††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work††thanks: These authors contributed
equally to this work

# Experimental signature of initial quantum coherence on entropy production

S. Hernández-Gómez  European Laboratory for Non-linear Spectroscopy (LENS),
Università di Firenze, I-50019 Sesto Fiorentino, Italy  Dipartimento di Fisica
e Astronomia, Università di Firenze, I-50019, Sesto Fiorentino, Italy
Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO),
I-50019 Sesto Fiorentino, Italy  S. Gherardini  CNR-INO, Area Science Park,
Basovizza, I-34149 Trieste, Italy  European Laboratory for Non-linear
Spectroscopy (LENS), Università di Firenze, I-50019 Sesto Fiorentino, Italy
A. Belenchia  Institut für Theoretische Physik, Eberhard-Karls-Universität
Tübingen, 72076 Tübingen, German  School of Mathematics and Physics, Queen’s
University Belfast, Belfast BT7 1NN, United Kingdom  A. Trombettoni
Department of Physics, University of Trieste, Strada Costiera 11, I-34151
Trieste, Italy  SISSA and INFN, Sezione di Trieste, Via Bonomea 265, I-34136
Trieste, Italy  CNR-IOM DEMOCRITOS Simulation Center and SISSA, Via Bonomea
265, I-34136 Trieste, Italy  M. Paternostro  School of Mathematics and
Physics, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom  N.
Fabbri  European Laboratory for Non-linear Spectroscopy (LENS), Università di
Firenze, I-50019 Sesto Fiorentino, Italy  Istituto Nazionale di Ottica del
Consiglio Nazionale delle Ricerche (CNR-INO), I-50019 Sesto Fiorentino, Italy

###### Abstract

We report the experimental quantification of the contribution to non-
equilibrium entropy production that stems from the quantum coherence content
in the initial state of a qubit exposed to both coherent driving and
dissipation. Our experimental demonstration builds on the exquisite
experimental control of the spin state of a nitrogen-vacancy defect in diamond
and is underpinned, theoretically, by the formulation of a generalized
fluctuation theorem designed to track the effects of quantum coherence. Our
results provide significant evidence of the possibility to pinpoint the
genuinely quantum mechanical contributions to the thermodynamics of non-
equilibrium quantum processes.

The irreversible character of most physical processes is, apparently, at odds
with the inherent reversibility of the fundamental laws of physics. The way
time-reversible quantum laws governing the interactions of microscopic systems
gives rise to the irreversible nature of macroscopic phenomena is a very open
field of investigation [1]. In this regard, a breakthrough has been provided
by the extension of the second law of thermodynamics into the quantum realm
through the so-called fluctuation theorems [2, 3, 4, 5]. A celebrated instance
of this is the integral fluctuation theorem, which stems from Jarzynski’s
identity [2] and Crooks’ relation [3], and connects the non-equilibrium energy
fluctuation statistics of unital processes with the corresponding free-energy
changes. Operationally, the standard approach to the quantification of energy
and entropy fluctuations in non-equilibrium contexts is the use of the
celebrated two-point measurement (TPM) scheme [6], which requires two
projective measurements, at the beginning and at the end of the dynamical
process under scrutiny [7, 8].

Despite the clear success of the TPM scheme, evidenced by successful
experimental verification in nuclear magnetic resonance [9, 10], trapped-ion
[11, 12, 13], superconducting-qubit [14], nitrogen-vacancy (NV) centers [15,
16] and linear optics settings [17, 18, 19], the scheme has significant
limitations when considering the role played by quantum features in the
statistics of energy fluctuations. In fact, any quantum coherence in the
initial state of the system, and expressed in the measurement basis, is washed
away. As a result, the system dynamics following the first measurement is
strongly affected. This has motivated recent efforts aimed at modifying the
TPM scheme to take into account the presence of quantum features, particularly
coherence, in non-equilibrium processes [20, 21, 22, 23, 24, 25, 26, 27, 28,
29]. Ref. [30] introduced an end-point measurement (EPM) approach, where the
initial statistics of energy fluctuations is inferred from the knowledge of
the initial state and the Hamiltonian of the system.

Here, we show the intrinsic operational nature of the EPM approach by
considering both the detailed and the integral form of the corresponding
fluctuation theorem, and using them to characterise experimentally the entropy
production associated to quantum coherence in an open quantum system’s states.
In particular, we experimentally make use of a qubit encoded in the spin of an
NV center in diamond, subjected to both a pulsed driving and environmental
effects [15, 16, 31]. We thus observe a significant increase of the
irreversibility of the resulting dissipative map that only originates from the
presence of quantum coherence in the initial state of the NV center. Moreover,
we show that measuring such a quantity provides a tight bound for the average
heat exchanged by the system with the environment.

While being valid in principle for arbitrary dynamics, our results establish
NV centers as valuable platforms for the exploration of energetics at the
quantum level, thus enlarging the already prominent domain of their
applications in quantum technologies [32, 33, 34, 35].

EPM-based fluctuation theorem.– We start by briefly reviewing the EPM scheme
as introduced in Ref. [30]. We thus consider a quantum system subjected to a
completely-positive trace preserving (CPTP) map $\Phi$. Let
$\rho_{0}=\mathcal{P}+\chi$ be its initial state, which we have decomposed in
its diagonal part $\mathcal{P}$ (expressed in the basis of the initial
Hamiltonian $H_{t_{i}}$) and the traceless component $\chi$ that accounts for
the quantum coherence. The EPM scheme prescribes to perform a single energy
measurement, at the end of the process, and to associate to it the stochastic
variables $\Delta E_{\rm i,f}\equiv E^{{\rm fin}}_{\rm f}-E^{{\rm in}}_{\rm
i}$ that encodes the energy fluctuations during the open dynamics. Here,
$E_{\rm k}^{{\rm in}({\rm fin})}$ denotes the eigenvalues of the initial
(final) Hamiltonian. We also introduce the probability distribution

$p_{\rm EPM}(\Delta E_{\rm i,f})=p(E^{{\rm in}}_{\rm i})p(E^{{\rm fin}}_{\rm
f})={\rm tr}(\Pi^{\rm in}_{\rm i}\rho_{0}){\rm tr}(\Pi^{\rm fin}_{\rm
f}\Phi(\rho_{0})),$ (1)

where $\Pi^{{\rm in}({\rm fin})}_{\rm j}$ is the projector on the $j$-th
initial (final) energy eigenstate. Computing the characteristic function of
the probability distribution in Eq. (1) leads immediately to an integral
fluctuation theorem. In fact, let us take ${\cal P}=\rho_{\rm{th}}\equiv
Z_{i}^{-1}\exp[-\beta H_{t_{i}}]$, i.e., a thermal state of the initial
Hamiltonian with inverse temperature $\beta$ and partition function
$Z_{i}\equiv{\rm tr}(\exp[-\beta H_{t_{i}}])$. One can then show that

$\langle e^{-\beta(\Delta E-\Delta F)}\rangle=d\left[{\rm tr}\left(\rho^{\rm
fin}_{\rm th}\,\Phi(\rho^{\rm in}_{\rm th})\right)+{\rm tr}\left(\rho^{\rm
fin}_{\rm th}\,\Phi(\chi)\right)\right],$ (2)

where $\rho^{\rm fin/in}_{\rm th}\equiv e^{-\beta H_{t_{f/i}}}/Z_{f/i}$ with
$Z_{f}\equiv{\rm tr}(\exp[-\beta H_{t_{f}}])$, and $d$ denoting the dimension
of the system’s Hilbert space. The second term in the right-hand-side of Eq.
(2) showcases the contribution coming from the initial quantum coherence,
while the first term represents a classical deviation from the Jarzynski’s
equality. Such deviation would be present also in the absence of initial
coherence due to the non-linearity of the EPM’s probability distribution for
convex combination of states [30].

Consider now a map $\Phi$ that admits a (non-singular) fixed point
$\rho^{*}=\Phi(\rho^{*})$. Using the results in [36], we can define the
corresponding backwards dynamics [see [37] (Part A)]. Then, one can compare
the joint probabilities ${\rm P}_{\Gamma}({\rm i,f})=p(E_{\rm i}^{\rm
in})p(E_{\rm f}^{\rm fin})\equiv p_{\rm i}^{\rm in}p_{\rm f}^{\rm fin}$ and
${\rm P}_{\widetilde{\Gamma}}({\rm f,i})={\rm tr}(\Pi^{\rm fin}_{\rm
f}\rho^{\rm in}_{\rm B}){\rm tr}(\Pi^{\rm in}_{\rm
i}\widetilde{{\Phi}}(\rho^{\rm in}_{\rm B}))\equiv\widetilde{p}^{\rm in}_{\rm
f}\widetilde{p}^{\rm fin}_{\rm i}$ for measuring the energy of the system in
the forward and backward trajectories, $\Gamma$ and $\widetilde{\Gamma}$
respectively. Note that, in ${\rm P}_{\widetilde{\Gamma}}({\rm f,i})$,
$\rho_{\rm in}^{\rm B}$ denotes the initial state of $\widetilde{\Gamma}$, and
$\widetilde{\Phi}$ is the time-reversed map [37] (Part A).

By extending the derivation of the Jarzynski equality [2] by means of the
Crooks’ formalism [3], we consider the case in which _(i)_ the initial quantum
state of the forward dynamics is $\rho_{0}=\rho_{\rm th}^{\rm
in}(\beta)+\chi$, i.e., $\rho_{0}$ is written as the sum of a thermal state at
inverse temperature $\beta$ and the traceless component $\chi$ encoding the
initial coherence in the energy basis, and _(ii)_ the initial state of the
backward quantum dynamics is the state that is thermal in the final
Hamiltonian at the same inverse temperature $\beta$, $\rho^{\rm in}_{\rm
B}=Z_{f}^{-1}\exp[-\beta H_{t_{f}}]$. These assumptions allow to write the
balance equation

$\frac{{\rm P}_{\Gamma}(\rm i,f)}{{\rm P}_{\widetilde{\Gamma}}(\rm
f,i)}=\exp\left[\beta(\Delta E_{\rm i,f}-\Delta F)+\Delta\sigma_{\rm
i,f}+\Delta\Sigma_{\rm i,f}\right],$ (3)

where

$\displaystyle\Delta\sigma_{\rm i,f}\equiv\ln\frac{p_{\rm f}^{\rm
fin}(\rho^{\rm in}_{\rm th}(\beta))}{\widetilde{p}^{\rm fin}_{\rm
i}(\widetilde{\Phi}(\rho^{\rm in}_{\rm B}))},~{}\Delta\Sigma_{\rm
i,f}\equiv\ln\left[1+\frac{p^{\rm fin}_{\rm f}(\chi)}{p^{\rm fin}_{\rm
f}(\rho_{\rm th}^{\rm in}(\beta))}\right]$ (4)

and $p^{\rm fin}_{\rm f}(\chi)\equiv{\rm tr}(\Pi^{\rm fin}_{\rm f}\Phi(\chi))$
(cf. Part A of Ref. [37]). In this way, averaging Eq. (3) over the forward
probability distribution, one obtains

$\left\langle e^{-\beta\Delta
E-(\Delta\sigma+\Delta\Sigma)}\right\rangle_{\Gamma}=e^{-\beta\Delta F}\,.$
(5)

Eq. (3) and Eq. (5) are the _detailed_ and the _integral_ forms of the EPM’s
fluctuation theorem, respectively.

(a) (b)

Figure 1:  Scheme of the protocol applied to the NV spin qubit. The dynamics
stems from a combination of coherent driving of the qubit (near-resonance
microwave (mw) radiation) and a train of short laser pulses that “open” the
system [panel (a)]. Specifically, the coherent drive couples the states
$\left|{0}\right\rangle_{S}$ and $\left|{+1}\right\rangle_{S}$. Instead, the
short laser pulses act as a dissipation, by first projecting the system into
the $S_{z}$ eigenstates and then optically pumping part of the populations
towards the $\left|{0}\right\rangle_{S}$ state [panel (b)].

It is crucial to notice that, thanks to the assumption _(ii)_ , the quantity
$\Delta\Sigma_{\rm i,f}$ depends only on the forward dynamics and it satisfies
a fluctuation theorem by its own (cf. Part B of Ref. [37]), i.e.,

$\left\langle e^{-\Delta\Sigma}\right\rangle_{\Gamma}=1\,.$ (6)

Thus, resorting to the Jensen’s inequality, we have
$\left\langle\Delta\Sigma\right\rangle\geq 0$. $\Delta\Sigma$ encodes the
entropic contribution of the initial quantum coherence of the system, and we
thus identify it as the coherence-affected irreversible entropy production for
a non-equilibrium dynamical process. At the same time, the quantity
$\Delta\sigma$ represents a completely classical contribution to the entropy
production that comes from adopting the EPM formalism, namely from the extra
uncertainty implied by the factorization condition in Eq. (1) (see also
discussion in [30]).

The experimental system.– We consider the spin qubit associated to a
negatively charged NV center –a localized impurity in a diamond lattice based
on a nitrogen substitutional atom next to a vacancy– which forms an electronic
spin $S=1$ in its orbital ground state [38, 39, 40, 41]. A magnetic bias field
aligned with $S_{z}$ removes the degeneracy of the spin eigenstates, so as to
allow for the selective coherent manipulation of the transition
$|0\rangle_{S}\leftrightarrow\left|{+1}\right\rangle_{S}$. The Hamiltonian $H$
of this effective two-level system is determined by a continuous nearly
resonant microwave field and, in the frame rotating at the microwave
frequency, the Hamiltonian is
$H=\hbar\omega(\cos\alpha\,\tilde{\sigma}_{z}-\sin\alpha\,\tilde{\sigma}_{x})/2$,
where $\omega=\sqrt{\Omega^{2}+\delta^{2}}$ and $\tan\alpha=-\Omega/\delta$,
$\Omega$ denotes the bare Rabi frequency and $\delta\in[0,\Omega]$ is the
microwave detuning. Note that we have used the tilde for the Pauli matrices in
view of the change of basis to the Hamiltonian eigenstates, i.e.,
$\\{|0\rangle,|1\rangle\\}\equiv\\{\cos\frac{\alpha}{2}\,\left|{0}\right\rangle_{S}-\sin\frac{\alpha}{2}\,\left|{+1}\right\rangle_{S},\sin\frac{\alpha}{2}\,\left|{0}\right\rangle_{S}+\cos\frac{\alpha}{2}\,\left|{+1}\right\rangle_{S}\\}$
with eigenvalues $\pm\hbar\omega/2$. In this new basis, the Hamiltonian
becomes $H=\omega\sigma_{z}/2$, where $\hbar$ is set to $1$ from here on.

The qubit is governed by an alternated sequence of unitary and non-unitary
(controlled-dissipative) dynamics, as follows. The system is repeatedly
subjected to a sequence of pulses, occurring regularly at time intervals
$\tau$. Among two consecutive pulses, the evolution of the NV center is
unitary and described by the operator $U\equiv\exp[-iH\tau]$. As depicted in
Fig. 1, the NV spin is subjected to open dynamics due to its interaction with
a train of short laser pulses with a duration $t_{L}$ that is much shorter
than the characteristic time-scale of the unitary dynamics ($t_{L}\ll
2\pi/\omega$). The short laser pulses trigger cycles of spin preserving and
non-preserving transitions between different orbital levels [15]. This entails
non-unitary dynamics that project the state of the system into the eigenstates
of $\tilde{\sigma}_{z}$ and partially transfer the spin population to
$|0\rangle_{S}$. Such _spin amplitude damping_ along the $\tilde{\sigma}_{z}$
axis, also known as optical pumping, is caused by the spin non-preserving
transitions, and can be modelled as a controlled dissipative channel toward
$|0\rangle_{S}$ [15, 31]. The overall dynamics takes the NV center into an
asymptotic fixed point $\rho^{*}$ that we can use to define the backward
dynamics [37] (Parts A,C).

In the experiments shown throughout the letter, we set $\alpha=\pi/4$ (i.e.,
$\delta=-\Omega$), $\tau\omega\simeq(2\pi)0.9$, and $\tau=190$ ns.

Coherence-affected entropy production.– We now show the results obtained with
the NV qubit subjected to the dissipative dynamics introduced above. Here, our
aim is to characterize the thermodynamic role of initial quantum coherence in
terms of the coherence-affected entropy production.

Figure 2: Panel (a): Experimental verification of the fluctuation theorem
$\left\langle e^{-\Delta\Sigma}\right\rangle_{\Gamma}=1$, Eq. (6), for the
coherence-affected irreversible entropy production, as a function of the
number $N$ of pulses used to drive the dynamics of the NV center. Panel (b):
Average experimental coherence-affected entropy production as a function of
$N$ (blue circles). In both panels, the experimental data are plotted against
the predictions from the numerical simulations that are obtained by taking
$|+\rangle_{y}=(\left|{0}\right\rangle+i\left|{1}\right\rangle)/\sqrt{2}$
(i.e., $\rho_{0}$ with $p=0$) as the initial quantum state. For such an
initial state, the coherence-affected entropy production is nearly extremal.

We have thus performed a series of experiments to determine both the EPM
probability distribution for the energy fluctuations statistics as well as the
usual TPM one.

At the beginning of each experimental realization, the electronic spin is
initialized into the $|0\rangle_{S}$ eigenstate of the spin operator $S_{z}$
via optical spin pumping under long laser excitation. Then, the system is
brought in each of the four different pure states that correspond to the
eigenvectors of $\sigma_{z}$ and $\sigma_{y}$, by applying rotation gates (on-
resonant microwave pulse) to the state $|0\rangle_{S}$. After $n$ short laser
pulses, with $n\in[0,N]$, we measure the energy of the system in the
Hamiltonian basis $\sigma_{z}$. To achieve this, we first apply another
rotation gate (on-resonant microwave pulse) such that
$\sigma_{z}\rightarrow\tilde{\sigma}_{z}$, and then we measure the photo-
luminescence (PL). The PL intensity determines the probability for the system
to be in the eigenstates $|0\rangle_{S}$ or $|+1\rangle_{S}$. The experiment
is repeated $10^{6}$-times from the beginning for each value of the pulses
number. In this way, all the probabilities required to obtain the EPM and TPM
statistics are obtained.

As we collect data from several experiments (in which the system is
initialized in one of the four pure qubit-states
$\\{|0\rangle,|1\rangle,|+\rangle_{y},|-\rangle_{y}\\}$), it is a matter of
data processing to compute the EPM and TPM statistics for every classical
mixture of such states [37] (Part C). In particular, looking at a Bloch sphere
representation for the qubit, this implies that we are able to obtain the
energy statistics of our quantum process initialized in one of the states,
which are included in the $y-z$ equatorial plane of the Bloch sphere and
provided by a convex combination of the states
$|0\rangle,|1\rangle,|+\rangle_{y},|-\rangle_{y}$. Specifically, in the
following we show experimental results corresponding to the initial states
(expressed in the Hamiltonian basis)

$\rho_{0}=\frac{1}{2}\begin{pmatrix}1+p&-(1-p)i\\\
(1-p)i&1-p\end{pmatrix}\qquad(p\in[0,1])$ (7)

such that $\rho_{0}$ is the convex mixture of $|1\rangle\\!\langle 1|$ and
$|+\rangle_{y}\\!\langle+|$ with probability $p$ and $1-p$, respectively.

The first quantity we are interested in characterising experimentally is the
coherence-affected entropy production encoded in the average of $\Delta\Sigma$
as given in Eq. (4). Note that this average is defined solely in terms of the
forward trajectory probability, so that we can fully characterize it by
resorting to the experimental data acquired during the forward dynamics. The
results are shown in Fig. 2: in the left panel we show the experimental
verification of the fluctuation theorem in Eq. (6). Instead, in the right
panel, the behaviour of $\langle\Delta\Sigma\rangle$ is plotted as a function
of the number of laser pulses. While the error bars are quite large, it can be
observed how the corresponding experimental points nicely follow the
theoretical predictions and how the experimental data show a positive
coherence-affected entropy production.

Figure 3: Panel (a) shows the three quantities in Eq. (8). The dashed red
curve stands for $\langle\Delta E\rangle_{\rm EPM}$, the dot-dashed black one
for $\langle\Delta E\rangle_{\rm TPM}$, and the solid blue line shows the
contribution of quantum coherence to the exchanged heat $\sum_{\rm f}{\rm
tr}[\Pi^{\rm fin}_{\rm f}\chi]E_{\rm f}$. The curves are obtained for an
initial state parameterized as in Eq. (7) with $p=0.2$. This choice guarantees
a good visibility of all the contributions. Panel (b): Experimental
verification of the integral form of the EPM fluctuation theorem in Eq. (5).
In this panel, we have taken Eq. (7) with $p=0.38$.

Another quantity that can be investigated directly from the available data on
the forward dynamics is the average of the energy fluctuations $\Delta E$. In
this regard, it is worth noting that this quantity is identified with the
average work when considering time-dependent unitary processes like in the
Jarzynski’s original work [2]. In our case, as the Hamiltonian is time-
independent, we can unambiguously interpret this quantity as the average
_heat_ that the system exchanges with its environment due to the open dynamics
to which the NV center is subjected. The average of the stochastic variable
$\Delta E_{\rm i,f}=E_{\rm f}-E_{\rm i}$ in the EPM approach is related to the
TPM scheme via the following relation:

$\langle\Delta E\rangle_{\rm EPM}=\langle\Delta E\rangle_{\rm TPM}+\sum_{\rm
f}{\rm tr}(\Pi^{\rm fin}_{\rm f}\chi)E_{\rm f}\,.$ (8)

The second term on the right-hand side of Eq. (8) represents a contribution to
the average energy-change ascribable to the quantum coherence of the initial
state, which is not deleted by a first energy measurement of $\rho_{0}$. Fig.
3 displays the three quantities entering in Eq. (8), by comparing theoretical
expectations with the experimental results. It is shown that the experimental
data are able to discern the coherence contribution to the heat exchanged
between the system and the environment due to the pulsed dynamics. Among the
protocols that allows to account for quantum features in energy fluctuations
[20, 21, 22, 23, 25, 26, 27, 30, 28], the EPM scheme requires only a final
energy measurement at the time $t$ and does not require any knowledge of the
quantum map $\Phi$ that models the forward dynamics. This makes the method
particularly suitable for application in open quantum systems.

Finally, we have also verified the validity of the full EPM fluctuation
theorem in Eq. (5) as well as its consequences for the expectation values of
the involved thermodynamic quantities. In principle, this requires having
access to the backward trajectories of the system, so as to characterize
$\Delta\sigma_{\rm i,f}$ in Eq. (4). This can be problematic due to the
presence of non-unitary dynamics: while the backward trajectories can be
easily simulated numerically, implementing them at the experimental level is
not currently possible with our set-up. However, for the range of experimental
parameters and the choice of the initial state of the backward process, the
backward dynamics are such that $\widetilde{p}^{\rm fin}_{\rm
j}(\widetilde{\Phi}(\rho^{\rm in}_{\rm B}))=p_{\rm j}^{\rm fin}(\rho^{\rm
in}_{\rm th}(\beta))$. Therefore, by assuming that this property holds and
thus that we can estimate $\langle\Delta\sigma\rangle$ from the sole data of
the forward trajectories, in Fig. 3 (b) we show the experimental verification
of the integral form of the EPM fluctuation theorem in Eq. (5).

Figure 4:  The dashed red curve corresponds to the average EPM energy-change
$\beta\langle\Delta E\rangle$. The dot-dashed magenta and solid blue solid
curves show the behavior of the right-hand-sides of the inequalities in Eq.
(9). The initial state is taken as in Eq. (7) with $p=0.38$.

The application of the Jensen inequality to Eqs. (2) and (5) leads to

$\displaystyle\beta\langle\Delta E\rangle\geq\begin{cases}-\ln({\cal G}_{\rm
EPM}),\\\ -(\langle\Delta\sigma\rangle+\langle\Delta\Sigma\rangle)\end{cases}$
(9)

with ${\cal G}_{\rm EPM}=2{\rm tr}\left(\rho^{\rm fin}_{\rm th}\Phi(\rho^{\rm
in}_{\rm th}+\chi)\right)$ the EPM characteristic function [30]. In Fig. 4,
the theoretical expectations of these quantities are compared with the
corresponding experimental results. For the initial state $\rho_{0}$
considered for this figure (i.e., the state in Eq. (7) with $p=0.38$), one can
observe that the results obtained by using the characteristic function of the
EPM approach are quite distinct from the ones coming from the integral
fluctuation theorem (5). Specifically, for the number of pulses explored
experimentally and for the initial states accessible from the experimental
data, the bound on the average energy-change derived from the integral
fluctuation theorem is _tighter_ than the one resulting from the EPM
characteristic function. In addition, we can also see in Fig. 4 that the
inequality in Eq. (9) is almost saturated, so that the EPM average energy
offers even a good estimate of the sum
$\langle\Delta\Sigma+\Delta\sigma\rangle$ in the regime experimentally
analyzed.

Conclusions.– We have used the EPM approach, for the characterization of
energy fluctuations arising from a non-equilibrium process, with the aim to
quantify the contribution to the entropy production that is originated by the
presence of quantum coherence in the initial density matrix of the quantum
system under scrutiny. The operational nature of our approach has enabled a
successful experimental quantification of such coherence-affected entropy
production in a solid state platform where an NV spin qubit undergoes
controlled dissipative dynamics.

Our study grounds the EPM approach as a powerful framework for the assessment
of quantum coherence in the energetics of quantum systems and devices.
Specifically, our findings add a crucial ingredient for the analysis of the
thermodynamic role of quantum coherence and allow the characterization of the
newly-introduced entropy production due to quantum coherence.

Acknowledgments.– We acknowledge support from the European Union’s Horizon
2020 FET-Open project TEQ (766900), the Leverhulme Trust Research Project
Grant UltraQuTe (grant RGP-2018-266), the Royal Society Wolfson Fellowship
(RSWF/R3/183013), the UK EPSRC (grant EP/T028424/1), the Department for the
Economy Northern Ireland under the US-Ireland R&D Partnership Programme, the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project
number BR 5221/4-1, the Blanceflor Foundation through the project “The
theRmodynamics behInd thE meaSuremenT postulate of quantum mEchanics
(TRIESTE)”, the CNR-FOE-LENS-2020, and from the European Commission under GA
n. 101070546–MUQUABIS.

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Supplementary Material:
Experimental signature of initial quantum coherence on entropy production

##  Appendix A Part A: Derivation of the detailed fluctuation theorem

Let us consider a CPTP map, written as
$\Phi(\bullet)=\sum_{\alpha}K_{\alpha}\bullet K_{\alpha}^{\dagger}$ with
$\\{K_{\alpha}\\}$ the set of Kraus operators, that allows a (non-singular)
fixed point $\rho^{*}$ such that $\rho^{*}=\Phi(\rho^{*})$. Following Crooks’
formalism [3], we define the time-reversed map

$\widetilde{\Phi}(\bullet)=\sum_{\alpha}[(\rho^{*})^{1/2}K^{\dagger}_{\alpha}(\rho^{*})^{-1/2}]\bullet[(\rho^{*})^{-1/2}K^{\dagger}_{\alpha}(\rho^{*})^{1/2}],$
(A-1)

where
$\widetilde{K}_{\alpha}\equiv(\rho^{*})^{1/2}K^{\dagger}_{\alpha}(\rho^{*})^{-1/2}$.

Afterwards, in accordance with the EPM formalism [30], we can introduce the
expression of the joint probabilities ${\rm P}_{\Gamma}(E^{\rm in}_{\rm
n},E^{\rm fin}_{\rm m})\equiv{\rm P}_{\Gamma}(n,m)$ and ${\rm
P}_{\widetilde{\Gamma}}(E^{\rm fin}_{\ell},E^{\rm in}_{k})\equiv{\rm
P}_{\widetilde{\Gamma}}(\ell,k)$, respectively, for the quantum trajectories
$\Gamma$ and $\widetilde{\Gamma}$. Let us observe that the latter is obtained
by reversing the arrow of time by means of a transformation that obeys the
time-reversal symmetry. One has

${\rm P}_{\Gamma}(n,m)={\rm tr}\left(\Pi_{n}^{\rm in}\rho_{0}\right){\rm
tr}\left(\Pi_{m}^{\rm fin}\rho^{\rm fin}\right)\equiv p_{n}^{\rm in}p_{m}^{\rm
fin}$ (A-2)

and

${\rm P}_{\widetilde{\Gamma}}(\ell,k)={\rm tr}\left(\Pi_{\ell}^{\rm
fin}\rho_{\rm B}^{\rm in}\right){\rm tr}\left(\Pi_{k}^{\rm in}\rho_{\rm
B}^{\rm fin}\right)\equiv\widetilde{p}_{\ell}^{\rm in}\widetilde{p}_{k}^{\rm
fin}$ (A-3)

where, we recall, $\rho^{\rm fin}\equiv\Phi[\rho_{0}]$ and $\rho_{\rm B}^{\rm
fin}\equiv\widetilde{\Phi}[\rho_{\rm B}^{\rm in}]$ with $\rho_{\rm B}^{\rm
in}$ denoting the initial state of the backward process and, without loss of
generality, we have assumed that each projector on the energy eigenstates is
invariant to the application of the time-reversal transformation. As a result,
the expression of the _detailed balance equation_ for the trajectories
$\Gamma$ and $\widetilde{\Gamma}$ is formally equal to

$\frac{{\rm P}_{\Gamma}(n,m)}{{\rm
P}_{\widetilde{\Gamma}}(\ell,k)}=\frac{p^{\rm in}_{n}p^{\rm
fin}_{m}}{\widetilde{p}^{\rm in}_{\ell}\widetilde{p}^{\rm fin}_{k}}\,.$ (A-4)

Notice that, for a generic open system dynamics, the state originating from
applying a time-reversal transformation on $\rho^{\rm fin}$ is at most in the
neighborhood of the initial state $\rho_{\rm B}^{\rm in}$ of the backward
process. Such a discrepancy along the time-reversal trajectory
$\widetilde{\Gamma}$ in the space of density operators necessarily entails
quantum entropy production. The latter is the main signature of thermodynamic
irreversibility due to both the open system dynamics and the procedure we use
to characterize energy-change fluctuations.

Before proceeding, it is worth observing that if (i) we apply the EPM scheme
for characterising energy fluctuations and (ii) we choose the initial state of
the backward process as the time-reversal of the final state of the forward
process, then $\widetilde{p}^{\rm in}_{\ell}=p^{\rm fin}_{m}$ for $\ell=m$. As
a matter of fact,

$\widetilde{p}^{\rm in}_{\ell}={\rm tr}\left(\rho^{\rm in}_{\rm B}\Pi^{\rm
fin}_{\ell}\right)={\rm tr}\left(\Theta\rho^{\rm
fin}\Theta^{\dagger}\Theta\Pi^{\rm fin}_{\ell}\Theta^{\dagger}\right)={\rm
tr}\left(\rho^{\rm fin}\Pi^{\rm fin}_{\ell}\right)=p^{\rm fin}_{\ell}$ (A-5)

where $\widetilde{(\bullet)}\equiv\Theta(\bullet)\Theta^{\dagger}$ with
$\Theta$ denoting the time-reversal operator, which by construction is anti-
unitary, i.e., it is an anti-linear operator and satisfies the relations
$\Theta^{\dagger}\Theta=\Theta\Theta^{\dagger}=\mathbbm{I}$. In this context,
we obtain the fluctuation relation

$\frac{{\rm P}_{\Gamma}}{{\rm
P}_{\widetilde{\Gamma}}}(n,m)=e^{-\Delta\overline{\sigma}^{\rm in}_{n}}$ (A-6)

where we have identified

$\Delta\overline{\sigma}^{\rm in}_{n}=-\ln\frac{{\rm tr}\left(\rho_{0}\Pi^{\rm
in}_{n}\right)}{{\rm tr}\left(\widetilde{\Phi}(\rho^{\rm in}_{\rm B})\Pi^{\rm
in}_{n}\right)}\,.$ (A-7)

This relation encodes information only on the initial stochastic quantum
entropy production due to the open system dynamics. In fact, for the special
case in which the dynamics is _unitary_ , and under the assumption of micro-
reversibility, i.e.,
$\Theta\,\mathcal{U}(\lambda)\Theta^{\dagger}=\mathcal{U}^{\dagger}(\widetilde{\lambda})$
with $\lambda(t)$ generic time-dependent transformation such that the system
Hamiltonian is invariant under time reversal, one finds that
$e^{-\Delta\overline{\sigma}^{\rm in}_{n}}=1\,\,\forall n$, i.e.,

${\rm P}_{\Gamma}(n,m)={\rm P}_{\widetilde{\Gamma}}(n,m)\,.$ (A-8)

Note also that, in determining the integral form of Eq. (A-6), one gets
$\left\langle e^{\Delta\overline{\sigma}^{\rm in}}\right\rangle_{\Gamma}=1$
with

$\left\langle\Delta\overline{\sigma}^{\rm
in}\right\rangle_{\Gamma}=\sum_{n}p^{\rm in}_{n}\ln\frac{\widetilde{p}^{\rm
fin}_{n}}{p^{\rm in}_{n}}=-S(p||\widetilde{p})$ (A-9)

where, we recall, $p^{\rm in}_{n}$ and $\widetilde{p}^{\rm fin}_{n}$ are the
probabilities to measure the $n$-th energy value of the system, respectively,
at the initial and final time instants of the forward and backward process.
Here, $S(q||p)$ denotes the _classical relative entropy_ between the two
probability distributions $q$ and $p$, and thus naturally corresponds to a
measure of how far is the final state of the inverse quantum dynamics from the
initial quantum state.

Coming back to the derivation of Eq. (3) of the main text, let us assume _(i)_
that the initial quantum state $\rho_{0}$ of the forward dynamics has thermal
populations but also non-zero coherence terms (in the system energy basis),
and _(ii)_ that the initial quantum state $\rho^{\rm in}_{\rm B}$ of the
backward quantum dynamics has thermal populations (with respect to the final
Hamiltonian) and once again non-vanishing off-diagonal elements. In other
terms,

$\displaystyle\rho_{0}=\rho_{\rm th}^{\rm in}(\beta)+\chi^{\rm
in}\equiv\frac{e^{-\beta H_{t_{i}}}}{Z_{i,\beta}}+\chi^{\rm in}$ (A-10)
$\displaystyle\rho^{\rm in}_{\rm B}=\rho^{\rm fin}_{\rm th}(\beta)+\chi^{\rm
fin}\equiv\frac{e^{-\beta H_{t_{f}}}}{Z_{f,\beta}}+\chi^{\rm fin}\,,$ (A-11)

where ${\rm tr}(\chi^{\rm in})={\rm tr}(\chi^{\rm fin})=0$, and the
Hamiltonian $H$ of the system is not necessarily assumed as a time-independent
operator. By substituting Eqs. (A-10) and (A-11) into (A-4) with $k=n=i$ and
$\ell=m=f$, one finds

$\frac{{\rm P}_{\Gamma}({\rm i,f})}{{\rm P}_{\widetilde{\Gamma}}({\rm
f,i})}=\exp\left[\beta(\Delta E_{\rm i,f}-\Delta F)+\sigma^{\rm fin}_{\rm
f}\left(\rho^{\rm in}_{\rm th}(\beta)\right)+\Sigma^{\rm fin}_{\rm
f}\left(\chi^{\rm in}\right)-\widetilde{\sigma}^{\rm fin}_{\rm
i}\left(\rho^{\rm fin}_{\rm th}(\beta)\right)-\widetilde{\Sigma}^{\rm
fin}_{\rm i}\left(\chi^{\rm fin}\right)\right],$ (A-12)

where

$\displaystyle\sigma^{\rm fin}_{\rm f}\left(\rho^{\rm in}_{\rm
th}(\beta)\right)\equiv\ln p^{\rm fin}_{\rm f}\left(\rho^{\rm in}_{\rm
th}(\beta)\right),$ (A-13) $\displaystyle\Sigma^{\rm fin}_{\rm
f}\left(\chi^{\rm in}\right)\equiv\ln\left(1+\frac{p^{\rm fin}_{\rm
f}(\chi^{\rm in})}{p^{\rm fin}_{\rm f}\left(\rho_{\rm th}^{\rm
in}(\beta)\right)}\right),$ (A-14) $\displaystyle\widetilde{\sigma}^{\rm
fin}_{\rm i}\left(\rho^{\rm fin}_{\rm
th}(\beta)\right)\equiv\ln\widetilde{p}^{\rm fin}_{\rm i}\left(\rho_{\rm
th}^{\rm fin}(\beta)\right),$ (A-15) $\displaystyle\widetilde{\Sigma}^{\rm
fin}_{\rm i}\left(\chi^{\rm
fin}\right)\equiv\ln\left(1+\frac{\widetilde{p}^{\rm fin}_{\rm i}(\chi^{\rm
fin})}{\widetilde{p}^{\rm fin}_{\rm i}\left(\rho_{\rm th}^{\rm
fin}(\beta)\right)}\right),$ (A-16)

$p^{\rm fin}_{\rm f}(\mathcal{A})\equiv\rm{tr}\left(\Pi^{\rm fin}_{\rm
f}\Phi(\mathcal{A})\right)$, and $\widetilde{p}^{\rm fin}_{\rm
i}(\mathcal{A})\equiv\rm{tr}\left(\Pi^{\rm fin}_{\rm
i}\widetilde{\Phi}(\mathcal{A})\right)$ with $\mathcal{A}$ a generic linear
operator. Note that $p^{\rm fin}_{\rm f}(\mathcal{A})$ and $\widetilde{p}^{\rm
fin}_{\rm i}(\mathcal{A})$ denote the probability to measure the ${\rm f}$-th
and ${\rm i}$-th final energy values of the quantum system in the forward and
backward process, respectively, conditioned to have evolved the thermal
contribution of the initial state (without coherence terms in the energy
eigenbasis). Finally, Eq. (A-12) can be simplified as

$\frac{{\rm P}_{\Gamma}({\rm i,f})}{{\rm P}_{\widetilde{\Gamma}}({\rm
f,i})}=\exp\left[\beta(\Delta E_{\rm i,f}-\Delta F)+\Delta\sigma_{\rm
i,f}+\Delta\Sigma_{\rm i,f}\right]$ (A-17)

by introducing the quantities

$\Delta\sigma_{\rm i,f}\equiv\sigma^{\rm fin}_{\rm f}\left(\rho^{\rm in}_{\rm
th}(\beta)\right)-\widetilde{\sigma}^{\rm fin}_{\rm i}\left(\rho^{\rm
fin}_{\rm
th}(\beta)\right)\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,\Delta\Sigma_{\rm
i,f}\equiv\Sigma^{\rm fin}_{\rm f}\left(\chi^{\rm
in}\right)-\widetilde{\Sigma}^{\rm fin}_{\rm i}\left(\chi^{\rm fin}\right).$
(A-18)

It is worth observing that, if no quantum coherence is present neither in
$\rho_{0}$ nor in $\rho^{\rm in}_{\rm B}$ (i.e., $\chi^{\rm in}=\chi^{\rm
fin}=0$), then $\Delta\Sigma_{\rm i,f}=0$. Thus, $\Delta\Sigma_{\rm i,f}$ can
be considered as a _correction_ , due to initial coherence in the energy basis
of the system, to the entropy difference $\Delta\sigma_{\rm i,f}$ obtained by
propagating initial thermal states in the forward and backward processes,
respectively.

The form of the detailed fluctuation theorem used in the main text is a
particular case of Eq. (A-17) where $\chi^{\rm fin}$ is assumed to vanish.
This choice is motivated by our aim to consider the minimal modification to
the “Jarzynski set-up”, respect to which only coherence in the initial state
of the forward dynamics is added.

##  Appendix B Part B: Derivation of the integral fluctuation theorem for
$\Delta\Sigma$

From Eq. (A-17) the integral fluctuation theorem of Eq. (5) can be easily
obtained.

Instead, for what concerns the integral fluctuation theorem involving the sole
coherence induced entropy production, let us consider the case in which
$\chi^{\rm fin}=0$. Upon substitution, one has

$\exp\left[-\Delta\Sigma_{\rm i,f}\right]=\frac{p^{\rm fin}_{\rm
f}\left(\rho_{\rm th}^{\rm in}(\beta)\right)}{p^{\rm fin}_{\rm
f}\left(\rho_{\rm th}^{\rm in}(\beta)\right)+p^{\rm fin}_{\rm f}(\chi^{\rm
in})}\,.$ (B-1)

Thus, taking the average over the EPM probability distribution of the forward
process $\Gamma$, we can conclude that

$\left\langle\exp\left[-\Delta\Sigma_{\rm
i,f}\right]\right\rangle_{\Gamma}=\sum_{\rm f}p^{\rm fin}_{\rm
f}\left(\rho_{\rm th}^{\rm in}(\beta)\right)=1\,,$ (B-2)

where we have used the fact that $p^{\rm fin}_{\rm f}\left(\rho_{\rm th}^{\rm
in}(\beta)\right)$ is itself a probability distribution normalized to $1$.

##  Appendix C Part C: Photodynamics of the NV-center

###  C.1 Theoretical modelling

In our experimental set-up, the photodynamics of the NV center is well
described by a seven-level quantum model [42, 43, 15], which, however, can be
effectively reduced to a two-level quantum system as detailed in the
Supplemental Material of Refs. [15]. Here, for completeness of exposition, we
report the modelling of this effective two-level open dynamics of the NV
center, by resorting to the super-operator formalism [44] and working in the
energy eigenbasis.

Let us thus model the NV center as a two-level quantum system subject to a
dissipative dynamics. The dissipative dynamics is induced by the laser pulses
and it is described by the linear super-operators ${\bf
S}\in\mathbb{C}^{4\times 4}$ acting directly on the column vector ${\rm
col}[\rho_{t}]\in\mathbb{C}^{4\times 1}$, with ${\rm col}[\rho]$ denoting the
vectorization of the density operator $\rho\in\mathbb{C}^{2\times 2}$. The
super-operator ${\bf S}$ is explicitly given by

${\bf S}=\frac{1}{2}\begin{pmatrix}2-p_{\rm
abs}(k_{c}-p_{d}\cos\alpha)&\displaystyle{p_{\rm
abs}k_{sc}}&\displaystyle{p_{\rm abs}k_{sc}}&p_{\rm
abs}(p_{d}\cos\alpha+k_{c})\\\ p_{\rm
abs}(k_{sc}+p_{d}\sin\alpha)&2-\displaystyle{p_{\rm
abs}(1+k_{s})}&-\displaystyle{p_{\rm abs}(k_{s}-1)}&p_{\rm
abs}(p_{d}\sin\alpha-k_{sc})\\\ p_{\rm
abs}(k_{sc}+p_{d}\sin\alpha)&-\displaystyle{p_{\rm
abs}(k_{s}-1)}&2-\displaystyle{p_{\rm abs}(1+k_{s})}&p_{\rm
abs}(p_{d}\sin\alpha-k_{sc})\\\ p_{\rm
abs}(k_{c}-p_{d}\cos\alpha)&-\displaystyle{p_{\rm
abs}k_{sc}}&-\displaystyle{p_{\rm abs}k_{sc}}&2-p_{\rm
abs}(p_{d}\cos\alpha+k_{c})\end{pmatrix},$ (C-1)

where

$k_{c}=1-(1-p_{d})(\cos\alpha)^{2},\qquad
k_{s}=1-(1-p_{d})(\sin\alpha)^{2},\qquad
k_{sc}=(1-p_{d})\sin\alpha\cos\alpha,$ (C-2)

$p_{\rm abs}$ is the absorption probability, $p_{d}$ is the probability of
population transfer to $|0\rangle_{S}$, and $\alpha\in[0,\pi/2]$.

The unitary dynamics in between two consecutive pulses is instead described by
the linear operator

${\bf U}=\exp\left(-i\tau\left(H\otimes\mathbbm{I}_{2}-\mathbbm{I}_{2}\otimes
H^{\ast}\right)\right)$ (C-3)

with $\hbar$ set to $1$, $\mathbbm{I}_{2}$ denoting the $2\times 2$ identity
matrix, and $H=\omega\,\sigma_{z}/2$ the Hamiltonian of the effective two-
level system.

Considering a total number of pulses $N$ we thus have

${\rm col}[\rho_{t}]={\bf L}\,{\rm col}[\rho_{0}]\quad\text{with}\quad{\bf
L}\equiv\left({\bf S}{\bf U}\right)^{N}.$ (C-4)

For a fixed set of values of the parameters $(p_{\rm
abs},p_{d},\alpha,\tau,\omega)$, the super-operator ${\bf L}$ governing the
open dynamics of the NV center possesses a unique steady-state $\rho^{*}$
which is characterised by a non vanishing coherence in the energy basis.

The Kraus representation for ${\bf L}$ can be uniquely determined by
diagonalizing its Choi matrix

${\bf T}\equiv\sum_{k,j=0}^{1}\left(E_{kj}\otimes\mathbbm{I}_{2}\right){\bf
L}\left(\mathbbm{I}_{2}\otimes E_{kj}\right)=\sum_{\ell=0}^{3}\xi_{\ell}{\bf
u}_{\ell}{\bf u}_{\ell}^{\dagger}$ (C-5)

where $E_{kj}\equiv|k\rangle\\!\langle j|$, with $|0\rangle\equiv(1,0)^{T}$
and $|1\rangle\equiv(0,1)^{T}$, and $(\xi_{\ell},{\bf u}_{\ell})$ denotes the
$j$-th pair (eigenvalue, eigenvector) resulting from the eigenvector
decomposition of ${\bf T}$. The Kraus operators $\\{K_{\ell}\\}$ associated
with the open map ${\bf L}$ are thus implicitly provided by the following
relation:

${\rm col}\left[K_{\ell}\right]=\sqrt{\xi_{\ell}}{\bf u}_{\ell}\quad\text{such
that}\quad\rho_{t}=\sum_{\ell=0}^{3}K_{\ell}\,\rho_{\rm
i}K_{\ell}^{\dagger}\,.$ (C-6)

One can easily determine that the open dynamics of the NV center is correctly
described by three Kraus operators $K_{\ell}$. These Kraus operators can then
be considered for the derivation of the backward dynamics as discussed in the
main text.

###  C.2 Experimental data and simulation

As described in the main text, in our experiments we measure the EPM
probability $p^{\rm fin}_{\rm f}(\rho)$, i.e., the probability of obtaining
$E^{\rm fin}_{\rm f}$ when measuring the energy of the system at the final
time $t_{\rm f}=N\tau$ (where $N$ is the number of laser pulses and $\tau$ is
the time between them), assuming that the system is initialized into the state
$\rho$. In particular, we performed four independent experiments for each of
the four initial states: $|0\rangle$, $|1\rangle$, $|+\rangle_{y}$, and
$|-\rangle_{y}$. The results of such measurements are shown in Fig. S1. Notice
that we only measure the probability $p^{\rm fin}_{{\rm f}=1}$ associated to
the final excited state $|1\rangle$. This is because, for a two level system,
the remaining probability is obtained as $p^{\rm fin}_{{\rm f}=0}=1-p^{\rm
fin}_{{\rm f}=1}$. As mentioned in the main text, the classical mixtures we
are interested in are a convex combination of these four pure states. For
example, the EPM probability associated with the initial thermal state
$\rho^{\rm in}_{\rm th}=e^{-\beta H_{t_{i}}}/Z_{i}$ is obtained as $p^{\rm
fin}_{\rm f}(\rho^{\rm in}_{\rm th})=p(E_{0}^{\rm in})p^{\rm fin}_{\rm
f}(|0\rangle\langle 0|)+p(E_{1}^{\rm in})p^{\rm fin}_{\rm f}(|1\rangle\langle
1|)$, where $p(E_{\rm i}^{\rm in})=e^{-\beta E_{\rm i}}/Z_{i}$. Similarly, the
probability $p^{\rm fin}_{\rm f}(\rho_{0})$ for a given initial state (see
main text) $\rho_{0}=p|1\rangle\\!\langle 1|+(1-p)|+\rangle_{y}\\!\langle+|$,
with $p\in[0,1]$, is obtained as $p^{\rm fin}_{\rm f}(\rho_{0})=pp^{\rm
fin}_{\rm f}(|1\rangle\\!\langle 1|)+(1-p)p^{\rm fin}_{\rm
f}(|+\rangle_{y}\\!\langle+|)$.

In Fig. S1 we compare the experimental data with the numerical simulation of
the dynamics, using the model described in the previous section. The values of
the parameters $p_{\rm abs}$, and $p_{d}$ are selected by minimizing the sum
of the squares of the residuals between data and simulation.

In Fig. 2 of the main text we show the results of the irreversible entropy
production $\Delta\Sigma$, obtained for an initial state $|+\rangle_{y}$. In
Fig. S2 we present similar results but for an initial state
$|-\rangle_{y}=(\left|{0}\right\rangle-i\left|{1}\right\rangle)/\sqrt{2}$.

Figure S1: Measurements of the EPM protocol as a function of the number $N$ of
pulses for the initial pure states
$\\{|0\rangle,|1\rangle,|+\rangle_{y},|-\rangle_{y}\\}$. The crosses with
error bars are the experimental data (dotted line is a guide to the eye).
Solid lines represent the numerical simulation of the dynamics described in
the previous section, using the parameters $p_{\rm abs}=0.700$, $p_{d}=0.255$.
The other parameters are, as mentioned in the main text, $\alpha=\pi/4$ (i.e.,
$\delta=-\Omega$), $\tau\omega\simeq(2\pi)0.9$, and $\tau=190$ ns.  Figure S2:
Panel (a): Experimental values (markers with error bars) of the irreversible
entropy production. Black squares are $\Delta\Sigma_{1,1}=\Delta\Sigma_{0,1}$,
and orange bullets are $\Delta\Sigma_{1,0}=\Delta\Sigma_{0,0}$. Dashed black
line and dotted orange line are the corresponding numerical simulations. Panel
(b): Experimental verification of the fluctuation theorem $\left\langle
e^{-\Delta\Sigma}\right\rangle_{\Gamma}=1$, Eq. (6), for the coherence-
affected irreversible entropy production, as a function of the number $N$ of
pulses used to drive the dynamics of the NV center. Panel (c): Average
experimental coherence-affected entropy production as a function of $N$ (blue
circles). In all panels, the experimental data are plotted against the
predictions from the numerical simulations that are obtained by taking
$|-\rangle_{y}=(\left|{0}\right\rangle-i\left|{1}\right\rangle)/\sqrt{2}$ as
the initial quantum state.