# Thermodynamics of Quantum Causal Models: An Inclusive, Hamiltonian Approach

Philipp Strasberg  Física Teòrica: Informació i Fenòmens Quàntics, Departament
de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona),
Spain

###### Abstract

Operational quantum stochastic thermodynamics is a recently proposed theory to
study the thermodynamics of open systems based on the rigorous notion of a
quantum stochastic process or quantum causal model. In there, a stochastic
trajectory is defined solely in terms of experimentally accessible measurement
results, which serve as the basis to define the corresponding thermodynamic
quantities. In contrast to this observer-dependent point of view, a ‘black
box’, which evolves unitarily and can simulate a quantum causal model, is
constructed here. The quantum thermodynamics of this big isolated system can
then be studied using widely accepted arguments from statistical mechanics. It
is shown that the resulting definitions of internal energy, heat, work, and
entropy have a natural extension to the trajectory level. The canonical choice
of them coincides with the proclaimed definitions of operational quantum
stochastic thermodynamics, thereby providing strong support in favour of that
novel framework. However, a few remaining ambiguities in the definition of
stochastic work and heat are also discovered and in light of these findings
some other proposals are reconsidered. Finally, it is demonstrated that the
first and second law hold for an even wider range of scenarios than previously
thought, covering a large class of quantum causal models based solely on a
single assumption about the initial system-bath state.

##  1 Introduction

The success of the classical framework of stochastic thermodynamics is
undeniable. It pushes the validity of the laws of thermodynamics far beyond
their original scope, it allows to consistently describe the thermodynamics of
small fluctuating out-of-equilibrium systems, even along a single trajectory,
and many of its predictions have been verified experimentally [1, 2, 3, 4, 5,
6].

In contrast, how to describe the thermodynamics of small quantum systems along
a single ‘trajectory’ remains a subject of debate since 20 years. Obviously,
the reason is the measurement backaction of an external observer, who
manipulates a small quantum system and thereby _changes_ the process. This
implies that any theory of quantum stochastic thermodynamics should be able to
consistently treat the measurement backaction and is necessarily different
from its classical counterpart [7]. Over the past, many different approaches
have been put forward, often differing in their predictions and lacking either
an experimentally feasible way to verify them or the ability to describe
quantum effects. Recently, based on a rigorous notion of a quantum stochastic
process or quantum causal model [8, 9, 10, 11, 12, 13, 14, 15, 16], an
‘operational’ approach to quantum stochastic thermodynamics was constructed
[17, 18, 19]. It puts the experimenter in the foreground by explicitly
including all external interventions (state preparation, measurements,
feedback operations, etc.) in the description. A ‘stochastic trajectory’ is
defined solely in terms of experimentally available (classical) measurement
results, on which the corresponding thermodynamic quantities are built. The
formalism is free from many restrictive and previously used assumptions (e.g.,
perfect measurements, continous measurements, detailed control about the bath
degrees of freedom, no feedback control, use of ambiguous notions for time-
reversed trajectories, etc.) and can be readily applied to analyse a multitude
of experiments including Refs. [20, 21].

Nevertheless, the definitions used in Refs. [17, 18, 19] were derived from an
observer-dependent point of view, involving quantum measurement theory,
subjective choices of the ‘Heisenberg cut’, and certain classicality
assumptions. To circumvent the use of any such elements, this paper rederives
the framework of operational quantum stochastic thermodynamics based on an
inclusive, Hamiltonian (‘autonomous’) approach. By using only arguments from
nonequilibrium statistical mechanics of isolated systems, we provide a solid
and independent justification for the definitions of Refs. [17, 18, 19]. To
the best of the authors’ knowledge, this distinguishes the operational
approach from other proposals in quantum stochastic thermodynamics.

The idea to model everything autonomously is not novel and has been used very
successfully to understand the physics of Maxwell’s demon [22, 23, 24, 25, 26,
27, 28, 29] or the thermodynamics of various forms of information processing
[30, 31, 32]. Of particular inspiration in our context is the approach by
Deffner and Jarzynski [31], hence it is also worthwhile to distinguish our
approach from it. First and most importantly, they did not explicitly connect
their autonomous approach to an observer-dependent point of view to obtain the
corresponding thermodynamic definitions at the trajectory level. Second and
more an issue of technicalities, their approach was classical, used certain
weak coupling assumptions, and they treated information and entropy
differently by excluding correlations, which turn out to be crucial for our
purposes. By overcoming all these assumptions, we do not only justify the
framework of Refs. [17, 18, 19], but we provide a general and promising tool
to study the emergence of thermodynamic quantities at the trajectory level
_without_ making explicit use of quantum measurements. This opens up a novel
possibility to derive the laws of thermodynamics at the trajectory level even
beyond the present considerations.

A short summary together with an outline of the paper reads as follows. First,
in Sec. 2 we briefly review the essential of a quantum causal model or quantum
stochastic process as far as it is needed for the following. Afterwards in
Sec. 3, we carefully construct the corresponding autonomous model, whose
dynamical equivalence to a quantum causal model is proven in Sec. 4. The
central part of this paper is Sec. 5. In there, we study the thermodynamics of
our autonomous model using arguments from statistical mechanics and we
demonstrate that it naturally induces definitions at the trajectory level in
accordance with operational quantum stochastic thermodynamics (apart from one
minor and typically negligible difference). Furthermore, we also discover that
our autonomous approach leaves room for some remaining ambiguities in the
definition of stochastic heat and work. In light of this freedom, we show that
the definition of stochastic work in the ‘two-point projective measurement
scheme’ [33, 34] provides one possible consistent choice within our autonomous
approach. However, further arguments show that it is only valid for _isolated_
, but not for open systems (in which we are primarily interested here). On the
other hand, the concept of “quantum heat”, at least as originally introduced
in Ref. [35], does not have any theoretical foundation within our autonomous
approach. Finally, the paper ends with some additional noteworthy remarks in
Sec. 6.

##  2 Quantum causal models

Albeit there are some differences in the detailed mathematical description of
a quantum causal model or quantum stochastic process [8, 9, 10, 11, 12, 13,
14, 15, 16], the common idea is that the primary entity in an experiment is
the control operation or intervention performed on the system, but not the
state (i.e., density operator) of the system itself. By shifting one level
higher from states to operations, a quantum causal model can be represented by
a multi-linear map from the set of interventions (applied at different times)
to a final output state. While being quite abstract at first place, it offers
many conceptual advantages, for instance, to optimize quantum circuits [36,
37], to rigorously define quantum non-Markovianity [38] or classicality [39,
40] in quantum processes, as well as to design multi-time resource theories
[41]. If the system is classical, the approach reduces to classical causal
modeling [42], which allows to go beyond the standard description of a
classical stochastic process, which is based only on passive and perfect
observations. We here follow closely Refs. [15, 16], which has a clear
interpretation in terms of stochastic trajectories, see also Ref. [43]. We
note that we do not attempt to review the approach in its full generality.

To begin with, we briefly repeat the essential of quantum operations,
instruments and interventions [44, 45]. At any time any such operation is
described by a completely positive map ${\cal{A}}(r)$, where $r$ denotes the
measurement result associated to this intervention. This could be the result
of a standard projective measurement or a more general measurement [46, 47].
Its action on the density operator $\rho_{S}$ of the system is denoted as
$\tilde{\rho}_{S}(r)={\cal{A}}(r)\rho_{S}$, where we used a ‘tilde’ to denote
a non-normalized state $\tilde{\rho}_{S}(r)$. The probability to obtain
outcome $r$ is encoded in its trace
$p(r)=\mbox{tr}_{S}\\{\tilde{\rho}_{S}(r)\\}$. A set of completely positive
maps ${\cal{A}}(r)$ forms an instrument if its average effect
${\cal{A}}\equiv\sum_{r}{\cal{A}}(r)$ is described by the completely positive
and trace-preserving map ${\cal{A}}$. It can be written in the familiar
operator-sum representation
${\cal{A}}\rho_{S}=\sum_{i}K_{i}\rho_{S}K_{i}^{\dagger}$ with
$\sum_{i}K_{i}^{\dagger}K_{i}=1_{S}$, where it is also known as a Kraus map.
Here, $1_{S}$ denotes the identity in the system Hilbert space.

By generalizing to multiple times, we now allow that the external agent
interrupts the time-evolution of the system at arbitrary times
$t_{n}>\dots>t_{0}$ by arbitrary interventions characterized by an instrument
$\\{{\cal{A}}_{k}(r_{k})\\}$. Here, the subscripts indicate the time $t_{k}$
at which the intervention happens. Note that at each time $t_{k}$ we can
choose a different instrument with, possibly, a different set of measurement
results associated to it. Given a sequence of measurement results, denoted by
$\textbf{r}_{n}=(r_{0},\dots,r_{n})$, the non-normalized state of the system
at a time $t>t_{n}$ can be formally written as

$\tilde{\rho}_{S}(t,\textbf{r}_{n})=\mathfrak{T}[{\cal{A}}_{n}(r_{n}),\dots,{\cal{A}}_{0}(r_{0})].$
(1)

Here, the so-called ‘process tensor’ [15, 16, 38] $\mathfrak{T}$ presents a
multi-linear map from the set of control operations to the final system state.
Note that we only indicated the dependence on the control operations
${\cal{A}}_{k}(r_{k})$ because those are the objects we assume to be
controllable in an experiment. In contrast, for instance, we did not indicate
the dependence on the initial system state, which is assumed to be arbitrary
but fixed [albeit it can be manipulated via ${\cal{A}}_{0}(r_{0})$]. In
particular, in case of system-environment correlations the process tensor does
not depend linearly on the initial system state. Furthermore, we remark that
the process tensor can be tomographically reconstructed by measuring the final
output state many times in response to the chosen set of control operations
and hence, it is experimentally a well-defined object. Finally, the
probability to get the results $\textbf{r}_{n}$ is given via
$p(\textbf{r}_{n})=\mbox{tr}_{S}\\{\tilde{\rho}_{S}(t,\textbf{r}_{n})\\}$.

Microscopically, the process tensor arises from the following picture. Let

$H_{SB}(\lambda_{t})=H_{S}(\lambda_{t})+H_{B}+V_{SB}$ (2)

denote an arbitrary system-bath Hamiltonian, where $H_{S}$ ($H_{B}$) is the
bare system (bath) Hamiltonian and $V_{SB}$ their mutual interaction.
Furthermore, in view of the thermodynamic framework considered later on, we
already introduced some time-dependent driving protocol $\lambda_{t}$ (e.g.,
an electric or magnetic field), which can change the energies of the system.
For the present section, however, this is of minor relevance. The global
unitary time evolution from $t$ to $t^{\prime}$ is described by the
superoperator

$\begin{split}\rho_{SB}(t^{\prime})&={\cal{U}}_{SB}(t^{\prime},t)\rho_{SB}(t)\\\
&\equiv
U_{SB}(t^{\prime},t)\rho_{SB}(t)U_{SB}^{\dagger}(t^{\prime},t),\end{split}$
(3)

where
$U_{SB}(t^{\prime},t)={\cal{T}}_{+}\exp\left[-i\int_{t}^{t^{\prime}}dsH_{SB}(\lambda_{s})/\hbar\right]$
with the time-ordering operator ${\cal{T}}_{+}$. Then, the process tensor can
be microscopically expressed as

$\begin{split}\tilde{\rho}_{S}(t,\textbf{r}_{n})=&\leavevmode\nobreak\
\mathfrak{T}[{\cal{A}}_{n}(r_{n}),\dots,{\cal{A}}_{0}(r_{0})]\\\
=&\leavevmode\nobreak\
\mbox{tr}_{B}\left\\{{\cal{U}}_{SB}(t,t_{n}){\cal{A}}_{n}(r_{n})\dots\right.\\\
&\left.\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\dots\cal{U}_{SB}(t_{1},t_{0}){\cal{A}}_{0}(r_{0})\rho_{SB}(t_{0}^{-})\right\\}.\end{split}$
(4)

Here, $\rho_{SB}(t_{0}^{-})$ denotes the global system-bath state, which can
be arbitrary at the moment, prior to the first intervention, which happens at
time $t_{0}$. Note that we use in general the notation $t^{\pm}$ to denote a
point in time just before or after $t$. Furthermore, the multi-linearity of
the process tensor is evident from Eq. (4). Finally, remember that each
${\cal{A}}_{k}(r_{k})\equiv{\cal{A}}_{k}(r_{k})\otimes{\cal{I}}_{B}$ acts non-
trivially only on the system (we suppress any identity operations ${\cal{I}}$
as well as many tensor products in the notation).

It turns out [10, 11, 12, 13, 14, 15, 16] that the framework can be even
further generalized. Remember that ${\mathfrak{T}}$ is a process _tensor_
acting multi-linearly on a sequence of interventions. Equivalently, the
process tensor can be seen as an object that acts on the tensor product of
spaces ${\cal{L}}({\cal{H}}_{S}\otimes{\cal{H}}_{S})$, where
${\cal{L}}({\cal{H}}_{S}\otimes{\cal{H}}_{S})$ denotes the vector space of
linear maps acting on ${\cal{H}}_{S}\otimes{\cal{H}}_{S}$ with the system
Hilbert space ${\cal{H}}_{S}$. Thus, if we denote by
$\textbf{A}_{n:0}(\textbf{r}_{n})\equiv{\cal{A}}_{n}(r_{n})\otimes\dots\otimes{\cal{A}}_{0}(r_{0})$
an element of that space, we can write Eq. (1) in short as
$\tilde{\rho}_{S}(t,\textbf{r}_{n})={\mathfrak{T}}\textbf{A}_{n:0}(\textbf{r}_{n})$.
Now, due to linearity, it is possible to consider any sequence of control
operations $\textbf{A}^{\prime}$, not only those that are decorrelated as
$\textbf{A}_{n:0}(\textbf{r}_{n})$ is. This happens, for instance, when one
considers the average effect of classical feedback control where
${\cal{A}}_{k}(r_{k})={\cal{A}}_{k}(r_{k}|\textbf{r}_{k-1})$ depends on
previous measurement results. Note that also the driving protocol
$\lambda_{t}=\lambda_{t}(\textbf{r}_{k-1})$ is allowed to depend on previous
measurement results. This generality captures any conceivable feedback
scenario, but for notational simplicity we suppress the possible dependence on
$\textbf{r}_{k-1}$ most of the times. Furthermore, it is even possible to
consider quantum correlated operations. This goes beyond classical feedback
control and can result in interventions that can no longer be written as a
completely positive map at a single time (the overall process tensor
nevertheless preserves complete positivity). It will become clear from the
exposition below that we can also include this into our autonomous framework,
but for ease of presentation we refrain from discussing the most general
scenario with all its details. Finally, within the framework of quantum causal
models it is even possible to consider interventions ${\cal{A}}_{k}(r_{k})$,
where the input and output spaces are different (for instance, by adding or
discarding ancillas to or from the system), or space-like separated
interventions, which happen at different laboratories. Again, we find that the
benefits added by the greater generality do not outweigh the drawbacks of a
more hampered presentation here.

##  3 Autonomous model

In this section we construct the autonomous model, which simulates a quantum
stochastic process of the form (1) if it is finally subjected to an
appropriate measurement giving result $\textbf{r}_{n}$. That this is in
principle possible is not new, see Refs. [10, 48, 15]. Our discussion is,
however, less abstract and more ‘physics’-oriented by explicitly specifying
Hamiltonians. This is needed later on to formulate a theory of thermodynamics.
We will guide our construction along the experimental setup sketched in Fig.
1. We proceed in two steps.

Figure 1: A system $S$ is in contact with a bath, which – in view of the
thermodynamic framework considered later on – is sketched as a heat bath with
initial temperature $T$. A preparation apparatus $P$ sequentially produces
ancillas $A(k)$, $k=0,1,\dots$, which interact with the system when they enter
the shaded grey area and thereby implement a control operation. Afterwards,
these ancillas are detected giving rise to a measurement outcome $r_{k}$,
which is stored in a memory $M$. As indicated by the feedback loop, the
external agent can decide to change, e.g., the state of each ancilla (sketched
with different colors) or the Hamiltonian of the system or the system-ancilla
interaction via the protocol $\lambda_{t}$ (not explicitly sketched)
conditioned on all previous outcomes.

First, we only consider the unconditional or unmeasured dynamics. This means
that the external agent only deterministically implements control operations
${\cal{A}}_{k}$ at time $t_{k}$ described by completely positive and trace-
preserving maps, which do not depend on any measurement result $r_{k}$.
Pictorially speaking, we ignore the right hand side of Fig. 1 (the detector,
the memory, and the feedback loop). Then, the main insight to get an
autonomous Hamiltonian model for this situations rests on the unitary dilation
theorem, first proven by Stinespring [49] (see also Refs. [44, 45]). It states
that any control operation can be written as the reduced dynamics of a unitary
interaction with an external ancilla system:

${\cal{A}}_{k}\rho_{S}=\mbox{tr}_{A(k)}\\{U_{SA(k)}\rho_{S}\otimes\rho_{A(k)}U_{SA(k)}^{\dagger}\\}.$
(5)

Here, $U_{SA(k)}$ denotes the unitary operator resulting from the system-
ancilla interaction and $\rho_{A(k)}$ the initial state of the $k$th ancilla,
which was prepared in a preparation apparatus $P$. Note that the unitary and
the initial state are allowed to depend on $k$ such that, in general,
${\cal{A}}_{k}\neq{\cal{A}}_{\ell}$ for $k\neq\ell$. The Hamiltonian
associated to this ‘unconditional’ setup therefore reads

$H_{SBPA}(\lambda_{t})=H_{SB}(\lambda_{t})+H_{PA}(\lambda_{t})+H_{SA}(\lambda_{t}).$
(6)

In detail, it consists of the following parts:

A. System-bath part $H_{SB}(\lambda_{t})$. This is the same as in Eq. (2)
describing the system, bath and their interaction ignoring any external
influence.

B. Ancilla preparation $H_{PA}(\lambda_{t})$. In this part the different
ancillas are produced by implementing a unitary $U_{PA(k)}$ prior to the
interaction of ancilla $A(k)$ with the system. By fixing a suitable initial
state $\rho_{P}(t_{0}^{-})$ of the preparation apparatus, we can – due to Eq.
(5) by choosing an appropriate $U_{PA(k)}$ – implement any operation we want
on the ancilla. Hence, we can prepare any ancilla state we like [50]. Due to
this, the initial state of the ancillas $\rho_{A}(t_{0}^{-})$ can be in
principle arbitrary, albeit in any experiment there are certain restrictions
imposed on the preparation of the initial ancilla states, see, e.g., Refs.
[20, 21]. Note that we use $A$ to denote the totality of all ancillas
$A(0),A(1),\dots,A(n)$ and that $n$ can be an arbitrary large number.

C. System-ancilla part $H_{SA}(\lambda_{t})$. This Hamiltonian reads in detail

$H_{SA}(\lambda_{t})=\sum_{k=0}^{n}\left[H_{A(k)}+V_{SA(k)}(\lambda_{t})\right]$
(7)

and describes the bare Hamiltonian $H_{A(k)}$ of each ancilla $A(k)$ as well
as its interaction $V_{SA(k)}(\lambda_{t})$ with the system. Each $H_{A(k)}$
can be different and in principle even time-dependent, albeit this is
typically not the case and therefore, we omitted it for notational simplicity.
In contrast, the time-dependence of $V_{SA(k)}(\lambda_{t})$, which can be
again different for each $A(k)$, is crucial. Later on in Sec. 4 we will design
it in such a way that it implements the unitary $U_{SA(k)}$ in Eq. (5). At the
moment, however, we are more relaxed and only assume that
$V_{SA(k)}(\lambda_{t})$ is zero outside the ‘interaction zone’ with the
system (the shaded grey area in Fig. 1). Especially, it is zero when the
ancilla gets prepared in $P$ or measured afterwards (see below).

D. Work reservoir $\lambda_{t}$. We still allow for an external time-dependent
field $\lambda_{t}$, which is responsible for, e.g., changing the system
Hamiltonian $H_{S}(\lambda_{t})$ or switching on and off the system-ancilla
interactions $V_{SA(k)}(\lambda_{t})$. This means that we model the driving,
which will be later on identified with the work supplied to the setup, semi-
classically. While this is not fully autonomous (in the sense of a completely
time-independent model), the resulting dynamics are nevertheless unitary. Note
that the ideal limit needed to generate a time-dependent Hamiltonian out of a
time-independent one is understood [31]. As the purpose of this paper is not
to understand the detailed autonomous modeling of work reservoirs, we stick
throughout to this semi-classical picture for ease of presentation.

We remark that the setup specified so far is identical to the framework of
repeated interactions or collisional models as considered in Refs. [51, 52,
53, 19]. Next, we want to explicitly include measurements and conditioning in
the description. Here, the key mathematical ingredient to autonomously model
the observer is an extension of Eq. (5). In fact, every possible intervention
${\cal{A}}_{k}(r_{k})$ can be implemented as [45, 54]

$\begin{split}{\cal{A}}&{}_{k}(r_{k})\rho_{S}=\\\
&\mbox{tr}_{A(k)}\left\\{P(r_{k})U_{SA(k)}\rho_{S}\otimes\rho_{A(k)}U_{SA(k)}^{\dagger}\right\\},\end{split}$
(8)

where $P(r_{k})$ is some orthogonal resolution of the identity in the ancilla
Hilbert space, $\sum_{r_{k}}P(r_{k})=1_{A(k)}$. Note that the average effect
of the intervention (8) is described by Eq. (5), i.e.,
$\sum_{r_{k}}{\cal{A}}_{k}(r_{k})={\cal{A}}_{k}$. To implement Eq. (8), we
need additional degrees of freedom. They will turn out to describe an
idealized classical memory responsible for performing the measurement of the
ancilla and for storing the measurement result $r_{k}$. Finally, we also need
to implement the feedback loop as sketched in Fig. 1 in an autonomous way, but
this does not need any additional physical degrees of freedom. Thus, the
Hamiltonian (6) is generalized to

$\begin{split}H_{\text{tot}}(\lambda_{t})=&\leavevmode\nobreak\
H_{M}(\lambda_{t})+V_{AM}(\lambda_{t})\\\
&+\sum_{\textbf{r}_{n}}H_{SBPA}(\lambda_{t},\textbf{r}_{n})|\textbf{r}_{n}{\rangle\langle}\textbf{r}_{n}|.\end{split}$
(9)

We now study its terms again separately in detail.

E. Memory part $H_{M}(\lambda_{t})$. Following the tradition of the
thermodynamics of computation [55], we split the memory in informational
degrees of freedom (IDF) $I$ and non-informational degrees of freedom (NIDF)
$N$, which are here responsible for dephasing the IDF (see also Ref. [31]).
Strictly speaking, the NIDF are not necessary for the following, but we keep
them as they simplify the algebra and argumentation at some places and, in
particular, including them seems more realistic from a physical perspective.
Thus, the Hamiltonian of the memory is split as

$H_{M}(\lambda_{t})=H_{I}+H_{N}+V_{IN}(\lambda_{t}).$ (10)

The Hilbert space of the IDF is spanned by the vectors
$|\textbf{r}_{n}\rangle=|r_{n}\rangle\otimes\dots\otimes|r_{0}\rangle$
encoding the measurement results. As customarily done, we assume that these
states are energetically degenerate, i.e., $H_{I}\sim 1_{I}$. Furthermore, the
IDF are initially in a standard reference state
$\rho_{I}(t_{0}^{-})=|\textbf{1}_{n}{\rangle\langle}\textbf{1}_{n}|=|1{\rangle\langle}1|\otimes\dots\otimes|1{\rangle\langle}1|$
decorrelated from the rest. We assume that the NIDF act like a pure dephasing
bath such that the information stored in $I$ is classical meaning that, after
tracing out the NIDF, the IDF are only classically correlated with the rest:

$\begin{split}\rho_{SBPAI}(t)&=\sum_{\textbf{r}_{n}}\tilde{\rho}_{SBPA}(t,\textbf{r}_{n})\otimes|\textbf{r}_{n}{\rangle\langle}\textbf{r}_{n}|\\\
&=\sum_{\textbf{r}_{n}}p(\textbf{r}_{n})\rho_{SBPA}(t,\textbf{r}_{n})\otimes|\textbf{r}_{n}{\rangle\langle}\textbf{r}_{n}|.\end{split}$
(11)

The dephasing can be implemented in various ways and, in principle, does not
entail any energetic cost. An explicit example works as follows111This part
can be skipped by readers, who know how to implement a dephasing operation in
a unitary way without energy cost.: let $r_{k}\in\\{1,\dots,d(k)\\}$ label the
in total $d(k)$ different measurement results at time $t_{k}$. Then, let
$H_{N}$ describe a set of $n$ non-interacting and energetically degenerate
entities, which are prepared in a maximally mixed state of dimension $d(k)$
respectively, $\rho_{N}(t_{0}^{-})=1_{d(0)}/d(0)\otimes\dots\otimes
1_{d(n)}/d(n)$ (note that a maximally mixed state is identical to a Gibbs
state for degenerate energies). Then, let $V_{IN}(\lambda_{t})$ implement a
short unitary evolution between $I(k)$ and $N(k)$, which happens right after
the $k$th measurement and has the form
$U_{IN(k)}=\sum_{r_{k}}|r_{k}{\rangle\langle}r_{k}|\otimes\sum_{i=1}^{d(k)}|i+r_{k}{\rangle\langle}i|$
[we interpret $|i+r_{k}\rangle$ modulo $d(k)$ if $i+r_{k}>d(k)$]. Due to the
degeneracy it is obvious that $[U_{IN(k)},H_{I}+H_{N}]=0$ and thus, the
unitary has no energetic cost. Furthermore, straightforward algebra shows that

$\begin{split}{\cal{D}}^{(k)}\rho_{I}&\equiv\mbox{tr}_{N}\left\\{U_{IN(k)}\rho_{I}\otimes\rho_{N}(t_{0}^{-})U_{IN(k)}^{\dagger}\right\\}\\\
&=\sum_{r_{k}}|r_{k}{\rangle\langle}r_{k}|\rho_{I}|r_{k}{\rangle\langle}r_{k}|.\end{split}$
(12)

Thus, we implemented a dephasing operation at zero energetic cost, as desired.

F. Ancilla-memory part $V_{AM}(\lambda_{t})$. This part is responsible for the
actual measurement of the ancilla by correlating its state with the IDF, i.e.,
$V_{AM}(\lambda_{t})=V_{AI}(\lambda_{t})=\sum_{k=0}^{n}V_{AI(k)}(\lambda_{t})$,
where we assumed that the $k$th IDF is responsible for the measurement of
ancilla $A(k)$. The desired unitary reads
$U_{AI(k)}=\sum_{r_{k}}P(r_{k})\otimes\sum_{i}|i+r_{k}-1{\rangle\langle}i|_{I(k)}$
such that for any $\rho^{\prime}_{A(k)}$ [we use a primed notation to
distinguish it from the state $\rho_{A(k)}$ appearing in Eqs. (5) and (8)]

$\begin{split}U_{AI(k)}&\rho^{\prime}_{A(k)}|1{\rangle\langle}1|U^{\dagger}_{AI(k)}=\\\
&\sum_{r_{k},r^{\prime}_{k}}P(r_{k})\rho^{\prime}_{A(k)}P(r^{\prime}_{k})|r_{k}{\rangle\langle}r^{\prime}_{k}|.\end{split}$
(13)

Thus, if we measure the IDF in state $|r_{k}\rangle$, the conditional state of
the ancilla is $P(r_{k})\rho^{\prime}_{A(k)}P(r_{k})$, which eventually gives
rise to Eq. (8). Note that the time-dependence of $V_{AI(k)}(\lambda_{t})$ is
such that the measurement happens _after_ the interaction between the system
and the $k$th ancilla as implemented by Eq. (7), but _before_ the dephasing
operation (12).

G. Conditional (feedback) part. So far, the external agent can implement
arbitrary control operations ${\cal{A}}_{k}(r_{k})$ at an arbitrary set of
discrete times $t_{k}$. However, in the most general case, the external agent
is also allowed to use the available information in the memory to condition
the future dynamics after time $t>t_{k}$ on the so far available measurement
results $\textbf{r}_{k}$. This is implemented by the last part of Eq. (9),
$\sum_{\textbf{r}_{n}}H_{SBPA}(\lambda_{t},\textbf{r}_{n})|\textbf{r}_{n}{\rangle\langle}\textbf{r}_{n}|$,
which applies a different ‘unconditional’ Hamiltonian (6) depending on the
state of the memory $|\textbf{r}_{n}{\rangle\langle}\textbf{r}_{n}|$. In fact,
due to Eq. (11) the evolution from time $t_{k}^{+}$ to $t_{k+1}^{-}$ is given
by

$\displaystyle\rho_{SBPAI}(t_{k+1}^{-})=$ (14)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{\textbf{r}_{k}}U_{SBPA}(\textbf{r}_{k})\rho_{SBPA}(\textbf{r}_{k},t_{k}^{+})U_{SBPA}^{\dagger}(\textbf{r}_{k})|\textbf{r}_{k}{\rangle\langle}\textbf{r}_{k}|,$

where

$\begin{split}U&{}_{SBPA}(\textbf{r}_{k})=\\\
&{\cal{T}}_{+}\exp\left[-i\int_{t_{k}}^{t_{k+1}}H_{SBPA}(\lambda_{s},\textbf{r}_{k})ds\right].\end{split}$
(15)

Here, we excluded the results $r_{\ell}$ for $\ell>k$ because we naturally
assume that for $t<t_{\ell}$
$H_{SBPA}(\lambda_{t},\textbf{r}_{n})=H_{SBPA}(\lambda_{t},\textbf{r}_{\ell-1})$
depends only on the so far obtained measurement results. To conclude, for each
measurement trajectory $\textbf{r}_{k}$ we can apply a different Hamiltonian
affecting any possible part of Eq. (6) and hence, allowing full control about
the system and the ancillas. If we do not perform feedback, then
$H_{SBPA}(\lambda_{t},\textbf{r}_{n})=H_{SBPA}(\lambda_{t})$ for all
$\textbf{r}_{n}$. Note that we could even change the time of the measurements
during the experiment by conditioning the memory Hamiltonian
$H_{M}(\lambda_{t})$ on previous measurement results too. For ease of
presentation we refrained from writing down the most general case. Finally, we
remark that the present construction can be seen as a general form of coherent
feedback control [56, 57, 58]. It was already used to study the thermodynamics
of feedback control in Refs. [22, 52].

We repeat that the temporal order of the dynamics is essential (see also Fig.
2): the preparation happens before the actual control operation (the system-
ancilla interaction), which happens before the measurement of the ancilla,
which happens before the final dephasing of the memory. Apart from this order
the time-dependence of all interactions is so far arbitrary.

Finally, the time evolution is fully fixed by specifying the global initial
state, which reads

$\rho_{\text{tot}}(t_{0}^{-})=\rho_{SB}(t_{0}^{-})\rho_{P}(t_{0}^{-})\rho_{A}(t_{0}^{-})\rho_{M}(t_{0}^{-}).$
(16)

Here, $\rho_{SB}(t_{0}^{-})$ is an arbitrary initial system-bath state, which
we will need to restrict in Sec. 5, $\rho_{P}(t_{0}^{-})$ is a suitable chosen
initial state of the preparation apparatus, $\rho_{A}(t_{0}^{-})$ is an
arbitrary initial ancilla state, and finally, the initial state of the memory
is chosen as
$\rho_{M}(t_{0}^{-})=|\textbf{1}_{n}{\rangle\langle}\textbf{1}_{n}|_{I}\rho_{N}(t_{0}^{-})$
with a suitable initial state for the NIDF as discussed above.

##  4 Dynamical equivalence with a quantum causal model

We now show that our autonomous model captures the dynamics of a quantum
causal model as described in Sec. 2. For that purpose we need to implement the
control operations instantaneously. Ideally, this requires that the
interaction between the system and the $k$th ancilla can be written as

$V_{SA(k)}(\lambda_{t})=\delta(t-t_{k})\ln(iU_{SA(k)}),$ (17)

where $\delta(t-t_{k})$ denotes the Dirac delta function. This implements an
instantaneous unitary evolution $U_{SA(k)}$ at time $t_{k}$.

Figure 2: This figure illustrates the temporal order of the setup, in which a
control operation is implemented within our autonomous setup. First, the $k$th
ancilla gets prepared due to the interaction $V_{PA(k)}(\lambda_{t})$
ultimately resulting in the operation ${\cal{U}}_{\text{prep}}^{(k)}$ (see
Sec. 4). Then, the ancilla interacts with the system via
$V_{SA(k)}(\lambda_{t})$ and, in the limit where this interactions happens
instantaneously (also see Sec. 4), creates the operation
${\cal{U}}_{\text{ctrl}}^{(k)}$. Afterwards, the ancilla gets ‘detected’ via
the interaction $V_{AI(k)}(\lambda_{t})$ with the IDF as described by the
operation ${\cal{U}}_{\text{meas}}^{(k)}$. Finally, the NIDF dephase the IDF
via $V_{IN(k)}(\lambda_{t})$ in turn creating the operation
${\cal{U}}_{\text{deph}}^{(k)}$. We remark that there is some freedom of how
to fix the time $t_{k}$, when the intervention ‘happens’. Here, it is
indicated as the time when the system-ancilla interaction takes place, which
is well-defined in the limit where this interaction is instantaneous as
assumed in Sec. 4. Note that we excluded the feedback loop from Fig. 1 for a
simplified graphical presentation only.

Starting from the initial state (16), the time evolution of the global state
can be iteratively constructed via

$\rho_{\text{tot}}(t_{k+1}^{-})={\cal{U}}_{SB}^{(k)}{\cal{U}}^{(k)}_{\text{deph}}{\cal{U}}^{(k)}_{\text{meas}}{\cal{U}}^{(k)}_{\text{ctrl}}{\cal{U}}^{(k)}_{\text{prep}}\rho_{\text{tot}}(t_{k}^{-}).$
(18)

Here, ${\cal{U}}_{SB}^{(k)}$ is the unitary system-bath evolution from $t_{k}$
to $t_{k+1}$ [cf. Eq. (3)], and ${\cal{U}}^{(k)}_{\text{prep}}$,
${\cal{U}}^{(k)}_{\text{ctrl}}$, ${\cal{U}}^{(k)}_{\text{meas}}$, and
${\cal{U}}^{(k)}_{\text{deph}}$ denote the operations resulting from the
preparation of the $k$th ancilla, its interaction with the system, its
measurement, and the final dephasing of the memory, respectively (see also
Fig. 2). While their temporal order is important, it is not necessary that
${\cal{U}}^{(k)}_{\text{deph}}$, ${\cal{U}}^{(k)}_{\text{meas}}$ or
${\cal{U}}^{(k)}_{\text{prep}}$ happen instantaneously before or after the
control operation ${\cal{U}}^{(k)}_{\text{ctrl}}$ since they commute with
${\cal{U}}_{SB}^{(k)}$. In fact, in an actual experiment delays are
unavoidable and preparations and measurements can take a finite time [20, 21].

After tracing out the NIDF as well as all ancillas, which are no longer
participating in the interaction and which we denote by $A_{\text{out}}$, we
write Eq. (18) as

$\begin{split}\mbox{tr}&{}_{A_{\text{out}}N}\\{\rho_{\text{tot}}(t_{k+1}^{-})\\}=\\\
&{\cal{U}}_{SB}^{(k)}{\cal{D}}^{(k)}{\cal{U}}^{(k)}_{\text{meas}}{\cal{U}}^{(k)}_{\text{ctrl}}{\cal{U}}^{(k)}_{\text{prep}}\rho_{SBPA(k)}(t_{k}^{-})\end{split}$
(19)

Notice that we have replaced ${\cal{U}}^{(k)}_{\text{deph}}$ by the dephasing
map (12) and due to Eq. (11) we have

$\displaystyle\rho_{SBPA(k)}(t_{k}^{-})=$ (20)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{\textbf{r}_{k-1}}\tilde{\rho}_{SBPA(k)}(t_{k}^{-},\textbf{r}_{k-1})\otimes|1,\textbf{r}_{k-1}{\rangle\langle}1,\textbf{r}_{k-1}|.$

Here, $|1,\textbf{r}_{k-1}\rangle$ describes the state of the IDF before the
measurement, where the $k$th register is still set to its standard state ‘1’.
Furthermore, we assumed that only ancilla $A(k)$ is participating in the $k$th
interaction, in principle more general scenarios are conceivable.222For
instance, the present framework also allows to ‘recycle’ an old ancilla and to
let it interact again with the system. This could implement a quantum
correlated operation as mentioned at the end of Sec. 2. For ease of
presentation we refrain from discussing the most general scenario with all its
details. Now, we use the preparation apparatus $P$ to prepare any ancilla
state $\rho_{A(k)}=\rho_{A(k)}(\textbf{r}_{k-1})$ we like using
${\cal{U}}^{(k)}_{\text{prep}}$. Due to Eq. (9) this preparation procedure is
allowed to depend on the previous measurement results $\textbf{r}_{k-1}$,
which we typically suppress in the notation. After tracing out $P$, we get

$\begin{split}\mbox{tr}&{}_{PA_{\text{out}}N}\\{\rho_{\text{tot}}(t_{k+1}^{-})\\}={\cal{U}}_{SB}^{(k)}{\cal{D}}^{(k)}{\cal{U}}^{(k)}_{\text{meas}}{\cal{U}}^{(k)}_{\text{ctrl}}\\\
&\times\sum_{\textbf{r}_{k-1}}\tilde{\rho}_{SB}(t_{k}^{-},\textbf{r}_{k-1})\rho_{A(k)}|1,\textbf{r}_{k-1}{\rangle\langle}1,\textbf{r}_{k-1}|.\end{split}$
(21)

Next, due to Eq. (13) the action of the ancilla measurement reads explicitly

$\displaystyle\mbox{tr}_{PA_{\text{out}}N}\\{\rho_{\text{tot}}(t_{k+1}^{-})\\}=\sum_{\textbf{r}_{k},r^{\prime}_{k}}{\cal{U}}_{SB}^{(k)}{\cal{D}}^{(k)}$
(22) $\displaystyle\leavevmode\nobreak\ \times
P(r_{k})\left[{\cal{U}}^{(k)}_{\text{ctrl}}\tilde{\rho}_{SB}(t_{k}^{-},\textbf{r}_{k-1})\rho_{A(k)}\right]P(r^{\prime}_{k})|\textbf{r}_{k}{\rangle\langle}r^{\prime}_{k},\textbf{r}_{k-1}|.$

Note that
${\cal{U}}^{(k)}_{\text{ctrl}}={\cal{U}}^{(k)}_{\text{ctrl}}(\textbf{r}_{k-1})$
can be conditioned on all previous measurement results due to Eq. (9). The
second line of Eq. (22) describes a giant Schrödinger cat state with respect
to the different superpositions of the measurement results $r_{k}$. This cat
is killed by the dephasing operation:

tr ${}_{PA_{\text{out}}N}\\{\rho_{\text{tot}}(t_{k+1}^{-})\\}=$ (23)
$\displaystyle{\cal{U}}_{SB}^{(k)}\sum_{\textbf{r}_{k}}{\cal{P}}(r_{k})\left[{\cal{U}}^{(k)}_{\text{ctrl}}\tilde{\rho}_{SB}(t_{k}^{-},\textbf{r}_{k-1})\rho_{A(k)}\right]|\textbf{r}_{k}{\rangle\langle}\textbf{r}_{k}|,$

where we introduced the superoperator ${\cal{P}}(r_{k})\rho_{A(k)}\equiv
P(r_{k})\rho_{A(k)}P(r_{k})$ corresponding to the measurement result $r_{k}$.

Equation (23) describes the state of the system, the bath, the $k$th ancilla,
and the IDF of our autonomous black box model at the $k$th time step. To
verify its equivalence with a quantum causal model, we imagine an external
‘super-observer’ (who has engineered the black box), who reads out the IDF by
performing a projective measurement. If the super-observer finds the results
$\textbf{r}_{k}$, the (non-normalized) conditional state of the bath, system
and $k$th ancilla of the black box is according to Eq. (23)

$\begin{split}\tilde{\rho}_{SBA(k)}&(t_{k+1}^{-},\textbf{r}_{k})=\\\
&{\cal{U}}_{SB}^{(k)}{\cal{P}}(r_{k}){\cal{U}}_{\text{ctrl}}^{(k)}\tilde{\rho}_{SB}(t_{k}^{-},\textbf{r}_{k-1})\rho_{A(k)}.\end{split}$
(24)

After tracing out the bath and the ancilla and using Eq. (8), we are left with

$\tilde{\rho}_{S}(t_{k+1}^{-},\textbf{r}_{k})=\mbox{tr}_{B}\left\\{{\cal{U}}_{SB}^{(k)}{\cal{A}}_{k}(r_{k})\tilde{\rho}_{SB}(t_{k}^{-},\textbf{r}_{k-1})\right\\}.$
(25)

If we iterate this, we arrive at Eq. (4). This shows that our autonomous setup
_conditioned_ on obtaining the measurement results $\textbf{r}_{k}$ simulates
any quantum causal model as introduced in Sec. 2.

##  5 Thermodynamic equivalence with the operational framework

In this central section we derive thermodynamic definitions at the
‘unmeasured’ level for our autonomous black box model (Sec. 5.2) and show that
they naturally imply corresponding thermodynamic definitions at the trajectory
level, which coincide with the definitions of Refs. [17, 18, 19] apart from
one minor exception (Sec. 5.3). However, we also discuss possible ambiguities
at the stochastic level (Sec. 5.4) and reconsider two other choices in the
literature in light of our findings (Sec. 5.5). We start with some agreements
though.

###  5.1 Agreements

The observer-dependent thermodynamic framework of Refs. [17, 18, 19] was
derived under certain idealized assumptions, which we summarize here:

I. Initial state. The global form of the initial state (16) remains, but we
assume that the initial system-bath state is described by a Gibbs ensemble
denoted by $\pi$, i.e., $\rho_{SB}(t_{0}^{-})=\pi_{SB}(\lambda_{0})\equiv
e^{-\beta H_{SB}(\lambda_{0})}/{\cal{Z}}_{SB}(\lambda_{0})$. Note that this is
in general a correlated state. We also assume that the NIDF are initially
described by a Gibbs state as specified in Point E above. They are initially
decorrelated from the rest.

II. Classical and fast memory. The IDF are treated as an ideal classical
memory. This implies that the IDF quickly dephase and, for all practically
relevant times, are only classically correlated with the system and the
ancillas, see Eq. (11). Furthermore, as already specified in Point E above,
the IDF are energetically degenerate and the dephasing operation is
implemented without energetic cost. Finally, the measurement of the ancilla
modeled by the interaction $V_{AI}(\lambda_{t})$ is idealized to be infinitely
fast, i.e., of the form
$V_{AI}(\lambda_{t})=\sum_{k}\delta(t-t^{\prime}_{k})\ln(iU_{AI(k)})$, where
$t^{\prime}_{k}$ denotes some time after the system-ancilla interaction.

III. Preparation apparatus. In principle, the preparation of the ancillas can
have a thermodynamic cost. However, the goal of the repeated interaction
framework is to include ancillas in an arbitrary nonequilibrium state into a
consistent thermodynamic framework, regardless of how they were prepared [51,
52, 53]. Consequently, also Refs. [17, 18, 19] ignored the preparation costs
of the ancillas. In our context, it suffices to point out that, at least in
principle, it is possible that the preparation has zero thermodynamic cost
(for instance, by implementing the preparation reversibly). Since we are not
interested in practical realization of our autonomous model, but rather in the
theoretical foundations of quantum stochastic thermodynamics, we neglect in
the following any discussion about the thermodynamic cost of the preparation
and simply assume that it provides us with the desired ancillas.

Finally, in this section we do _not_ assume that the system-ancilla
interaction $V_{SA(k)}(\lambda_{t})$ happens instantaneously, but it can
instead take a finite time as also considered in Refs. [52, 19]. In this sense
we are more general here than in Sec. 4. Indeed, we discuss at the end that an
instantaneous, delta-like interaction causes a subtle difference in the
thermodynamic description.

###  5.2 Thermodynamics at the unmeasured level

Our autonomous setup describes one big ‘supersystem’ $SAI$, which consists of
the system $S$, the ancillas $A$ and the IDF $I$, and which we label for the
moment as $X=SAI$. It is connected to two heat baths: first, the bath $B$ in
direct contact with the system $S$ and second, the NIDF $N$ responsible for
dephasing the memory. The overall setup can therefore be recast in form of the
Hamiltonian
$H_{\text{tot}}(\lambda_{t})=H_{X}(\lambda_{t})+H_{B}+H_{N}+V_{XB}+V_{XN}(\lambda_{t})$.
The following results are based on two recent advances in strong coupling
thermodynamics. First, we use the quantum version [59] of the ‘Hamiltonian of
mean force’ framework [60] (see also Refs. [61, 62, 63, 64, 65] for related
research in this direction). Then, we combine it with the framework of Refs.
[66, 67] to take into account the initially decorrelated dephasing bath. A
detailed calculation how to combine the two frameworks can be found in the
Supplement of Ref. [19] and therefore we here only present its essential
elements.

We start with the conventional definition of mechanical work, which quantifies
the global change in internal energy, i.e.,
$W(t)=\mbox{tr}\\{H_{\text{tot}}(\lambda_{t})\rho_{\text{tot}}(t)\\}-\mbox{tr}\\{H_{\text{tot}}(\lambda_{0})\rho_{\text{tot}}(t_{0})\\}$.
Furthermore, by construction the interaction $V_{XN}(\lambda_{t})$ caused by
the dephasing bath does not have any overall work cost, see Point E above.
Therefore, we can identify the total work with the work done on the
supersystem $X$. It can be expressed as as an integral over the instantaneous
supplied power:

$W(t)=\int_{t_{0}}^{t}ds\mbox{tr}\left\\{\frac{\partial
H_{X}(\lambda_{s})}{\partial s}\rho_{X}(s)\right\\}.$ (26)

Note that, whenever it will be clear from context, we will suppress the
subscript on the trace operation in the following.

Next, we turn to the internal energy. To define it, we need the concept of the
Hamiltonian of mean force, which is defined via the reduced equilibrium state
of a global canonical Gibbs state. Specifically, with respect to an arbitrary
system $X$ coupled to the bath $B$ we define

$\pi_{X}^{*}\equiv\mbox{tr}_{B}\\{\pi_{XB}\\}\equiv\frac{e^{-\beta
H_{X}^{*}}}{{\cal{Z}}_{X}^{*}},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
{\cal{Z}}_{X}^{*}\equiv\frac{{\cal{Z}}_{XB}}{{\cal{Z}}_{X}}.$ (27)

This implicitly defines the Hamiltonian of mean force $H_{X}^{*}$. Note that
$\pi_{X}^{*}\neq\pi_{X}$ in general. In addition, $H_{X}^{*}$ depends on the
inverse temperature $\beta$ and the control parameter $\lambda_{t}$.
Classically, it can be seen as an effective free energy landscape for the
system, which is different from the bare energy $H_{X}$ due to the strong
system-bath coupling. For readers unfamiliar with the framework of strong
coupling thermodynamics, it might be easier to follow the rest of the paper by
replacing the Hamiltonian of mean force $H_{X}^{*}$ with the standard
Hamiltonian $H_{X}$, which amounts to assuming a weakly coupled heat bath. In
fact, the main contribution of this paper is to provide a recipe to deduce
trajectory-dependent thermodynamic definition from an autonomous picture
without explicit measurements. With which thermodynamic definitions one starts
at the unmeasured level is of rather minor relevance here. We only choose the
strong coupling approach for the sake of generality to make clear that the
resulting framework of operational quantum stochastic thermodynamics does not
rely on the commonly used weak coupling or Markovian approximations.

We now define the internal energy of $X$ as

$\begin{split}U(t)\equiv&\leavevmode\nobreak\
\mbox{tr}\left\\{(H_{X}^{*}+\beta\partial_{\beta}H_{X}^{*})\rho_{X}(t)\right\\}\\\
&+\mbox{tr}\\{V_{XN}(\lambda_{t})\rho_{XN}(t)\\},\end{split}$ (28)

where $\partial_{\beta}$ denotes a partial derivative with respect to the
inverse temperature. The first line coincides with the standard definition
within the Hamiltonian of mean force framework [60, 59] and describes
deviations from the weak coupling definition given by
$\mbox{tr}\\{H_{X}\rho_{X}(t)\\}$. The second line in Eq. (28) needs to be
added to take into account the initially decoupled second bath [66, 19].
However, this expression can be simplified since the interaction
$V_{XN}(\lambda_{t})=V_{IN}(\lambda_{t})$ responsible for the dephasing of the
IDF is expected to act only very shortly after each measurement and hence, for
practically all times we can set
$\mbox{tr}\\{V_{XN}(\lambda_{t})\rho_{XN}(t)\\}=0$.333Alternatively, one could
imagine a permanently but weakly coupled dephasing bath. Then, it also follows
that $\mbox{tr}\\{V_{XN}(\lambda_{t})\rho_{XN}(t)\\}\approx 0$. Hence,

$U(t)=\mbox{tr}\left\\{(H_{X}^{*}+\beta\partial_{\beta}H_{X}^{*})\rho_{X}(t)\right\\}.$
(29)

Because we now have a definition for work and internal energy, this
automatically fixes the heat via the first law

$Q(t)\equiv\Delta U(t)-W(t),$ (30)

where $\Delta U(t)=U(t)-U(t_{0}^{-})$ denotes the change in internal energy.

Let us now turn to the second law. First, we define the thermodynamic entropy
of the supersystem $X$

$S(t)\equiv
S_{\text{vN}}[\rho_{X}(t)]+\beta^{2}\mbox{tr}\\{(\partial_{\beta}H_{X}^{*})\rho_{X}(t)\\}.$
(31)

Here, $S_{\text{vN}}(\rho)\equiv-\mbox{tr}\\{\rho\ln\rho\\}$ denotes the von
Neumann entropy and the second term is again a strong coupling correction [60,
59]. Then, the second law of nonequilibrium thermodynamics states that the
entropy production $\Sigma$ is always positive, which can be expressed as
($k_{B}\equiv 1$)

$\begin{split}\Sigma(t)&=\Delta S(t)-\beta Q(t)\\\ &=\beta[W(t)-\Delta
F(t)]\geq 0.\end{split}$ (32)

Here, we defined the nonequilibrium free energy

$\begin{split}F(t)&\equiv U-TS\\\
&=\mbox{tr}_{X}\\{H_{X}^{*}\rho_{X}(t)\\}-TS_{\text{vN}}[\rho_{X}(t)].\end{split}$
(33)

It differs from the conventional weak coupling definition solely by the
replacement of $H_{X}$ with $H_{X}^{*}$. The positivity of entropy production
follows from monotonicity of relative entropy [68, 69] since

$\begin{split}\Sigma(t)=&\leavevmode\nobreak\
D[\rho_{\text{tot}}(t)\|\pi_{XB}(\lambda_{t})\otimes\pi_{N}]\\\
&-D[\rho_{X}(t)\|\pi_{X}^{*}(\lambda_{t})],\end{split}$ (34)

where $D[\rho\|\sigma]\equiv\mbox{tr}\\{\rho(\ln\rho-\ln\sigma)\\}$ denotes
the quantum relative entropy. Showing the equivalence of Eqs. (32) and (34) is
tedious, but follows only standard steps, see the Supplement of Ref. [19].

We now investigate the definitions above in detail by making extensive use of
Eq. (11). First, the work (26) originates from the three time-dependent terms
$H_{S}(\lambda_{t})$, $V_{SA}(\lambda_{t})$, and $V_{AI}(\lambda_{t})$. The
first two contributions can be written as

$\displaystyle W_{S}(t)=$ (35) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{\textbf{r}_{n}}p(\textbf{r}_{n})\int_{t_{0}}^{t}ds\mbox{tr}\left\\{\frac{\partial
H_{S}(\lambda_{s},\textbf{r}_{n})}{\partial
s}\rho_{S}(s,\textbf{r}_{n})\right\\},$ $\displaystyle W_{SA}(t)=$ (36)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{\textbf{r}_{n}}p(\textbf{r}_{n})\int_{t_{0}}^{t}ds\mbox{tr}\left\\{\frac{\partial
V_{SA}(\lambda_{s},\textbf{r}_{n})}{\partial
s}\rho_{SA}(s,\textbf{r}_{n})\right\\}.$

The third contribution due to $V_{AI}(\lambda_{t})$ can be simplified by
noting that the ancilla and IDF are isolated during the measurement such that
we simply have to add up the changes in the ancilla energies (remember that
the IDF are energetically degenerate). Thus, let
$\rho^{\prime}_{A(k)}(\textbf{r}_{k-1})$ denote the state of the $k$th ancilla
after the interaction with the system but before the measurement (which can
depend on $\textbf{r}_{k-1}$) and let
$\rho^{\prime\prime}_{A(k)}(\textbf{r}_{k})$ denote its state after the
measurement conditioned on finding the IDF in state $|r_{k}\rangle$. Then, if
we split the work $W_{AI}(t)=\sum_{k}W_{AI(k)}(t)$ into its contributions due
to the $k$th control step, we find that

$\displaystyle W_{AI(k)}(t)=$ (37) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\
\sum_{\textbf{r}_{k}}p(\textbf{r}_{k})\mbox{tr}\left\\{H_{A(k)}[\rho^{\prime\prime}_{A(k)}(\textbf{r}_{k})-\rho^{\prime}_{A(k)}(\textbf{r}_{k-1})]\right\\}.$

This equation is derived in detail in Sec. 5.4.

Next, we turn to the internal energy and first notice that the Hamiltonian of
mean force can be simplified to

$H_{X}^{*}(\lambda_{t})=\sum_{\textbf{r}_{n}}H_{SA}^{*}(\lambda_{t},\textbf{r}_{n})|\textbf{r}_{n}{\rangle\langle}\textbf{r}_{n}|+V_{AI}(\lambda_{t}).$
(38)

This follows from the facts that the IDF are energetically degenerate and that
the measurement of the ancilla happens after the interaction with the system.
That is, at any given time the $k$th ancilla is either in contact with the
system [and then $V_{A(k)I}(\lambda_{t})=0$] or not, in which case
$H_{A(k)}+V_{A(k)I}(\lambda_{t})$ commutes with the rest of the Hamiltonian.
Since there is also at most one ancilla in contact with the system at a given
time (say again the $k$th ancilla), we can also conclude that
$H_{SA}^{*}(\lambda_{t},\textbf{r}_{n})=H_{SA(k)}^{*}(\lambda_{t},\textbf{r}_{n})+\sum_{i\neq
k}H_{A(i)}$. In the case of a causal model as considered in Secs. 2 and 4
(described by an instantaneous system-ancilla interaction) we can even set
$H_{SA}^{*}(\lambda_{t},\textbf{r}_{n})=H_{S}^{*}(\lambda_{t},\textbf{r}_{n})+\sum_{k}H_{A(k)}$.
The splitting (38) together with Eq. (11) implies for the internal energy
[denoting $H^{*}_{SA}=H_{SA}^{*}(\lambda_{t},\textbf{r}_{n})$ for simplicity]

$U(t)=\sum_{\textbf{r}_{n}}p(\textbf{r}_{n})\mbox{tr}\left\\{(H_{SA}^{*}+\partial_{\beta}H_{SA}^{*})\rho_{SA}(t,\textbf{r}_{n})\right\\}.$
(39)

Similarly to the term $V_{XN}(\lambda_{t})$ in Eq. (28), we have also here
neglected the interaction term $V_{AI}(\lambda_{t})$: it describes a very fast
process, whose temporary resolution is unimportant for us, i.e., for most
times $V_{AI}(\lambda_{t})=0$, see Point II above in Sec. 5.1. The energetic
change due to the measurement is nevertheless fully captured by the work (37).

Finally, we look at the definition of entropy, Eq. (31). Due to Eqs. (11) and
(38) this can be written as

$\begin{split}S(t)=&\leavevmode\nobreak\
\sum_{\textbf{r}_{n}}p(\textbf{r}_{n})\\{S_{\text{vN}}[\rho_{SA}(t,\textbf{r}_{n})]-\ln
p(\textbf{r}_{n})\\}\\\
&+\sum_{\textbf{r}_{n}}p(\textbf{r}_{n})\beta^{2}\mbox{tr}\\{(\partial_{\beta}H_{SA}^{*})\rho_{SA}(t,\textbf{r}_{n})\\}.\end{split}$
(40)

Similarly, the nonequilibrium free energy (33) becomes

$\displaystyle F(t)=$
$\displaystyle\sum_{\textbf{r}_{n}}p(\textbf{r}_{n})\mbox{tr}\\{H_{SA}^{*}\rho_{SA}(t,\textbf{r}_{n})\\}$
(41) $\displaystyle+T\sum_{\textbf{r}_{n}}p(\textbf{r}_{n})\\{\ln
p(\textbf{r}_{n})-S_{\text{vN}}[\rho_{SA}(t,\textbf{r}_{n})]\\}.$

###  5.3 Conditional thermodynamics: the canonical choice

Let us repeat our philosophy so far: We started with a quantum causal model
and constructed an autonomous model, which simulates it. The unitary dilation
theorem (5) as well as its extension (8) to non-deterministic interventions
naturally forced us to introduce a stream of ancillas and a classical memory
into the picture. Then, we studied the thermodynamics of the isolated
autonomous model by combining recently developed tools in strong coupling
thermodynamics [66, 67, 60, 59, 19] and simplified the resulting expression as
much as possible. Now, we imagine the same situation as in Sec. 4 where an
external super-observer measures the memory and obtains outcome
$\textbf{r}_{n}$. What is the internal energy and system entropy as well as
the work supplied and the heat flow conditioned on this outcome?

Above, we already wrote down all thermodynamic quantities in a suggestive way
as an ensemble average over $\textbf{r}_{n}$ via

$X(t)=\sum_{\textbf{r}_{n}}p(\textbf{r}_{n})x(\textbf{r}_{n},t),$ (42)

where $X$ is a placeholder for $W,U,Q$ and $S$. Therefore, to get the right
thermodynamic quantity $X(t)$ on average, $x(\textbf{r}_{n},t)$ presents its
stochastic counterpart (denoted by a small letter as customarily done in
stochastic thermodynamics). For instance, the stochastic work at the
trajectory level follows from Eqs. (35), (36) and (37) as

$\displaystyle w_{S}(t,\textbf{r}_{n})=$ (43)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\int_{t_{0}}^{t}ds\mbox{tr}\left\\{\frac{\partial H_{S}(\lambda_{s})}{\partial
s}\rho_{S}(s,\textbf{r}_{n})\right\\},$ $\displaystyle
w_{SA}(t,\textbf{r}_{n})=$ (44) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \int_{t_{0}}^{t}ds\mbox{tr}\left\\{\frac{\partial
V_{SA}(\lambda_{s})}{\partial s}\rho_{SA}(s,\textbf{r}_{n})\right\\},$
$\displaystyle w_{AI}(t,\textbf{r}_{n})=$ (45)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{k=0}^{n}\mbox{tr}\left\\{H_{A(k)}[\rho^{\prime\prime}_{A(k)}(\textbf{r}_{k})-\rho^{\prime}_{A(k)}(\textbf{r}_{k-1})]\right\\}.$

Likewise, the internal energy (39) and heat (30) at the trajectory level
become

$\displaystyle u(t,\textbf{r}_{n})=$ $\displaystyle\leavevmode\nobreak\
\mbox{tr}\\{(H_{SA}^{*}+\beta\partial_{\beta}H_{SA}^{*})\rho_{SA}(t,\textbf{r}_{n})\\},$
(46) $\displaystyle q(t,\textbf{r}_{n})=$ $\displaystyle\leavevmode\nobreak\
u(t,\textbf{r}_{n})-w(t,\textbf{r}_{n}),$ (47)

where
$w(t,\textbf{r}_{n})=w_{S}(t,\textbf{r}_{n})+w_{SA}(t,\textbf{r}_{n})+w_{AI}(t,\textbf{r}_{n})$.
Finally, the entropy and nonequilibrium free energy follow from Eqs. (40) and
(41):

$\displaystyle s(t,\textbf{r}_{n})=$ $\displaystyle-\ln
p(\textbf{r}_{n})+S_{\text{vN}}[\rho_{SA}(t,\textbf{r}_{n})]$ (48)
$\displaystyle+\beta^{2}\mbox{tr}\\{(\partial_{\beta}H_{SA}^{*})\rho_{SA}(t,\textbf{r}_{n}),$
$\displaystyle f(t,\textbf{r}_{n})=$ $\displaystyle\leavevmode\nobreak\
\mbox{tr}\\{H_{SA}^{*}\rho_{SA}(t,\textbf{r}_{n})\\}$ (49) $\displaystyle+T\ln
p(\textbf{r}_{n})-TS_{\text{vN}}[\rho_{SA}(t,\textbf{r}_{n})].$

These quantities, which were derived from an inclusive, Hamiltonian approach,
can now be compared with the proposed definitions in Refs. [17, 19] (Ref. [18]
deals with the classical counterpart). To compare them, one has to keep in
mind that the definitions in Ref. [17] were proposed for the weak coupling
regime. This implies $H_{X}^{*}=H_{X}$ and in particular
$\partial_{\beta}H_{X}^{*}=0$. Furthermore, the ancillas were called ‘units’
in Refs. [17, 19]. The $k$th unit was denoted by $U(k)$ and the entire string
of units was denoted $U(\textbf{n})$ instead of $A$.

Apart from one minor exception, _all definitions coincide_. Therefore, the
question raised in Ref. [17] “whether there exist good _a priori_ arguments”
(in contrast to the many _a posteriori_ justifications given in Refs. [17, 18,
19]) to justify the definitions used in operational quantum stochastic
thermodynamics can be unequivocally be answered with “Yes!”

The exception concerns Eq. (45), which was previously interpreted as a _heat_
exchange of the ancilla during the control operation, see, e.g., Eq. (31) in
Ref. [17] or Eq. (13) in the Supplement of Ref. [19]. Within our autonomous
approach we now recognize it actually as a _work_ cost, see also below for
more details. Interestingly, somewhat anticipating this case, this term was
already _excluded_ from the second law in Refs. [17, 19]. Therefore, no major
conclusion has to be changed apart from relabeling one term as work instead of
heat. In fact, typically this term is of minor relevance as it vanishes, for
instance, if the ancillas are energetically neutral or if the final
measurement of them happens in their energy eigenbasis as in Refs. [20, 21].

###  5.4 Ambiguities in stochastic work and heat

We first catch up on the promised derivation of Eq. (37) by focusing on the
measurement of the $k$th ancilla. During that measurement, described by the
interaction Hamiltonian $V_{AI(k)}(\lambda_{t})$, the ancilla $A(k)$ and the
IDF are isolated. The change in their internal energy is therefore identical
to the work supplied to them, i.e.,

$W_{AI(k)}=\sum_{\textbf{r}_{k-1}}p(\textbf{r}_{k-1})\mbox{tr}\\{H_{A(k)}[{\cal{U}}_{\text{meas}}^{(k)}-{\cal{I}}]\rho^{\prime}_{A(k)}|1,\textbf{r}_{k-1}{\rangle\langle}1,\textbf{r}_{k-1}|\\}.$
(50)

Here, we used Eq. (11) and that the IDF are energetically degenerate such that
we only have to track the change in expectation value of $H_{A(k)}$. Remember
that $\rho^{\prime}_{A(k)}=\rho^{\prime}_{A(k)}(\textbf{r}_{k-1})$ denotes the
state of the $k$th ancilla after the interaction with the system, which can
depend on $\textbf{r}_{k-1}$. Next, we use Eq. (13) and take the trace over
the IDF to infer that

$W_{AI(k)}=\sum_{\textbf{r}_{k}}p(\textbf{r}_{k-1})\mbox{tr}\left\\{H_{A(k)}P(r_{k})\rho^{\prime}_{A(k)}P(r_{k})\right\\}-\sum_{\textbf{r}_{k-1}}p(\textbf{r}_{k-1})\mbox{tr}_{A(k)}\left\\{H_{A(k)}\rho^{\prime}_{A(k)}\right\\}.$
(51)

Notice that
$P(r_{k})\rho^{\prime}_{A(k)}P(r_{k})=P(r_{k})\rho^{\prime}_{A(k)}(\textbf{r}_{k-1})P(r_{k})$
is a non-normalized state and its norm is the probability
$p(r_{k}|\textbf{r}_{k-1})$ to obtain result $r_{k}$ given the previous
results $\textbf{r}_{k-1}$. Thus, by writing
$P(r_{k})\rho^{\prime}_{A(k)}(\textbf{r}_{k-1})P(r_{k})=p(r_{k}|\textbf{r}_{k-1})\rho^{\prime\prime}_{A(k)}(\textbf{r}_{k})$,
where $\rho^{\prime\prime}_{A(k)}(\textbf{r}_{k})$ denotes the normalized
state of the $k$th ancilla after the measurement conditioned on
$\textbf{r}_{k}$, we obtain

$W_{AI(k)}=\sum_{\textbf{r}_{k}}p(\textbf{r}_{k})\mbox{tr}\left\\{H_{A(k)}[\rho^{\prime\prime}_{A(k)}(\textbf{r}_{k})-\rho^{\prime}_{A(k)}(\textbf{r}_{k-1})]\right\\}.$
(52)

Here, we also used the elementary rules of probability theory
$p(\textbf{r}_{k})=p(r_{k}|\textbf{r}_{k-1})p(\textbf{r}_{k-1})$ and
$\sum_{r_{k}}p(\textbf{r}_{k})=p(\textbf{r}_{k-1})$. This concludes the
derivation of Eq. (37). Consequently, the stochastic work (45) was identified
with the term following $p(\textbf{r}_{k})$ in Eq. (37) and the heat (47) is
indirectly defined via the first law.

We are now in a position, where we can see the origin of the ambiguity in
assigning heat and work at the trajectory level. Imagine we start with Eq.
(50) again, but we express it as

$W_{AI(k)}=\sum_{\textbf{r}_{k-1}}p(\textbf{r}_{k-1})\mbox{tr}\\{H_{SA}[{\cal{U}}_{\text{meas}}^{(k)}-{\cal{I}}]\rho^{\prime}_{SA}|1,\textbf{r}_{k-1}{\rangle\langle}1,\textbf{r}_{k-1}|\\},$
(53)

where $H_{SA}=H_{S}(\lambda_{t})+H_{A}$ is the Hamiltonian of the system and
all ancillas. This is possible since the operation
${\cal{U}}_{\text{meas}}^{(k)}$ acts only non-trivially on the $k$th ancilla
and the IDF and hence, the expectation value remains unchanged when including
additional degrees of freedom. If we then follow the same steps as above, we
end up with

$W_{AI(k)}=\sum_{\textbf{r}_{k}}p(\textbf{r}_{k})\mbox{tr}\left\\{[H_{S}(\lambda_{t})+H_{A}][\rho^{\prime\prime}_{SA}(\textbf{r}_{k})-\rho^{\prime}_{SA}(\textbf{r}_{k-1})]\right\\}.$
(54)

Since this expression is still correct, it allows us to confirm by comparison
with Eq. (50) that the _average_ work injected into the system or the
remaining ancillas is zero as expected. However, if we now follow the strategy
$X(t)=\sum_{\textbf{r}_{n}}p(\textbf{r}_{n})x(\textbf{r}_{n},t)$ to identify
the stochastic work, we obtain the definition

$\displaystyle\tilde{w}_{AI(k)}(\textbf{r}_{k})=$ (55)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{tr}\left\\{[H_{S}(\lambda_{t})+H_{A}][\rho^{\prime\prime}_{SA}(\textbf{r}_{k})-\rho^{\prime}_{SA}(\textbf{r}_{k-1})]\right\\}.$

Now, the _stochastic_ work injected into the system or the remaining ancillas
is not zero since our state of knowledge about those entities changes when
receiving the measurement result $r_{k}$. Hence, if we sum this over all
measurements $k$, we do not get back Eq. (45). Consequently, via the first law
we also get a different expression for the stochastic heat (47).

Note that this ambiguity of assigning stochastic heat and work _only_ happens
during the measurement step of the ancilla, i.e., Eqs. (43) and (44) remain
_unchanged_ , and it also does _not_ affect the definitions of state functions
such as stochastic internal energy or entropy.

###  5.5 Comparison with other choices in the literature

Together with the section above we are now in a position to reconsider other
choices in the literature. In particular, the question of how to
thermodynamically describe a projective measurement of a quantum system has
gained a lot of attention. For that particular class of interventions it is
actually superfluous to consider the stream of ancillas and one could directly
look at an interaction between the system and the $k$th IDF to implement a
projective measurement as described in Point F of Sec. 3. On the other hand,
nothing will change in our conclusions if we keep the ancilla but simply
assume that it is energetically degenerate, i.e., $H_{A}\sim 1_{A}$ for the
rest of this section.

We start with the two-point projective measurement scheme [33, 34], which is a
theoretically successful approach to derive quantum fluctuation theorems. In
there, one considers an isolated system subjected to two projective
measurements of the energy at the beginning and at the end of the protocol.
The difference in the measurement outcomes is interpreted as the stochastic
work in this framework. This stochastic work includes two terms. One term is
due to changing the system Hamiltonian $H_{S}(\lambda_{t})$ in time, which is
fully captured by Eq. (43). The other term interpretes the change in energy
caused by updating our state of knowledge due to the final projective
measurement as work, which corresponds to the alternative choice (55). Adding
these two contributions, results in

$\begin{split}w_{S}(t,&\textbf{r}_{n-1})+\tilde{w}_{AI(k)}(\textbf{r}_{n})\\\
&=\int_{t_{0}}^{t}ds\mbox{tr}\left\\{\frac{\partial
H_{S}(\lambda_{t})}{\partial s}\rho_{S}(s,\textbf{r}_{n-1})\right\\}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\mbox{tr}\\{H_{S}(\lambda_{t})[\rho^{\prime\prime}_{S}(\textbf{r}_{n})-\rho^{\prime}_{S}(\textbf{r}_{n-1})]\\}.\end{split}$
(56)

Now, we specialize to the two-point projective measurement scheme, where
$\textbf{r}_{n}=(E_{0},E_{1})$ only denotes the two results of the initial and
final projective measurement. The corresponding eigenstates of the initial and
final Hamiltonian are denoted as $|E_{0}\rangle$ and $|E_{1}\rangle$ and we
identify $\rho^{\prime\prime}_{S}(E_{0},E_{1})=|E_{1}{\rangle\langle}E_{1}|$
and
$\rho^{\prime}_{S}(E_{0})=U_{S}(t_{1},t_{0})|E_{0}{\rangle\langle}E_{0}|U_{S}^{\dagger}(t_{1},t_{0})$
denotes the unitarily evolved system state prior to the final measurement.
Since the system is assumed to be isolated here, it follows that

$w_{S}(t,\textbf{r}_{n-1})+\tilde{w}_{AI(k)}(\textbf{r}_{n})=E_{1}-E_{0}.$
(57)

Therefore, Eq. (57) reproduces the work statistics of the two-point projective
measurement approach.

Does this imply that Eq. (55) is the natural choice instead of Eq. (45)?
Notice that in this paper we were mainly interested in an _open_ system
coupled to a heat bath. Now, suppose we were to follow the ideology of the
two-point projective measurement approach and consider the following example.
At some initial time $t_{0}$ we have prepared a two-level system with energy
gap $\Omega$ in its excited state, $\rho_{S}(t_{0})=|e{\rangle\langle}e|$,
which then evolves in time while being in contact with a heat bath (which, for
the sake of simplicity, is considered to be an ideal weakly coupled Markovian
heat bath here). Then, we perform at time $t_{1}>t_{0}$ a measurement of its
energy and find it in the ground state $|g\rangle$. If we do not drive the
system ($\lambda_{t}=$ constant), its change in internal energy is simply

$\Delta
u={\left\langle{g|H_{S}|g}\right\rangle}-{\left\langle{e|H_{S}|e}\right\rangle}=-\Omega.$
(58)

Clearly, a natural interpretation of this situation would suggest to identify
$\Delta u$ with the _heat_ exchanged with the bath, which induced at some
unknown time $t\in(t_{0},t_{1})$ a jump from the excited to the ground state.
Instead, the two-point projective measurement approach would identify parts of
$\Delta u$ as work, namely the part of energy change caused by a change of its
state from $\rho^{\prime}_{S}(t_{1})$ (the state prior to the measurement at
$t_{1}$) to $|g{\rangle\langle}g|$ (the post-measurement state), cf. Eq. (55).
For open quantum systems, the two-point projective measurement approach
therefore does not reproduce our classical intuition about heat exchanges
induced by stochastic transitions from one state to another, which are
revealed by updating our state of knowledge. In fact, one can show that the
canonical choice of Sec. 5.3 reduces to the conventional definitions used in
classical stochastic thermodynamics [2, 4, 5] when considering ideal continous
measurements of an open classical system [17]. Furthermore, note that Eq. (57)
excludes the energetic cost of the first measurement yielding result $E_{0}$.
However, if the system was prior to the measurement in weak contact with a
heat bath and only afterwards isolated, Eq. (57) is the correct stochastic
work if one adapts the convention that Eq. (45) is the correct choice for open
quantum systems.

An opposite interpretation to the two-point projective measurement approach
was suggested in Ref. [35], where the change in energy of an isolated system
due to a projective measurement of an arbitrary observable was identified as
heat. This heat does not appear in any second law and it was called “quantum
heat”. While we see that our canonical choice in Sec. 5.3 allows to identify
parts of the changes in energy due to a projective measurement as heat, _on
average_ it predicts that any change in energy due to a measurement is due to
work, which follows from Eq. (54). This average, derived within our inclusive,
Hamiltonian approach, agrees with the two-point projective measurement
approach on average and coincides with the “switching work” known from the
repeated interaction framework [70], see also Ref. [52]. Therefore, the
concept of “quantum heat”, at least as originally introduced in Ref. [35],
does not have any theoretical foundation within our autonomous approach.

##  6 Final remarks

The main message of this paper is a very positive one. After 20 years of
debate, the present paper shows that there exists a straightforward way to
derive the definitions of quantum stochastic thermodynamics by starting from
unambiguous notions at the unmeasured level. A certain amount of freedom in
defining heat and work at the stochastic level remains, but additional
arguments can be invoked in favour of one or the other. In particular, the
most consistent choice might depend on the question whether the considered
system is open or isolated. That this can give rise to different thermodynamic
definitions should not be too surprising as this is the same in classical
thermodynamics. Furthermore, the resulting definitions turn out to be
surprisingly simple and mostly follow from what was known (since a long time)
at the unmeasured level _if_ one correctly takes into account the measurement
results $\textbf{r}_{n}$. This basically means that one has to replace
$\rho_{S}(t)$ by the correct state of knowledge $\rho_{S}(t,\textbf{r}_{n})$
to compute, e.g., the stochastic work or internal energy.

Furthermore, it cannot be overemphasized that the operational framework of
quantum stochastic thermodynamics equips a large class of quantum causal
models with a consistent thermodynamic interpretation, even along a single
trajectory. The main physical assumptions is an initially equilibrated system-
bath state, the remaining assumptions listed at the end of Sec. 2 are rather
of minor relevance for current practical purposes in quantum thermodynamics.
In particular, the present paper shows that the strong coupling definitions
even hold in case of real-time feedback control, which could not be
established in Ref. [19]. Thus, operational quantum stochastic thermodynamics
opens up the possibility to analyse the thermodynamics of almost every quantum
experiment, even beyond average quantities, and its thermodynamic consistency
is guaranteed by virtue of the results reported here.

There is one caveat, however, which is not linked to the framework of
operational quantum stochastic thermodynamics _per se_ but rather to the limit
in which a quantum causal model or quantum stochastic process is defined. As
long as the system-ancilla interaction is not instantaneous, a clear advantage
of operational quantum stochastic thermodynamics is that it allows to define
thermodynamic quantities, even along a single trajectory, solely in terms of
experimentally available information. Everything can be computed based on
knowledge of the conditional system-ancilla state
$\rho_{SA}(t,\textbf{r}_{n})$ given a trajectory of measurement results
$\textbf{r}_{n}$. In this sense, the theory is fully ‘operational’. But, quite
ironically, this is no longer true in the peculiar limit, where the system-
ancilla interaction $V_{SA(k)}(\lambda_{t})$ is idealized as a delta-peak [see
Eq. (17)]. This implements a unitary ${\cal{U}}_{\text{ctrl}}^{(k)}$ on the
system-ancilla space, whose energetic change is work. But if the system-bath
coupling $V_{SB}$ is not negligible, the work $W_{SA}(t_{k})$ invested in the
$k$th control operation becomes

$\displaystyle W_{SA}(t_{k})$ (59) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
=\mbox{tr}\left\\{H_{\text{tot}}(\lambda_{k})({\cal{U}}_{\text{ctrl}}^{(k)}-{\cal{I}})\rho_{\text{tot}}(t_{k})\right\\}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
=\mbox{tr}\left\\{[H_{S}(\lambda_{k})+V_{SB}+H_{A(k)}]({\cal{U}}_{\text{ctrl}}^{(k)}-{\cal{I}})\rho_{SBA}(t_{k})\right\\}.$

This shows that one has to eventually evaluate the term
$\mbox{tr}_{SBA}\\{V_{SB}({\cal{U}}_{\text{ctrl}}^{(k)}-{\cal{I}})\rho_{SBA}(t_{k})\\}$,
which requires explicit knowledge about the bath degrees of freedom, albeit
for any smooth, non-singular time-dependence of $V_{SA(k)}(\lambda_{t})$ this
is never necessary, see Eq. (44). Thus, beyond the weak coupling regime, the
strict limit of a quantum causal model makes the operational approach no
longer fully operational. However, at least for typical open quantum systems
linearly coupled to a quadratic bath, Eq. (59) can be still efficiently
computed using reaction coordinate master equations as explicitly demonstrated
in, e.g., Refs. [71, 72, 27].

_Acknowlegdements._ I am grateful to Jens Eisert for raising the question
whether the operational framework of quantum stochastic thermodynamics can be
derived from an autonomous perspective. This research was financially
supported by the DFG (project STR 1505/2-1) and the Spanish MINECO
FIS2016-80681-P (AEI-FEDER, UE).

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