# Activation of thermal states by quantum SWITCH-driven thermalization and its
limits

Tamal Guha  QICI Quantum Information and Computation Initiative, Department of
Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong
Saptarshi Roy  QICI Quantum Information and Computation Initiative, Department
of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong
Kyrylo Simonov  s7 rail technology GmbH, Lastenstraße 36, 4020 Linz, Austria
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090
Vienna, Austria  Zoltán Zimborás  Wigner Research Centre for Physics, H-1525,
P.O.Box 49, Budapest, Hungary  Faculty of Informatics, Eötvös Loránd
University, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary

(30th August 2024)

###### Abstract

Coherently controlled thermalization via the quantum SWITCH allows to activate
the state of a work medium and, hence, extract a non-zero work. However, such
thermalization puts certain constraints on extractable work with respect to
the temperatures of the work medium and reservoirs. We analyze these
constraints and the possible ways to go beyond them. We characterize the
impact that entanglement between target and control has on ergotropy and
highlight the role of the coherence present in the state of the work medium.
Last but not least, some of the results are generalized to the case of
$N$-switch.

##  I Introduction

The second law of thermodynamics establishes the celebrated time’s arrow by
directing the spontaneous evolution of a system undergoing any thermodynamic
process. It is irreversible: for example, when interacting with a thermal bath
of infinite heat capacity, the system eventually adopts the thermal
distribution of the same bath reaching equilibrium with it and thereby maximum
thermodynamic entropy [1]. This phenomenon, generally referred as
thermalization, is not limited by the macroscopic scale but also can be
revealed in microscopic systems, whose dynamics is governed by the laws of
quantum mechanics [2, 3]. Adapting the description of an open quantum system’s
dynamics, the thermalization process with respect to a bath at temperature $T$
can be associated with a quantum operation (thermal map) represented as a
completely positive trace-preserving (CPTP) map that, regardless of the input
state of the system, produces a canonical thermal state of temperature $T$.
Evidently, sequential interaction of the thermodynamic system with multiple
thermal baths at same temperature has no distinct feature other than producing
the same thermal state and so does sequential action of the corresponding
thermal maps on the input state of the system.

However, quantum theory allows in principle multiple quantum operations to
occur in a way not compatible with any underlying causal structure. A
particular example is provided by the quantum SWITCH, a higher-order quantum
process that executes the operations in a superposition of their possible
relative orders making thereby the corresponding causal order indefinite.
Compared with operations occurring in a well-defined order, the advent of
quantum SWITCH has offered advantages in a plethora of information processing
tasks like channel discrimination [4], computation [5], classical and quantum
communication [6, 7, 8, 9], and others. In a similar spirit, the relevance of
the quantum SWITCH in the domain of quantum thermodynamics has recently been
observed in the framework of quantum cooling [10, 11, 12, 13], boosted quantum
battery charging [14], and thermodynamic work extraction games [15, 16]. As
regards the latter, quantum SWITCH of two thermal maps has been considered for
extraction of the free energy amount of work from thermal states [15] as well
as with respect to work extraction under a unitary cyclic evolution of a
quantum system [16]. Indeed, amount of work that can be extracted from a
quantum system under a unitary cycle is given by ergotropy [17] that
quantifies the energy difference between the concerned state and the
corresponding passive state, i.e., the equal entropic state with minimum
energy [18, 19, 20, 21]. Evidently, the energy of a single passive state
cannot be lowered further, and no work can be extracted from it under unitary
cyclic evolution. However, possessing multiple copies of a passive state can
in principle lead to a non-zero amount of extractable work when a global
unitary is performed [22]. Interestingly, thermal states are an only class of
passive states (dubbed as completely passive states) that do not allow to
extract a non-zero ergotropic work even from infinitely many copies of them
[18].

Since a sequential action of two identical thermal maps produces a thermal and
thereby completely passive state, no extractable work is stored at the output
end. Interestingly, relaxing a definite causal order of the maps’ occurrence
by superposing their sequential application orders via the quantum SWITCH can
activate a non-zero ergotropy at the output end even if the input state is
thermal. However, for a thermal input state, the non-zero ergotropy is
activated only if its temperature is below a certain threshold bound $T_{C}$
[16]. In the present paper, we consider a generic framework for ergotropic
work extraction under the sequential occurrence of thermal maps and question
whether the temperature bound $T_{C}$ can be overcome. Firstly, we generalize
the setup to occurrence of asymptotically many identical thermal maps in an
indefinite causal order via the $N$-SWITCH: in this case, the amount of
ergotropy stored in the output state can be doubled compared with the scenario
of two thermal maps controlled by the quantum SWITCH (i.e., $2$-SWITCH).
However, increasing the number of switched thermal maps still does not violate
the temperature bound $T_{C}$, with a restricted classical communication
channel between the control and the sender.

Thereafter, we move towards a more general structure of the quantum SWITCH and
allow the locally thermal input to be correlated initially with the control
qubit. Notably, the local and global ergotropic difference for locally thermal
correlated quantum states have important consequences to identify the nature
of shared correlation amongst them [23, 24, 25]. However, thermal maps break
entanglement, and the signature of the correlation between the parts of the
joint system vanishes whenever one part evolves under such operations.
Interestingly, the signature of initial correlation can outlast further if
these thermal interactions are controlled via the quantum SWITCH and
accordingly the potential ergotropy of the final output can supersede that of
the uncorrelated quantum SWITCH. Moreover, we have further strengthen the
power of entangled causal order by violating the threshold temperature bound
$T_{C}$ in an entangled $2$-SWITCH settings only.

The remainder of the paper is organized as follows. In Section II, we
introduce the necessary framework and recall the concepts of ergotropy and
thermal maps, and the quantum SWITCH. In Section III, we discuss action of the
quantum $N$-SWITCH on identical thermal maps and recover the threshold
temperature bound $T_{C}$ for the activation of ergotropy. In Section IV, we
generalize the quantum SWITCH setup by allowing correlations between the input
thermal state and the control qubit and exhibit the possible configuration to
overcome the temperature bound. Last but not least, in Section V, we draw the
conclusions and possible future directions as well as applications of the
results.

##  II Framework

###  II.1 Ergotropy of a quantum state

It is well known that internal energy of an isolated (quantum) system $S$
evolving under certain Hamiltonian $H$ does not change. However, it is
possible to lower it by coupling the system to a macroscopic source
represented by properly chosen external time-dependent potential $V(t)$ [17].
Such potential varies cyclically the Hamiltonian of the system during a
certain time interval $t\in[0,\tau]$. The evolution of the system’s state
$\rho(t)$ is thus governed by the effective Hamiltonian $H^{\prime}(t)=H+V(t)$
with $V(0)=V(\tau)=0$.

In particular, the lowest energy can be reached by an appropriate choice of
the external potential $V(t)$ that leads to a passive state of the system at
$t=\tau$. Indeed, let us assume that the initial state of the system is given
by $\rho(0)=\sum_{k}p_{k}\ket{\psi_{k}}\bra{\psi_{k}}$, while its initial
(“isolated”) Hamiltonian can be expanded as
$H=\sum_{i}\epsilon_{i}\ket{\epsilon_{i}}\bra{\epsilon_{i}}$ with increasing
energies $\epsilon_{i}\leq\epsilon_{i+1}$. Then the effective Hamiltonian
$H^{\prime}(t)$ with the optimal choice of the external potential generates a
unitary evolution under the operator $U_{p}$ that produces the passive state

$\rho(\tau)=U_{p}\rho(0)U_{p}^{\dagger}=\sum_{i}p_{i}\ket{\epsilon_{i}}\bra{\epsilon_{i}},$
(1)

with decreasing populations $p_{i}\geq p_{i+1}$. The amount of extractable
work under such setting is maximum work that can be extracted from
$\rho(0)\equiv\rho$ via unitary cycles and termed as ergotropy,

$\displaystyle W(\rho)$ $\displaystyle=$
$\displaystyle\max_{U\in\mathcal{U}}\text{Tr}[(\rho-U\rho U^{\dagger})H]$ (2)
$\displaystyle=$ $\displaystyle\text{Tr}[(\rho-U_{p}\rho U_{p}^{\dagger})H]$
$\displaystyle=$
$\displaystyle\sum_{i,j}p_{i}\epsilon_{j}(|\bra{\psi_{i}}\epsilon_{j}\rangle|^{2}-\delta_{ij}),$

where $\mathcal{U}$ is the set of all unitary transformations generated from
the effective time-dependent Hamiltonian $H+V(t)$.

Note that the passive states are diagonal in the Hamiltonian eigenbasis, along
with an inverse population distribution with respect to the energy levels
[18]. Therefore, the ergotropy $W(\rho)$ can originate from two different
consecutive contributions [26] (for the sake of simplicity, we consider a two-
dimensional system that allows one to provide them with simple analytic
expressions):

  1. 1)

incoherent counterpart $W_{i}(\rho)$ which can be extracted from $\rho$ by
reorientation of the diagonal elements $p_{i}$ (i.e., probabilities) inverse
to the energy levels,

$W_{i}(\rho)=\max\\{0,\delta\rho\\},$ (3)

where $\delta\rho=\rho_{22}-\rho_{11}$ is the population imbalance,

  2. 2)

coherent ergotropy $W_{c}(\rho)$ which can be extracted by the transformation
of basis $\\{\ket{\psi_{i}}\\}\to\\{\ket{\epsilon_{i}}\\}$, after the
probabilities have been permutated and $W_{i}(\rho)$ has been extracted,

$W_{c}(\rho)=\frac{1}{2}\left(\eta-\sqrt{\eta^{2}-4|\rho_{12}|^{2}}\right),$
(4)

where $\eta=\sqrt{2P(\rho)-1}$ depends on the purity
$P(\rho)=\operatorname{Tr}(\rho^{2})$ of the system’s state, and $\rho_{12}$
is its off-diagonal element.

Obviously, this also implies that a single copy of $\rho$ has zero ergotropy
if it is passive in nature.

For the multipartite quantum systems, there are different aspects of ergotropy
depending upon the restrictions imposed on the unitary operation $U$. When the
constituents of a multipartite quantum states are allowed to apply only a
local unitary, i.e., $\mathcal{U}:=\bigotimes\mathcal{U}_{k}$ in Eq. (2), then
the amount of extractable work is referred as local ergotropy ($W_{l}$). On
the other hand, considering the set of all unitary acting on the state jointly
will give rise to the global ergotropy ($W_{g}$). Obviously, $W_{g}\geq W_{l}$
and also there are quantum states for which $W_{l}=0$ while $W_{g}\neq 0$ [23,
25]. Such states are generally termed as locally passive, or more strongly
locally thermal quantum states. However, there can be an intermediate concept
of locally extractable ergotropy, namely the daemonic ergotropy ($W_{d}$)
[24], which involves a measurement on a subsystem $A$ of the multipartite
quantum state of $SA$ and, depending upon the outcome, the other part is used
to extract work unitarily. Indeed, it is defined as an average ergotropy of
post-selected states of the system with respect to the optimal projective
measurement $\\{\Pi_{a}\\}$ on $A$,

$W_{d}(\rho_{SA})=\max_{\\{\Pi_{a}\\}}\Bigl{(}\sum_{a}p_{a}W(\rho_{S|a})\Bigr{)},$
(5)

where $W(\rho_{S|a})=p_{a}^{-1}\operatorname{Tr}_{A}(\Pi_{a}\rho_{SA})$ is the
post-selected state of $S$ associated to the measurement outcome $a$ obtained
with the probability $p_{a}=\operatorname{Tr}(\Pi_{a}\rho_{SA})$. On a
slightly different note, while the difference of $W_{g}$ and $W_{l}$ is a
potential candidate to witness the presence of entanglement in the quantum
state [25], the presence of quantum discord as well as genuinely classical-
classical correlations between $S$ and $A$ implies a gap between $W_{d}$ and
$W_{l}$ [24].

###  II.2 Thermalizing maps

Any quantum operation on a quantum system $S$ can be obtained as local
dynamics of a unitary interaction between the system and environment. In
particular, a general thermal operation $\mathcal{T}$ can be modelled by an
energy-preserving unitary interaction of $S$ with a thermal bath $E$ at
temperature $T$ [27]. Indeed, it is given by

$\mathcal{T}(\rho)=\text{Tr}_{E}[U_{SE}(\rho\otimes\tau_{\beta})U_{SE}^{\dagger}],$
(6)

where $\tau_{\beta}$ is the state of thermal bath $E$, i.e., the thermal state
following Gibbs probability distribution,

$\tau_{\beta}=\frac{1}{Z_{\beta}}e^{-\beta
H}=\frac{1}{Z_{\beta}}\sum_{j}e^{-\beta\epsilon_{j}}\ket{j}\bra{j},$ (7)

where $Z_{\beta}=\text{Tr}[e^{-\beta H}]=\sum_{k}e^{-\beta\epsilon_{k}}$ is
the canonical partition function, $H$ is the governing Hamiltonian,
$\beta=(k_{B}T)^{-1}$, and $k_{B}$ is the Boltzmann constant. It is clear from
that thermal states follow inverse population distribution with the energy
levels and hence can be identified as passive states, which possess no
ergotropy.

In turn, we identify a particular class $\Lambda_{\beta}$ of thermal
operations dubbed thermalizing maps which produce a thermal state
$\tau_{\beta}$,

$\Lambda_{\beta}[\rho]=\tau_{\beta},$ (8)

for any input state $\rho$ of $S$, i.e., are pin maps. This operation can be
realized if the energy-conserving unitary $U_{SE}$ of the thermal operation
$\mathcal{T}$ is a SWAP operator

$U_{SE}=\sum_{i,j}\ket{ij}\bra{ji},$ (9)

which interchanges the states of the system and the bath, i.e.,
$U_{SE}(\rho\otimes\tau_{\beta})U_{SE}^{\dagger}=\tau_{\beta}\otimes\rho$. For
a chosen thermal state $\tau_{\beta}$, the corresponding qudit thermalizing
map can be realized as a convex combination of $d$ pin maps $\mathcal{E}_{k}$,
each having a fixed output, namely, one of the elements of the energy
eigenbasis $\\{\ket{\epsilon_{k}}\\}_{k=0}^{d-1}$,

$\Lambda_{\beta}[\rho]=\sum_{k}\frac{e^{-\beta\epsilon_{k}}}{Z}\mathcal{E}_{k}[\rho].$
(10)

Alike every quantum CPTP operations, thermalizing maps can also be expressed
in terms of Kraus operators

$\Lambda_{\beta}(\rho)=\sum_{k=0}^{d^{2}-1}E_{k}\rho E_{k}^{\dagger},$ (11)

where $E_{k}=\sqrt{e^{-\beta\epsilon_{[k/d]}}}\ket{[k/d]}\bra{k\mod d}$, and,
evidently, $\sum_{k}E_{k}^{\dagger}E_{k}=\mathbb{I}$. In particular, assuming
that the system is a qubit (hence, $d=2$) whose dynamics is governed by a
Hamiltonian with $\epsilon_{0}=0$ and $\epsilon_{1}=\epsilon$, the Kraus
operators can be given in the following form,

$\displaystyle E_{0}$ $\displaystyle=$ $\displaystyle\sqrt{p}\ket{0}\bra{0},$
$\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle\sqrt{p}\ket{0}\bra{1},$
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\sqrt{1-p}\ket{1}\bra{0},$ $\displaystyle E_{3}$
$\displaystyle=$ $\displaystyle\sqrt{1-p}\ket{1}\bra{1},$ (12)

where $p=\frac{1}{1+\exp(-\beta\epsilon)}$.

###  II.3 Controlled quantum operations

Quantum SWITCH is a particular example of a higher-order operation mapping two
or more quantum operations to another quantum operation that is not compatible
with any well-defined order of their occurrence [5]. It can be interpreted as
coherent control of the order of the operations’ occurrence by an external
quantum system (control system $C$). For example, considering $N$ such
operations $\\{\Lambda_{1},\cdots,\Lambda_{N}\\}$, quantum SWITCH assigns them
a CPTP map $\mathcal{S}(\Lambda_{1},{\cdots},\Lambda_{N})$ given by

$\mathcal{S}(\Lambda_{1},{\cdots},\Lambda_{N})[\rho_{\text{T}}\otimes\omega_{\text{C}}]={\sum_{{i_{1}},{\cdots},{i_{N}}}}{S_{i_{1}{\cdots}i_{N}}}(\rho_{\text{T}}\otimes\omega_{\text{C}})S_{i_{1}{\cdots}i_{N}}^{\dagger},$
(13)

where $\rho_{\text{T}}$ is the input state of the thermodynamic system $S$
(target), and $\omega_{\text{C}}$ is the input state of control system. Note
that, if all possible cyclic permutations of these $N$ channels have been
considered, then the quantum capacity of the side channel should be $\log N$
bits. The Kraus operators $S_{i_{1}{\cdots}i_{N}}$ corresponding to Eq. (13)
will take the form,

$S_{i_{1}{\cdots}i_{N}}=\sum_{j=0}^{N-1}\mathcal{P}_{j}\Bigl{(}E_{i_{1}}^{(1)}{\cdots}E_{i_{N}}^{(N)}\Bigr{)}_{\text{T}}\otimes\ket{j}\bra{j}_{\text{C}},$
(14)

where $E_{l}^{(k)}$ denotes the $l^{th}$-Kraus element of the operation
$\Lambda_{k}$, while $\mathcal{P}_{j}$ represents the $j^{th}$ cyclic
permutation of Kraus elements of different operations. In particular, the
usual quantum SWITCH for two quantum operations is represented by the Kraus
operators

$S_{ij}=E_{i}^{(1)}E_{j}^{(2)}\otimes\ket{0}\bra{0}_{\text{C}}+E_{j}^{(2)}E_{i}^{(1)}\otimes\ket{1}\bra{1}_{\text{C}}.$
(15)

It is further important to observe that the action of quantum SWITCH in Eq.
(13) does not depend on the correlations that can be initially shared between
the target and control system. This allows for a further generalization,

$\displaystyle\mathcal{S}(\Lambda_{1},\cdots,\Lambda_{N})[\rho_{\text{TC}}]=\sum_{i_{1},\cdots,i_{N}}S_{i_{1}\cdots
i_{N}}(\rho_{\text{TC}})S_{i_{1}\cdots i_{N}}^{\dagger}.$ (16)

where $\rho_{\text{TC}}$ is the joint state of the target and control system,
and the Kraus operators $S_{i_{1}\cdots i_{N}}$ coincide with ones as in Eq.
(14).

##  III Controlled configuration of thermal maps

Let us first illustrate a thermodynamic framework for distant work transfer.
We consider the scenario where Alice, the sender, wishes to transfer a work
storage, viz., a quantum battery to Bob, at a distant location. Obviously, if
they share perfect quantum channel then Alice is able to transport exact
amount of work to the receivers end. But, if the channel is a thermal map,
then the output, being a fixed thermal state, possesses no ergotropic work
content. Moreover, if the input state at Alice’s end is initially a thermal
state, then even a perfect quantum channel will not be helpful for work
transfer. However, a conjunction of these two seemingly impossible scenario
gives rise to a possibility of work transportation, using the complete
quantumness of the associated dynamics. The scenario specifically assumes that
the distant parties Alice and Bob are connected via a noisy transmission line,
consists of a series of thermal maps and also there is perfect side channel,
whose input and output is accessed by a third party, Charlie. While Charlie
can share any prior correlation with Alice, he is only allowed to communicate
a bit of classical information to Bob. Importantly, the input system at
Alice’s lab contains no local ergotropy, i.e., locally thermal in nature and
both her and Bob’s systems are governed by the two-level Hamiltonian
$H_{\text{A (B)}}=\ket{1}\bra{1}$. On the other hand, we associate a trivial
Hamiltonian for Charlie’s quantum system, i.e., $H_{\text{C}}=\mathbb{I}$.

Before presenting the results with quantum control, we will first consider the
causally definite sequence of thermal maps. Note that, the thermal map
$\Lambda_{\beta}$ acts as a pin-map which produces a fixed output
$\tau_{\beta}$ irrespective of the input quantum state. Hence, if there are
$N$-different thermal maps
$\Lambda_{\beta_{1}},\Lambda_{\beta_{2}},\cdots,\Lambda_{\beta_{N}}$, then for
every possible causally ordered sequence of these maps one of the thermal
states $\tau_{\beta_{k}},\leavevmode\nobreak\ k\in\\{1,2,\cdots,N\\}$ will be
transferred at the receivers end. Now, every causally separable configuration
of these thermal maps can be identified as a subjective ignorance (i.e.,
convex combination) of all these causally ordered sequences. As a consequence,
the final state will be

$\rho_{\text{def}}=\sum_{k=1}^{N}p_{k}\tau_{\beta_{k}}$

where, $p_{k}$ is the probability for all possible causal configuration of
these thermal maps for which the final action is $\Lambda_{\beta_{k}}$. Now,
since the convex combination of any thermal states are always passive in
nature [23, 28], the extractable ergotropy will vanish at the receivers’ end.

Also importantly, the action of a pin-map destroys any possible correlation
hold by the input quantum state, which further implies that after the causally
separable configuration of all these $N$ thermal channels, the output state at
the receivers end becomes completely uncorrelated with the rest of the
environment. Hence, it possesses no more daemonic ergotropy, i.e., a
ergotropic work assisted by the classical communication from an initially
correlated ancillary quantum system.

Figure 1: Schematic representation of the protocol: the target and control
system are initially in a joint state $\rho_{TC}$, and the target is locally
at temperature $\beta_{in}$, i.e.,
$\operatorname{Tr}_{C}[\rho_{TC}]=\tau_{\beta_{in}}$. The target undergoes
action of two thermalizing maps of temperature $\beta$ whose application order
is controlled by the quantum SWITCH. Thereafter, the control system is
measured, and the ergotropic work is extracted from the resulting state of the
target.

###  III.1 Quantum-controlled thermalization

Now let us assume that the order of maps $\Lambda_{\beta}$ at the same inverse
temperature $\beta$ is controlled coherently by an external quantum control
system within the quantum SWITCH (see Fig. 1). For example, when a qubit
target system is thermalized with two baths at inverse temperature $\beta$ in
an indefinite causal order via the quantum SWITCH, its final state is not
necessarily the thermal state $\tau_{\beta}$, so that certain amount of work
can be extracted from it [15, 16]. Indeed, when the target stays initially in
a thermal state of inverse temperature $\beta_{in}$, whereas the control qubit
is in the state $\ket{+}=\frac{\ket{0}+\ket{1}}{\sqrt{2}}$, i.e.,
$\rho_{TC}=\tau_{\beta_{in}}\otimes\ket{+}\bra{+}$ is a product state, the
quantum SWITCH outputs the joint state

$\mathcal{S}(\Lambda_{\beta},\Lambda_{\beta})[\tau_{\beta_{in}}\otimes\ket{+}\bra{+}]=\frac{1}{2}\Bigl{(}(\tau_{\beta}+\tau_{\beta}\tau_{\beta_{in}}\tau_{\beta})\otimes\ket{+}\bra{+}+(\tau_{\beta}-\tau_{\beta}\tau_{\beta_{in}}\tau_{\beta})\otimes\ket{-}\bra{-}\Bigl{)}.$
(17)

This means that if the control qubit is measured in the basis
$\\{\ket{\pm}\\}=\\{\frac{\ket{0}\pm\ket{1}}{\sqrt{2}}\\}$ after the system is
thermalized with the baths, the resulting (un-normalized) state of the target
system

$\tilde{\rho}_{\pm}=\frac{1}{2Z_{\beta}}\begin{pmatrix}1\pm\frac{1}{Z_{\beta}Z_{\beta_{in}}}&0\\\
0&e^{-\beta}\Bigl{[}1\mp\frac{e^{-(\beta+\beta^{in})}}{Z_{\beta}Z_{\beta_{in}}}\Bigr{]}\end{pmatrix},$
(18)

generally speaking, differs from the thermal state $\tau_{\beta}$ obtained
when the thermal baths are placed in a classical configuration and can carry
some amount fo ergotropy. Both states do not carry any coherence with respect
to the energetic eigenbasis, hence, their ergotropy is of genuinely incoherent
origin and can be extracted by permutating the energetic populations. Indeed,
overall daemonic ergotropy of (17) is given by [16]

$W_{d}(\sigma_{2})=\frac{1}{2Z_{\beta}^{2}Z_{\beta_{in}}}\operatorname{max}\\{0,e^{-2\beta}-e^{-\beta^{in}}\\},$
(19)

where
$\sigma_{2}=\mathcal{S}(\Lambda_{\beta},\Lambda_{\beta})[\tau_{\beta_{in}}\otimes\ket{+}\bra{+}]$
is the output of the quantum SWITCH. In this way, if a qubit system at inverse
temperatute $\beta_{in}$ gets thermalized with two thermal baths at
temperature $\beta$ in an indefinite causal order via the quantum SWITCH,
there exists a temperature bound

$\beta_{in}>2\beta,$ (20)

at which a non-zero ergotropic work can be extracted from the final state of
the target system. A similar temperature bound
$\beta_{in}>\beta_{1}+\beta_{2}$ can be derived for quantum-controlled thermal
baths at different inverse temperatures $\beta_{1,2}$. Nevertheless, the
optimal value of ergotropy is obtained for the case of baths at equal
temperatures, $\beta_{1}=\beta_{2}=\beta$, therefore, in what follows, we
stick to this setting.

A natural question arises whether the temperature bound can be shifted or
overcome. Firstly, we check whether it can be changed by increasing the number
of thermal baths controlled by the quantum SWITCH. The target qubit system
initially remains in the same thermal state $\tau_{\beta_{in}}$, and we assume
that is thermalized with $N$ baths at inverse temperature $\beta$. Their
occurrence is controlled by the generalized $N$-SWITCH which entangles an
$N$-dimensional system (playing the role of the control system) with the
cyclic orders of the thermalizing maps. Indeed, when the control system stays
initially in the state
$\ket{\gamma_{+}}=\frac{1}{\sqrt{N}}\sum_{i=0}^{N-1}\ket{i}$, we get the
following state out of the $N$-SWITCH,

$\mathcal{S}(\Lambda_{\beta},...,\Lambda_{\beta})[\tau_{\beta_{in}}\otimes\ket{\gamma_{+}}\bra{\gamma_{+}}]=\frac{1}{N}\Bigl{(}(\tau_{\beta}+(N-1)\tau_{\beta}\tau_{\beta_{in}}\tau_{\beta})\otimes\ket{\gamma_{+}}\bra{\gamma_{+}}+(N-1)(\tau_{\beta}-\tau_{\beta}\tau_{\beta_{in}}\tau_{\beta})\otimes(\mathds{1}-\ket{\gamma_{+}}\bra{\gamma_{+}})\Bigr{)}.$
(21)

Hence, when a yes/no measurement of the control with respect to the state
$\ket{\gamma_{+}}$ is performed, we can easily get the corresponding post-
measurement states

$\displaystyle\tilde{\rho}_{+}$ $\displaystyle=$
$\displaystyle\frac{1}{NZ_{\beta}}\begin{pmatrix}1+\frac{N-1}{Z_{\beta}Z_{\beta_{in}}}&0\\\
0&e^{-\beta}\Bigl{[}1+\frac{N-1}{Z_{\beta}Z_{\beta_{in}}}e^{-(\beta+\beta^{in})}\Bigr{]}\end{pmatrix},$
(22) $\displaystyle\tilde{\rho}_{-}$ $\displaystyle=$
$\displaystyle\frac{N-1}{NZ_{\beta}}\begin{pmatrix}1-\frac{1}{Z_{\beta}Z_{\beta_{in}}}&0\\\
0&e^{-\beta}\Bigl{[}1-\frac{e^{-(\beta+\beta^{in})}}{Z_{\beta}Z_{\beta_{in}}}\Bigr{]}\end{pmatrix},$
(23)

whose ergotropy, as the one of (18) for 2-SWITCH, is incoherent. The resulting
daemonic ergotropy is given by

$W_{d}(\sigma_{N})=\frac{N-1}{NZ_{\beta}^{2}Z_{\beta_{in}}}\operatorname{max}\\{0,e^{-2\beta}-e^{-\beta_{in}}\\},$
(24)

where
$\sigma_{N}=\mathcal{S}(\Lambda_{\beta},...,\Lambda_{\beta})[\tau_{\beta_{in}}\otimes\ket{\gamma_{+}}\bra{\gamma_{+}}]$
is the output of the $N$-SWITCH. This means that for the initial temperatures
$\beta_{in}$ satisfying the bound (20), local ergotropy extraction can be
improved by increasing the number of thermal baths $N$,

$W_{d}(\sigma_{N})=2\Bigl{(}1-\frac{1}{N}\Bigr{)}W_{d}(\sigma_{2}),$ (25)

up to doubling the work which can be extracted in the usual 2-SWITCH scenario.
However, the temperature bound (20) remains unchanged and cannot be beaten in
this manner.

##  IV Advantages of correlated causal order

As the results obtained in the previous Section suggest, despite enhanced
daemonic ergotropy, increasing the number of thermalizing maps in coherently-
controlled order does not allow to overcome the bound (20). In this way, we
focus again on the quantum SWITCH controlling two thermalizing maps and allow
the target and control systems to share some initial correlations. Hence, the
assumption of an initial product state
$\rho_{TC}=\tau_{\beta_{in}}\otimes\ket{+}\bra{+}$ is relaxed, and only the
marginal state of the target is demanded to remain thermal, i.e.,

$\operatorname{Tr}_{C}(\rho_{TC})=\tau_{\beta_{in}}.$ (26)

Crucially, sharing a priori correlations (classical as well as quantum)
between the target and control systems, as for separable $\rho_{TC}$, does not
allow one to locally extract ergotropy from the target system when the maps
act consecutively or in mixture of their causal orders (for example, when the
control system is discarded). However, as we will see in the following, it
allows one to gain ergotropy beyond the temperature bound (20) when the
application order of the thermalizing maps is put into a superposition by the
quantum SWITCH.

Figure 2: Daemonic ergotropy from the output of thermalizing maps in the case
of classically controlled target and control. It can be seen that at inverse
maps’ temperature $\beta=0.4$ the temperature bound is vanished while at
$\beta=0.1$ is only shifted.

###  IV.1 Classical correlations between target and control

We start the analysis with the case of target and control systems sharing
prior classical correlations. Importantly, we exclude quantum discord, which
is known to positively contribute to ergotropy [29], and assume classical-
classical correlations only [30, 31],

$\displaystyle\rho_{TC}=\sum_{i}p_{i}\Pi_{i}\otimes\Theta_{i},$ (27)

where $p_{i}$ are probabilities, i.e., $\sum_{i}p_{i}=1$, and $\\{\Pi_{i}\\}$
and $\\{\Theta_{i}\\}$ are projectors onto certain orthonormal bases in the
respective subsystems. The condition (26) of local thermality puts constraints
on the possible expansions (27) of the joint state $\rho_{TC}$, namely,

$\displaystyle\rho_{TC}=\frac{1}{Z_{\beta_{in}}}\Bigl{(}\ket{0}\bra{0}\otimes\ket{\psi}\bra{\psi}+e^{-\beta_{in}}\ket{1}\bra{1}\otimes\ket{\psi^{\perp}}\bra{\psi^{\perp}}\Bigr{)},$

where $\\{\Theta_{i}\\}=\\{\ket{\psi},\ket{\psi^{\perp}}\\}$ constitute an
arbitrary orthonormal basis for the control qubit. After the target is
thermalized with the quantum-controlled baths, a joint output state
$\sigma_{CC}=\mathcal{S}(\Lambda_{\beta},\Lambda_{\beta})[\rho_{TC}]$ is
produced by the quantum SWITCH reading

$\displaystyle\sigma_{CC}$ $\displaystyle=$
$\displaystyle\frac{1}{2Z_{\beta_{in}}}\Bigl{(}(\tau_{\beta}+\tau_{\beta}\ket{0}\bra{0}\tau_{\beta})\otimes\ket{\psi}\bra{\psi}+(\tau_{\beta}-\tau_{\beta}\ket{0}\bra{0}\tau_{\beta})\otimes
Z\ket{\psi}\bra{\psi}Z$ (29) $\displaystyle+$ $\displaystyle
e^{-\beta_{in}}\Bigr{(}(\tau_{\beta}+\tau_{\beta}\ket{1}\bra{1}\tau_{\beta})\otimes\ket{\psi^{\perp}}\bra{\psi^{\perp}}+(\tau_{\beta}-\tau_{\beta}\ket{1}\bra{1}\tau_{\beta})\otimes
Z\ket{\psi^{\perp}}\bra{\psi^{\perp}}Z\Bigr{)}\Bigr{)},$

where $Z$ is the Pauli Z-operator. Evidently, $\sigma_{CC}$ has block-diagonal
form and, therefore, do not carry any coherence between target and control. On
the other hand, the target component of each summand in t (29) carries no
coherence as well. Therefore, daemonic ergotropy that can be extracted from
the quantum SWITCH with classically correlated input is of purely incoherent
nature.

Figure 3: Comparison of regions of non-zero ergotropy in the space of
$(\beta_{in},\beta)$-pairs for the initially uncorrelated and clasically
correlated target and control.

It can be demonstrated that, similar to the case of uncorrelated initial
state, the maximal daemonic ergotropy can be achieved for
$\Theta_{i}=\ket{\pm}$. Indeed, after measurement of the control in the
optimal $\\{\ket{\pm}\\}$ basis, the (un-normalized) post-measurement states
read as

$\displaystyle\tilde{\rho}_{\pm}=\frac{1}{2Z_{\beta}}\begin{pmatrix}1\pm\frac{1}{Z_{\beta}Z_{\beta_{in}}}&0\\\
0&e^{-\beta}\Bigl{(}1\mp\frac{1}{Z_{\beta}Z_{\beta_{in}}}e^{-(\beta+\beta_{in})}\Bigr{)}\end{pmatrix},$
(30)

carrying the incoherent ergotropy

$W_{d}(\sigma_{CC})=\frac{1}{2Z_{\beta}^{2}Z_{\beta_{in}}}\operatorname{max}\\{0,e^{-2\beta}-e^{-\beta^{in}}+2e^{-(2\beta+\beta_{in})}\\},$
(31)

where $\sigma_{CC}=\mathcal{S}(\Lambda_{\beta},\Lambda_{\beta})[\rho_{TC}]$ is
the output of the quantum SWITCH. Hence, initial classical correlations
between target and control alters the temperature bound (20) for the non-zero
ergotropy,

$\displaystyle\ln(e^{\beta_{in}}+2)>2\beta.$ (32)

This means that initial classical correlations improve work extraction from
the quantum SWITCH by shifting the temperature bound towards smaller
$\beta_{in}$ (see Fig. 2). Moreover, it can easily seen that the left-hand
side of (32) is bounded below by $\ln(3)$ corresponding to the infinite
initial temperature (i.e., $\beta_{in}=0$). Hence, the inverse temperature
$\tilde{\beta}=\frac{\ln(3)}{2}$ of the maps can be seen as a critical value
where the temperature bound (32) becomes trivial. Indeed, for
$\beta<\tilde{\beta}$, classical correlations can activate ergotropy
extraction in the entire range of initial temperatures $\beta_{in}$, so that
the temperature bound is overcome completely. On the other hand, for
$\beta\geq\frac{1}{2}\ln 3$, there still exists a finite range
$\beta_{in}\in[0,\ln(e^{2\beta}-2)]$ with no possible ergotropy extraction.
Therefore, we can conclude that prior classical correlations allow one to beat
the temperature bound (20) only partially. Indeed, the temperature bound (32)
from classically correlated target and control improves non-zero work
extraction only in a tiny region of small inverse temperatures $\beta_{in}$ of
initial state of the target and $\beta$ of the maps (see Fig. 3). In the
following section, we assume quantum nature of the initial correlations
between target and control and ask what are the advantages it can additionally
offer. In particular, we examine whether the temperature bound (20) can be
completely reduced in this setting.

###  IV.2 Do quantum correlations perform better?

Now, we allow prior quantum correlations between target and control and
identify maximal work that can be extracted from the output of the quantum
SWITCH. Importantly, it can be shown that maximal ergotropy from target
initially sharing quantum discord with control does not exceed one from target
initially entangled with control. Therefore, for the sake of simplicity, we
focus on entanglement in what follows. As in the case of initial classical
correlations, we take into account that the target is initially in a thermal
state $\tau_{\beta_{in}}$ due to condition (26). Hence, the joint state
$\rho_{TC}$ can be seen as a purification
$|T_{\alpha}(\beta_{in})\rangle\langle T_{\alpha}(\beta_{in})|$ of
$\tau_{\beta_{in}}$, where

$|T_{\alpha}(\beta_{in})\rangle=\frac{1}{\sqrt{Z_{\beta_{in}}}}\Bigl{(}|0\psi\rangle+e^{-\frac{\beta_{in}}{2}}|1\psi^{\perp}\rangle\Bigr{)},$
(33)

is the initial pure joint state where the basis states of the target are
entangled with the states
$|\psi\rangle=\sqrt{\alpha}|0\rangle+e^{i\phi}\sqrt{1-\alpha}|1\rangle$ and
$|\psi^{\perp}\rangle=e^{-i\phi}\sqrt{1-\alpha}|0\rangle-\sqrt{\alpha}|1\rangle$
of the control qubit that constitute an arbitrary orthonormal basis with
$\alpha\in[0,1]$ and $\phi\in[0,2\pi]$. The output of the quantum SWITCH (see
Appendix A for its precise form)

$\sigma_{QC}=\mathcal{S}(\Lambda_{\beta},\Lambda_{\beta})[\ket{T_{\alpha}(\beta_{in})}\bra{T_{\alpha}(\beta_{in})}],$
(34)

in contrast to (29), can carry coherence between target and control and, in
turn, allows for a non-zero coherent contribution to ergotropy. This can be
seen from the post-selected states of the target when the control system is
measured in the $\\{\ket{\pm}\\}$ basis,

$\displaystyle\tilde{\rho}_{\pm}=\frac{1}{2Z_{\beta}}\begin{pmatrix}1\pm\frac{2\cos\phi\sqrt{\alpha(1-\alpha)}}{Z^{2}_{\beta}Z_{\beta_{in}}}&\pm\frac{e^{2i\phi}(1-\alpha)-\alpha}{Z_{\beta}Z_{\beta_{in}}}e^{-\beta}e^{-\frac{\beta_{in}}{2}}\\\
\pm\frac{e^{-2i\phi}(1-\alpha)-\alpha}{Z_{\beta}Z_{\beta_{in}}}e^{-\beta}e^{-\frac{\beta_{in}}{2}}&e^{-\beta}\Bigl{(}1\mp\frac{2\cos\phi\sqrt{\alpha(1-\alpha)}}{Z^{2}_{\beta}Z_{\beta_{in}}}e^{-(\beta+\beta_{in})}\Bigr{)}\end{pmatrix}$

For a given purification $(\alpha,\phi)$ of the initial thermal state
$\tau_{\beta_{in}}$, the corresponding incoherent ergotropy reads

$\displaystyle
W_{d,i}(\sigma_{QC};\alpha,\phi)=\frac{1}{2Z^{2}_{\beta}Z_{\beta_{in}}}\max\Bigl{\\{}0,e^{-2\beta}-e^{-\beta_{in}}+\Bigl{(}1+2\cos\phi\sqrt{\alpha(1-\alpha)}\Bigr{)}e^{-(2\beta+\beta_{in})}-\Bigl{(}1-2\cos\phi\sqrt{\alpha(1-\alpha)}\Bigr{)}\Bigr{\\}},$
(36)

while the expression for coherent ergotropy $W_{d,c}(\alpha,\phi)$ is omitted
for the sake of brevity. Given the $(\beta_{in},\beta)$-pair, we report the
optimal ergotropy maximizing over all possible purifications

$\displaystyle\mathcal{W}_{d}(\sigma_{QC})=\max_{\alpha,\phi}W_{d}(\sigma_{QC};\alpha,\phi).$
(37)

Importantly, the optimal purification that yields the maximal ergotropy is
expectedly insensitive of the phase angle $\phi$ and determined solely by
$\alpha$. Therefore, without loss of generality, we fix $\phi=0$ and find that
the optimal purification $\alpha_{\text{opt}}$ depends crucially on relation
between the temperatures $\beta_{in}$ of the initial thermal state and $\beta$
of the maps,

$\displaystyle\alpha_{\text{opt}}(\beta_{in},\beta)=\left\\{\begin{array}[]{cc}0\text{
or }1,&\text{for}\leavevmode\nobreak\ \beta_{in}\leq 2\beta\\\
\frac{1}{2},&\text{for}\leavevmode\nobreak\
\beta_{in}>2\beta\end{array}\right..$ (40)

Indeed, taking into account the temperature bound (20) affecting the ergotropy
for the initially uncorrelated target and control, we conclude that, given the
temperature $\beta$ of the maps, initial temperatures violating (20) demand
purification in the computational basis of the control for the maximal
ergotropy extraction, whereas, for ones violating the temperature bound, the
optimal purification is in the $\\{|\pm\rangle\\}$ basis.

For the purification on the computational basis, the resulting ergotropy

$\mathcal{W}_{d}(\sigma_{QC})=\frac{1}{2}\tanh\Bigl{(}\frac{\beta}{2}\Bigr{)}\Biggl{(}\sqrt{1+\frac{1}{4}\sinh^{-2}(\beta)\cosh^{-2}\Bigl{(}\frac{\beta_{in}}{2}\Bigr{)}}-1\Biggr{)}$
(41)

is non-zero for almost any ($\beta_{in},\beta$)-pair, except for extreme cases
of infinite inverse temperatures, and is constituted only by the coherent
counterpart. This stands in stark contrast to prior classical correlations
which allow one for exclusively incoherent ergotropy extraction only within a
certain region of temperature pairs. Interestingly, in the range
$0\leq\beta_{in}\leq 2\beta$, the amount of extractable work is larger for
higher input temperatures, i.e., smaller $\beta_{in}$ values (see Fig. 4).
This feature differs from behavior of ergotropy in the setups with initially
uncorrelated and classically correlated target and control, where, in the
regions of non-zero ergotropy, it increases while increasing $\beta_{in}$.
Indeed, coherent ergotropy varies monotonically with the entanglement between
the target and the control. Since the latter depends on the initial inverse
temperature as $(\cosh\frac{\beta_{in}}{2})^{-1}$, i.e., is its decreasing
function and, therefore, makes the extractable work decreasing while
increasing $\beta_{in}$.

On the other hand, for temperatures satisfying the temperature bound (20), we
find enhancement of ergotropy under initially entangled target and control
compared with uncorrelated ones. Interestingly, in this case, the maximal
ergotropy and its behavior coincide with the maximal ergotropy that can be
extracted when the target and control are correlated only classically. Indeed,
the sharp change in the optimal configuration at $\beta_{in}=2\beta$ makes the
type of extractable ergotropy from being coherent $(\beta_{in}\leq 2\beta)$ to
incoherent $(\beta_{in}>2\beta)$. This results in a non-monotonic behaviour of
the maximal extractable ergotropy $\mathcal{W}_{d}$ that decreases with
$\beta_{in}$ for $\beta_{in}\leq 2\beta$ while increasing with $\beta_{in}$
for $\beta_{in}>2\beta$, see Fig. 5.

Figure 4: Daemonic ergotropy from the output of thermalizing maps at inverse
temperature $\beta=1$ controlled by the quantum SWITCH depending on the
purification of the initial thermal state. For $\beta_{in}<2\beta$ we the
optimal $\alpha=0$ or $1$. For $\beta_{in}>2\beta$, the optimal
$\alpha=\frac{1}{2}$.

##  V Conclusions

Thermal states and channels are considered to be useless for work extraction
in thermodynamics. However, in the quantum domain, there is a possibility of
making the causal order of two processes indefinite. The quantum SWITCH is an
example of such a operation that can induce causal indefiniteness in the
action of two or more maps (channels) using a control system. Using the
quantum SWITCH, it was shown in [16] that a finite amount of ergotropic work
extraction is possible using the apparently thermodynamically useless setting
of an initial thermal state passing through two thermal maps. The quantum
SWITCH puts the thermal maps in a superposition of causal orders which
accounts for the positive work extraction.

However, the ergotropic gains reported in [16] are conditional and occurs only
when the temperature of the thermal maps should be less than twice the
temperature of the input thermal state. In this work, we attempt to lift these
restrictions on work extraction. In earlier works, the state of the control
was taken to be in the Fourier basis (c.f. [32]) that induces superposition of
causal order. Interestingly, the quantum SWITCH can support more general
causal orderings when the target and the control pre-share some correlation.
In this generalized setting, we consider initially correlated target control
states such that the marginal target state is thermal. Note that in absence of
the SWITCH, no ergotropy can be extracted from the target system when it
passes through two thermal maps irrespective of the correlation that the
target share with any other system.

In this work we systematically analyze all possible correlations that the
target and control can share and explore its impact on local ergotropy
extraction from the target system. First we examine the situation when the
target and control share classical correlation. In this case we report
enhanced ergotropy extraction in the domain where the product target control
configuration with the control in the superposition state gave a finite
ergotropy extraction. In the region where no ergotropy extraction was possible
using the product setting, we observe two distinct features depending on the
temperature of the thermal maps. If the inverse temperature of the maps is
less than a critical value, the prior classical correlation enable extraction
of ergotropy for all input target temperatures: “complete activation”. In the
other case, the although the temperature bound gets shifted to include a
greater domain of input temperature for ergotropy extraction but there still
remains a forbidden zone: “partial activation”.

We then move on to the domain of quantum correlations and consider entangled
target control states. We find out that prior entanglement can induce
unconditional "activation" of ergotropy extraction in the entire forbidden
region. Inside the forbidden region, the amount of ergotropy obtained is also
greater compared to what can be extracted using only classical correlations.
Outside the forbidden region there is no difference in the enhancement offered
by prior classical correlation and discord over the initial product
configuration. Surprisingly, prior entanglement do not give any more
extractable ergotropy than offered by a separable discordant state.

Figure 5: Daemonic ergotropy from the output of thermalizing maps at inverse
temperature $\beta=1$ controlled by the quantum SWITCH depending on the
purification of the initial thermal state. For $\beta_{in}<2\beta$ we the
optimal $\alpha=0$ or $1$. For $\beta_{in}>2\beta$, the optimal
$\alpha=\frac{1}{2}$.

## Acknowledgement

We acknowledge support of the Hong Kong Research Grant Council through grants
17326616 and 17300918, and through the Senior Research Fellowship Scheme via
SRFS2021-7S02, and support of Grant No. 61466 from the John Templeton
Foundation as part of the QISS project.

##  Appendix A The state after the SWITCH action

The output state $\sigma$ in the computational basis $\\{\ket{ij}\\}$, where
$i,j\in\\{0,1\\}$ is given in the matrix form by

$\displaystyle\sigma_{Q}=\begin{bmatrix}p(1{-}\alpha{+}q(-1{+}2\alpha))&e^{-i\phi}p^{2}q\sqrt{\alpha(1{-}\alpha)}&0&-(1{-}p)p\alpha\sqrt{q(1{-}q)}\\\
e^{i\phi}p^{2}q\sqrt{\alpha(1{-}\alpha)}&p(\alpha{+}q(1{-}2\alpha))&e^{2i\phi}(1{-}p)p(1{-}\alpha)\sqrt{q(1{-}q)}&0\\\
0&e^{-2i\phi}(1{-}p)p(1{-}\alpha)\sqrt{q(1{-}q)}&(1{-}p)(1{-}\alpha{+}q(-1{+}2\alpha))&-e^{-i\phi}(1{-}p)^{2}(1{-}q)\sqrt{\alpha(1{-}\alpha)}\\\
-(1{-}p)p\alpha\sqrt{q(1{-}q)}&0&-e^{i\phi}(1{-}p)^{2}(1{-}q)\sqrt{\alpha(1{-}\alpha)}&(1{-}p)(\alpha{+}q(1{-}2\alpha))\end{bmatrix}.$

Here $p$ is related to the temperature of the thermal maps via
$\beta=\ln\frac{p}{1-p}$, while the temperature of the local thermal state is
given by $\beta_{in}=\ln\frac{q}{1-q}$.

##  Appendix B The post measurement states

The un-normalized post measurement states of the target system after the
control is measured in the $\\{\ket{+},\ket{-}\\}$-basis read as

$\displaystyle\tilde{\rho}_{\pm}=\frac{1}{2}\begin{bmatrix}p\pm
2\cos\phi\leavevmode\nobreak\ p^{2}q\sqrt{\alpha(1-\alpha)}&\pm
p(1-p)\sqrt{q(1-q)}\big{(}e^{2i\phi}(1-\alpha)-\alpha\big{)}\\\ \pm
p(1-p)\sqrt{q(1-q)}\big{(}e^{-2i\phi}(1-\alpha)-\alpha\big{)}&(1-p)\mp
2\cos\phi\leavevmode\nobreak\
(1-p)^{2}(1-q)\sqrt{\alpha(1-\alpha)}\end{bmatrix}$ (43)

The corresponding normalized states are
$\rho_{\pm}=\tilde{\rho}_{\pm}/p_{\pm}$, where
$p_{\pm}=\text{tr}\tilde{\rho}_{\pm}$. In this case also, $p$ is related to
the temperature of the thermal maps via $\beta=\ln\frac{p}{1-p}$, while
$\beta_{in}=\ln\frac{q}{1-q}$ is the temperature of the local thermal state.

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