# Two-point measurement of entropy production from the outcomes of a single
experiment with correlated photon pairs

Gabriel H. Aguilar  Federal University of Rio de Janeiro, Rio de Janeiro, RJ,
Brazil  Thaís L. Silva  Federal University of Rio de Janeiro, Rio de Janeiro,
RJ, Brazil  Thiago E. Guimarães  Federal University of Rio de Janeiro, Rio de
Janeiro, RJ, Brazil  Rodrigo S. Piera  Federal University of Rio de Janeiro,
Rio de Janeiro, RJ, Brazil  Lucas C. Céleri  Institute of Physics, Federal
University of Goiás, P. O. Box 131, 74001-970, Goiânia, Brazil  Gabriel T.
Landi<EMAIL_ADDRESS>Instituto de Física, Universidade de São Paulo, CEP
05314-970, São Paulo, Brazil

###### Abstract

Fluctuation theorems are one of the pillars of non-equilibrium thermodynamics.
Broadly speaking, they concern the statistical distribution of quantities such
as heat, work or entropy production. Quantum experiments, however, usually can
only assess these distributions indirectly. In this letter we provide an
experimental demonstration of a quantum fluctuation theorem where the
distribution of entropy production is obtained directly from the outcomes
(clicks) of an optical experiment. The setup consists of entangled photon
pairs, one of which is sent an interferometer emulating a finite temperature
amplitude damping device. Blocking specific paths of the interferometer is
tantamount to restricting the possible configurations of the reservoir. And by
measuring its entangled pair, we can directly implement the two-point
measurement scheme, thus avoiding the destructive nature of photo-detection.

_Introduction —_ In the micro- and mesoscopic domains, thermodynamic
quantities such as heat, work and entropy production may fluctuate
significantly. They must therefore be described by random variables, with an
associated probability distribution. This change in paradigm has led to
ground-breaking new insights in non-equilibrium thermodynamics. One reason, in
particular, was the discovery of Fluctuation Theorems (FTs), special
symmetries of these distributions which generalize the second law of
thermodynamics Gallavotti and Cohen (1995); Evans  _et al._ (1993); Crooks
(1998); Jarzynski (1999); Lebowitz and Spohn (1999). Thermodynamics, however,
deals with processes, not states. That is, quantities such as heat or work
depend on the _transformations_ a system undergoes. Assessing these in an
experiment therefore requires measuring the system in (at least) two instants
of time. For classical processes, this is generally not an issue Liphardt  _et
al._ (2002); Collin  _et al._ (2005); Gomez-Solano  _et al._ (2011). But in
the quantum domain, measurements become invasive.

The default protocol for estimating quantum thermodynamic quantities is the
two-point measurement (TPM) scheme Talkner  _et al._ (2007); Esposito  _et
al._ (2009); Campisi  _et al._ (2011) (Fig. 1(a)). It consists in measuring
the system in the energy eigenbasis, before and after the process. This yields
a stochastic trajectory of outcomes, $\gamma\to\gamma^{\prime}$, from which
the corresponding thermodynamic quantities can be computed. However,
experimentally implementing the two measurements _in sequence_ can be
extremely challenging: Quantum observables are often only inferred indirectly
or via a destructive process, such as photo-detection.

Instead, the distribution of outcomes $P(\gamma,\gamma^{\prime})$ is usually
written as $P(\gamma,\gamma^{\prime})=P(\gamma^{\prime}|\gamma)P(\gamma)$, and
the two terms are determined from separate experiments. This was the case, for
instance, in Ref. An  _et al._ (2014), which was the first to report on
quantum fluctuation theorems. In one experiment the system is prepared in some
initial state (usually thermal), and the initial probabilities $P(\gamma)$ are
measured. Then, in a second experiment one _prepares_ a given state $\gamma$,
run the process and then measure it in $\gamma^{\prime}$, thus obtaining the
transition probability $P(\gamma^{\prime}|\gamma)$. The TPM is then
reconstructed from post-processing, with the purpose of confirming the
validity of FTs. Similar issues are also found in other platforms Hernández-
Gómez  _et al._ (2020); de Araújo  _et al._ (2018); Talarico  _et al._ (2016).

Figure 1:  (a) The standard TPM scheme, which consists in measuring a quantum
system before and after a thermodynamic process. The first measurement, with
outcome $\gamma$, may invasively affect the process and, in general, is
difficult to implement experimentally in a single experimental run. (b) We
bypass these issues using two features. First, we distinguish the different
configurations of the bath (black and purple lines inside the gray rectangle)
by blocking specific arms of the interferometer. And second, we use entangled
photon pairs, where only one is sent through the thermodynamic process. Due to
the coincidences postselection, measuring the other photon yields information
about the initial state, before the process. (c) Experimental setup. Pairs of
entangled photons are created using SPDC. Photon in mode S is sent to a FTAD
channel, while photon in mode A is detected after a projective measurement in
its polarization. Coincidence counts are registered with detections in Det1
and Det2. (d) Interferometric implementation of the FTAD channel (see main
text and appendix for more details).

In the last years, there has been significant progress in developing
alternative implementations of the TPM. One approach, used e.g. in nuclear
magnetic resonance experiments Batalhão  _et al._ (2014); *Batalhao2015;
Camati  _et al._ (2016), is based on Ramsay interferometry to indirectly
estimate the characteristic function (from which the distribution can be
reconstructed via a Fourier transform) Mazzola  _et al._ (2013); Dorner  _et
al._ (2013). Another method is based on the interpretation of work as the
direct outcomes of a generalized quantum measurement (POVM), as put forth in
Roncaglia  _et al._ (2014). This was implemented in Ref. Cerisola  _et al._
(2017), to directly measure the work distribution in a quantum gas, without
having to resort first to the TPM. Continuous weak measurements have also been
used in superconducting circuits Naghiloo  _et al._ (2020); *Naghiloo2018, as
an alternative to the two-point nature of the TPM. To the best of our
knowledge, the only experiment directly performing the two measurements of the
TPM in the same experiment is the one in Ref. Masuyama  _et al._ (2018), where
nondemolition measurements on superconducting qubits were employed.

In this letter we report an experiment where both outcomes of the TPM are
directly associated with the clicks of a single experiment. We study the
entropy production in a system comprising an entangled photon pairs, where one
of the photons is sent to an interferometer implementing a finite temperature
amplitude damping (FTAD) channel, representing the interaction with a heat
bath Jarzynski and Wójcik (2004); Jevtic  _et al._ (2015); Manzano  _et al._
(2018); Santos  _et al._ (2019). To perform the TPM, we introduce two
features, which are unique of our setup (Fig. 1(b)). First, each path of the
interferometer is mapped into a different bath configuration. Hence, by
blocking all but one of the paths, we can measure the statistics conditioned
on a given bath configuration. Second, we use coincidence counts for measuring
both photons in the entangled pair. Detecting the polarization of the photon
that went through the interferometer yields $\gamma^{\prime}$. And the
outcomes of the entangled pair determines the configuration $\gamma$, before
the process occurred. An experiment similar in spirit, but focused on a work
protocol, has just been reported in Solfanelli  _et al._ (2021).

_Experimental setup —_ The experimental setup is shown in Fig. 1(c). A He-Cd
laser pumps two beta-barium-borate (BBO) crystals in a crossed axis
configuration. By type-I spontaneous parametric down-conversion (SPDC),
photon-pairs in the entangled state
$|\psi_{SA}\rangle=\sqrt{\delta}|00\rangle_{SA}+\sqrt{1-\delta}|11\rangle_{SA}$
are produced in the modes S and A Kwiat  _et al._ (1999), where the
computational basis $|0\rangle_{S/A}$ and $|1\rangle_{S/A}$ corresponds to the
horizontal and vertical polarizations, respectively. The reduced state of S
thus reads $\rho_{S}=\delta|0\rangle_{S}\langle
0|+(1-\delta)|1\rangle_{S}\langle 1|$, which is a thermal state, with
occupation $\delta$.

The A-photons are detected in an avalanche photon-diode (Det2), after
polarization projective measurements, implemented by a half-wave plate (HWP)
and a polarized beam-splitter (PBS). Due to the correlated nature of
$|\psi_{SA}\rangle$ and the postselection of the detection of coincidences,
outcomes 0,1 for A imply initial states $\gamma=0,1$ for S, and occur with
probability $p_{\gamma}=\\{\delta,1-\delta\\}$.

The S-photons, on the other hand, pass through an interferometer which
implements an FTAD channel of the form
$\Lambda[\rho_{S}]=\sum_{j=1}^{4}E_{j}\rho_{S}E_{j}^{\dagger}$, with Kraus
operators (satisfying $\sum_{j}E_{j}^{\dagger}E_{j}=\mathbb{1}$)

$\displaystyle E_{1}=\sqrt{p}\left[\begin{array}[]{cc}1&0\\\
0&\sqrt{\eta}\end{array}\right]\qquad$ $\displaystyle
E_{2}=\sqrt{p}\left[\begin{array}[]{cc}0&\sqrt{1-\eta}\\\
0&0\end{array}\right]$ (5) $\displaystyle
E_{3}=\sqrt{1-p}\left[\begin{array}[]{cc}\sqrt{\eta}&0\\\
0&1\end{array}\right]\qquad$ $\displaystyle
E_{4}=\sqrt{1-p}\left[\begin{array}[]{cc}0&0\\\
\sqrt{1-\eta}&0\end{array}\right],$ (10)

which can be thought of as a combination of emission ($E_{1,2}$) and
excitation ($E_{3,4}$) processes with coupling strength $\eta\in[0,1]$ and
probabilities $p$ and $1-p$ Nielsen and Chuang (2000). The FTAD mimics a
thermal bath acting on the photon, with the parameter $p$ interpreted as the
thermal occupation probability. In fact, as discussed in Sup , each index
$j=1,2,3,4$ can also be associated to a pair of initial and final states of
the reservoir.

Figure 2: Experimentally determined probabilities and entropy production for
all possible stochastic trajectories. (a) Probabilities (11) for Kraus
channels $E_{1},\ldots,E_{4}$ and the system transitions
$\gamma\to\gamma^{\prime}$, with $\eta=0.7(2)$. (b)-(e) Same, but as a
function of $\eta$. Each corresponds to a different channel $E_{i}$. Each
curve refers to a transition $\gamma\to\gamma^{\prime}$, as labeled in image
(b). (f)-(i) Corresponding stochastic entropy production
$\sigma[\gamma,\gamma^{\prime},j]$, Eq. (12). Points correspond to the
experimental data, while the curves refer to the theoretical predictions.
Error bars are calculated from a Poissonian statistics for each outcome. In
all curves $p=0.19(1)$ and $\delta=0.77(1)$.

The FTAD interferometer is implemented by entangling the photon polarization
with path degrees of freedom, as sketched in Fig. 1(d). The S-photons are
first sent to a birrefringent calcite beam displacer BD1 that transmits
(deviates) the vertically (horizontally) polarized photons, creating two
spatial modes (up “$u$" and down “$d$"), which we will call transversal modes
(TMs). The polarization of photons in mode $d$ is rotated by a HWP at 45∘ and
both TMs pass through a HWP, whose axis is rotated by $\phi/2$. The photons
are thus reflected (transmitted) in PBS1 with probability $p=\sin^{2}\phi$
($1-p=\cos^{2}\phi$) into the longitudinal mode (LM) long “$l_{i}$" (short
“$s_{i}$"), where the set of Kraus operators $\\{E_{1},E_{2}\\}$
($\\{E_{3},E_{4}\\}$) are implemented. In our setup, we tune the coupling
strength of the channel with the angle $\theta/2$ of two HWPs, such that
$\eta=\cos^{2}\theta$. Hence, both parameters of the FTAD are adjusted at
will. Finally, the photons are sent to PBS2, which incoherently combines both
LMs, and splits the photons in the two final states of the reservoir, $s_{f}$
and $l_{f}$. All HPWs in 0∘ serve to compensate the path length, allowing a
coherent superposition of the TMs at BD2 and BD3, necessary for the
implementation $E_{1}$ and $E_{3}$. BD4 plays two roles in our setup: (i) it
traces out the TM and, in conjunction with the plate “T”, (ii) it selects
photons coming from $l_{f}$ or $s_{f}$, performing a projective measurement in
the computational basis of the reservoir. The optical elements in the pink box
are responsible for performing projective polarization measurements, yielding
final outcomes $\gamma^{\prime}=0,1$. The joint distribution
$P(\gamma,\gamma^{\prime})$ is then obtained by registering coincidence counts
between Det1 and Det2.

As discussed above, and formally demonstrated in the supplemental material Sup
, each Kraus operator $E_{j}$ is associated to a specific optical path.
Crucially, by blocking 3 out of the 4 paths, we can thus also determine
$P(\gamma,\gamma^{\prime},j)$, which represents the joint probability that the
system undergoes a transition from $\gamma\to\gamma^{\prime}$ in path
$j=1,2,3,4$, which is given by

$P(\gamma,\gamma^{\prime},j)=p_{\gamma}|\langle\gamma^{\prime}|E_{j}|\gamma\rangle|^{2}.$
(11)

Thermodynamically, this would be tantamount to the joint TPM distribution of
both system and bath, where the index $j$ collectively describes the initial
and final state of the bath. In our case, this bath is represented instead by
the path degrees of freedom of the interferometer.

The experimentally obtained probabilities are shown in Fig. 2 for fixed
$p=0.19(1)$ and $\delta=0.77(1)$. Image (a) summarizes all probabilities which
are non-zero, with fixed $\eta=0.7(2)$. In fact, out of the possible $16$
outcomes $(\gamma,\gamma^{\prime},j)$, only 6 are not zero. This is because
$E_{1}$ and $E_{3}$ do not generate jumps, and hence allow only for the
transitions $0\to 0$ and $1\to 1$ in the system. Conversely, $E_{2}$ and
$E_{4}$ necessarily cause the system to jump, from $1\to 0$ and from $0\to 1$,
respectively. This is a manifestation of the preservation of the total number
of excitations in system and bath. Figs. 2(b)-(e) depicts the probabilities as
a function of $\eta$, each plot corresponding to a different Kraus channel
$E_{j}$. For comparison, the theoretical predictions from Eq. (11) are also
shown. In all cases, the fidelity (Bhattacharyya distance) between theory and
experiment was above 0.98, therefore confirming that the TPM protocol was
successfully implemented.

_Stochastic entropy production —_ The stochastic entropy production along a
trajectory $(\gamma,\gamma^{\prime},j)$ reads Breuer (2003); Manzano  _et al._
(2018); Landi and Paternostro (2020)

$\sigma(\gamma,\gamma^{\prime},j)=\ln(p_{\gamma}/\tilde{p}_{\gamma^{\prime}})+\Phi_{j},$
(12)

where $\tilde{p}_{\gamma^{\prime}}=\sum_{\gamma,j}P(\gamma,\gamma^{\prime},j)$
is the probability associated to the final state of S. The first term in (12)
represents the stochastic variation in the system’s entropy. Indeed, averaging
over Eq. (11) leads to $\langle\ln
p_{\gamma}/\tilde{p}_{\gamma^{\prime}}\rangle=S(\Lambda[\rho_{S}])-S(\rho_{S})$,
where $S(\rho)=-\operatorname{tr}(\rho\ln\rho)$ is the von Neumann entropy.
The second term in Eq. (12) is the entropy flux $\Phi_{j}$ to the FTAD
interferometer. It therefore depends only on the path $j$ the photon undergoes
(and not on $\gamma,\gamma^{\prime}$). These fluxes are associated to quantum
jumps in the system Breuer (2003). Since $E_{1}$ and $E_{3}$ involve no jumps,
we have $\Phi_{1}=\Phi_{3}=0$. Conversely, paths $2$ and $4$ must be
accompanied by finite fluxes $\Phi_{2}$ and $\Phi_{4}$, since they involve
jumps from $1\to 0$ and $0\to 1$, respectively. To determine $\Phi_{2}$ and
$\Phi_{4}$ we use the fact that the time-reversed Kraus operators
$\tilde{E}_{j}$ should be related to the forward Kraus operators according to
$\tilde{E}_{j}=e^{-\Phi_{j}/2}E_{j}^{\dagger}$ Crooks (2008); Manzano  _et
al._ (2018). For the FTAD this yields the unique solution
$\Phi_{2}=-\Phi_{4}=\ln p/(1-p)$. With these expressions for $\Phi_{j}$, it
then follows from Eq. (12) that $\sigma$ satisfies the integral fluctuation
theorem

$\langle e^{-\sigma}\rangle=1.$ (13)

The experimentally determined values of the stochastic entropy production (12)
are shown in Fig. 2(f)-(i) as a function of $\eta$ for fixed $p=0.19(1)$ and
$\delta=0.77(1)$. Each plot corresponds to a different channel $j$ of the
FTAD. For some trajectories the corresponding probabilities (Fig. 2(b)-(e))
are identically zero, so that these values will not contribute to any
averages. For instance, because of Fig. 2(b), in Fig. 2(f) the only values of
$\sigma$ which will actually play a non-trivial role are those corresponding
to the trajectories $0\to 0$ and $1\to 1$ (gray and red curves).
Experimentally, we can still determine $\sigma(\gamma,\gamma^{\prime},j)$ even
when $P(\gamma,\gamma^{\prime},j)\equiv 0$. This follows from Eq. (12), since
the probability $\tilde{p}_{\gamma^{\prime}}$ is an average over all
trajectories.

As can be seen in Fig. 2(f)-(i), at the stochastic level some of the entropy
productions can be negative. This does not contradict the second law, which
holds only at the level of averages. Indeed, from Eq. (13) and Jensen’s
inequality, it follows that $\langle\sigma\rangle\geqslant 0$. This means that
negative values are less likely, which can be observed by comparing the
probabilities in Fig. 2(b)-(e). The average entropy production
$\langle\sigma\rangle=\sum_{\gamma,\gamma^{\prime},j}\sigma(\gamma,\gamma^{\prime},j)P(\gamma,\gamma^{\prime},j)$
is shown in Fig. 3(a), together with the fluctuation theorem (13) in Fig.
3(b). In both cases, the experiment confirms the experimental predictions:
$\langle\sigma\rangle\geqslant 0$ and $\langle e^{-\sigma}\rangle=1$.

Figure 3:  (a) Average entropy production $\langle\sigma\rangle$ and (b)
fluctuation theorem $\langle e^{-\sigma}\rangle$ as a function fo $\eta$, for
$p=0.19(1)$.

_Significance -_ The key feature of our setup is the ability to obtain the
full TPM statistics directly from the clicks of an experiment. This was
accomplished by combining two features. First, the configurations of the
reservoir are characterized by a set of Kraus operators, each corresponding to
a different optical path (see the appendix). Hence, by blocking 3 out of the 4
paths, we can directly study the statistics conditioned on each bath
configuration. Second, we use an entangled photon pair to non-destructively
measure the initial system configuration. Combined with the final measurement,
associated to the photon that went through the interferometer, this yields the
full TPM statistics.

Thermodynamic quantities, such as heat, work or entropy production,
characterize transformations (i.e. processes) that the system undergoes. But
directly assessing processes in quantum systems is notoriously difficult. Our
setup offers a platform for overcoming this.

As a proof-of-principle, we have focused on the entropy production in a heat
exchange process. This is convenient since entropy production is a fully
information-theoretic quantity, and hence avoids entering into issues about
the energetics of photon processes. However, our approach is highly flexible
and can be extended to various other thermodynamic protocols, including
systems with initial coherences. The latter, in particular, would be an
interesting direction of future research, since the TPM becomes invasive in
the presence of energetic coherences Aberg (2016); Perarnau-Llobet  _et al._
(2017); Levy and Lostaglio (2020); Gherardini  _et al._ (2020); Micadei  _et
al._ (2020); Deffner  _et al._ (2016); Sone  _et al._ (2020). Our framework
could then be used to compare the TPM with other alternatives that have
recently been proposed to overcome this invasiveness, such as Bayesian
networks Micadei  _et al._ (2020) and quasiprobabilities Levy and Lostaglio
(2020); Yunger Halpern  _et al._ (2018).

###### Acknowledgements.

We thank Stephen Walborn for valuable discussions. The authors acknowledge
financial support from the Brazilian agencies CNPq (PQ Grants No.
307058/2017-4m No. and INCT-IQ) GTL acknowledge the financial support of the
Sao Paulo Funding Agency FAPESP (Grants No. 2017 /50304-7, 2017/07973-5 and
2018/12813-0). The authors acknowledge the financial support of the Brazilian
funding agency CNPq (PQ Grants No. 305740/2016-4, 307058/2017-4, 309020/2020-4
and INCT-IQ 246569/2014-0). TLS would like to thank Serrapilheira Institute
(Grant No. Serra-1709-17173). This work was realized as part of the
CAPES/PROCAD program.

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Supplemental Material

In this supplemental material we provide additional details on the
implementation of the Finite Temperature Amplitude Damping (FTAD) channel used
in the main text [Eq. (1)]. We first discuss the connection between the
channel and the interaction between a system and a thermal bath. Then we give
additional details on how this FTAD emerges from the interferometer used in
our experiment.

##  I FTAD and interaction with a thermal reservoir

We consider a system $S$ interacting with an environment $E$, prepared in an
arbitrary uncorrelated state $\rho_{S}\otimes\rho_{E}$. The two are assumed to
evolve unitarily according to

$\rho_{SE}^{\prime}=U\rho_{S}\otimes\rho_{E}U^{\dagger},$ (S1)

The transformation over the system state is recovered by tracing out the
environmental degrees of freedom

$\rho^{\prime}_{S}=\sum_{k}\left\langle{k}\right|U\rho_{S}\otimes\rho_{E}U^{\dagger}\left|{k}\right\rangle,$
(S2)

where $\\{\left|{k}\right\rangle\\}$ is an orthonormal basis of the
environment Nielsen and Chuang (2000). If $\rho_{E}$ is a pure state, say
$\rho_{e}=|0\rangle\langle 0|$, then we can identify the Kraus operators with
$E_{k}=\left\langle{k}\right|U\left|{0}\right\rangle$. Thus, the allowed
number of independent Kraus operators is equal to the dimension of the Hilbert
space of the environment. For example, a FTAD channel would require a four-
dimensional environment. Instead, we can assume that the environment is
initially in a mixed state, which allows us to mimic the channel using an
environment with dimension 2, which is half of the number of Kraus operators
Horodecki  _et al._ (1999). It is not always possible to reconstruct a channel
in this fashion Terhal  _et al._ (1999), but particularly for the FTAD it is.

Thinking about the physical interpretation of the FTAD channel as an
interaction of a system with a thermal bath, it makes sense to consider the
initial state of the environment as a thermal state
$\rho_{E}=p|0\rangle\langle 0|+(1-p)|1\rangle\langle 1|$, whose temperature is
determined by the channel parameter $p$. Plugging this into equation (S2)
leads to

$\rho^{\prime}_{S}=\sum_{k=0,1}\left[\sqrt{p}\left\langle{k}\right|U\left|{0}\right\rangle\right]\rho_{S}\left[\sqrt{p}\left\langle{0}\right|U^{\dagger}\left|{k}\right\rangle\right]+\left[\sqrt{1-p}\left\langle{k}\right|U\left|{1}\right\rangle\right]\rho_{S}\left[\sqrt{1-p}\left\langle{1}\right|U^{\dagger}\left|{k}\right\rangle\right].$
(S3)

This is a completely positive and trace preserving map with Kraus operators

$\begin{split}E_{1}=\sqrt{p}\left\langle{0}\right|U\left|{0}\right\rangle&\qquad
E_{2}=\sqrt{p}\left\langle{1}\right|U\left|{0}\right\rangle\\\
E_{3}=\sqrt{1-p}\left\langle{1}\right|U\left|{1}\right\rangle&\qquad
E_{4}=\sqrt{1-p}\left\langle{0}\right|U\left|{1}\right\rangle.\end{split}$
(S4)

One can see that the Kraus operators in Eq. (5) are recovered by considering
the following two-qubit unitary transformation

$U=\left[\begin{array}[]{cccc}1&0&0&0\\\ 0&\sqrt{\eta}&-\sqrt{1-\eta}&0\\\
0&\sqrt{1-\eta}&\sqrt{\eta}&0\\\ 0&0&0&1\end{array}\right],$ (S5)

where the first and the second qubit are the system and the reservoir,
respectively. This is a partial swap unitary, which preserves the number of
excitations in the global system and the coupling $\eta$ gives the probability
that the system decays, emitting one excitation to the reservoir.

##  II Detailed explanation of the experimental setup

Figure S1:  Interferometer used in our experiment to implement the FTAD
channel.

In this appendix we describe the net transformation of the interferometer
shown in Fig. 1(d), and reproduced in Fig. S1. We will also demonstrate that
each path of the photons corresponds to a trajectory where only one of the
Kraus operators is applied. For this, let us consider, without any loss of
generality, that the polarization of the S-photons is prepared in a pure state
$\left|{\psi_{0}}\right\rangle=a\left|{0}\right\rangle+b\left|{1}\right\rangle$.
After passing through the BD1, this state is transformed into

$\left|{\psi_{1}}\right\rangle=a\left|{0}\right\rangle\left|{u}\right\rangle+b\left|{1}\right\rangle\left|{d}\right\rangle,$
(S6)

where $\left|{u}\right\rangle$ and $\left|{d}\right\rangle$ are the TM up and
down, respectively. Then, the polarization of photons in mode $d$ is rotated
by a HWP at 45∘. This is equivalent to a Pauli $X$-gate in the polarization
qubit controlled by the TM qubit, which transfers the polarization state to
the TM state, resulting in

$\left|{\psi_{2}}\right\rangle=\left|{0}\right\rangle\left(a\left|{u}\right\rangle+b\left|{d}\right\rangle\right).$
(S7)

Both TMs pass through a HWP, whose axis is rotated by an angle $\phi/2$,
transforming the photon’s state to

$\left|{\psi_{3}}\right\rangle=(\cos{\phi}\left|{0}\right\rangle+\sin{\phi}\left|{1}\right\rangle)\left(a\left|{u}\right\rangle+b\left|{d}\right\rangle\right).$
(S8)

Afterwards, the PBS1 creates two new paths, the longitudinal short “$s_{i}$"
and long “$l_{i}$" paths, by transmitting horizontally polarized photons while
reflecting the vertically polarized. The state of the photons can be written
as:

$\left|{\psi_{4}}\right\rangle=\left(\cos{\phi}\left|{0}\right\rangle\left|{s_{i}}\right\rangle+\sin{\phi}\left|{1}\right\rangle\left|{l_{i}}\right\rangle\right)\left(a\left|{u}\right\rangle+b\left|{d}\right\rangle\right)$
(S9)

We now identify, within the TM and LM, which are the modes that actually
constitute the degrees of freedom of the environment. By tracing out the
polarization, we obtain a thermal state in LM, whose temperature is controlled
by the angle $\phi$, multiplied by the state $\left|{\psi_{0}}\right\rangle$,
transferred to the TM. This is exactly the initial product state in Eq. (S1),
meaning that the initial excited and ground state of the reservoir are related
with the longitudinal modes $s_{i}$ and $l_{i}$, respectively. As mentioned in
the main text, the photons will go through the $l_{i}$ or the $s_{i}$ path
with probabilities $p=\sin^{2}\phi$ and $1-p=\cos^{2}\phi$. Given that the
coherence length of the SPDC photons is much smaller than the size of the path
difference between $s_{i}$ and $l_{i}$, then there is no coherence between
$s_{i}$ and $l_{i}$ at PBS2. The same happens after PBS2 when the photons can
follow two final LM $s_{f}$ and $l_{f}$ with different lengths, being
incorehently recombined in PBS3. In this last case, each path is associated
with one of the computational basis states of the bath after the interaction
with the system. Therefore, after tracing out the corresponding path, each
combination of $\\{s_{i},l_{i}\\}$ and $\\{s_{f},l_{f}\\}$ paths performs the
transformation of one of the Kraus operators composing the FTAD channel. In
what follows, we demonstrate this by considering each path separately. It is
worth noting that we will write sub-normalized states in each step such that
all the parameters of the Kraus operator appear. The correct normalizations
comes when considering the total transformation.

Long $l_{i}$ path. The unnormalized state of the photons that are reflected by
PBS1 can be written as
$\left|{\psi_{4l_{i}}}\right\rangle=\sin\phi\left|{1}\right\rangle(a\left|{u}\right\rangle+b\left|{d}\right\rangle)$.
This is transformed by a set of HWPs into the state

$\left|{\psi_{5l_{i}}}\right\rangle=\sin\phi\left[-a\left|{0}\right\rangle\left|{u}\right\rangle+b\left(-\cos\theta\left|{0}\right\rangle+\sin\theta\left|{1}\right\rangle\right)\left|{d}\right\rangle\right].$
(S10)

Given that the coupling strength of the channel is related with the angle
$\theta$, one can notice that only the photons in $d$ interact with the bath,
which is related to the initial state $\left|{1}\right\rangle$ of the
polarization (see $\left|{\psi_{0}}\right\rangle$). Provided that the FTAD is
an operation that preserves the number of excitations, the longitudinal mode
$l_{i}$ represents the initial state $\left|{0}\right\rangle$ of the
reservoir.

After reaching PBS2, the photons can follow $s_{f}$ or $l_{f}$, depending on
its polarization. Its final state in each path after BD2 and BD3 reads:

  * •

Short $s_{f}$ path:

$\left|{\psi_{6l_{i}s_{f}}}\right\rangle=\sin\phi\left(b\sin\theta\left|{1}\right\rangle\right)\left|{d}\right\rangle\left|{s_{f}}\right\rangle$
(S11)

  * •

Long $l_{f}$ path:

$\left|{\psi_{6l_{i}l_{f}}}\right\rangle=-\sin\phi\left(a\left|{1}\right\rangle+b\cos\theta\left|{0}\right\rangle\right)\left|{d}\right\rangle\left|{l_{f}}\right\rangle.$
(S12)

One can see that the role of BD2 and BD3 along with the preceding HWPs is to
transfer back the state from the TMs to the polarization. The TMs are just
auxiliary degrees of freedom used to properly implement the FTAD in the
polarization of the photons.

Short $s_{i}$ path. The photons transmitted in PBS1 are in a unnormalized
state
$\left|{\psi_{4s_{i}}}\right\rangle=\cos\phi\left|{0}\right\rangle(a\left|{u}\right\rangle+b\left|{d}\right\rangle)$,
which is transformed by the set of HWPs into

$\left|{\psi_{5s_{i}}}\right\rangle=\cos\phi\left[a\left(\cos\theta\left|{0}\right\rangle+\sin\theta\left|{1}\right\rangle\right)\left|{u}\right\rangle+b\left|{0}\right\rangle\left|{d}\right\rangle\right].$
(S13)

Akin to what happens in the $l_{i}$ path, here the bath interacts only with
photons in mode $u$, which is related with the initial state
$\left|{0}\right\rangle$ of the polarization. This means that the $s_{i}$ mode
is associated with the initial state $\left|{1}\right\rangle$ of the
reservoir. Again, we can write the photon state in each path $s_{f}$ and
$l_{f}$ after BD2 and BD3 as:

  * •

Short $s_{f}$ path:

$\left|{\psi_{6s_{i}s_{f}}}\right\rangle=\cos\phi\left(a\cos\theta\left|{1}\right\rangle+b\left|{0}\right\rangle\right)\left|{u}\right\rangle\left|{s_{f}}\right\rangle$
(S14)

  * •

Long $l_{f}$ path:

$\left|{\psi_{6s_{i}l_{f}}}\right\rangle=\cos\phi\left(a\sin\theta\left|{0}\right\rangle\right)\left|{u}\right\rangle\left|{l_{f}}\right\rangle.$
(S15)

The very final polarization state obtained after recombining incoherently all
the longitudinal and transversal modes is given as

$\rho_{f}=\rm
Tr_{TM,LM}\left[|\psi_{6s_{i}s_{f}}\rangle\langle\psi_{6s_{i}s_{f}}|+|\psi_{6s_{i}l_{f}}\rangle\langle\psi_{6s_{i}l_{f}}|+|\psi_{6l_{i}s_{f}}\rangle\langle\psi_{6l_{i}s_{f}}|+|\psi_{6l_{i}l_{f}}\rangle\langle\psi_{6l_{i}l_{f}}|\right].$
(S16)

Using Eqs. (5), (S11), (S12), (S14), and (S15), and taking $p=\sin^{2}\phi$
and $\eta=\cos^{2}\theta$, one can see that the trajectories $l_{i}l_{f}$,
$l_{i}s_{f}$, $s_{i}s_{f}$ and $s_{i}l_{f}$ can be identified with the terms
$E_{1}|\psi_{0}\rangle\langle\psi_{0}|E_{1}^{\dagger}$,
$E_{2}|\psi_{0}\rangle\langle\psi_{0}|E_{2}^{\dagger}$,
$E_{3}|\psi_{0}\rangle\langle\psi_{0}|E_{3}^{\dagger}$, and
$E_{4}|\psi_{0}\rangle\langle\psi_{0}|E_{4}^{\dagger}$, respectively, up to a
Pauli-X gate interchanging the roles of $\left|{0}\right\rangle$ and
$\left|{1}\right\rangle$.

Instead of recombining all modes together and then performing polarization
measurements, we place a HWP in the end of each $s_{f}$ and $l_{f}$ modes,
which together with PBS3, performs any desired projection onto the
polarization. BD4 and the wave plates "T" and "R" (fixed at 45∘) plays two
roles in our setup: (i) to implement an incoherent combination of the TM
modes, which is equivalent to tracing out this degree of freedom, and (ii) by
mapping the information of the LM $s_{f}$ and $l_{f}$ before PBS3 to
polarization after this optical element, projective measurement onto the final
$s_{f}$ and $l_{f}$ are performend by simply changing the angle of the plate
T.