# Thermodynamics of quantum information in noisy polarizers

Maxwell Aifer<EMAIL_ADDRESS>Department of Physics, University of Maryland,
Baltimore County, Baltimore, MD 21250, USA  Nathan M. Myers<EMAIL_ADDRESS>Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA  Sebastian
Deffner<EMAIL_ADDRESS>Department of Physics, University of Maryland,
Baltimore County, Baltimore, MD 21250, USA

###### Abstract

Among the emerging technologies with prophesied quantum advantage, quantum
communications has already led to fascinating demonstrations—including quantum
teleportation to and from satellites. However, all optical communication
necessitates the use of optical devices, and their comprehensive quantum
thermodynamic description is still severely lacking. In the present analysis
we prove several versions of Landauer’s principle for noisy polarizers, namely
absorbing linear polarizers and polarizing beamsplitters. As main results we
obtain statements of the second law quantifying the minimal amount of heat
that is dissipated in the creating of linearly polarized light. Our findings
are illustrated with an experimentally tractable example, namely the
temperature dependence of a quantum eraser.

##  I Introduction

There are generally three applications in which quantum advantage is expected
to be of technological significance—computation, sensing, and communication
[1]. Despite the distinct and unique technological challenges of each area,
devices designed for each of these applications can be considered quantum
devices that process information. Hence, they can be described by means of
quantum thermodynamics [2].

The development of classical thermodynamics of information [3] originated in
the formulation of Landauer’s principle [4]. This statement of the second law
asserts that any computational task requiring the erasure of information must
result in dissipation of heat, and the amount of heat produced is at least
$k_{B}T\ln(2)$ times the number of logical bits erased. Recent years have seen
intense research efforts in generalizing the bound to a wide variety of
physical situations, such as classical systems with both discrete and
continuous state spaces [5, 6, 7, 8, 9], as well as quantum systems undergoing
Markovian and non-Markovian dynamics [10, 11, 12, 13]. Landauer’s principle
has even been experimentally verified in microscale systems using an
overdamped colloidal particle in a double-well potential [14]. However, given
the rather heuristic nature of the original formulation [4], it is still being
debated whether the original statement can really be shown in all generality
[15, 16, 17]. Notably, some authors have recently interpreted certain non-
Markovian processes as violations of Landauer’s principle [18, 19, 20, 21,
22].

Curiously, most of the current discussion is focused on statements of
Landauer’s principle for computation. Yet, the communication of quantum
information also obviously incurs thermodynamic costs, which can be determined
with versions of the Landauer bound. Such “dynamical” formulations of
Landauer’s principle can be traced back to Bremermann [23], who proposed that
any computational device must obey the fundamental laws of physics namely
_special relativity_ , _quantum mechanics_ , and _thermodynamics_. Then,
identifying Shannon’s noise energy with the $\Delta E$ in Heisenberg’s
uncertainty relation [24] for energy and time, $\Delta E\Delta t\geq\hbar$, he
found an upper bound on the rate with which information can be communicated. A
more rigorous argument was put forward by Bekenstein [25] in the context of
black hole thermodynamics [26, 27, 28, 29, 30]. However, the Bremermann-
Bekenstein bound does not seem to enjoy the same prominence as Landauer’s
principle, and in fact many different statements for the maximal rate with
which entropy and information can be communicated have been formulated [31,
32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44].

In the present work, we will be analyzing the thermodynamics of some aspects
of quantum optical communication. This is motivated by the fact that light is
an attractive physical platform, given that it can carry information at the
greatest possible speed and that it interacts relatively weakly with the
environment [45, 46]. The most significant technical challenge in creating
optical quantum networks is overcoming attenuation [47, 48]. Entanglement
swapping schemes can extend the range of quantum communication [49], but the
performance of this method is still limited by the amount of dissipation
caused by the individual optical components used [50, 51, 52]. Hence, a
comprehensive thermodynamic characterization of noisy optical elements appears
urgently needed.

Most communication schemes employ linearly polarized light [53]. Such light
can be produced by sending unpolarized light through a linear polarizer, which
absorbs one component of the electric field and transmits the component
perpendicular to it. We will call such optical elements absorbing linear
polarizers (ALP). It is also possible to produce linearly polarized light
using a polarizing beamsplitter (PBS), which transmits one component of the
electric field and reflects the component which is perpendicular to it. While
it is clear that some heat must be generated by the ALP, it is less obvious
that there is any minimum amount of dissipation which is caused by the PBS
[54, 55]. In what follows we will show that both devices in fact are
responsible for dissipation of heat when an assumption of locality is made,
which can be understood through the concept of modularity dissipation [9].
Most light sources found in nature can be accurately described by the
classical laws of electrodynamics, and their description does not benefit from
quantization of the field. However, there are engineered sources available
which can produce single or entangled pairs of photons on demand, and the
results of experiments done with these sources cannot always be explained by
classical electrodynamics [56, 57, 58]. In our analysis, we will first address
the case of classical light. The “classicality” of light can be characterized
in a number of ways, such as by the negativity of the Wigner function [59, 59,
60]. For our purposes, we will consider classical light to be a statistical
mixture of approximate coherent states at large photon number. A derivation is
given of a version of Landauer’s principle for ALPs acting on classical
sources followed by a derivation of a similar result applicable to to PBSs,
assuming the reflected light is inaccessible. Then, we will provide insight
into more exotic sources with nonclassical behavior. We show that the findings
for the PBS acting on classical light can be seen as stemming from local
nonconservation of globally conserved quantities, and that an analogous
dissipation cost applies to quantum information processing tasks. Having
generalized the PBS result to the quantum case, we investigate the ALP acting
on quantum sources, and argue that an ALP can be modeled as a collection of
PBSs strung together with intervening thermal reservoirs. Using this model we
are able to probe the dynamics of the polarization process, rather than just
its end result. Interestingly, our model becomes formally equivalent to a
repeated interaction scheme (or collision model) which has received attention
recently in the field of open quantum systems [61] including in the context of
Landauer’s principle [62], and we leverage these existing results to quantify
the dissipation which occurs as (possibly nonclassical) light propagates
through the ALP. Notably, our analysis reveals a discontinuity in the slope of
the energy-entropy curve as temperature approaches zero, which agrees with
previous investigations of the random scattering of light [63]. The collision
model approach also allows us to describe the dependence of the ALP’s behavior
on its temperature, both in terms of optical extinction and decoherence. This
permits us to make predictions concerning the relationship between temperature
and the signature of entanglement observed in the quantum eraser [64, 65, 66,
67]. We predict that at higher temperatures the observed restoration of
interference caused by measurement is suppressed.

##  II Landauer’s principle for linear polarizers

In the following, we will be deriving various statements of Landauer’s
principle. The principle expresses that a decrease in Shannon entropy of an
information-bearing degree of freedom is accompanied by the dissipation of
heat [4, 68, 5]

${\mathchar 22\relax\mkern-12.0mu{\rm d}}Q\geq k_{B}T\,(-ds),$ (1)

where $s$ is the differential Shannon entropy [69]

$s=-\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}f_{X}(x)\ln{\left(f_{X}(x)\right)}.$
(2)

Here, $f_{X}(x)$ denotes the probability density function (PDF) of a discrete
or continuous random variable $X$.

In our case, $X$ is the electric field, in which information is encoded. Then,
an ALP has the effect of erasing this information stored in one component of
the electric field. Therefore, by Landauer’s principle the ALP should be
required to dissipate heat. This is in fact the case, and we now provide the
corresponding statement of Landauer’s principle.

###  II.1 Classical absorbing linear polarizer

The ALP transmits horizontally polarized light and absorbs vertically
polarized light. Consider a single mode of the electric field of frequency
$\omega$. The vertical component of the electric field is [70]

$E_{\text{v}}(z,t)=iE_{0\omega}\left[\alpha_{\text{v}}(\omega)e^{i(\omega
t-kz)}-\alpha_{\text{v}}(\omega)^{*}e^{i(kz-\omega t)}\right],$ (3)

where the complex amplitude $\alpha_{\text{v}}(\omega)$ will be considered a
random variable with PDF $f_{\alpha}$ and $E_{0\omega}$ is an arbitrary
constant electric field strength. The distribution is assumed to have finite
variance (and therefore finite energy) but is otherwise unspecified. The
differential Shannon entropy of $\alpha_{\text{v}}(\omega)$ is

$s[\alpha_{\text{v}}(\omega)]=-\int
d^{2}\alpha_{\text{v}}\,f_{\alpha}(\alpha_{\text{v}})\ln{\left(f_{\alpha}(\alpha_{\text{v}})\right)},$
(4)

where the integral is over the entire complex plane. We assume the field
decays rapidly outside of some region with volume $V$, and so the ensemble-
averaged energy content of the field’s vertical component is

$E=\frac{1}{2}V\epsilon_{0}E_{0\omega}^{2}\braket{}{\alpha_{\text{v}}}{{}^{2}}.$
(5)

After the light has passed through the polarizer, we assume that the state of
the field afterwards is a thermal state, given by the canonical ensemble,
$f^{\mathrm{can}}\propto\exp{\left(-E/k_{B}T_{P}\right)}$, where $T_{P}$ is
the temperature of the polarizer. Although it is well known that this
assumption leads to an infinite total energy in the completely classical case
when all frequencies are present [71], this does not affect the present
argument, which only concerns a single mode of the field. This thermal state
has entropy $s^{\mathrm{can}}$, and so the total erasure is simply the
difference

$-\Delta s=s-s^{\mathrm{can}}.$ (6)

We show in Appendix A that the following statement of Landauer’s principle
holds

$Q\geq k_{B}T_{P}\left(e^{-\Delta s}-1\right).$ (7)

Expanding about zero for small $\Delta s$, we obtain the Landauer bound

${\mathchar 22\relax\mkern-12.0mu{\rm d}}Q\geq k_{B}T_{P}\left(-ds\right).$
(8)

Equations (7) and (8) are our first main results, asserting Landauer’s
principle for the ALP. The assumption of finite variance of the field was
necessary to avoid having a divergent amount of heat dissipated, and similarly
it was necessary to assume that the ALP does not perfectly polarize the field
(a thermal distribution at temperature $T_{P}$ remains in the vertical
component) to avoid a divergent change in differential entropy. The assumption
that all energy is converted to heat is justified on the basis that the ALP is
a passive component with no mechanism for storing energy. If there are
multiple frequencies $\omega_{j}$ in the source, we may treat the amplitude at
each frequency as a random variable $\alpha_{\text{v}}(\omega_{j})$, and we
assume these are not correlated with each other. In this case due to the
additivity of the Shannon entropy and the energy, the same result holds (in
fact this argument holds even for a continuum of modes). We leave the case of
correlated amplitudes for future work. It should also be noted that the
quantity $-ds$ should not be interpreted as an amount of data which is erased,
but simply as a decrease in Shannon entropy. There is a subtle distinction
between these concepts given that it is possible that there is still some
finite Shannon entropy even if all of the original data has been erased, if
the original content of the signal has been replaced by random thermal
fluctuations.

###  II.2 Quantum absorbing linear polarizer

The somewhat natural question is how things change if the light is treated
quantum mechanically. In the quantum case the amplitude $\alpha$ is no longer
an observable, and thus it does not have a probability density function.
Instead we consider the dimensionless quadratures [72]

$\hat{q}=\frac{1}{\sqrt{2}}\left(\hat{a}^{\dagger}+\hat{a}\right),$ (9)

and

$\hat{p}=\frac{i}{\sqrt{2}}\left(\hat{a}^{\dagger}-\hat{a}\right),$ (10)

in terms of which the Hamiltonian is expressed as

$\hat{H}=\frac{1}{2}\hbar\omega\left(\hat{q}^{2}+\hat{p}^{2}\right).$ (11)

Therefore the average energy is

$E=\frac{1}{2}\hbar\omega\left(\braket{q^{2}}+\braket{p^{2}}\right)$ (12)

Since $\hat{q}$ and $\hat{p}$ do not commute, the interpretation of a joint
probability distribution over these variables is ambiguous, so it is not
immediately clear how to define the entropy.

Especially in quantum optics [72] it has proven particularly useful to express
quantum states in their Wigner representation

$W(q,p)=\frac{1}{\pi\hbar}\,\int
dy\,\bra{q+y}\rho\ket{q-y}\,e^{-2ip\,y/\hbar}\,.$ (13)

Correspondingly, a Wigner entropy can be defined as [73]

$s_{\text{W}}[W]=-\int dqdp\,W(q,p)\ln{\left(W(q,p)\right)},$ (14)

which is nothing but the Weierstrass transform of the Wehrl entropy. The
latter has been shown to be thermodynamically significant [74]. However, the
Wigner entropy is only defined for states with strictly positive Wigner
functions. Nonetheless, we are able to define a “maximal Wigner entropy” for
arbitrary states by the following argument: the sub-additivity of classical
Shannon entropy dictates that for a joint distribution $f_{X,Y}$ with marginal
distributions $f_{X}$ and $f_{Y}$, [75]

$s[f_{X,Y}]\leq s[f_{X}]+s[f_{Y}].$ (15)

When the Wigner entropy is defined, it is the same as the Shannon entropy over
a joint distribution. Also, recall that the marginals of the Wigner
distribution are the distributions for the quadratures $q$ and $p$ [72].
Therefore, the following sub-additivity rule is obeyed

$s_{\text{W}}[W]\leq s[f_{q}]+s[f_{p}].$ (16)

Even when $s_{\text{W}}[W]$ is not defined, we can still evaluate $s[f_{q}]$
and $s[f_{p}]$, and we define the maximal Wigner entropy as

$s_{\text{WM}}[W]=s[f_{q}]+s[f_{p}].$ (17)

Then erasure is bounded above by

$-\Delta s_{\text{W}}\leq s_{\text{WM}}[W]-s_{\text{W}}[W^{\prime}].$ (18)

It is shown in Appendix B that the following version of Landauer’s principle
holds

$Q\geq\frac{\hbar\omega}{2}\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)\left(e^{-\Delta
s_{\text{W}}}-1\right).$ (19)

Again, expanding for small $\Delta s$ we have

${\mathchar 22\relax\mkern-12.0mu{\rm
d}}Q\geq\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)\left(-ds_{\text{W}}\right),$
(20)

In Fig. 1 we plot the two Landauer bounds (8) and (20) as a function of
temperature. Observe that they agree at high temperatures, and at low
temperature (when $\hbar\omega\approx k_{B}T)$, the quantum heat cost per bit
asymptotically approaches a constant value $\ln(2)/2$ while the classical heat
cost per bit goes to zero.

Figure 1: Comparison of the classical, Eq. (8), and quantum, Eq. (20),
Landauer bounds for the ALP.

##  III Landauer’s principle for polarizing beamsplitters

Another optical element that is subject to dissipation is a polarizing
beamsplitters (PBS). Note that it is possible to produce linearly polarized
light with a PBS which transmits horizontally polarized light and reflects
vertically polarized light in a different direction. In this case, rather than
being absorbed, the vertically polarized output is simply sent elsewhere, cf.
Fig. 2. It is not clear that there is any nonzero amount of heat which must be
generated in this process, which is often assumed to be non-dissipative [76].
In what follows we show that there is a minimal amount of dissipation which
scales with temperature.

###  III.1 Semi-classical description

Figure 2: Schematic representation of a polarizing beamsplitter. The incident
beam is separated into into its horizontal and vertical polarization
components.

We first treat the PBS semi-classically, meaning we do not involve the quantum
state of light, although we consider it to be made up of pointlike particles;
in other words we are using the Newtonian corpuscular model [77]. Each photon
is assumed to have either horizontal or vertical polarization, with each
photon’s polarization an independent random variable, have probability one
half of being vertically polarized and probability one half of being
horizontally polarized. The PBS itself is assumed to have mass $m$ and
temperature $T$, and the photons are assigned a frequency $\omega=E/\hbar$. We
show (see Appendix C), under the assumption that the reflected photons are
locally unavailable, that the following lower bound on dissipation holds

${\mathchar 22\relax\mkern-12.0mu{\rm
d}}Q\geq\frac{\hbar^{2}\omega^{2}}{2m^{2}c^{4}}\,k_{B}T(-ds).$ (21)

Equation (21) relates logical information processed to dissipation cost for
the semiclassical PBS.

This result is similar to Landauer’s principle in that it is a lower bound on
heat generation which is proportional to the amount of logical information
processed. However it differs in that it does not require any information to
be “erased” per se. In our treatment we have assumed that light which is
reflected by the beamsplitter is no longer accessible locally, which leads to
dissipation. This is an instance of the more general phenomenon of modularity
dissipation [78], whereby locally accessible information is transformed into
global correlations across different parts of a system, which cannot be
exploited due to physical constraints.

###  III.2 Quantum non-conservation cost

The result of the previous section was based on the global conservation of
momentum and on the assumption of modularity, which prevents the exploitation
of global correlations. This reasoning can be generalized to any information
processing scheme which locally does not conserve quantities which are
globally conserved, even if the desired logical operation is invertible. Here
we give an example of this non-conservation cost in a quantum system which is
equivalent to the PBS.

Consider the action of the polarizing beamsplitter (PBS) in the one photon
subspace. The basis states of this subspace are

$\ket{\text{h1}},\ket{\text{h2}},\ket{\text{v1}},\ket{\text{v2}}.$ (22)

With this basis ordering the PBS unitary is then given by the following matrix

$U_{\text{PBS}}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix}.$ (23)

The PBS can therefore be thought of as a CNOT acting on a two qubit system,
whose control qubit is the polarization degree of freedom and whose target
degree of freedom is the path, $U_{\text{PBS}}=U_{\text{CN}}$ for the one-
photon subspace [79]. However, since the components of angular momentum are
globally conserved, it is not possible to implement this operation
deterministically unless there is access to a bath which can absorb the change
in angular momentum. It has been realized that global conservation laws
constrain the possible fidelity of quantum gates [80, 81], although to our
knowledge this has not been applied to finding the thermodynamic cost of
quantum gates under an assumption of modularity.

We assume that the two-qubit register and the bath are initially separable.
The target qubit is known to be in the zero state, the control qubit is in the
completely mixed state, and the bath is in an arbitrary state

$\rho=\frac{1}{2}(\ket{0}\bra{0}+\ket{1}\bra{1})\otimes\ket{0}\bra{0}\otimes\rho_{B},$
(24)

where the bath is a register of $N$ qubits. As it is not possible to actually
implement the CNOT exactly, by the previous arguments, we will assume it is
implemented with some error bound $\epsilon$. This is similar to the C-maybe
interaction that has been studied in the context of quantum Darwinism [82, 83,
84]. Explicitly, we have

$\sqrt{\mathrm{tr}\left\\{\left(\rho_{S}^{\prime}-U_{\text{CN}}\,\rho_{S}\,U_{\text{CN}}^{\dagger}\right)^{2}\right\\}}\leq\epsilon.$
(25)

We show in Appendix D that the loss of purity on the bath is bounded below by

$\mathrm{tr}\left\\{\rho_{B}^{\prime
2}\right\\}\leq\text{tr}\\{\rho_{B}^{2}\\}-\frac{1}{N2^{N}}\,\left(1-2\sqrt{2}\epsilon\right)^{2}.$
(26)

This implies that for error $\epsilon<(2\sqrt{2})^{-1}$ there is a nonzero
loss of purity on the bath as a result of the CNOT operation, and the minimal
loss of purity decreases exponentially with the size of the bath.

Figure 3: Comparison of the classical, Eq. (21), and quantum, Eq. (26),
dissipation bounds for the PBS.

Figure 3 gives a comparison of the bounds given by Eqs. (21) and (26), plotted
as a function of system size. It is assumed that in the quantum case the
register is made up of electrons, so the number of spins is simply $m/m_{e}$.
However, the two equations give lower bounds on the change in different
quantities (heat and purity), so they cannot be compared directly. We still
may get an idea of how the order of magnitude of the quantum and classical
bounds compare at different system sizes. The plot shows that there for
$m/m_{e}>100$ the classical bound dominates and for $m/m_{e}<100$ the quantum
bound becomes more relevant.

##  IV Quantum Master Equation for Linear Polarizers

The above analysis makes it apparent that a more rigorous treatment of optical
elements as genuinely quantum devices is required. To this end, we now
consider the propagation of light through an ALP, and model this as a
dynamical process using a quantum master equation. The ALP is conceptualized
as a series of layers through which the light propagates, interacting with
each layer in turn. Each layer is modeled as a PBS, which allows for noise
photons to enter from the environment as well as for losses of photons to the
environment due to attenuation. To set up this model, we first give the
quantum description of the PBS.

###  IV.1 Quantum polarizing beamsplitter

The PBS has two input ports, each with two polarization modes. We label the
two polarization modes of one input port $a_{\text{h}}$ an $a_{\text{v}}$, and
similarly use $b_{\text{h}}$ and $b_{\text{v}}$ for the other input port
(these will later be the annihilation operators for the input ports). The
corresponding output ports are labeled $a_{\text{h}}^{\prime}$,
$a_{\text{v}}^{\prime}$, $b_{\text{h}}^{\prime}$, and $b_{\text{v}}^{\prime}$
(See Fig. 4).

Figure 4: Port labelling convention for the polarizing beamsplitter.

The behavior of the PBS is easily described in the Heisenberg picture, where
the operators are transformed by a scattering matrix $S$

$\begin{pmatrix}a_{\text{h}}^{\prime}\\\ a_{\text{v}}^{\prime}\\\
b_{\text{h}}^{\prime}\\\
b_{\text{v}}^{\prime}\end{pmatrix}=S\begin{pmatrix}a_{\text{h}}\\\
a_{\text{v}}\\\ b_{\text{h}}\\\ b_{\text{v}}\end{pmatrix}.$ (27)

The PBS scattering matrix for the PBS may be taken to be

$S=\begin{pmatrix}1&0&0&0\\\ 0&t&0&r\\\ 0&0&1&0\\\ 0&-r&0&t\end{pmatrix},$
(28)

where $t$ and $r$ are the transmission and reflection coefficients, satisfying
$t^{2}+r^{2}=1$. We will therefore use the parameterization $t=\cos(\phi)$,
$r=\sin(\phi)$. We can also describe PBS whose transmission and reflection
axes are rotated by an angle $\theta$. In this case the scattering matrix is

$S_{\theta}=R_{\theta}SR_{\theta}^{\dagger},$ (29)

where

$R_{\theta}=\begin{pmatrix}\cos(\theta)&-\sin(\theta)&0&0\\\
\sin(\theta)&\cos(\theta)&0&0\\\ 0&0&\cos(\theta)&-\sin(\theta)\\\
0&0&\sin(\theta)&\cos(\theta)\end{pmatrix}.$ (30)

As is shown in Appendix F, the Schrödinger picture evolution operator on the
Hilbert space of quantum states can be evaluated as

$U=\exp\left(\sum_{ij}\ln(S)_{ij}a_{i}^{\dagger}a_{j}\right).$ (31)

In this case we find

$U(\theta,\phi)=\exp\left(\phi[\sin^{2}(\theta)S_{\text{h}}+\cos^{2}(\theta)S_{\text{v}}+\sin(2\theta)S_{c}]\right),$
(32)

where

$S_{\text{h}}=a_{\text{h}}^{\dagger}b_{\text{h}}-b_{\text{h}}^{\dagger}a_{\text{h}},$
(33)

$S_{\text{v}}=a_{\text{v}}^{\dagger}b_{\text{v}}-b_{\text{v}}^{\dagger}a_{\text{v}},$
(34)

and

$S_{c}=\frac{1}{2}(b_{\text{v}}^{\dagger}a_{\text{h}}+b_{\text{h}}^{\dagger}a_{\text{v}}-a_{\text{v}}^{\dagger}b_{\text{h}}-a_{\text{h}}^{\dagger}b_{\text{v}}).$
(35)

The density matrix is then transformed by a unitary map

$\rho^{\prime}=U(\theta,\phi)\,\rho\,U^{\dagger}(\theta,\phi).$ (36)

Finally, the quantum state exiting the PBS is given formally by

$\rho^{\prime}=F_{\theta}(\rho)=\mathrm{tr_{b}}\left\\{U_{\theta}\rho\otimes\eta_{T}U_{\theta}^{\dagger}\right\\},$
(37)

where $\eta_{T}$ is the Gibbs state at the temperature $T$ of the environment.

###  IV.2 Multilayer model of the linear polarizer

Figure 5: A polarizer conceptualized as a sequence of planes with varying
polarization axes. Figure 6: A linear polarizer modeled as a chain of
polarizing beamsplitters.

We model the linear polarizer as a sequence of PBS elements at randomized
angles which are drawn from a thermal distribution. These PBS elements
represent physical objects (for example, the nanoparticles in a nanoparticle-
based polarizer), which are in uncertain configurations due to thermal energy.
Initially the system is in some mixed state $\rho_{0}$ of the form of Eq.
(43). At each layer the state is transformed using Eq. (37). We also allow the
angle of each polarizer to be different, so the $n$th step of the evolution is
given by

$\rho_{n+1}=\mathrm{tr_{b_{n}}}\left\\{U_{n}\rho_{n}\otimes\eta_{T}U_{n}^{\dagger}\right\\}=F_{n}(\rho_{n})$
(38)

Equation (38) is of the form of a so-called collision model or repeated
interaction scheme [61, 85, 86].

We see from Eq. (32) that there is a continuum limit as long as $\phi$ is
taken to zero as $dt$ goes to zero. We set $\phi=c_{\phi}dt$. Then to first
order in $dt$, we have

$U_{\theta}=1+c_{\phi}dt(\sin^{2}(\theta)S_{\text{h}}+\cos^{2}(\theta)S_{\text{v}}+\sin(2\theta)S_{\text{c}})$
(39)

This leads to the following master equation

$\frac{d\rho}{dt}=c_{\phi}\,\mathrm{tr_{b}}\left\\{[\mathcal{H}_{\theta},\rho\otimes\eta_{T}]\right\\},$
(40)

where

$\mathcal{H}_{\theta}=\sin^{2}(\theta)S_{\text{h}}+\cos^{2}(\theta)S_{\text{v}}+\sin(2\theta)S_{\text{c}}.$
(41)

It has been shown by Lorenzo et al. [11] that for collision models of this
form, the following form of Landauer’s principle is observed

${\mathchar 22\relax\mkern-12.0mu{\rm d}}Q\geq k_{B}T(-ds),$ (42)

which is again the desired result.

###  IV.3 Single-photon states at low temperature

To make contact with experimentally realistic scenarios we are concerned with
photons of optical frequency ($\sim 10^{14}$ Hz), so at room temperature
($k_{B}T\approx 25$ meV) the probability of finding $n$ photons in a given
mode is proportional to $\exp(-17n)$. We therefore assume that the number of
photons in the vacuum ports is always $0$ initially. We are effectively
treating the environment as being at zero temperature for the purposes of
thermal radiation. However, we will not assume zero temperature for the
mechanical degrees of freedom of the polarizer itself which contribute to
uncertainty in the transmission axis.

Suppose that incident on the input port of a PBS there is a general mixed
state belonging to the subspace with at most one photon,

$\rho=\sum_{\alpha,\alpha^{\prime}\in\\{0,\text{h},\text{v}\\}}c_{\alpha\alpha^{\prime}}\ket{\alpha}\bra{\alpha^{\prime}}.$
(43)

The state of the two input ports $a$ and $b$ combined is then

$\sigma=\rho\otimes\ket{0}\bra{0}$ (44)

We are concerned with the reduced density matrix for the input mode, $\rho$.
The evolution of $\rho$ is given by

$\rho^{\prime}=\mathrm{tr_{b}}\left\\{U_{\theta}\rho\otimes\ket{0}\bra{0}U_{\theta}^{\dagger}\right\\}\,.$
(45)

The evolution given by Eq. (37) is a CPTP map [87], and therefore can be given
in terms of Krauss operators. For this particular case, it can be expressed in
terms of two Krauss operators which depend on $\theta$, $K_{1\theta}$ and
$K_{2\theta}$

$\rho^{\prime}=F_{\theta}(\rho)=K_{1\theta}^{\dagger}\rho
K_{1\theta}+K_{2\theta}^{\dagger}\rho K_{2\theta}.$ (46)

Explicitly, the Krauss operators are,

$K_{1\theta}=\begin{pmatrix}1&0&0\\\
0&t\cos^{2}(\theta)+\sin^{2}(\theta)&-(1-t)\sin(\theta)\cos(\theta)\\\
0&-(1-t)\sin(\theta)\cos(\theta)&\cos^{2}(\theta)+t\sin^{2}(\theta)\end{pmatrix}$
(47)

and,

$K_{2\theta}=\begin{pmatrix}0&r\cos(\theta)&r\sin(\theta)\\\ 0&0&0\\\
0&0&0\end{pmatrix}$ (48)

Equations (46), (47), and (48) completely specify the evolution of the density
matrix as the single photon propagates through a single layer of the polarizer
at low temperature. To simulate the evolution in the multilayer model, the
angle $\theta$ for each polarizer is drawn from a thermal distribution

$f(\theta)=\frac{1}{\sqrt{2\pi
k_{B}T/\kappa}}\exp\left(-\frac{1}{2}\frac{\theta^{2}}{k_{B}T/\kappa}\right),$
(49)

where $\kappa$ is a physical parameter of the polarizing material which
characterizes the energy required to change the orientation of polarizing
elements from their (mechanical) equilibrium positions. The resulting maps
$F_{\theta}$ are applied one after another. For $\theta=0$ we get a
particularly simple evolution

$F_{0}(\rho)=\begin{pmatrix}\rho_{00}+r^{2}\rho_{11}&t\rho_{01}&\rho_{02}\\\ \
t\rho_{10}&t^{2}\rho_{11}&t\rho_{12}\\\
\rho_{20}&t\rho_{21}&\rho_{22}\end{pmatrix}$ (50)

In Appendix E, we consider the Heisenberg picture description of a single
application of the map $F_{\theta}$ with a random angle drawn from the
distribution (49). It is found that an adjoint map can be defined which gives
evolved operators satisfying a modified commutation relation

$[a_{\text{h}}^{\prime},a_{\text{h}}^{\prime{\dagger}}]=[a_{\text{v}}^{\prime},a_{\text{v}}^{\prime{\dagger}}]=1-2(1-t)(\chi-\chi^{2}).$
(51)

where $\chi=\frac{1}{2}\left[1-\exp(-2k_{B}T/\kappa)\right]$.

In Eq. (51) we observe the direct impact of the noise in the PBS on the
optical signal. The quantum mechanical communtation relations “degrade” as a
function of temperature, which is nothing but another way of looking at
effects of decoherence.

###  IV.4 Numerical study of the low-temperature limit

Using the above formalism we simulate the evolution of the quantum state of
light, limited to the $\leq 1$ photon subspace, and assuming low temperature,
that is $k_{B}T\ll\hbar\omega$. This simulation allows us to discuss some
qualitative aspects of the polarization process. We initialize the system in
the state

$\rho_{0}=\begin{pmatrix}1/3&1/3&1/3\\\ 1/3&1/3&1/3\\\
1/3&1/3&1/3\end{pmatrix}.$ (52)

Then the CPTP map $F_{0}$ is iterated for $N=10,000$ layers with $t=0.9$. A
range of temperatures are chosen in the interval
$\sqrt{k_{B}T/\kappa}\in[0.05,0.3]$. We find that the population
$p_{\text{v}}=\rho_{\text{vv}}$ decays very quickly, as expected, falling to
near zero after less than one hundred layers. The horizontal population
$p_{\text{h}}$ takes a longer time to decay. As temperature increases,
$p_{\text{v}}$ decays more slowly and $p_{\text{h}}$ decays more quickly,
until the two polarizations behave near identically for
$\sqrt{k_{B}T/\kappa}\approx 1$. At low temperature, due to the lag between
absorption of vertically and horizontally polarized light, the vacuum
population has a plateau at intermediate times (see Fig. 7).

Figure 7: Populations $p_{0}=\rho_{00}$ (solid lines),
$p_{\text{h}}=\rho_{11}$ (dotted lines), and $p_{\text{v}}=\rho_{22}$ (dashed
lines). The vertical population $p_{\text{v}}$ rapidly decays to zero but the
horizontal component takes longer. As a result the vacuum population has a
plateau that is flatter at lower temperatures.

We see from Fig. 8 that the coherences $|\rho_{\text{v0}}|^{2}$ and
$|\rho_{\text{hv}}|^{2}$ decay exponentially, with the horizontal-vacuum
coherence decaying faster. The decay of coherence occurs faster at higher
temperatures. This result is in line with our expectations, given that
decoherence has been found to be enhanced at higher temperatures in a variety
of physical systems [88, 89, 90].

Figure 8: Coherences $|\rho_{\text{hv}}|^{2}$ (thick lines) and
$|\rho_{\text{v0}}|^{2}$ (thin lines). The coherences both decay
exponentially, but $|\rho_{\text{v0}}|^{2}$ decays at a faster rate at all
temperatures. Both coherences decay faster at higher temperatures.

The Shannon entropy is evaluated at all times, and it is found to decrease
monotonically. The energy is proportional to $1-p_{0}$, and also decreases
monotonically. Figure 9 shows the time dependence of the energy and Shannon
entropy, with both display the plateau feature at low temperatures discussed
earlier.

Figure 9: The time dependence of Shannon entropy (solid lines) and energy
$1-p_{0}$ (dashed lines). Both display plateaus at intermediate times which
are made flatter at lower temperatures. This is due to the difference in
timescales between absorption of horizontally and vertically polarized light.
Both functions are decreasing, and their relative slopes are different before
and after the plateau.

Landauer’s principle concerns the change in heat relative to the change in
Shannon entropy, which can be expressed as the derivative ${\mathchar
22\relax\mkern-12.0mu{\rm d}}Q/(-ds)$ along a system trajectory. Since we
assume ${\mathchar 22\relax\mkern-12.0mu{\rm d}}Q=-dE$, we are therefore
interested in the derivative $dE/ds$ along some system trajectory. Although
this ratio is related to temperature, it is not given by any function of
temperature given that the system is out of equilibrium. We plot the energy
against the Shannon entropy for 100 realizations of the random process at
various temperatures. For each realization, the angles of the polarizer layers
have been drawn from a Boltzmann distribution at the given temperature, and
the initial conditions are always those given by Eq. (52).

Figure 10: One hundred realizations of the polarization process are plotted
with the same initial conditions but randomized transmission angles for the
individual layers drawn from the Boltzmann distribution. Energy is
monotonically increasing with entropy, but displays a discontinuity in its
slope at zero temperature.

Figure 10 shows that the energy always increases monotonically with Shannon
entropy, as expected. Interestingly, as temperature approaches zero, a
discontinuity of the slope appears near $s=0.6$. This can be explained by the
plateaus appearing in Fig. 9 at low temperature. On the plateaus, both the
entropy and energy are nearly constant, although many layers of the polarizer
are traversed. After the plateau, the relative slope of the energy and entropy
is different than what it was before, and so a discontinuity of the slope
$dE/ds$ appears when the energy and Shannon entropy are plotted against each
other.

##  V Temperature-dependent quantum eraser

Figure 11: Setup for the temperature-dependent quantum eraser experiment.

We conclude the analysis with an experimentally testable consequence of our
findings. We have developed a theoretical model for the effect of temperature
on the evolution of a single-photon state through a polarizing medium, and our
model predicts that at higher temperature there is enhanced decoherence. We
now propose an experimental test of this, which is a modified quantum eraser
experiment where the linear polarizer used to measure one of the photon’s
polarization is in contact with a heat reservoir at temperature $T$. In the
quantum eraser experiment [64, 67, 91, 66, 92, 65] there are two photons,
called the signal and idler photons. The signal photon may be either
horizontally or vertically polarized, or it may be absorbed by the polarizer
so it lives in a three-dimensional Hilbert space spanned by the vectors
$\ket{\text{0}},\ket{\text{h}},\ket{\text{v}}$. The idler photon may be either
horizontally or vertically polarized, and it may go down path $1$ or path $2$
of the interferometer, so it lives in a four-dimensional Hilbert space spanned
by the vectors
$\ket{\text{h1}},\ket{\text{v1}},\ket{\text{h2}},\ket{\text{v2}}$. So the
state of the two photons together belongs to a $12$-dimensional Hilbert space
spanned by the following states

$\displaystyle\begin{split}&\ket{\text{0h1}},\ket{\text{hh1}},\ket{\text{vh1}},\ket{\text{0v1}},\ket{\text{hv1}},\ket{\text{vv1}},\\\
&\ket{\text{0h2}},\ket{\text{hh2}},\ket{\text{vh2}},\ket{\text{0v2}},\ket{\text{hv2}},\ket{\text{vv2}}\end{split}$
(53)

These states can be expressed as $\ket{ijk}$, where
$i\in\\{0,\text{h},\text{v}\\}$, $j\in\\{\text{h},\text{v}\\}$, and
$k\in\\{1,2\\}$. Then in general we have a density matrix of the form

$\sigma=\sum_{ijk}c_{ijki^{\prime}j^{\prime}k^{\prime}}\ket{ijk}\bra{i^{\prime}j^{\prime}k^{\prime}}.$
(54)

The full density matrix can be written as a $4\times 4$ array of $3\times 3$
submatrices, where the indices in the $4\times 4$ array correspond to the
idler photon and the indices of each submatrix correspond to the signal
photon.

$\sigma=\begin{pmatrix}\rho_{\text{v1,v1}}&\rho_{\text{v1,h1}}&\rho_{\text{v1,v2}}&\rho_{\text{v1,h2}}\\\
\rho_{\text{h1,v1}}&\rho_{\text{h1,h1}}&\rho_{\text{h1,v2}}&\rho_{\text{h1,h2}}\\\
\rho_{\text{v2,v1}}&\rho_{\text{v2,h1}}&\rho_{\text{v2,v2}}&\rho_{\text{v2,h2}}\\\
\rho_{\text{h2,v1}}&\rho_{\text{h2,h1}}&\rho_{\text{h2,v2}}&\rho_{\text{h2,h2}}\end{pmatrix}$
(55)

To evolve the density matrix forward by one step, we use Eq. (37). This gives

$\sigma^{\prime}=G_{\theta}(\sigma)=\begin{pmatrix}F_{\theta}(\rho_{\text{v1,v1}})&\dots&F_{\theta}(\rho_{\text{v1,h2}})\\\
\\\ \\\
F_{\theta}(\rho_{\text{h2,v1}})&\dots&F_{\theta}(\rho_{\text{h2,h2}})\end{pmatrix}$
(56)

The initial condition for the quantum eraser experiment is a Bell state in the
polarization basis for the two photons:

$\ket{\psi_{0}}=\frac{1}{\sqrt{2}}(\ket{\text{hh1}}+\ket{\text{vv1}})$ (57)

First, a beamsplitter is applied to the idler photon. There are two distinct
configurations for the interferometer, one in which the paths are “marked” and
the other the paths “unmarked”. For the marked configuration there is a
quarter wave plate (QWP) in each path of the interferometer (see Fig. 11). One
of the QWPs is at an angle of $\pi/4$ radians, the other at an angle of
$-\pi/4$ radians. For the unmarked configuration there are no QWPs. First we
simulate what happens in the unmarked configuration when there is no
measurement made on the signal photon, and recover the standard result of a
simple Mach-Zehnder interferometer experiment (see Fig. 12) [93].

Figure 12: Probability for the photon to exit each port as a function of
relative phase difference for the two arms in the unmarked case.

Next we look at the marked configuration, again with no measurement made on
the signal photon, the result of which is shown in Fig. 13. We see that the
interference pattern is now lost.

Figure 13: Probability for the photon to exit each port as a function of
relative phase difference for the two arms in the marked case.

We then make a measurement on the signal photon with a linear polarizer at an
angle $\theta$ from the vertical, resulting in a restoration of the
interference pattern.

Figure 14: Probability for the photon to exit each port as a function of
relative phase difference for the two arms in the marked case, with a
measurement made on the signal photon by a LP at angle $\theta$ from vertical.

We see that the interference pattern is gone when $\theta=0$, and is
completely restored for $\theta=\pi/4$, which is the main result of the
traditional quantum eraser experiment (see Fig. 14). The variation in the
interference pattern is most pronounced at a path phase difference of
$\phi=\pi/2$.

Figure 15: Probability for the photon to each port as a function of
temperature, with a relative phase of $\pi/2$ between the paths. Ensemble
average of 1000 realizations.

We see from Fig. 15 that the effect of quantum erasing is temperature
dependent. In particular, as temperature is increased, the ability of the
polarizer to restore interference is suppressed, and eventually goes away
entirely.

##  VI Concluding remarks

Our analysis has provided insight into the (quantum) thermodynamics of linear
polarizers; we have given explicit forms of Landauer’s principle for both
absorbing linear polarizers and polarizing beamsplitters. We have also
developed a formalism that incorporates the thermal energy contained in
mechanical degrees of freedom of a polarizer, and used this model to
investigate the time-dependent dynamics of the polarization process. We have
provided a qualitative description of the dependence of this process on
temperature, and proposed an experiment to test the temperature-dependence of
decoherence via a quantum eraser apparatus.

Some questions have been left unanswered which warrant further analysis. A
full solution of the master equation derived for the quantum absorbing
polarizer, even in the low temperature limit and the single photon subspace is
lacking. We also have not yet written a master equation valid when $k_{B}T$ is
on the order of at least $\hbar\omega$. Answering these and other questions
will provide guidance for optimal control of optical polarization states. This
is expected to be particularly challenging task when $k_{B}T$ is nearly of the
order of $\hbar\omega$, since then environmental noise becomes significant.
Therefore we hope to extend our results with special attention paid to
deriving a master equation valid in the high temperature limit. It is then
desirable to find bounds on the minimum dissipation required to perform
various tasks in such an environment, including the version of Landauer’s
principle in Eq. (42).

###### Acknowledgements.

We would like to thank Todd Pittman, Alejandro Rodriguez Perez, Steve
Campbell, and Pat Eblen for enlighting discussions. M.A. gratefully
acknowledges support from Harry Shaw of NASA Goddard Space Flight Center.
N.M.M. acknowledges support from AFOSR (FA2386-21-1-4081, FA9550-19-1-0272,
FA9550-23-1-0034) and ARO (W911NF2210247, W911NF2010013). S.D. acknowledges
support from the John Templeton Foundation under Grant No. 62422.

##  Appendix A Derivation of classical Landauer’s principle

Recall that for a random vector of dimension $D$, and with covariance matrix
$\Sigma$, the entropy is upper bounded by that of the normal distribution [94,
95]

$s[f]\leq\frac{1}{2}\ln{\left((2\pi e)^{D}\left|\Sigma\right|\right)},$ (58)

We regard $f_{\alpha}$ as a bivariate distribution over both the real and
imaginary parts of $\alpha_{\text{v}}$. Note that in general
$\sqrt{\left|\Sigma\right|}\leq\braket{}{\alpha_{\text{v}}}{{}^{2}}/2$, so

$s[f_{\alpha}]\leq\ln{\left(\pi
e\braket{}{\alpha_{\text{v}}}{{}^{2}}\right)}=\ln{\left(\frac{2e\pi
E}{V\epsilon_{0}E_{0\omega}^{2}}\right)}.$ (59)

We assume that after the light has passed through the polarizer, it is in a
thermal distribution with temperature $T_{P}$, which is associated with the
polarizer itself. Therefore the energy is $E^{\prime}=k_{B}T$ and the entropy
is

$s[f_{\alpha}^{\prime}]=\ln{\left(\frac{2e\pi
E^{\prime}}{V\epsilon_{0}E_{0\omega}^{2}}\right)}.$ (60)

Consequently, the decrease in Shannon entropy is at most

$-\Delta s\leq\ln{\left(\frac{2e\pi
E}{V\epsilon_{0}E_{0\omega}^{2}}\right)}-\ln{\left(\frac{2e\pi
E^{\prime}}{V\epsilon_{0}E_{0\omega}^{2}}\right)}=\ln{\left(\frac{E}{E^{\prime}}\right)}.$
(61)

Since there is no work reservoir, all lost energy is dissipated as heat, and
we have

$Q=E-E^{\prime}=E^{\prime}\left(\frac{E}{E^{\prime}}-1\right).$ (62)

Using $E^{\prime}=k_{B}T_{P}$ and the upper bound on $\Delta s$ then gives

$Q\geq k_{B}T_{P}\left(e^{-\Delta s}-1\right).$ (63)

##  Appendix B Derivation of quantum Landauer’s principle

Again using Eq. (58), we find that

$s_{\text{WM}}[W]=s[f_{q}]+s[f_{p}]\leq\ln{\left(2\pi
e\sqrt{\braket{q^{2}}\braket{p^{2}}}\right)}.$ (64)

Due to Eq. (12) this is bounded above by

$s_{\text{WM}}[W]\leq\ln{\left(\frac{2\pi eE}{\hbar\omega}\right)}.$ (65)

We again assume that the light is in a thermal state after leaving the
polarizer. The thermal state has Wigner function [96, 97]

$W^{\prime}(q,p)=\frac{\hbar\omega}{\pi
E^{\prime}}\exp\left(-\frac{\hbar\omega(q^{2}+p^{2})}{2E^{\prime}}\right),$
(66)

where $E^{\prime}$ is given by

$E^{\prime}(T)=\hbar\omega\left(\frac{1}{\exp(\hbar\omega/k_{B}T)-1}+\frac{1}{2}\right),$
(67)

or alternatively

$E^{\prime}(T)=\frac{\hbar\omega}{2}\coth\left(\frac{\hbar\omega}{2k_{B}T}\right).$
(68)

Since the thermal Wigner function (66) is positive, the final Wigner entropy
$s_{W}[W^{\prime}]$ is defined, and is given by

$s_{\text{W}}[W^{\prime}]=\ln{\left(\frac{2\pi
eE^{\prime}}{\hbar\omega}\right)}.$ (69)

We can write

$-\Delta s_{\text{W}}\leq
s_{\text{WM}}[W]-s_{\text{W}}[W^{\prime}]\leq\ln\left(\frac{E}{E^{\prime}}\right)$
(70)

We once again equate the heat to the lost energy

$Q=E^{\prime}\left(\frac{E}{E^{\prime}}-1\right),$ (71)

and hence

$Q\geq\frac{\hbar\omega}{2}\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)\left(e^{-\Delta
s_{\text{W}}}-1\right).$ (72)

##  Appendix C Minimal dissipation for the semi-classical PBS

Let $\hat{n}$ be the unit vector which is perpendicular to the PBS surface and
is pointing _into_ the bulk. The PBS and the photon have momenta $\bm{p}_{b}$
and $\bm{p}$ respectively in the lab frame. We define
$p_{bn}=\hat{n}\cdot\bm{p}_{b}$ and $p_{n}=\hat{n}\cdot\bm{p}$ as the inward
normal components of $\bm{p}_{b}$ and $\bm{p}$. Using a Lorentz
transformation, this component of the photon’s momentum in the rest frame of
the PBS is then [98]

$p_{n}^{\text{(PBS)}}=\gamma(p_{n}-\hbar\omega p_{bn}/mc^{2}),$ (73)

where $\gamma$ is the Lorentz factor and $\omega$ is the frequency in the lab
frame. Because the PBS momentum is nonrelativistic we set $\gamma=1$. We
assume that in the rest frame of the PBS, the photon’s angle of reflection is
the same as the angle of incidence, although this is only approximately true
due to the transfer of energy from the photon to the PBS. That is, we set

$p^{\prime\text{(PBS)}}_{n}=-p^{\text{(PBS)}}_{n}.$ (74)

Thus in the lab frame, the photon’s momentum after the collision differs from
its initial momentum by an amount

$\delta p_{n}=p_{n}^{\prime}-p_{n}=-2p_{n}+2\hbar\omega p_{bn}/mc^{2}.$ (75)

To ensure that a reflected photon can be distinguished from a transmitted
photon, the difference in momentum of the two paths should be greater than the
uncertainty in momentum of the reflected photon, that is

$\left|\braket{\delta p_{n}}\right|\geq\sigma[\delta p_{n}].$ (76)

This condition is in effect a restriction on the statistical distance (as
defined by Wootters [99]) between the momenta of the transmitted and reflected
photons. And so from Eq. (75), and the fact that $\sigma[p_{n}]=0$ we have

$\left|\delta p_{n}\right|\geq 2\hbar\omega\sigma[p_{bn}]/mc^{2},$ (77)

Because the PBS is at finite temperature $T$, we assume its momentum is
initially given by the canonical distribution

$p_{bn}\sim\mathcal{N}(0,mk_{B}T),$ (78)

and therefore

$\left|\delta
p_{n}\right|\geq\frac{2\hbar\omega}{c^{2}}\sqrt{\frac{k_{B}T}{m}}.$ (79)

Next note that for momentum to be conserved, the momentum of the PBS must also
change by an amount $\delta p_{bn}=-\delta p_{n}$. In our convention $\delta
p_{bn}$ is positive by definition, so

$\delta p_{bn}\geq\frac{2\hbar\omega}{c^{2}}\sqrt{\frac{k_{B}T}{m}}.$ (80)

Many photons are incident on the device, one after another. For each photon,
the initial momentum $p_{n}$ is the same, and so $\delta p_{bn}$ is very
nearly the same for each reflected photon, so we treat it as constant. We
assume that each photon is vertically polarized with probability one half and
horizontally polarized with probability one half. So with probability one half
$p_{bn}$ is increased by $\delta p_{bn}$ as each photon passes. Therefore
after $N$ photons have come, the total change in the momentum of the PBS is
given by a binomial distribution

$\frac{p_{bn}(t)-p_{bn}(0)}{\delta p_{bn}}\sim B(N,1/2)$ (81)

By the de Moivre-Laplace theorem [100], for large $N$ the distribution
approaches a Gaussian with variance $\sigma^{2}=N/4$. Therefore,

$\sigma^{2}[p_{bn}(t)-p_{bn}(0)]=\delta p_{bn}^{2}N/4.$ (82)

Recall that the sum of two Gaussian random variables is Gaussian, with
variance equal to the sum of the two original distributions’ variances. Using
the variance of $p_{bn}(0)$ then gives

$\sigma^{2}[p_{bn}(t)]=\delta p_{bn}^{2}N/4+mk_{B}T.$ (83)

Then, using the formula for the Shannon entropy of a Gaussian, we have

$s(t)-s(0)=\frac{1}{2}\ln{\left(\frac{\delta
p_{bn}^{2}N/4+mk_{B}T}{mk_{B}T}\right)}$ (84)
$s(t)-s(0)\geq\frac{1}{2}\ln{\left(\frac{\hbar^{2}\omega^{2}}{m^{2}c^{4}}N+1\right)}.$
(85)

Our approximation is only valid for $\hbar\omega\ll mc^{2}$, and in this limit
we have

$s(t)-s(0)\geq\frac{1}{2}\frac{\hbar^{2}\omega^{2}}{m^{2}c^{4}}N.$ (86)

After many photons have passed, $p_{n}$ is in an equilibrium distribution, so
we can equate the Shannon entropy with the thermal entropy. Then the heat
dissipated as a single photon passes is

$Q\geq\frac{\hbar^{2}\omega^{2}}{2m^{2}c^{4}}k_{B}T.$ (87)

The information carried by the photon is one bit, as per our assumption, so we
may write this as

${\mathchar 22\relax\mkern-12.0mu{\rm
d}}Q\geq\frac{\hbar^{2}\omega^{2}}{2m^{2}c^{4}}\,k_{B}T(-ds).$ (88)

##  Appendix D Minimal loss of purity for quantum PBS

The joint state of the system and the bath is acted on by a unitary operator
$U$. Define the quantum state

$\rho_{B}^{i}=\mathrm{tr_{S}}\left\\{U[\ket{i0}\bra{i0}\otimes\rho_{B}]U^{\dagger}\right\\},$
(89)

and

$\rho_{S}^{i}=\mathrm{tr_{B}}\left\\{U[\ket{i0}\bra{i0}\otimes\rho_{B}]U^{\dagger}\right\\},$
(90)

where $i\in\\{0,1\\}$. Therefore, we write

$\rho_{B}^{\prime}=\frac{1}{2}\left(\rho_{B}^{0}+\rho_{B}^{1}\right).$ (91)

Now suppose that the bath is composed of $N$ qubits. For conservation of
angular momentum to hold, we require that

$\begin{split}&\mathrm{tr}\left\\{\rho_{B}^{i}\sum_{j=1}^{N}Z_{B}^{j}\right\\}+\mathrm{tr}\left\\{\rho_{S}^{i}(Z_{S}^{1}+Z_{S}^{2})\right\\}=\\\
&\mathrm{tr}\left\\{\rho_{B}\sum_{j=1}^{N}Z_{B}^{j}\right\\}+\mathrm{tr}\left\\{\ket{i0}\bra{i0}(Z_{S}^{1}+Z_{S}^{2})\right\\},\end{split}$
(92)

where $Z$ is the Pauli Z-matrix. The noisy CNOT is implemented on the system
qubits within accuracy $\epsilon$, meaning

$\sqrt{\mathrm{tr}\left\\{(\rho_{S}^{\prime}-U_{\text{CN}}\rho_{S}U^{\dagger}_{\text{CN}})^{2}\right\\}}\leq\epsilon\,.$
(93)

Employing the Cauchy-Schwarz inequality we obtain

$\begin{split}&|\mathrm{tr}\left\\{(\rho_{S}^{\prime}-U_{\text{CN}}\rho_{S}U^{\dagger}_{\text{CN}})(Z_{S}^{1}+Z_{S}^{2})\right\\}|\leq\\\
&\quad\epsilon\sqrt{\mathrm{tr}\left\\{(Z_{1}+Z_{2})^{2}\right\\}}=2\sqrt{2}\epsilon\end{split}$
(94)

and hence

$|\mathrm{tr}\left\\{\rho_{S}^{i}(Z_{S}^{1}+Z_{S}^{2})\right\\}-2(-1)^{i}|\leq
2\sqrt{2}\epsilon,$ (95)

and

$\mathrm{tr}\left\\{\ket{i0}\bra{i0}(Z_{S}^{1}+Z_{S}^{2})\right\\}=(-1)^{i}-1.$
(96)

Finally, we can then write

$\left|\mathrm{tr}\left\\{\left(\rho_{B}^{i}-\rho_{B}\right)\sum_{j}Z_{B}^{j}\right\\}-[-1-(-1)^{i}]\right|\leq
2\sqrt{2}\epsilon\,,$ (97)

which further leads to

$\mathrm{tr}\left\\{\left(\rho_{B}^{1}-\rho_{B}^{0}\right)\sum_{j}Z_{B}^{j}\right\\}\geq
2-4\sqrt{2}\epsilon.$ (98)

Thus, the purity of the final state $\rho_{B}^{\prime}$ is

$\mathrm{tr}\left\\{\rho_{B}^{\prime
2}\right\\}=\frac{1}{4}\left(\mathrm{tr}\left\\{(\rho_{B}^{0})^{2}\right\\}+\mathrm{tr}\left\\{(\rho_{B}^{1})^{2}\right\\}+2\mathrm{tr}\left\\{\rho_{B}^{0}\rho_{B}^{1}\right\\}\right).$
(99)

Note that the last term can also be expressed as

$\begin{split}&2\,\mathrm{tr}\left\\{\rho_{B}^{0}\rho_{B}^{1}\right\\}=\mathrm{tr}\left\\{(\rho_{B}^{0})^{2}\right\\}+\mathrm{tr}\left\\{(\rho_{B}^{1})^{2}\right\\}\\\
&\qquad-\mathrm{tr}\left\\{(\rho_{B}^{1}-\rho_{B}^{0})^{2}\right\\}.\end{split}$
(100)

Now, since the purity of the states $\rho_{B}^{0}$ and $\rho_{B}^{1}$ cannot
exceed the purity of $\rho_{B}$, we have

$2\,\mathrm{tr}\left\\{\rho_{B}^{0}\rho_{B}^{1}\right\\}\leq
2\mathrm{tr}\left\\{\rho_{B}^{2}\right\\}-\mathrm{tr}\left\\{(\rho_{B}^{1}-\rho_{B}^{0})^{2}\right\\}.$
(101)

Using the Cauchy-Schwarz inequality again we obtain

$\begin{split}&\mathrm{tr}\left\\{\left(\rho_{B}^{1}-\rho_{B}^{0}\right)\sum_{j=1}^{N}Z_{B}^{j}\right\\}^{2}\leq\\\
&\qquad\mathrm{tr}\left\\{\left(\rho_{B}^{1}-\rho_{B}^{0}\right)^{2}\right\\}\mathrm{tr}\left\\{\left(\sum_{j=1}^{N}Z_{B}^{j}\right)^{2}\right\\}\end{split}$
(102)

and with $\text{tr}\\{Z_{j}Z_{i}\\}=0$ for $i\neq j$ we can have

$\mathrm{tr}\left\\{\left(\sum_{j=1}^{N}Z_{j}\right)^{2}\right\\}=\mathrm{tr}\left\\{\sum_{j=1}^{N}Z_{j}^{2}\right\\}=N2^{N}\,.$
(103)

Collecting expressions we can further write

$\begin{split}&\mathrm{tr}\left\\{\left(\rho_{B}^{1}-\rho_{B}^{0}\right)^{2}\right\\}\geq\frac{1}{N2^{N}}\mathrm{tr}\left\\{\left(\rho_{B}^{1}-\rho_{B}^{0}\right)\sum_{j}Z_{B}^{j}\right\\}^{2}\\\
&\qquad\geq\frac{(2-4\sqrt{2}\epsilon)^{2}}{N2^{N}}\end{split}$ (104)

and finally

$2\,\mathrm{tr}\left\\{\rho_{B}^{0}\rho_{B}^{1}\right\\}\leq
2\,\mathrm{tr}\left\\{\rho_{B}^{2}\right\\}-\frac{(2-4\sqrt{2}\epsilon)^{2}}{N2^{N}},$
(105)

which we rewrite (as in the main text) as

$\mathrm{tr}\left\\{\rho_{B}^{\prime
2}\right\\}\leq\mathrm{tr}\left\\{\rho_{B}^{2}\right\\}-\frac{1}{N2^{N}}\left(1-2\sqrt{2}\epsilon\right)^{2}.$
(106)

##  Appendix E Derivation of modified commutation relation

Consider the case where a PBS is at an unknown angle $\theta$, with some
probability density $p(\theta)$. If the state of the two input modes is given
some density matrix $\sigma_{0}=\rho_{0}\otimes\eta_{0}$, then the output
density matrix is

$\sigma_{1}=\int_{-\infty}^{\infty}d\theta\,p(\theta)U_{\theta}\sigma_{0}U_{\theta}^{\dagger}=\mathcal{E}(\sigma_{0}),$
(107)

where we define $\mathcal{E}$ as the quantum channel which propagates the
density matrix in the Schrödinger picture. There is an adjoint map
$\mathcal{E}^{\dagger}$ which can be used to propagate operators in the
Heisenberg picture, and satisfies [101]

$\mathrm{tr}\left\\{\rho\,\mathcal{E}^{\dagger}(X)\right\\}=\mathrm{tr}\left\\{\mathcal{E}(\rho)\,X\right\\},$
(108)

for all density matrices $\rho$ and operators $X$. In fact it is easily seen
that $\mathcal{E}^{\dagger}$ is given by

$\mathcal{E}^{\dagger}(X)=\int_{-\infty}^{\infty}d\theta\,f(\theta)U_{\theta}^{\dagger}XU_{\theta}.$
(109)

Then using Eq. (27) we see that the adjoint map acting on the annihilation
operators $x_{i}$ can be expressed as

$\mathcal{E}^{\dagger}(x_{i})=(\bar{S}\bm{x})_{i},$ (110)

where we have defined the ensemble-averaged scattering matrix as

$\bar{S}=\int_{-\infty}^{\infty}d\theta\,f(\theta)S_{\theta}.$ (111)

Note that the evolved operators $\mathcal{E}(x_{i})$ do not necessarily obey
the standard Dirac commutation relation anymore since $\bar{J}$ need not be
unitary. While we can evaluate expectation values using Eq. (108), we cannot
build new operators or states using the evolved operators. However, many
useful operators are expressed in terms of products of annihilation operators.
We will consider the class of operators which can be written as

$X=\sum_{ij}X_{ij}x^{\dagger}_{i}x_{j}.$ (112)

To see how such an operator evolves under these dynamics, we first note

$U_{\theta}^{\dagger}x_{i}^{\dagger}x_{j}U_{\theta}=\sum_{k}S_{\theta
ik}^{*}x_{k}^{\dagger}\sum_{\ell}S_{\theta j\ell}x_{\ell}\
=\sum_{k\ell}\mathcal{S}^{ij}_{k\ell}(\theta)x^{\dagger}_{k}x_{\ell}\,,$ (113)

where we defined

$\mathcal{S}^{ij}_{k\ell}(\theta)=S_{\theta ik}^{*}S_{\theta j\ell}.$ (114)

Therefore by Eq. (109) we have

$\mathcal{E}^{\dagger}(X)=\sum_{ijk\ell}X_{ij}\overline{\mathcal{S}}^{ij}_{k\ell}x^{\dagger}_{k}x_{\ell},$
(115)

where

$\overline{\mathcal{S}}^{ij}_{k\ell}=\int_{-\infty}^{\infty}d\theta\,p(\theta)\mathcal{S}^{ij}_{k\ell}(\theta).$
(116)

We will now assume that $f(\theta)$ is Gaussian, with some temperature $T$

$f(\theta)=\frac{1}{\sqrt{2\pi
k_{B}T/\kappa}}\exp\left(-\frac{1}{2}\frac{\theta^{2}}{k_{B}T/\kappa}\right).$
(117)

Then evaluating Eq. (111), we find that

$\bar{S}=\begin{pmatrix}1-(1-t)\chi&0&r\chi&0\\\ 0&t+(1-t)\chi&0&r-r\chi\\\
-r\chi&0&1-(1-t)\chi&0\\\ 0&r\chi-r&0&t+(1-t)\chi\end{pmatrix},$ (118)

where $\chi=\frac{1}{2}\left(1-\exp(-2k_{B}T/\kappa)\right)$. We can then
express the primed annihilation operators as

$a_{\text{h}}^{\prime}=\left(1-(1-t)\chi\right)a_{\text{h}}-r\chi\,b_{\text{h}}$
(119)

and

$a_{\text{v}}^{\prime}=\left(t+(1-t)\chi\right)a_{\text{v}}+r\left(\chi-1\right)b_{\text{v}}.$
(120)

Note that $\chi$ is a monotonically increasing function of temperature and
$\chi=0$ at $T=0$, and hence Eqs. (119) and (120) clearly show the influence
on increasing temperature on the propagation of the annihilation operators.
For zero temperature we have

$\lim_{T\to
0}a_{\text{h}}^{\prime}=a_{\text{h}}\quad\mathrm{and}\quad\lim_{T\to
0}a_{\text{v}}^{\prime}=ta_{\text{v}}-rb_{\text{v}}$ (121)

meaning the model behaves as a PBS at definite angle, as expected. In the
limit of infinite temperature $\chi\to 1/2$, and we find that

$\lim_{T\to\infty}a_{\text{h}}^{\prime}=\frac{1+t}{2}a_{\text{h}}-\frac{r}{2}b_{\text{h}},$
(122)

and

$\lim_{T\to\infty}a_{\text{v}}^{\prime}=\frac{1+t}{2}a_{\text{v}}-\frac{r}{2}b_{\text{v}},$
(123)

meaning that horizontally polarized and vertically polarized light are not
distinguished by the device. The primed annihilation operators do not obey the
standard commutation relations. Instead we find that

$[a_{\text{h}}^{\prime},a_{\text{h}}^{\prime{\dagger}}]=[a_{\text{v}}^{\prime},a_{\text{v}}^{\prime{\dagger}}]=1-2(1-t)(\chi-\chi^{2}).$
(124)

##  Appendix F Construction of the Evolution Operator

In this last appendix, we show that the two treatments we gave in the
Heisenberg and Schrödinger picture are in fact equivalent. This formulation is
not a new result but we have included it for convenience (for similar
calculations see e.g. [102, 103]). Recall that the evolution operator was
obtained as follows, for $t=1$

$U(t)=\exp\left(t\sum_{ij}\braket{i}{\ln(S)}{j}a_{i}^{\dagger}a_{j}\right),$
(125)

where $S$ is the Unitary scattering matrix. Then

$\partial_{t}a_{k}^{\prime}=(\partial_{t}U^{\dagger})a_{k}U+U^{\dagger}a_{k}(\partial_{t}U),$
(126)

and

$\partial_{t}a_{k}^{\prime}=U^{\dagger}\sum_{ij}\braket{j}{\ln(S)^{\dagger}}{i}a_{j}^{\dagger}a_{i}a_{k}U+U^{\dagger}\sum_{ij}\braket{i}{\ln(S)}{j}a_{k}a_{i}^{\dagger}a_{j}U.$
(127)

Since $S$ is unitary, $\ln(S)$ is anti-Hermitian. Using this fact and
interchanging $i$ and $j$ in the first summation, we get

$\partial_{t}a_{k}^{\prime}=U^{\dagger}\left(-\sum_{ij}\braket{i}{\ln(S)}{j}a_{i}^{\dagger}a_{j}a_{k}+\sum_{ij}\braket{i}{\ln(S)}{j}a_{k}a_{i}^{\dagger}a_{j}\right)U,$
(128)

and

$\partial_{t}a_{k}^{\prime}=U^{\dagger}\sum_{ij}\braket{i}{\ln(S)}{j}[a_{k},a_{i}^{\dagger}a_{j}]U=U^{\dagger}\sum_{j}\braket{k}{\ln(S)}{j}a_{j}U\,.$
(129)

Thus, we have

$\partial_{t}a_{k}^{\prime}=\sum_{j}\braket{k}{\ln(S)}{j}a_{j}^{\prime}.$
(130)

The unique solution to this equation is

$a_{k}^{\prime}=\sum_{kj}\braket{k}{\exp(t\ln S)}{j}a_{j}^{\prime}\,.$ (131)

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