)]$ (58)

in the above equation and using the cyclicity of the Trace.

##  Appendix B Nonequilibrium Green’s functions in the quasi-static limit

###  B.1 Key definitions and relations

Consider the Hamiltonian defined in Eq. (34). The general system lesser two-
time Green’s function in that case is defined as

$G_{nm}^{<}(t,t^{\prime})=i\langle
d_{m}^{\dagger}(t^{\prime})d_{n}(t)\rangle\,.$ (59)

The advanced system, tunneling, and reservoir Green’s function are defined,
respectively, as

$G^{A}_{nm}(t,t^{\prime})=i\theta(t^{\prime}-t)\langle\\{d_{n}(t),d^{\dagger}_{m}(t^{\prime})\\}\rangle\,,$
(60)
$G^{A}_{nk}(t,t^{\prime})=i\theta(t^{\prime}-t)\langle\\{d_{n}(t),c^{\dagger}_{k}(t^{\prime})\\}\rangle\,,$
(61)

and

$G^{A}_{kk^{\prime}}(t,t^{\prime})=i\theta(t^{\prime}-t)\langle\\{c_{k}(t),c^{\dagger}_{k^{\prime}}(t^{\prime})\\}\rangle\,.$
(62)

The analysis of quasi-static driving is facilitated by performing the so-
called Wigner transformation [51, 52] of the time-dependent quantities defined
by the substitutions

$\tau=t-t^{\prime}\,,$ (63)

and

$\bar{t}=\frac{t+t^{\prime}}{2}\,,$ (64)

where $\tau$ is the fast dynamical quantum time-scale while $\bar{t}$ is the
slow time-scale of the quasi-static drive. We denote the Wigner transformed
lesser Green’s function as $\widetilde{G}(\bar{t},\tau)$, and its Fourier
transform with respect to the quantum time $\tau$ is defined as

$\mathcal{G}(\bar{t},\epsilon):=\int
d\tau\,e^{i\epsilon\tau}\widetilde{G}(\bar{t},\tau)\,,$ (65)

with the inverse transform is given by

$\widetilde{G}(\bar{t},\tau)=\int\frac{d\epsilon}{2\pi}\,e^{-i\epsilon\tau}\mathcal{G}(\bar{t},\epsilon)\,.$
(66)

The equal-time Green’s function is obtained by setting $\tau=0$ and
$\bar{t}=t$, so that

$G^{<}_{nm}(t,t)=\widetilde{G}_{nm}^{<}(\bar{t}=t,0)=\int\frac{d\epsilon}{2\pi}\,\mathcal{G}_{nm}^{<}(\bar{t}=t,\epsilon)\,.$
(67)

Derivatives of observables with respect to the quasi-static timescale
$\bar{t}$ are assumed to be infinitessimal, and a series expansion of the
Green’s function in powers of $\partial/\partial\bar{t}$ can be developed [51]

$\widetilde{G}(\bar{t},\tau)=\widetilde{G}^{(0)}(\bar{t},\tau)+\widetilde{G}^{(1)}(\bar{t},\tau)+\widetilde{G}^{(2)}(\bar{t},\tau)+...\,,$
(68)

where the superscript $(n)$ denotes the power of $\partial/\partial\bar{t}$.
Only the lowest-order term in the above expansion is needed to describe quasi-
static driving. In particular,

$\mathcal{G}^{<(0)}(\bar{t},\epsilon)=2\pi
if(\epsilon)A^{(0)}(\bar{t},\epsilon),$ (69)

where the quasi-static spectral function is
$A^{(0)}(\bar{t},\epsilon)=\delta(\epsilon-h(\bar{t}))$.

The submatrix of $A^{(0)}$ within the system Hilbert-space may be expressed as

$A^{(0)}(\bar{t},\epsilon)=\frac{1}{2\pi
i}[\mathcal{G}^{A(0)}(\bar{t},\epsilon)-\mathcal{G}^{R(0)}(\bar{t},\epsilon)]\,,$
(70)

where $\mathcal{G}^{A(0)}$ and $\mathcal{G}^{R(0)}$ are the Wigner-transformed
advanced and retarded system Green’s functions, respectively, in the quasi-
static limit

$\mathcal{G}^{A/R(0)}(\bar{t},\epsilon)=\left[1\epsilon-h(\bar{t})-\Sigma^{A/R}(\bar{t},\epsilon)\right]^{-1},$
(71)

and the (advanced) self-energy in the quasi-static limit is

$\Sigma^{A}_{nm}(\bar{t},\epsilon)=\sum_{k}V_{kn}(\bar{t})\mathcal{G}^{A(0)}_{kk}(\epsilon)V^{*}_{km}(\bar{t})\,.$
(72)

For the resonant-level model considered in Sec. V.4,

$\Sigma^{A}(\bar{t},\epsilon)=\Lambda(\bar{t},\epsilon)+i\frac{\Gamma(\bar{t},\epsilon)}{2}$
(73)

is simply a complex number whose real and imaginary parts are shown in Fig. 7
and may be calculated explicitly as [57]

$\Lambda(\bar{t},\epsilon)=\frac{V^{2}(\bar{t})}{2t_{0}^{2}}\begin{cases}(\epsilon-\epsilon_{0})+[(\epsilon-\epsilon_{0})^{2}-4t_{0}^{2})]^{\frac{1}{2}},&\epsilon-\epsilon_{0}<-2t_{0},\\\
(\epsilon-\epsilon_{0}),&|\epsilon-\epsilon_{0}|<2t_{0},\\\
(\epsilon-\epsilon_{0})-[(\epsilon-\epsilon_{0})^{2}-4t_{0}^{2})]^{\frac{1}{2}},&\epsilon-\epsilon_{0}>2t_{0},\end{cases}$
(74)
$\Gamma(\bar{t},\epsilon)=\begin{cases}\frac{2V^{2}(\bar{t})}{t_{0}}\Big{(}1-\Big{(}\frac{\epsilon-\epsilon_{0}}{2t_{0}}\Big{)}^{2}\Big{)}^{\frac{1}{2}},&|\epsilon-\epsilon_{0}|<2t_{0},\\\
0,&\text{otherwise}.\end{cases}$ (75)

Figure 7: The real and imaginary parts of the advanced self-energy
$\Sigma^{A}(\epsilon)=\Lambda(\epsilon)+i\Gamma(\epsilon)/2$ for the resonant-
level model as a function of energy for $V=0.6$ and $t_{0}=1.25$.

###  B.2 Relation of thermodynamic quantities to density of states [Eqs. (6),
(7), and (9)]

From Eqs. (3) and (5) it follows that

$U^{(0)}(t)=\int
d\epsilon\,f(\epsilon)\mathbb{Tr}\\{h(t)A^{(0)}(t,\epsilon)\\}\,.$ (76)

From the identity $h(t)A^{(0)}(t,\epsilon)=\epsilon A^{(0)}(t,\epsilon)$ (from
the spectral theorem), and the fact that the quasi-static density of states
$g(t,\epsilon)=\mathbb{Tr}\\{A^{(0)}(t,\epsilon)\\}$, Eq. (6) follows
immediately. The analogous expressions for quasi-static power [Eq. (7)],
entropy [Eq. (8)], and particle number [Eq. (9)] can be derived similarly. See
Appendix G for further details on the entropy.

##  Appendix C Global Quasi-static dynamics and thermodynamics

The quantum adiabatic theorem [58] tells us that the state of a system with an
adiabatically varying Hamiltonian $h(t)$ is given by a linear combination of
vectors of the form

$|\psi_{\nu}(t)\rangle=e^{i\theta_{\nu}(t)}e^{i\gamma_{\nu}(t)}|\nu(t)\rangle\,,$
(77)

where
$\theta_{\nu}(t)=-\frac{1}{\hbar}\int_{0}^{t}dt^{\prime}\,\epsilon_{\nu}(t^{\prime})$
is the so-called dynamical phase,
$\gamma_{\nu}(t)=i\int_{0}^{t}dt^{\prime}\,\langle\psi_{\nu}(t^{\prime})|\frac{\partial}{\partial
t^{\prime}}\psi_{\nu}(t^{\prime})\rangle$ is the so-called geometrical phase,
and $|\nu(t)\rangle$ solves the instantaneous Hamiltonian eigenvalue equation
$h(t)|\nu(t)\rangle=\epsilon_{\nu}(t)|\nu(t)\rangle$.

###  C.1 Derivation of the Global Fundamental Thermodynamic Identity [Eq.
(12)]

Using $g(t,\epsilon)=\sum_{\nu}\delta(\epsilon-\epsilon_{\nu}(t))$, where
$\epsilon_{\nu}(t)$ are the instantaneous eigenvalues of $h(t)$, and it is
understood that the sum becomes an integral for continuous spectra, the
thermodynamic quantities [Eqs. (6), (8), (9), and (11)], may be written

$U(t)=\sum_{\nu}f_{\nu}(t)\epsilon_{\nu}(t)\,,$ (78)
$S(t)=-\sum_{\nu}\left\\{f_{\nu}(t)\ln
f_{\nu}(t)+(1-f_{\nu}(t))\ln[1-f_{\nu}(t)]\right\\}\,,$ (79)
$N(t)=\sum_{\nu}f_{\nu}(t)\,,$ (80)

and

$\Omega(t)=-\frac{1}{\beta}\sum_{\nu}\ln[1+e^{-\beta(\epsilon_{\nu}(t)-\mu)}]\,,$
(81)

where $f_{\nu}(t)\equiv f(\epsilon_{\nu}(t))$.

It follows by the chain rule that the first variations can be computed as

$\dot{U}^{(1)}(t)=\sum_{\nu}(\dot{f}_{\nu}(t)\epsilon_{\nu}(t)+f_{\nu}(t)\dot{\epsilon}_{\nu}(t))\,,$
(82)
$\dot{S}^{(1)}(t)=\beta\sum_{\nu}(\epsilon_{\nu}(t)-\mu)\dot{f}_{\nu}(t)\,,$
(83) $\dot{N}^{(1)}(t)=\sum_{\nu}\dot{f}_{\nu}(t)\,,$ (84)

and

$\dot{\Omega}^{(1)}(t)=\sum_{\nu}f_{\nu}(t)\dot{\epsilon}_{\nu}(t)\,.$ (85)

Eq. (12) follows straightforwardly from the addition and rearrangement of the
last four equations.

###  C.2 Derivation of Eq. (13)

Eq. (56) can be written in the instantaneous eigenbasis of the Hamiltonian as

$\dot{W}_{ext}(t)=\sum_{\nu}f(\epsilon_{\nu}(t))\langle\nu(t)|\dot{h}(t)|\nu(t)\rangle\,.$
(86)

The instantaneous eigenstates and eigenvalues of $h(t)$ satisfy [58]

$\langle\nu(t)|\dot{h}(t)|\mu(t)\rangle=\dot{\epsilon}_{\nu}(t)\delta_{\mu\nu}+(\epsilon_{\mu}(t)-\epsilon_{\nu}(t))\langle\nu(t)|\dot{\mu}(t)\rangle\,,$
(87)

using which we may write

$\dot{W}_{ext}(t)=\sum_{\nu}f(\epsilon_{\nu}(t))\dot{\epsilon}_{\nu}(t)\,,$
(88)

where the RHS is identical to that of Eq. (85), thus establishing Eq. (13).

##  Appendix D Motivation for the operator Hilbert-space partition [Eq. (20)]

We begin by noting that the LDOS [Eq. (14)] can be rewritten as

$g_{\gamma}(t,\epsilon):=\mathbb{Tr}\\{\Pi_{\gamma}A^{(0)}(t,\epsilon)\\}=\int_{x\in\gamma}dx\,\langle
x|\delta(\epsilon-h(t)|x\rangle\,,$ (89)

which makes clear that the LDOS is simply the integral of the local spectrum
over subspace $\gamma$.

To motivate the Hilbert-space partition of a quantum observable [31], let us
consider a dynamical observable $O$ that is compatible with the Hamiltonian
$H(t)$. Then the corresponding Hilbert-space operator $o$ commutes with
$h(t)$, and $o|\epsilon\rangle=o(\epsilon)|\epsilon\rangle$, where
$h(t)|\epsilon\rangle=\epsilon|\epsilon\rangle$ and $o(\epsilon)$ is an
eigenvalue of $o$. In analogy with Eqs. (15) and (17), define

$O_{\gamma}^{(0)}(t):=\int
d\epsilon\,g_{\gamma}(t,\epsilon)f(\epsilon)o(\epsilon)=\int
d\epsilon\,\mathbb{Tr}\\{\Pi_{\gamma}\delta(\epsilon-h(t))\\}f(\epsilon)o(\epsilon).$
(90)

Because $[o,h(t)]=0$, the operator $o$ can be brought inside the trace so that

$O_{\gamma}^{(0)}(t)=\int
d\epsilon\,\mathbb{Tr}\\{\Pi_{\gamma}\delta(\epsilon-h(t))o\\}f(\epsilon).$
(91)

Finally, from the cyclicity of the trace and the the compatibility of $o$ and
$h(t)$, this can be expressed using Eq. (20) as

$O_{\gamma}^{(0)}(t)=\int
d\epsilon\,\mathbb{Tr}\\{o|_{\gamma}\,\delta(\epsilon-h(t))\\}f(\epsilon)=\langle
O|_{\gamma}(t)\rangle^{(0)}.$ (92)

This completes the demonstration that the partition of thermodynamic
quantities based on the local spectrum (LDOS) given in Eqs. (15) and (17)
corresponds to the quantum statistical average of the partitioned observables
defined by Eq. (20). This equivalence holds for dynamical observables
compatible with the Hamiltonian. For the application to statistical
observables, such as entropy, see Appendix G. For dynamical observables that
do not commute with the Hamiltonian, such as the power operator $\dot{H}(t)$,
Eq. (20) is taken as the definition of the partitioned observable, but the
quantum statistical average is no longer given by Eq. (90), but is instead
given by the more general Eq. (92).

##  Appendix E Derivation of the expression for partitioned power [Eq. (32)]

The quantum statistical average of the partitioned power operator in the
quasi-static limit is given by

$\displaystyle\langle\dot{H}|_{\gamma}(t)\rangle$ $\displaystyle=$
$\displaystyle\sum_{\nu}f(\epsilon_{\nu})\langle\nu|\dot{h}|_{\gamma}|\nu\rangle$
(93) $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\nu}f(\epsilon_{\nu})\langle\nu|\dot{h}\Pi_{\gamma}+\Pi_{\gamma}\dot{h}|\nu\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\mu,\nu}f(\epsilon_{\nu})\left[\langle\nu|\dot{h}|\mu\rangle\langle\mu|\Pi_{\gamma}|\nu\rangle+\langle\nu|\Pi_{\gamma}|\mu\rangle\langle\mu|\dot{h}|\nu\rangle\right]$
$\displaystyle=$
$\displaystyle\sum_{\mu,\nu}\frac{f(\epsilon_{\nu})+f(\epsilon_{\nu})}{2}\langle\nu|\dot{h}|\mu\rangle\langle\mu|\Pi_{\gamma}|\nu\rangle\,,$

from which Eq. (32) follows immediately, using for the diagonal term
$\langle\nu|\dot{h}(t)|\nu\rangle=\dot{\epsilon}_{\nu}(t)$.

##  Appendix F Quasi-static dynamics and thermodynamics of an open quantum
system

To prove the work sum rule for open systems [Eq. (39)], we start by evaluating
the power delivered by the external forces [Eq. (2)]. For quasi-static
driving, Eq. (7) gives

$\dot{W}_{ext}(t)=\int\frac{d\epsilon}{\pi}\,f(\epsilon)\,\mathbb{ImTr}\\{\dot{h}(t)\mathcal{G}^{A(0)}(t,\epsilon)\\}\,,$
(94)

where we have used Eq. (70). Treating the system and coupling terms in
$\dot{h}(t)=\dot{h}_{S}(t)+\dot{h}_{SR}(t)$ separately, and using the equation
of motion for the coupling Green’s function,
$\mathcal{G}^{A(0)}_{nk}(t,\epsilon)=\sum_{m}\mathcal{G}^{A(0)}_{nm}(t,\epsilon)V^{*}_{km}(t)\mathcal{G}^{A}_{kk}(\epsilon)$,
Eq. (94) can be written entirely in terms of the system Green’s function as

$\dot{W}_{ext}(t)=\int\frac{d\epsilon}{\pi}\,f(\epsilon)\,\mathbb{ImTr}\Bigg{\\{}(\dot{h}_{S}(t)+\dot{\Sigma}^{A}(t,\epsilon))\mathcal{G}^{A(0)}(t,\epsilon)\Bigg{\\}}\,.$
(95)

where
$\Sigma^{A}_{nm}(t,\epsilon)=\sum_{k}V_{kn}(t)\mathcal{G}^{A}_{kk}(\epsilon)V^{*}_{km}(t)$
is the advanced coupling self-energy.

To compute the time derivative of the grand canonical potential, we may write,
in analogy with the Friedel Sum Rule [50]

$\Omega^{(0)}_{S}(t)+\delta\Omega^{(0)}_{R}(t)=\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\Bigg{(}1-\frac{\partial\Sigma^{A}(t,\epsilon)}{\partial\epsilon}\Bigg{)}\mathcal{G}^{A(0)}(t,\epsilon)\Bigg{\\}}\omega(\epsilon)\,.$
(96)

The first variation of this is

$\dot{\Omega}^{(1)}(t)=\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\Bigg{(}-\frac{\partial^{2}\Sigma^{A}(t,\epsilon)}{\partial
t\partial\epsilon}\Bigg{)}\mathcal{G}^{A(0)}(t,\epsilon)+\Bigg{(}1-\frac{\partial\Sigma^{A}(t,\epsilon)}{\partial\epsilon}\Bigg{)}\frac{\partial\mathcal{G}^{A(0)}(t,\epsilon)}{\partial
t}\Bigg{\\}}\omega(\epsilon)\,,$ (97)

which gives the NEGF results of Eqs. (46) and (V.3).

To prove the work sum rule, we will show that Eqs. (95) and (97) are equal.
First, $\mathcal{G}^{A(0)}$ obeys the equation of motion

$\frac{\partial\mathcal{G}^{A(0)}(t,\epsilon)}{\partial
t}=\mathcal{G}^{A(0)}(t,\epsilon)[\dot{h}_{S}(t)+\dot{\Sigma}^{A}(t,\epsilon)]\mathcal{G}^{A(0)}(t,\epsilon)\,,$
(98)

which follows from the identity

$\frac{dM^{-1}(x)}{dx}=-M^{-1}(x)\frac{dM(x)}{dx}M^{-1}(x)\,,$ (99)

where $M(x)$ is an invertible matrix which is a function of a real scalar $x$.
Using Eq. (98) in the second term on the RHS of the Eq. (97), along with the
cyclicity of the trace, gives

$\dot{\Omega}^{(1)}(t)=\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\Bigg{(}-\frac{\partial^{2}\Sigma^{A}(t,\epsilon)}{\partial
t\partial\epsilon}\Bigg{)}\mathcal{G}^{A(0)}(t,\epsilon)+\mathcal{G}^{A(0)}(t,\epsilon)\Bigg{(}1-\frac{\partial\Sigma^{A}(t,\epsilon)}{\partial\epsilon}\Bigg{)}\mathcal{G}^{A(0)}(t,\epsilon)[\dot{h}_{S}(t)+\dot{\Sigma}^{A}(t,\epsilon)]\Bigg{\\}}\omega(\epsilon)\,.$
(100)

Finally, using the identity

$\frac{\partial\mathcal{G}^{A(0)}(t,\epsilon)}{\partial\epsilon}=-\mathcal{G}^{A(0)}(t,\epsilon)\Bigg{(}1-\frac{\partial\Sigma^{A}(t,\epsilon)}{\partial\epsilon}\Bigg{)}\mathcal{G}^{A(0)}(t,\epsilon)$
(101)

[obtained using Eq. (99)] in Eq. (100), we obtain

$\dot{\Omega}^{(1)}(t)=\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\Bigg{(}-\frac{\partial^{2}\Sigma^{A}(t,\epsilon)}{\partial
t\partial\epsilon}\Bigg{)}\mathcal{G}^{A(0)}(t,\epsilon)-\frac{\partial\mathcal{G}^{A(0)}(t,\epsilon)}{\partial\epsilon}[\dot{h}_{S}(t)+\dot{\Sigma}^{A}(t,\epsilon)]\Bigg{\\}}\omega(\epsilon)\,.$
(102)

Performing an integration by parts of the second term on the RHS of the above
equation, we get

$\displaystyle\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\frac{\partial\mathcal{G}^{A(0)}(t,\epsilon)}{\partial\epsilon}[\dot{h}_{S}(t)+\dot{\Sigma}^{A}(t,\epsilon)]\Bigg{\\}}\omega(\epsilon)$
$\displaystyle=$
$\displaystyle-\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\,f(\epsilon)\mathbb{ImTr}\Bigg{\\{}\mathcal{G}^{A(0)}(t,\epsilon)[\dot{h}_{S}(t)+\dot{\Sigma}^{A}(t,\epsilon)]\Bigg{\\}}$
$\displaystyle-\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\,\mathbb{ImTr}\Bigg{\\{}\Bigg{(}\frac{\partial^{2}\Sigma^{A}(t,\epsilon)}{\partial
t\partial\epsilon}\Bigg{)}\mathcal{G}^{A(0)}(t,\epsilon)\Bigg{\\}}\omega(\epsilon)\,,$

where the boundary term vanishes and we have used
$\partial\omega(\epsilon)/\partial\epsilon=f(\epsilon)$. Substituting this
back into Eq. (102), we recover the result on the RHS of Eq. (95), thus
proving the work sum rule,
$\dot{W}_{ext}^{(1)}=\dot{\Omega}_{S}^{(1)}+\delta\dot{\Omega}_{R}^{(1)}$.

In an analogous manner, the NEGF expressions for internal energy, entropy, and
particle currents can be derived. We simply state the results below.

$\dot{U}_{S}^{(1)}=\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\frac{\partial\mathcal{G}^{A(0)}(t,\epsilon)}{\partial
t}\Bigg{\\}}f(\epsilon)\epsilon\,,$ (104)
$\displaystyle\delta\dot{U}_{R}^{(1)}=$ $\displaystyle-$
$\displaystyle\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\Bigg{(}\frac{\partial\Sigma^{A}(t,\epsilon)}{\partial\epsilon}\Bigg{)}\Bigg{(}\frac{\partial\mathcal{G}^{A(0)}(t,\epsilon)}{\partial
t}\Bigg{)}\Bigg{\\}}f(\epsilon)\epsilon$ (105) $\displaystyle-$
$\displaystyle\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\frac{\partial^{2}\Sigma^{A}(t,\epsilon)}{\partial
t\partial\epsilon}\mathcal{G}^{A(0)}(t,\epsilon)\Bigg{\\}}f(\epsilon)\epsilon\,.$
$\dot{S}_{S}^{(1)}=\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\frac{\partial\mathcal{G}^{A(0)}(t,\epsilon)}{\partial
t}\Bigg{\\}}s(\epsilon)\,,$ (106) $\displaystyle\delta\dot{S}_{R}^{(1)}=$
$\displaystyle-$
$\displaystyle\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\Bigg{(}\frac{\partial\Sigma^{A}(t,\epsilon)}{\partial\epsilon}\Bigg{)}\Bigg{(}\frac{\partial\mathcal{G}^{A(0)}(t,\epsilon)}{\partial
t}\Bigg{)}\Bigg{\\}}s(\epsilon)$ (107) $\displaystyle-$
$\displaystyle\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\frac{\partial^{2}\Sigma^{A}(t,\epsilon)}{\partial
t\partial\epsilon}\mathcal{G}^{A(0)}(t,\epsilon)\Bigg{\\}}s(\epsilon)\,.$
$\dot{N}_{S}^{(1)}=\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\frac{\partial\mathcal{G}^{A(0)}(t,\epsilon)}{\partial
t}\Bigg{\\}}f(\epsilon)\,,$ (108) $\displaystyle\delta\dot{N}_{R}^{(1)}=$
$\displaystyle-$
$\displaystyle\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\Bigg{(}\frac{\partial\Sigma^{A}(t,\epsilon)}{\partial\epsilon}\Bigg{)}\Bigg{(}\frac{\partial\mathcal{G}^{A(0)}(t,\epsilon)}{\partial
t}\Bigg{)}\Bigg{\\}}f(\epsilon)$ (109) $\displaystyle-$
$\displaystyle\int_{-\infty}^{\infty}\,\frac{d\epsilon}{\pi}\mathbb{ImTr}\Bigg{\\{}\frac{\partial^{2}\Sigma^{A}(t,\epsilon)}{\partial
t\partial\epsilon}\mathcal{G}^{A(0)}(t,\epsilon)\Bigg{\\}}f(\epsilon)\,.$

##  Appendix G $\alpha$-Partition of the Entropy

In this appendix, Fock-space operators are denoted with hats. For a system of
independent fermions in the Grand Canonical Ensemble driven quasi-statically,
the density matrix has the instantaneous equilibrium form

$\hat{\rho}^{(0)}=\prod_{\nu}[f_{\nu}\psi_{\nu}^{\dagger}\psi_{\nu}+(1-f_{\nu})\psi_{\nu}\psi_{\nu}^{\dagger}],$
(110)

where $\psi_{\nu}^{\dagger}$ creates a fermion in an eigenstate of the
instantaneous Hamiltonian $h(t)$,
$\psi_{\nu}^{\dagger}|0\rangle=|\nu(t)\rangle$, where
$h(t)|\nu(t)\rangle=\varepsilon_{\nu}(t)|\nu(t)\rangle$ and
$f_{\nu}:=f(\varepsilon_{\nu}(t))$. For such a system, the entropy operator
has the form

$\hat{S}^{(0)}=-\ln\hat{\rho}^{(0)}=\sum_{\nu}[-\ln
f_{\nu}\psi_{\nu}^{\dagger}\psi_{\nu}-\ln(1-f_{\nu})\psi_{\nu}\psi_{\nu}^{\dagger}],$
(111)

which consists of a particle-like piece $\hat{S}^{n}$ and a hole-like piece
$\hat{S}^{p}$, whose corresponding operators on the single-particle Hilbert-
space are

$\displaystyle s^{n}$ $\displaystyle=$ $\displaystyle-\sum_{\nu}\ln
f_{\nu}|\nu\rangle\langle\nu|,$ (112) $\displaystyle s^{p}$ $\displaystyle=$
$\displaystyle-\sum_{\nu}\ln(1-f_{\nu})|\nu\rangle\langle\nu|.$ (113)

These can be partitioned according to Eq. (20) to obtain the Hilbert-space
partition of the entropy.

For the $\alpha$-partition of the entropy, we need to separate $s=s^{n}+s^{p}$
into diagonal and off-diagonal pieces

$s_{SS}:=\Pi_{S}(s^{n}+s^{p})\Pi_{S}$ (114)

and

$s_{SR}:=\Pi_{S}(s^{n}+s^{p})\Pi_{R}+\Pi_{R}(s^{n}+s^{p})\Pi_{S}.$ (115)

The $\alpha$-partition corresponds to the Hilbert-space operator

$s_{SS}+\alpha s_{SR}=(1-2\alpha)\Pi_{S}s\Pi_{S}+\alpha\\{\Pi_{S},s\\},$ (116)

where we have used $1=2\alpha+(1-2\alpha)$ and $\Pi_{S}+\Pi_{R}=1$. Clearly,
for $\alpha=1/2$, $s_{SS}+\alpha s_{SR}=s|_{S}$, as defined by Eq. (20).

The Fock-space entropy operator under $\alpha$-partition is

$\alpha\mbox{-}\hat{S}_{S}:=\hat{S}_{SS}+\alpha\hat{S}_{SR},$ (117)

and taking its quantum statistical average

$\alpha\mbox{-}S^{(0)}_{S}=\mathrm{Tr}\\{\hat{\rho}^{(0)}(\alpha\mbox{-}\hat{S}^{(0)}_{S})\\}$
(118)

yields Eq. (53) with $\gamma=S$. For additional details, see Ref. [31].