# Quantum switch as a thermodynamic resource in the context of passive states

Otavio A. D. Molitor<EMAIL_ADDRESS>International
Centre for Theory of Quantum Technologies (ICTQT), University of Gdańsk,
80-308 Gdańsk, Poland  Łukasz Rudnicki<EMAIL_ADDRESS>International Centre for Theory of Quantum Technologies (ICTQT), University of
Gdańsk, 80-308 Gdańsk, Poland

###### Abstract

In recent years many works have explored possible advantages of _indefinite
causal order_ , with the main focus on its controlled implementation known as
_quantum switch_. In this paper, we tackle advantages in quantum
thermodynamics, studying whether quantum switch is capable of activating a
passive state: either alone or with extra resources (active control state)
and/or operations (measurement of the control system). By disproving the first
possibility and confirming the second one, we show that quantum switch is not
a thermodynamic resource in the discussed context, though, it can facilitate
work extraction given external resources. We discuss our findings by
considering specific examples: a qubit system subject to rotations around the
$x$ and $y$ axes in the Bloch sphere, as well as general unitaries from the
$U(2)$ group; and the system as a quantum harmonic oscillator with
displacement operators, and with a combination of displacement and squeeze
operators.

##  I Introduction

A possibility that a superposition principle can be applied to quantum
operations, leading to so called _indefinite causal order_ , was for the first
time considered in 1990 [1]. The topic received a visible boost of attention
two decades later, when Chiribella _et al._ [2] and Oreshkov _et al._ [3]
introduced the concepts of non-classical causal structures and process
matrices. In this case, however, they took off from a previous work by Hardy
[4], which considers dynamical and indefinite causal structure in a potential
theory of quantum gravity. A recent review [5] comprehensively covers deep
theory background behind indefinite causal order treated as a quantum
resource.

The quantum switch (QS) [2] is the paramount toy model for considerations
involving indefinite causal order. It implements the controlled superposition
of orders in which two (or more) unitaries (or more generally, quantum
channels) are applied to a target system. Among others, it has been shown that
with the help of QS one can get computational advantages [2, 6, 7, 8],
communication advantages [9, 10, 11] or even super Heisenberg limit in
metrology [12, 13, 14, 15, 16]. For example, it has been predicted [10] that
completely depolarizing quantum channels (which have zero capacity, i.e. they
do not transmit any information), if superposed with the help of the QS, can
be used for information transmition. Moreover, such an “acausal” superposition
of noisy channels would behave as a perfect channel [17], providing a
possibility to reduce noise completely. Intriguingly, predicted enhancements
are indeed due to superposition of orders in time [17] and do not seem to
occur fully for superposition of paths in space [18]. Interestingly, these
effects seem to be more related to the sole resourcefulness [19, 20, 21] of
indefinite causal order, rather than to particular arrangements of the
involved channels. Moreover, as already mentioned, a first proposal for
metrology assisted by indefinite causal order implemented through the QS has
also recently been announced. In [12] the problem of estimating the product
between the average position and momentum displacements has been investigated.
While a basic parallel scheme with measurements of individual displacements
results in an error of estimation of the product compatible with the standard
quantum (shot noise) limit, a direct measurement of the two average
displacements (sequential layout) allows for a quadratic improvement of the
scaling – the famous Heisenberg limit. However, the QS-assisted measurement
protocol, which uses the fact that the parameter in question can be encoded in
the commutator of the displacements, leads to a further quadratic improvement,
the super-Heisenberg limit. Note that in this case the Weyl commutation
relation between position and momentum unitary displacement operators is
crucial, therefore, it is not just a bare resourcefulness of the quantum
switch playing the role. The topic of metrology using indefinite causal order
has further been investigated in a comprehensive way [22, 13, 15, 21].

For completeness let us also briefly report experimental effort directed
towards indefinite causal order, the topic in which a few proof-of-concept
demonstrations have been performed to date. The first experiment with
indefinite causal order implemented through QS was realised in a quantum
optics context [23]. In this work a superposition of gate orders was created
by considering additional degrees of freedom of photons to encode the involved
qubits, with an auxiliary qubit responsible for controlling the order in which
two given gates are applied to the qubit of interest. The sole concept of
causality was the subject of a different experiment [24], where an object
called a causal witness, postulated in [25], has been used to prove the
“acausality” of the process based on QS. Such causal witnesses serve a similar
purpose to entanglement witnesses. Moreover, the quantum switch was recently
demonstrated in other experiments [26, 27, 16]. Experiments with indefinite
causal order are now entering the phase in which certain theoretical proposals
mentioned above can be implemented: see [11] for communication complexity;
[28, 29] for enhancements in quantum communication; [30] for computational
advantages. Again on theory side, we can even observe an engineering
perspective (quantum internet [31]) entering the scene.

In the above brief review we often interchangeably treat indefinite causal
order and the quantum switch. This is mainly because virtually all theoretical
proposals and experimental demonstrations have to do, or even are fully
concerned with QS. However, indefinite causal order is more than that, since
QS does not even violate causal inequalities [25], the latter being a true
signature of acausality. On the other hand, as the metrology application
shows, the emergence of extraordinary improvements might need an additional
ingredient (Weyl commutation relations in this particular case), so it is not
granted that indefinite causal order (or just QS) is the source of quantum
advantages. These observations become particularly relevant in the context of
quantum thermodynamics, where first attempts to use QS have just appeared [32,
33, 34, 35]. Therefore, in the current contribution we scrutinize
resourcefulness of QS in light of activation of passive states in quantum
thermodynamics. While it was initially treated in [34], here the activation of
passive states by means of the QS is further explored and new situations are
explored. First, we study the activation with and without measuring and
tracing out the control degree-of-freedom, in which the latter depends on non-
diagonal elements in the Hamiltonian of the control. Second, in our case at
all times we treat the state of the control and its measurement basis in the
Bloch sphere representation, which experimentally speaking might have more
impact. Third, the conditions for state activation are derived by looking at
energy differences before and after applying the QS (and post-selection).
Hence, whenever this energy difference is negative, one certifies state
activation. Finally, when it comes to concrete examples of systems, we show
that for two-level systems (qubit) depending on the chosen measurement basis
no work can be extracted, even if the control was initially prepared with
quantum coherence. Moreover, we also consider a quantum harmonic oscillator as
the system, which has a whole set of peculiarities when compared to finite-
level systems (e.g. we can continuously set the unitaries to the identity
operator, which might lead to singularities in the renormalization after
measurement of the control). The case of applying the QS to systems with
infinite levels – to which the quantum harmonic oscillator is an example – was
to our best knowledge just treated previously in [36], which was done in a
very abstract manner using the process matrix formalism. In this paper we go
to a more concrete scenario, using well-known examples of unitaries for
continuous-variable systems (displacement and squeeze operators111This name is
not consistent in the literature, as it is as well called “squeezing operator”
by some authors. In this paper we stick to the term “squeeze operator”.).
Therefore, since electromagnetic modes can be treated as quantum harmonic
oscillators, the study here provided can have an impact in quantum optic
setups.

This paper is organized as follows. In Sec. II we expand the discussion
concerned with the interplay between quantum thermodynamics and the quantum
switch. We observe that, while QS can be used in a clever way to activate
[34], so called, passive states [37], the question about an origin of
necessary resources is, due to a special status of thermodynamics, perhaps
more relevant than in other scenarios mentioned above. Therefore, we pose a
very precise question pertaining to the problem of passive states subject to
QS, in order to figure out whether thermodynamic advantages in this context
can come from the sole resourcefulness of QS, or they rather come from the
ancillary degrees of freedom. In Sec. III we show that the latter scenario
applies to the setup under consideration. In order to see how the results work
in different cases, we apply the framework to specific examples in Sec. IV.
First, in Sec. IV.1 the situation considered is when the system is a qubit
(only two-levels) for different unitaries: (i) when they are rotations around
the $x$ and $y$ axes in the Bloch sphere (Sec. IV.1.1) and (ii) when they are
represented as general unitaries from the $U(2)$ group (Sec. IV.1.2). Then, we
study the case in which the system is a quantum harmonic oscillator (Sec.
IV.2) with two different combinations of unitaries: (i) both unitaries are
displacement operators (Sec. IV.2.1) and (ii) one unitary is a displacement
operator and the second is a squeeze operator (Sec. IV.2.2). Finally, we pass
to Sec. V, where we draw conclusions about this work.

##  II Quantum switch and passive states in thermodynamics

Whenever QS has been shown to provide quantum advantages, a similar scheme is
being exercised. One considers a state of the system, denoted by $\rho_{S}$,
and a control qubit, denoted by $\rho_{C}$. Initially, the state of the total
system is $\rho_{SC}=\rho_{S}\otimes\rho_{C}$, so by assumption there is no
correlation between the system and the control qubit at the initial time.

Given two unitaries $U_{1}$ and $U_{2}$ (or other quantum channels, as
everything naturally extends to Kraus decomposition involving more terms, see
[10] as a profound example), one performs the operation

$U_{\textrm{QS}}=U_{2}U_{1}\otimes\outerproduct{0}{0}_{C}+U_{1}U_{2}\otimes\outerproduct{1}{1}_{C}.$
(1)

Note that, from a physical point of view, in the above definition we give
meaning to the computational basis of the control qubit,
$\\{\ket{0},\ket{1}\\}$. Afterwards, one measures the control qubit in a
_suitable_ basis (most often in $\\{\ket{+},\ket{-}\\}$) and infers
conclusions about gains associated with the system.

Let us now consider the problem of passive states in thermodynamics. The state
$\rho_{S}$ is passive with respect to the system Hamiltonian $H_{S}$, if [38,
39, 40]

$\tr\left\\{\rho_{S}H_{S}\right\\}\leq\tr\left\\{U\rho_{S}U^{\dagger}H_{S}\right\\},$
(2)

for every unitary operation $U$. The notion closely related with passivity of
states is ergotropy [41, 42] as it measures extractable work. The definition
of ergotropy is the following: consider a quantum system whose state is
$\rho_{S}$ and Hamiltonian $H_{S}$, the ergotropy of such system – or
_extractable work_ – is expressed by

$W_{\text{max}}:=\tr\\{\rho_{S}H_{S}\\}-\underset{U}{\text{min}}\tr\\{U\rho_{S}U^{\dagger}H_{S}\\}$
(3)

where the minimization procedure is applied to all the unitary transformations
existing in the Hilbert space of the system $\mathcal{H}_{S}$. As shown in
[43], the ergotropy is upper bounded as the following:

$W_{\text{max}}\leq\tr\\{\rho_{S}H_{S}\\}-\tr\\{\sigma_{\beta}H_{S}\\}$ (4)

with

$\sigma_{\beta}=\frac{e^{-\beta H_{S}}}{Z_{S}}$ (5)

being the Gibbs thermal state at inverse temperature $\beta$, such that its
von Neumann entropy $S(\rho):=-\tr\\{\rho\ln\rho\\}$ is the same as of
$\rho_{S}$, i.e. $S(\rho_{S})=S(\sigma_{\beta})$. Moreover, $Z_{S}$ is the
partition function of the system, defined by $Z_{S}:=\tr\\{\text{exp}(-\beta
H_{S})\\}$. Therefore, it can be seen that the Gibbs thermal state sets a
limit on the amount of work that can be extracted from a quantum system. As a
matter of fact, having a system in a thermal state means that no work can be
extracted from it using any unitary $U$.

One then is led to a question: is it possible to extract work from passive
states with the help of QS? The natural way is to check whether starting from
a passive state one can obtain a state whose ergotropy is non-zero. In this
line, Simonov _et al._ [34] scrutinized gains in ergotropy due to application
of QS. As usual, it turded out that occurrence of the potential benefits
critically depends on the basis in which the control qubit is being measured.
This fact suggests the follow-up question. Is the increased ergotropy a
consequence of extra information (like in the Maxwell demon problem) which is
available after tailored measurements applied to the control qubit (i.e. some
resources associated with the control qubit), or is it rather the acausal
character of the quantum switch which plays the major role? Here, we are going
to discuss the second possibility.

To this end we resort to the fact, that the notion of passivity admits the
phenomenon of _superadditivity_. While two states can individually be passive,
its tensor product does not need to be such. Only Gibbs states are completely
passive, which means they do not admit superadditivity in that context.

##  III Results

We are in position to formalize our problem at hand. As before, we assume that
at an initial time the total state is not correlated
$\rho_{SC}=\rho_{S}\otimes\rho_{C}$, and moreover, both $\rho_{S}$ and
$\rho_{C}$ are individually passive with respect to their local Hamiltonians
$H_{S}$ and $H_{C}$. Clearly, due to the phenomenon of superadditivity, if we
admit any unitaries acting on the composite system, we will potentially be
able to extract work from $\rho_{SC}$, as the composite state is not
necessarily passive.

We however restrict the set of allowed global unitary operations to such which
are realized by QS involving two unitaries on the system side only. We are
then to check whether this setting is enough to activate the system, i.e. to
observe

$\displaystyle\Delta_{\text{QS}}$ $\displaystyle:=$
$\displaystyle\tr\left\\{U_{\text{QS}}\rho_{SC}U_{\text{QS}}^{\dagger}H_{SC}\right\\}-\tr\left\\{\rho_{SC}H_{SC}\right\\}$
(6) $\displaystyle=$ $\displaystyle E_{SC}^{\prime}-E_{SC}<0,$

with the total Hamiltonian
$H_{SC}=H_{S}\otimes\mathds{1}_{C}+\mathds{1}_{S}\otimes H_{C}$. Here we
consider that system and control do not interact, hence $E_{SC}=E_{S}+E_{C}$.

Since application of “causally separable” unitaries on the system side is not
enough for activation to occur, what we shall do is to check whether including
QS is already a sufficient resource for the discussed purpose. To this end we
can explicitly compute

$E_{SC}^{\prime}=\bra{0}\rho_{C}\ket{0}E_{12}+\bra{1}\rho_{C}\ket{1}E_{21}+\bra{0}\\!\rho_{C}\outerproduct{0}{0}H_{C}\ket{0}+\bra{1}\rho_{C}\outerproduct{1}{1}H_{C}\ket{1}+\chi\bra{0}\rho_{C}\outerproduct{1}{1}H_{C}\ket{0}+\chi^{*}\bra{1}\rho_{C}\outerproduct{0}{0}H_{C}\ket{1},$
(7)

with

$E_{12}:=\tr\left\\{U_{2}U_{1}\rho_{S}U_{1}^{\dagger}U_{2}^{\dagger}H_{S}\right\\},\qquad
E_{21}:=\tr\left\\{U_{1}U_{2}\rho_{S}U_{2}^{\dagger}U_{1}^{\dagger}H_{S}\right\\},$
(8)

and

$\chi=\tr\left\\{U_{2}U_{1}\rho_{S}U_{2}^{\dagger}U_{1}^{\dagger}\right\\}\equiv\left|\chi\right|e^{i\phi}.$
(9)

The complex number $\chi$ is connected to the unitary cross-map [34], encoding
correlations relevant when both unitaries do not commute. In the communing
case we trivially get $\chi=1$. Let us denote:

$U_{\phi,\pm}=\left(\begin{array}[]{cc}1&0\\\ 0&\pm
e^{-i\phi}\end{array}\right),$ (10)

and introduce states

$\tilde{\rho}_{C}=\frac{1+\left|\chi\right|}{2}U_{\phi,+}\rho_{C}U_{\phi,+}^{\dagger}+\frac{1-\left|\chi\right|}{2}U_{\phi,-}\rho_{C}U_{\phi,-}^{\dagger},$
(11)
$\tilde{\rho}_{S}=\bra{0}\rho_{C}\ket{0}U_{2}U_{1}\rho_{S}U_{1}^{\dagger}U_{2}^{\dagger}+\bra{1}\rho_{C}\ket{1}U_{1}U_{2}\rho_{S}U_{2}^{\dagger}U_{1}^{\dagger}.$
(12)

We find that

$E_{SC}^{\prime}=\tilde{E}_{S}+\tilde{E}_{C}.$ (13)

where by analogy we define

$\tilde{E}_{S}:=\tr\\{\tilde{\rho}_{S}H_{S}\\},\qquad\tilde{E}_{C}:=\tr\\{\tilde{\rho}_{C}H_{C}\\}.$
(14)

Since both $\tilde{\rho}_{S}$ and $\tilde{\rho}_{C}$ represent the result of
applying incoherent (convex) superpositions of local unitary operations to
$\rho_{S}$ and $\rho_{C}$ respectively, we conclude that

$E_{S}+E_{C}\leq\tilde{E}_{S}+\tilde{E}_{C}.$ (15)

Consequently

$\Delta_{\text{QS}}\geq 0$ (16)

since from individual passivity of $\tilde{\rho}_{S}$ and $\tilde{\rho}_{C}$
we know that

$\Delta_{S}:=\tilde{E}_{S}-E_{S}\geq
0,\qquad\Delta_{C}:=\tilde{E}_{C}-E_{C}\geq 0.$ (17)

As our first result, we find that the QS itself is incapable of performing an
activation of a passive state. While it has been expected that the passive
control state will not become activated (no unitaries are operating on these
degrees of freedom), the same is shown to be true for the reduced system state
itself, and as a mere consequence of linearity extends to the composite state
of system and control. Therefore, the predicted increase of the ergotropy of
the composite system seems to be associated with resources of the control
qubit and/or measurements performed on it, rather than the sole action of the
quantum switch.

To frame the first possibility we suppose that the control qubit is in a
generic state that does not need to be passive. As a direct calculation shows

$\Delta_{C}=2\real\\{\bra{0}\rho_{C}\outerproduct{1}{1}H_{C}\ket{0}(\chi-1)\\},$
(18)

where $\real\\{z\\}$ denotes the real part of the complex number $z$, so it is
straightforward to minimize this expression with respect to the state of the
qubit. To this end one needs $\bra{0}\rho_{C}\ket{1}=-e^{i\phi}/\sqrt{2}$,
where the phase $\phi$ is selected to cancel the phase of
$\bra{1}H_{C}\ket{0}(\chi-1)$. Consequently

$\min_{\rho_{C}}\Delta_{\text{C}}=-\sqrt{2}\left|\bra{1}H_{C}\ket{0}(\chi-1)\right|.$
(19)

The last expression does not only show an expected effect of activation for
the control, but also proves that the composite system plus control state
activation is possible if and only if $\chi\neq 1$ and the Hamiltonian $H_{C}$
has non-diagonal terms in the computational basis $\\{\ket{0},\ket{1}\\}$
defined by the action of the QS. Non-commuting unitaries $U_{1}$ and $U_{2}$
are essential for state activation, otherwise $\chi=1$. To get
$\Delta_{\text{QS}}\leq 0$ one needs a sufficiently large value of the control
Hamiltonian coherence in comparison with the energy scale of the system. We
stress that in this way it is impossible to activate just the system. The
activation can occur only for the composite state.

On the other hand, even if the requirement of Hamiltonian of the control
having non-diagonal terms in the computational basis is not fulfilled, it is
still possible to activate the state of the system by measuring the control
qubit. In order to show how it happens, consider that the state of the control
is pure $\rho_{C}=\outerproduct{\psi}{\psi}_{C}$ and parametrized in the
following way:

$\ket{\psi}_{C}=\cos(\frac{\theta_{C}}{2})\ket{0}_{C}+e^{i\varphi_{C}}\sin(\frac{\theta_{C}}{2})\ket{1}_{C}$
(20)

such that in the Bloch sphere it is represented by the point
$(\sin\theta_{C}\cos\varphi_{C},\sin\theta_{C}\sin\varphi_{C},\cos\theta_{C})$.
Then, the final joint state of system and control is equal to

$\displaystyle\rho_{SC}^{\prime}$
$\displaystyle=\cos^{2}\left(\frac{\theta_{C}}{2}\right)U_{2}U_{1}\rho_{S}U_{1}^{\dagger}U_{2}^{\dagger}\otimes\outerproduct{0}{0}_{C}+\frac{e^{i\varphi_{C}}}{2}\sin\theta_{C}\,U_{2}U_{1}\rho_{S}U_{2}^{\dagger}U_{1}^{\dagger}\otimes\outerproduct{0}{1}_{C}$
(21)
$\displaystyle+\frac{e^{-i\varphi_{C}}}{2}\sin\theta_{C}\,U_{1}U_{2}\rho_{S}U_{1}^{\dagger}U_{2}^{\dagger}\otimes\outerproduct{1}{0}_{C}+\sin^{2}\left(\frac{\theta_{C}}{2}\right)U_{1}U_{2}\rho_{S}U_{2}^{\dagger}U_{1}^{\dagger}\otimes\outerproduct{1}{1}_{C}.$

The measurement of the control is supposed to be done by means of a projective
measurement onto the state

$\ket{\psi_{M}}_{C}=\cos(\frac{\theta_{M}}{2})\ket{0}_{C}+e^{i\varphi_{M}}\sin(\frac{\theta_{M}}{2})\ket{1}_{C}$
(22)

so that the state of the system post-measurement of the control is

$\rho_{S,M}=\frac{(\mathds{1}_{S}\otimes\bra{\psi_{M}}_{C})\rho_{SC}^{\prime}(\mathds{1}_{S}\otimes\ket{\psi_{M}}_{C})}{\tr\\{(\mathds{1}_{S}\otimes\bra{\psi_{M}}_{C})\rho_{SC}^{\prime}(\mathds{1}_{S}\otimes\ket{\psi_{M}}_{C})\\}}.$
(23)

This state explicitly reads

$\displaystyle\rho_{S,M}$
$\displaystyle=\frac{1}{N_{M}}\bigg{(}\cos^{2}\left(\frac{\theta_{C}}{2}\right)\cos^{2}\left(\frac{\theta_{M}}{2}\right)\,U_{2}U_{1}\rho_{S}U_{1}^{\dagger}U_{2}^{\dagger}+\sin^{2}\left(\frac{\theta_{C}}{2}\right)\sin^{2}\left(\frac{\theta_{M}}{2}\right)\,U_{1}U_{2}\rho_{S}U_{2}^{\dagger}U_{1}^{\dagger}$
(24)
$\displaystyle+\frac{e^{-i(\varphi_{C}+\varphi_{M})}}{4}\sin\theta_{M}\sin\theta_{C}\,U_{1}U_{2}\rho_{S}U_{1}^{\dagger}U_{2}^{\dagger}+\frac{e^{i(\varphi_{C}+\varphi_{M})}}{4}\sin\theta_{M}\sin\theta_{C}\,U_{2}U_{1}\rho_{S}U_{2}^{\dagger}U_{1}^{\dagger}\bigg{)}$

with

$N_{M}=\frac{1}{2}(1+\cos\theta_{C}\cos\theta_{M}+\sin\theta_{C}\sin\theta_{M}\real\\{\chi
e^{i(\varphi_{C}+\varphi_{M})})\\}).$ (25)

As one can see, the state expressed by Eq. (24) contains not only the
incoherent terms (the ones associated with the diagonal elements of
$\rho_{C}$), but also coherences coming from the off-diagonal terms of the
state of the control. Hence, the average internal energy of the system post-
application of the QS and post-measurement of the control,

$E_{S,M}:=\tr\\{\rho_{S,M}H_{S}\\}$ (26)

might be inferior than the initial average internal energy. As in the pre-
measurement case, coherences in the control qubit are mandatory for state
activation. In fact, one can see that whenever the difference between the
final ($E_{S,M}$) and initial value is negative, that is

$\Delta_{S,M}:=E_{S,M}-E_{S}<0$ (27)

there is activation of the state of the system, and work can be extracted from
it.

By expanding the previous expression for $\Delta_{S,M}$, one finds that it is
equivalent to:

$\Delta_{S,M}=\frac{1}{N_{M}}\bigg{(}\cos^{2}\left(\frac{\theta_{C}}{2}\right)\cos^{2}\left(\frac{\theta_{M}}{2}\right)\Delta_{12}+\sin^{2}\left(\frac{\theta_{C}}{2}\right)\sin^{2}\left(\frac{\theta_{M}}{2}\right)\Delta_{21}+\frac{1}{2}\sin\theta_{C}\sin\theta_{M}\real\\{\Delta_{F}e^{i(\varphi_{C}+\varphi_{M})}\\}\bigg{)}$
(28)

where

$\Delta_{12}:=\tr\\{U_{2}U_{1}\rho_{S}U_{1}^{\dagger}U_{2}^{\dagger}\\}-E_{S},\qquad\Delta_{21}:=\tr\\{U_{1}U_{2}\rho_{S}U_{2}^{\dagger}U_{1}^{\dagger}\\}-E_{S}$
(29)

and

$\Delta_{F}:=F_{S}-\chi E_{S},\qquad
F_{S}:=\tr\\{U_{2}U_{1}\rho_{S}U_{2}^{\dagger}U_{1}^{\dagger}H_{S}\\}.$ (30)

Since $\Delta_{12}\geq 0$ and $\Delta_{21}\geq 0$, one finds the conditions
for a possible state activation after measuring the control: (i)
$\theta_{C}\neq 0,\pi$ and $\theta_{M}\neq 0,\pi$ (i.e. the states cannot be
either $\ket{0}$ or $\ket{1}$); (ii)
$\tan(\varphi_{C}+\varphi_{M})\neq\real\\{\Delta_{F}\\}/\imaginary\\{\Delta_{F}\\}$,
and (iii)
$\sin\theta_{C}\sin\theta_{M}\real\\{\Delta_{F}e^{i(\varphi_{C}+\varphi_{M})}\\}<0$.
These are necessary, but not sufficient conditions for state activation of the
system.

Figure 1: Plot of $\Delta_{\text{QS}}$ as a function of $\beta$, for fixed
$\omega=1.0$, $\alpha_{x}=\pi/2$, $\alpha_{y}=\pi$ and different values of
$t$, when the unitaries are rotations around the $x$ and $y$ axes. As it is
evidenced by the plots, for $|t|=0$ no activation occurs
($\Delta_{\text{QS}}\geq 0$). However, as $|t|$ increases, more energy can be
extracted from system plus control, and for a larger range of inverse
temperatures $\beta$.

##  IV Examples

Let us now test the above considerations with concrete scenarios. We start
with the situation in which the system has only two-levels (qubit) and the
unitaries are rotations around the $x$ and $y$ axes of the Bloch sphere (Sec.
IV.1.1). Still in the qubit scenario, we then consider general $U(2)$
unitaries in Sec. IV.1.2. In the continuation, we pass to the case in which
the system consists of a quantum harmonic oscillator in two different
combinations of unitaries: (i) both being displacement operators (Sec. IV.2.1)
and (ii) one unitary being the displacement operator and the other the squeeze
operator (Sec. IV.2.2). In all parts, units are such that $\hbar=k_{B}=1$.

###  IV.1 Two-level systems

Consider the case in which both system and control are represented by two-
level systems (qubits). In such a case, the individual Hamiltonian of the
system is written as

$H_{S}=\frac{\omega}{2}(\mathds{1}_{S}-\sigma^{z}_{S})$ (31)

where $\mathds{1}_{S}$ is the identity operator living in the Hilbert space of
the system , and $\sigma^{z}_{S}$ is the “z” Pauli matrix in
$\mathcal{H}_{S}$. The Hamiltonian of the control is similar, but it contains
a non-diagonal term,

$H_{C}=\frac{\omega}{2}(\mathds{1}_{C}-\sigma^{z}_{C})+t\outerproduct{0}{1}_{C}+t^{*}\outerproduct{1}{0}_{C}$
(32)

with the same identity and Pauli operators as before, but now living in
$\mathcal{H}_{C}$ and $t=|t|e^{i\theta}\in\mathbb{C}$ is connected to the
probability that the control qubit will jump from one state to the other. For
simplicity, we consider that system and control are resonant
($\omega_{S}=\omega_{C}=\omega$) and the total Hamiltonian is a non-
interacting one $H_{SC}=H_{S}\otimes\mathds{1}_{C}+\mathds{1}_{S}\otimes
H_{C}$. The initial state of the system is taken to be the Gibbs state,

$\rho_{S}=\frac{e^{-\beta
H_{S}}}{Z_{S}}=\begin{pmatrix}\frac{1}{1+e^{-\beta\omega}}&&0\\\
0&&1-\frac{1}{1+e^{-\beta\omega}}\end{pmatrix}$ (33)

with $Z_{S}=\tr\\{e^{-\beta H_{S}}\\}$ and $\beta$ is the inverse temperature
of the system. Moreover, the control is initially prepared in a generic pure
state $\rho_{C}=\outerproduct{\psi}{\psi}_{C}$, with
$\ket{\psi}_{C}=\cos(\theta_{C}/2)\ket{0}_{C}+e^{i\varphi_{C}}\sin(\theta_{C}/2)\ket{1}_{C}$,
and $\theta_{C}\in[0,\pi]$, $\varphi_{C}\in[0,2\pi]$. The initial joint state
is a non-correlated, product state $\rho_{SC}=\rho_{S}\otimes\rho_{C}$. We
then consider two different scenarios for the unitaries: (i) they correspond
to rotations around the $x$ and $y$ axes in the Bloch sphere and (ii) general
$U(2)$ unitaries (which themselves are decomposed as rotations in the Bloch
sphere).

Figure 2: Plots of the value of $\varphi_{M}$ (in radians) for which
$\Delta_{S,M}$ is minimum (left) and of $\Delta_{S,M}$ as a function of
$\varphi_{M}$ (in radians) for different $\alpha$ (right). In both we consider
the state of the control to be equal to $\outerproduct{+}{+}_{C}$
($\theta_{C}=\pi/2$, $\varphi_{C}=0$), $\omega=1.0$, $\theta_{M}=\pi/2$ and
$\beta\to 0$.

####  IV.1.1 Rotation operators

First, we start with unitaries as rotations around the $x$ and $y$ axes in the
Bloch sphere,

$U_{1}=R_{x}(\alpha_{x})=e^{-i\sigma_{x}\alpha_{x}/2},\qquad
U_{2}=R_{y}(\alpha_{y})=e^{-i\sigma_{y}\alpha_{y}/2}$ (34)

where $\sigma_{x}$ and $\sigma_{y}$ are Pauli matrices, and
$\alpha_{x},\alpha_{y}\in[0,2\pi]$ are the angles of rotation. The final state
after applying the QS is then denoted
$\rho_{SC}^{\prime}=U_{\text{QS}}\rho_{SC}U_{\text{QS}}^{\dagger}$, with
$U_{\text{QS}}$ as in Eq. (1). With all that it is possible to calculate

$\Delta_{\text{QS}}=\frac{\omega}{2}\Bigg{[}1-\cos\alpha_{x}\cos\alpha_{y}+\frac{|t|}{\omega}\sin\alpha_{x}\sin\alpha_{y}\sin\theta_{C}\sin(\theta+\varphi_{C})\Big{)}\tanh\left(\frac{\beta\omega}{2}\right)\Bigg{]}-2|t|\cos(\theta+\varphi_{C})\sin\theta_{C}\sin^{2}\left(\frac{\alpha_{x}}{2}\right)\sin^{2}\left(\frac{\alpha_{y}}{2}\right).$
(35)

This expression is plotted in Fig.1 in the case that the state of the control
corresponds to the pure state $\ket{+}=(\ket{0}+\ket{1})/\sqrt{2}$. As
expected from the calculations in Sec. III, when the Hamiltonian of the
control is diagonal in the computational basis ($|t|=0$), the energy of the
system plus control is always higher than the initial energy after applying
the quantum switch, for whatever inverse temperature $\beta$. As one increases
the value of $|t|$, it is possible to reach lower final energy for a range of
$\beta$, meaning that the state of system plus control is activated.

Now passing to the case that the control is measured by means of a projector
in the state
$\ket{\psi_{M}}_{C}=\sin(\theta_{M}/2)\ket{0}_{C}+e^{i\varphi_{M}}\sin(\theta_{M}/2)\ket{1}_{C}$,
we have the following necessary (but not sufficient) conditions for state
activation:

$\sin\theta_{C}\neq 0,\quad\sin\theta_{M}\neq 0$ (36)
$\tan(\varphi_{C}+\varphi_{M})\neq(\cot\alpha_{x}\cot\alpha_{y}-\csc\alpha_{x}\csc\alpha_{y})\sinh\beta\omega$
(37)

and

$\frac{\omega\sin\theta_{C}\sin\theta_{M}}{2(1+e^{\beta\omega})^{2}}((e^{2\beta\omega}-1)(1-\cos\alpha_{x}\cos\alpha_{y})\cos(\varphi_{C}+\varphi_{M})+2e^{\beta\omega}\sin\alpha_{x}\sin\alpha_{y}\sin(\varphi_{C}+\varphi_{M}))<0.$
(38)

These get simplified when $\beta\to 0$ (i.e. the Gibbs thermal state
corresponds to the maximally mixed state $\rho_{S}=\mathds{1}_{S}/2$) and
$\alpha_{x}=\alpha_{y}=\alpha$, then the last two conditions become:

$\tan(\varphi_{C}+\varphi_{M})\neq 0$ (39)

and

$\sin\theta_{C}\sin\theta_{M}\sin(\varphi_{C}+\varphi_{M})<0$ (40)

then, if we set, for example, $\theta_{C}=\pi/2$ and $\varphi_{C}=0$ – which
corresponds to the state $\outerproduct{+}{+}_{C}$ – one has that
$\varphi_{M}\neq 0,\pi$ and $\sin\theta_{M}\sin\varphi_{M}<0$, meaning that
one must have $\varphi_{M}\in\,\,]\pi,2\pi[$ for possible state activation.

In this simplified scenario (state of the control $\outerproduct{+}{+}_{C}$,
$\beta\to 0$ and $\alpha_{x}=\alpha_{y}=\alpha$), $\Delta_{S,M}$ becomes:

$\Delta_{S,M}=\frac{\omega\sin\varphi_{M}}{2\cos\varphi_{M}+4\cos\varphi_{M}\cot\alpha\csc\alpha+4\csc^{2}\alpha\csc\theta_{M}}$
(41)

which clearly has a minimum for $\theta_{M}=\pi/2$ (remember that
$\theta_{M}\in\,\,]0,\pi[\,\,$). Then we plot in Fig. 2 the angle
$\varphi_{M}$ for which we obtain minimum $\Delta_{S,M}$, as well as the
values of the latter as a function of $\varphi_{M}$ for different values of
$\alpha$. Here, as a matter of fact, the previous conditions for state
activation are not only necessary, but also sufficient, since when $\beta\to
0$ the “causally ordered” energy differences are equal to zero:
$\Delta_{12}=\Delta_{21}=0$.

Figure 3: Minimum values achieved by $\Delta_{\text{QS}}$ as a function of
$|t|$, for fixed $\omega=1.0$ and for different inverse temperatures and
angles $\theta$ ($\beta=0$ on the left, $\beta=0.1$ on the middle and
$\beta=0.2$ on the right). For each point, a different combination of
$\lambda_{k},\gamma_{k},\delta_{k}$ leads to the minimum value of
$\Delta_{\text{QS}}$. Here the state of the control is set to be the
$\outerproduct{+}{+}_{C}$ state ($\theta_{C}=\pi/2$, $\varphi_{C}=0$).
Curiously, the slopes of the dashed lines from the plots do not depend on
$\beta$, but solely on $\theta$.

####  IV.1.2 General U(2) unitaries

In the most general scenario, unitary operations in
$\mathcal{L}(\mathcal{H}_{2})$, with $\mathcal{H}_{2}$ being the Hilbert space
of dimension 2 (qubits), can be written in the generic $U(2)$ representation
group [44],

$U_{k}=e^{i\alpha_{k}}R_{z}(\lambda_{k})R_{y}(\gamma_{k})R_{z}(\delta_{k}),\quad
k=1,2$ (42)

with $\alpha_{k},\lambda_{k},\gamma_{k},\delta_{k}\in\mathbb{R}$ and
$R_{y},R_{z}$ are rotations around the $y$ and $z$ axis of the Bloch sphere,
respectively. Unfortunately, compact expressions cannot be obtained here, but
numerically we can try to find combinations of
$\alpha_{k},\lambda_{k},\gamma_{k},\delta_{k}$ that minimize
$\Delta_{\text{QS}}$ for a given combination of $\omega,\beta$ and $t$
(actually $\alpha_{k}$ do not matter in this case), given a certain state of
the control, which we take to be the $\outerproduct{+}{+}_{C}$ state. In Fig.
3, one finds the plots of the minimum value of $\Delta_{\text{QS}}$ as a
function of $|t|$ for a few inverse temperatures $\beta$ and angles $\theta$.
We see that for each $|t|>0$ it is always possible to reach negative
$\Delta_{\text{QS}}$ and curiously, all the points with the same angle
$\theta$ converge to the same line with a well defined slope, irrespective of
the inverse temperature $\beta$. Then, one might take the simplifying scenario
when $\beta\to 0$:

$\Delta_{\text{QS}}^{\beta\to
0}=-\left(\frac{-12+\epsilon(\lambda_{1},\gamma_{1},\delta_{1},\lambda_{2},\gamma_{2},\delta_{2})}{16}\right)\cos\theta\,|t|$
(43)

where

$\text{min}\,\epsilon(\lambda_{1},\gamma_{1},\delta_{1},\lambda_{2},\gamma_{2},\delta_{2})\equiv-20,$
(44)

such that the minimum value of the energy difference of system plus control is
totally determined by the off-diagonal term of the Hamiltonian of the control
$t$ (of course, it happens for different combinations of $\lambda_{k}$,
$\gamma_{k}$ and $\delta_{k}$).

Figure 4: Minimum value achieved by $\Delta_{S,M}$ as a function of
$\varphi_{M}$, when $\omega=1.0$, $\theta_{C}=\theta_{M}=\pi/2$ and
$\varphi_{C}=0$, for different $\beta$. For a better visualization, we plot
just a few points, in order to show that the minimum value of $\Delta_{S,M}$
is a constant, except for $\varphi_{M}=0,\pi$, when no state activation is
possible ($\text{min}\,\Delta_{S,M}=0$). Figure 5: Plots of
$\Delta_{\text{QS}}$ for unitaries being displacement operators as a function
of $|\alpha_{1}|$ and $|\alpha_{2}|$, for $\omega=1.0$, $\theta_{C}=\pi/2$,
$\varphi_{C}=0$ ($\rho_{C}=\outerproduct{+}{+}_{C}$), $\theta=0$,
$\phi_{1}-\phi_{2}=\pi/2$ and for different values of $|t|$. Dashed red lines
in the plots show whenever $\Delta_{\text{QS}}=0$, delimiting the borders of
the regions where state activation is possible and impossible.

On the other hand, one has the situation after measuring the control qubit.
Again, no simple analytical expression might be obtained. Nonetheless, the
numerical minimization of $\Delta_{S,M}$ is done depending on the state of the
control and of the measurement basis. For instance, consider the situation
when the state of the control is the $\outerproduct{+}{+}_{C}$ state
($\theta_{C}=\pi/2$ and $\varphi_{C}=0$) and the measurement state is on the
$xy$-plane ($\theta_{M}=\pi/2$). Then, the plot of the minimum value of
$\Delta_{S,M}$ for different inverse temperatures $\beta$, as a function of
$\varphi_{M}$ is found in Fig. 4. It shows that when we use the basis
$\\{\ket{+},\ket{-}\\}$, corresponding to $\varphi_{M}=0$ and
$\varphi_{M}=\pi$ respectively, to measure the control qubit, _no_ state
activation might be achieved, for whatever values of $\lambda_{k}$,
$\gamma_{k}$ and $\delta_{k}$. Moreover, since $\text{min}\,\Delta_{S,M}$
changes with the inverse temperature up to a constant, we take $\beta\to 0$,
then $\Delta_{S,M}$ becomes

$\Delta_{S,M}^{\beta\to
0}=\omega\frac{f(\lambda_{1},\gamma_{1},\delta_{1},\lambda_{2},\gamma_{2},\delta_{2})\sin\varphi_{M}}{32+g(\lambda_{1},\gamma_{1},\delta_{1},\lambda_{2},\gamma_{2},\delta_{2})\cos\varphi_{M}}$
(45)

with complicated functions
$f(\lambda_{1},\gamma_{1},\delta_{1},\lambda_{2},\gamma_{2},\delta_{2})$ and
$g(\lambda_{1},\gamma_{1},\delta_{1},\lambda_{2},\gamma_{2},\delta_{2})$.
Nonetheless, we know their minimum value to be:

$\text{min}\,f(\lambda_{1},\gamma_{1},\delta_{1},\lambda_{2},\gamma_{2},\delta_{2})=-16$
(46)

and

$\text{min}\,g(\lambda_{1},\gamma_{1},\delta_{1},\lambda_{2},\gamma_{2},\delta_{2})\sim-8.57.$
(47)

Here the $\sin\varphi_{M}$ on the numerator shows how $\Delta_{S,M}^{\beta\to
0}=0$ when $\varphi_{M}=0,\pi$.

###  IV.2 Quantum harmonic oscillator

Continuing, now we pass to the situation in which the system is a one-mode
quantum harmonic oscillator and the control is still a two-level system
(qubit). The Hamiltonian of the control is the same as in the previous
sections and the Hamiltonian of the system is

$H_{S}=\omega\left(a^{\dagger}a+\frac{\mathds{1}_{S}}{2}\right),$ (48)

where $a$ ($a^{\dagger}$) is the annihilation (creation) operator and as
before we consider that the system and the control are resonant (same
excitation energy $\omega$). The total Hamiltonian is simply the sum of the
individual Hamiltonians, as no interaction is assumed between system and
control. The initial state of the control is the coherent
$\rho_{C}=\outerproduct{+}{+}_{C}$ state ($\theta_{C}=\pi/2,\varphi_{C}=0$)
and the state of the system is the thermal Gibbs state:

$\rho_{S}=\frac{e^{-\beta
H_{S}}}{Z_{S}}=(1-e^{-\beta\omega})\sum_{n}e^{-\beta\omega
n}\outerproduct{n}{n}_{S}$ (49)

with $Z_{S}=\tr\\{e^{-\beta
H_{S}}\\}=1/(e^{\beta\omega/2}-e^{-\beta\omega/2})$ being the partition
function of the system and $\ket{n}_{S}$ is the energy eigenstate of the
system containing $n$ excitations. As usual, the initial state of system plus
control is the separable state $\rho_{SC}=\rho_{S}\otimes\rho_{C}$.

When it comes to the unitaries, we might consider different cases: (i) first
when both are displacement operators and (ii) second when one is a
displacement operator and the other the squeeze operator.

####  IV.2.1 Displacement operators

We start then with both unitaries being displacement operators, that is [45]

$U_{k}=D(\alpha_{k})=e^{\alpha_{k}a^{\dagger}-\alpha_{k}^{*}a},\text{for}\,\,k=1,2$
(50)

where $\alpha_{k}=|\alpha_{k}|e^{i\phi_{k}}\in\mathbb{C}$ as well as $a$ and
$a^{\dagger}$ annihilation and creation operators, respectively. We find that

$\Delta_{\text{QS}}=\omega|\alpha^{\prime}|^{2}+|t|\big{[}\text{cos}(\theta-\varphi_{C}+2|\alpha_{1}||\alpha_{2}|\sin(\phi_{1}-\phi_{2}))-\cos(\theta-\varphi_{C})\big{]}\sin\theta_{C}$
(51)

with $\alpha^{\prime}=\alpha_{1}+\alpha_{2}$ (detailed calculations can be
found in Appendix A). This result is interesting, as it shows that whenever
$\alpha_{1}$ and $\alpha_{2}$ are parallel/anti-parallel in phase-space
($\phi_{1}-\phi_{2}=\pi\,m$, $m\in\mathbb{Z}$), $\Delta_{\text{QS}}$ is always
non-negative even with coherences in the control state and $|t|>0$. It
reinforces the fact that these are indeed necessary, but not sufficient
conditions for state activation. Also, we see that $\Delta_{\text{QS}}$ does
not depend on the inverse temperature $\beta$, what is also something
surprising. Plots of the previous equation for specific values of the
parameters when $\phi_{1}-\phi_{2}=\pi(2m-1)/2,m\in\mathbb{Z}$ can be found in
Fig. 5. These plots show that the higher $|t|$, the lower $|\alpha_{1}|$ and
$|\alpha_{2}|$ are necessary for state activation.

The simplifying case in which $\phi_{1}-\phi_{2}=\pi(2k-1)/2,k\in\mathbb{Z}$,
$\theta=0$, $|\alpha_{1}|=|\alpha_{2}|=|\alpha|$ and $\theta_{C}=\pi/2$,
$\varphi_{C}=0$ ($\rho_{C}=\outerproduct{+}{+}_{C}$) leads to the following
expression for $\Delta_{\text{QS}}$:

$\Delta_{\text{QS}}=2\left(\omega|\alpha|^{2}-|t|\sin^{2}\left(|\alpha|^{2}\right)\right).$
(52)

The latter is plotted in Fig. 6. It shows that state activation happens for a
limited range of $|\alpha|$, depending on the value of $|t|$. Moreover, it is
possible to determine what is the value of $|\alpha|$ for which
$\Delta_{\text{QS}}$ is minimum. It corresponds to:

$|\alpha|_{\text{min}}=\sqrt{\frac{\pi-\arcsin(\omega/|t|)}{2}}$ (53)

and it points to the fact that there are only solutions for $\omega\leq|t|$
and when $|t|>>\omega$, $|\alpha|_{\text{min}}\sim\sqrt{\pi/2}$.

Figure 6: The plots of $\Delta_{\text{QS}}$ in the case that
$|\alpha_{1}|=|\alpha_{2}|=|\alpha_{2}|$, $\omega=1.0$, $\theta=\varphi_{C}=0$
and $\phi_{1}-\phi_{2}=\theta_{C}=\pi/2$, and for different $|t|$. Clearly one
sees that activation of the system plus control state is only possible for a
range of $|\alpha|$, with a specific $|\alpha|_{\text{min}}$ leading to
minimum $\Delta_{\text{QS}}$.

Now we check whether after measuring the control in the basis
$\ket{\psi_{M}}_{C}=\cos(\theta_{M}/2)\ket{0}_{C}+e^{i\varphi_{M}}\sin(\theta_{M}/2)\ket{1}_{C}$
the state of the system can be activated. Calculations (Appendix A) lead to

$\Delta_{S,M}=\omega|\alpha^{\prime}|^{2}\geq
0,\quad\forall\theta_{C},\varphi_{C},\theta_{M},\varphi_{M}$ (54)

meaning that for _any_ control state and _any_ measurement state, applying
displacement operators in a quantum switch setup does not activate _any_
passive state. This result is a consequence of the fact that displacement
operators have a particular commutation relation,

$[D(\alpha_{1}),D(\alpha_{2})]=(1-\chi)D(\alpha_{1})D(\alpha_{2})$ (55)

which shows that they almost commute, differing by the complex number $\chi$
(where $0\leq|\chi|\leq 1$). The physical implication of this mathematical
property is that, when applying displacement operators in different orders to
the thermal state, the final displaced state is the same in both cases up to a
global complex phase. This state is clearly passive as well and no matter what
post-selection is chosen, no work can be extracted from it. Previously it was
not the case, because the non-diagonal element of the Hamiltonian of the
control allowed the use of the coherence in the control for state activation.

Figure 7: Plots of $\Delta_{\text{QS}}$ for unitaries being the displacement
and squeeze operators as a function of $|\alpha|$ and $|z|$, in the case that
$\omega=\beta=1.0$, $\theta_{C}=\pi/2$, $\varphi_{C}=0$
($\rho_{C}=\outerproduct{+}{+}_{C}$), $\theta=\phi=\xi=0$ and for different
values of $|t|$: (left) $|t|=0$, (center) $|t|=20$ and (right) $|t|=30$.
Dashed red lines represent the situation when $\Delta_{\text{QS}}=0$,
delimiting the borders of the regions where state activation is possible and
impossible.

####  IV.2.2 Displacement operator and squeeze operator

Moreover, we might choose two different unitaries to compose our QS. One
unitary is taken to be the displacement operator and the other one the squeeze
operator [45]:

$U_{1}=D(\alpha)=e^{\alpha a^{\dagger}-\alpha^{*}a}$ (56)
$U_{2}=S(z)=e^{(za^{\dagger}a^{\dagger}-z^{*}a\,a)/2}$ (57)

where $\alpha=|\alpha|e^{i\phi},z=|z|e^{i\xi}\in\mathbb{C}$, and again $a$ and
$a^{\dagger}$ are annihilation and creation operators, respectively. Here the
calculations are even more lengthy and as previously the details are shown in
Appendix B. The final energy difference of system plus control is

$\Delta_{\text{QS}}=\frac{\omega}{2}+\omega|\alpha|^{2}+\frac{\omega\cosh(2|z|)}{2}+2\omega\langle
n\rangle_{\text{th}}\sinh^{2}|z|+\omega|\alpha|^{2}\cos^{2}\left(\frac{\theta_{C}}{2}\right)\cos(\xi-2\phi)\sinh(2|z|)+|t|\sin\theta_{C}\real\\{e^{i(\varphi_{C}-\theta)}(\chi-1)\\}.$
(58)

where

$\chi=\innerproduct{\gamma}{\alpha}e^{-\langle
n\rangle_{\text{th}}|\alpha-\gamma|^{2}}$ (59)

with $\langle n\rangle_{\text{th}}=1/(e^{\beta\omega}-1)$ being the thermal
boson occupation number, $\gamma$ arises from the combination of displacement
and squeeze operators

$\gamma=|\alpha|e^{i\phi}\cosh|z|-|\alpha|e^{i(\xi-\phi)}\sinh|z|$ (60)

and

$\innerproduct{\gamma}{\alpha}=e^{\gamma^{*}\alpha-|\alpha|^{2}/2-|\gamma|^{2}/2}$
(61)

The full expression is not easy to visualize, hence we show by plots how
$\Delta_{\text{QS}}$ behaves with the parameters assuming specific values
(Fig. 7). In comparison with the case of two displacement operators, one sees
here that in order to achieve activation of the Gibbs thermal state, the value
of $|t|$ must be considerably higher (one order of magnitude above $\omega$
and $\beta$).

After measuring the control qubit in the state
$\ket{\psi_{M}}_{C}=\cos(\theta_{M}/2)\ket{0}_{C}+e^{i\varphi_{M}}\sin(\theta_{M}/2)\ket{1}_{C}$,
the final energy difference of the system, as shown in Appendix B is equal to:

$\displaystyle\Delta_{S,M}$
$\displaystyle=\frac{1}{N_{M}}\Bigg{[}\frac{\omega}{4}(1+\cos\theta_{C}\cos\theta_{M})(2|\alpha|^{2}+(2\langle
n\rangle_{\text{th}}+1)(\cosh(2|z|)-1))$ (62)
$\displaystyle+\omega|\alpha|^{2}\cos^{2}\left(\frac{\theta_{C}}{2}\right)\cos^{2}\left(\frac{\theta_{M}}{2}\right)\cos(\xi-2\phi)\sinh(2|z|)+\frac{1}{2}\sin\theta_{C}\sin\theta_{M}\real\\{\Delta_{F}e^{i(\varphi_{C}+\varphi_{M})}\\}\Bigg{]}$

with

$\Delta_{F}=\omega\chi(\gamma^{*}\alpha+(2\gamma^{*}\alpha-|\gamma|^{2}-|\alpha|^{2})\langle
n\rangle_{\text{th}}-|\alpha-\gamma|^{2}\langle n\rangle^{2}_{\text{th}})$
(63)
$N_{M}=\frac{1}{2}(1+\cos\theta_{C}\cos\theta_{M}+\sin\theta_{C}\sin\theta_{M}\real\\{\chi
e^{i(\varphi_{C}+\varphi_{M})})\\}$ (64)

and $\chi$ the same as before (Eq. (59)). From now on we analyze two cases:
(i) $\xi-2\phi=0$:

$\displaystyle\Delta_{S,M}^{0}$
$\displaystyle=\frac{1}{N_{M}^{0}}\Bigg{[}\frac{\omega}{4}(1+\cos\theta_{C}\cos\theta_{M})(2|\alpha|^{2}+(2\langle
n\rangle_{\text{th}}+1)(\cosh(2|z|)-1))+\omega|\alpha|^{2}\cos^{2}\left(\frac{\theta_{C}}{2}\right)\cos^{2}\left(\frac{\theta_{M}}{2}\right)\sinh(2|z|)$
(65)
$\displaystyle-\frac{\omega|\alpha|^{2}}{2}\sin\theta_{C}\sin\theta_{M}e^{-2|z|-|\alpha|^{2}e^{-|z|}(2\langle
n\rangle_{\text{th}}+1)(\cosh|z|-1)}\bigg{(}\langle
n\rangle^{2}_{\text{th}}(e^{2|z|}-2e^{|z|}+1)+\langle
n\rangle_{\text{th}}(e^{2|z|}-2e^{|z|}+|\alpha|^{2}e^{-2|z|})-e^{|z|}\bigg{)}\cos(\varphi_{C}+\varphi_{M})\Bigg{]}$

where

$N_{M}^{0}=\frac{1}{2}\Bigg{(}1+\cos\theta_{C}\cos\theta_{M}+\sin\theta_{C}\sin\theta_{M}e^{-2|\alpha|^{2}\sinh^{2}(|z|/2)(\cosh|z|-\sinh|z|)}\cos(\varphi_{C}+\varphi_{M})\Bigg{)}$
(66)

and (ii) $\xi-2\phi=\pi$:

$\displaystyle\Delta_{S,M}^{\pi}$
$\displaystyle=\frac{1}{N_{M}^{\pi}}\Bigg{[}\frac{\omega}{4}(1+\cos\theta_{C}\cos\theta_{M})(2|\alpha|^{2}+(2\langle
n\rangle_{\text{th}}+1)(\cosh(2|z|)-1))-\omega|\alpha|^{2}\cos^{2}\left(\frac{\theta_{C}}{2}\right)\cos^{2}\left(\frac{\theta_{M}}{2}\right)\sinh(2|z|)$
(67)
$\displaystyle-\frac{\omega|\alpha|^{2}}{2}\sin\theta_{C}\sin\theta_{M}e^{-\frac{|\alpha|^{2}}{2}(e^{|z|}-1)^{2}(2\langle
n\rangle_{\text{th}}+1)}\bigg{(}\langle
n\rangle^{2}_{\text{th}}(e^{2|z|}-2e^{|z|}+1)+\langle
n\rangle_{\text{th}}(|\alpha|^{2}e^{4|z|}-2e^{|z|}+1)-e^{|z|}\bigg{)}\cos(\varphi_{C}+\varphi_{M})\Bigg{]}$

with

$N_{M}^{\pi}=\frac{1}{2}\Bigg{(}1+\cos\theta_{C}\cos\theta_{M}+\sin\theta_{C}\sin\theta_{M}e^{-\frac{|\alpha|^{2}}{2}(e^{|z|}-1)^{2}(2\langle
n\rangle_{\text{th}}+1)}\cos(\varphi_{C}+\varphi_{M})\Bigg{)}.$ (68)

The plots of these expressions can be found in Fig. 8. They show that, when
$\xi-2\phi=0$, one can activate the state of the system after performing the
measurement on the $\outerproduct{-}{-}$ state ($\varphi_{M}=\pi$). In this
case, one must be careful, as $|z|$ and $|\alpha|$ go to zero the denominator
$N_{M}^{0}$ tends to zero faster then the denominator and divergences occur.
When measuring with other angles $\varphi_{M}$, no state activation occurs. On
the other hand, for $\xi-2\phi=\pi$ it is possible to get
$\Delta_{S,M}^{\pi}<0$ for all $\varphi_{M}$ (here as before the
$\varphi_{M}=\pi$ case must be taken with care).

Figure 8: Plots of $\Delta_{S,M}^{0}$ (first row) and $\Delta_{S,M}^{\pi}$
(second row) as functions of $|\alpha|$ and $|z|$, for $\omega=\beta=1.0$,
$\theta_{C}=\pi/2$, $\varphi_{C}=0$ ($\rho_{C}=\outerproduct{+}{+}_{C}$),
$\theta_{M}=\pi/2$ and four different $\varphi_{M}:$ $0,\pi/2,\pi,3\pi/2$
(each column). As in previous plots, the dashed red lines indicate when
$\Delta_{S,M}^{0,\pi}=0$. Here one sees that when $\xi-2\phi=0$ (first row),
the only situation when state activation is achieved is when the measurement
state is the $\outerproduct{-}{-}$ state ($\varphi_{M}=\pi$). Nonetheless,
this is a delicate scenario, since for low $|\alpha|$ and $|z|$ the value of
the denominator converges to zero faster than the numerator (possible
divergence). On the other hand, when $\xi-2\phi=\pi$ (second row), it is
always possible to activate the state of the system, where caution must also
must be taken when $\varphi_{M}=\pi$.

As a last scenario to be evaluated, consider when $\beta\to\infty$, which
corresponds to:

$\langle n\rangle_{\text{th}}\to 0$ (69)

and the Gibbs thermal state tends asymptotically to the ground state
$\rho_{S}\to\outerproduct{0}{0}_{S}$. Then, Eqs. (65) and (67) are simplified
accordingly. The final expressions are plotted in Fig. 9 in the case that the
magnitude of the displacement and the squeeze are the same $|\alpha|=|z|$,
$\rho_{C}=\outerproduct{+}{+}_{C}$ and $\theta_{M}=\pi/2$. The plots show that
in this limit, it is impossible to achieve state activation for any chosen
measurement state when $\xi-2\phi=0$. Nevertheless, in the situation that
$\xi-2\phi=\pi$ one can get negative values of $\Delta_{S,M}^{\pi}$ for any of
the chosen measurement angles $\varphi_{M}$. Here, as before, when
$|\alpha|=|z|\to 0$ and one measures in the $\outerproduct{-}{-}$ state
($\varphi_{M}=\pi$), the values of $\Delta_{S,M}^{0,\pi}$ diverge. Finally, it
is noticeable that $\varphi_{M}=\pi/2$ and $\varphi_{M}=3\pi/2$ lead to the
same results.

Figure 9: Plots of $\Delta_{SM}^{0}$ (left) and $\Delta_{SM}^{\pi}$ (right)
when $\beta\to\infty$ as functions of $|\alpha|=|z|$, for $\omega=1.0$,
$\theta_{C}=\pi/2$ and $\varphi_{C}=0$ ($\rho_{C}=\outerproduct{+}{+}_{C}$),
$\theta_{M}=\pi/2$, as well as for different values of $\varphi_{M}$. In this
situation one sees that when $\xi-2\phi=0$ no state activation is possible,
for whatever choice of measurement state. On the other hand, when
$\xi-2\phi=\pi$ all the measurement choices enable state activation. For both,
the cases $\varphi_{M}=\pi/2$ and $\varphi_{M}=3\pi/2$ coincide and when
$|\alpha|=|z|\to 0$ there is divergence of $\Delta_{S,M}^{0,\pi}$.

##  V Conclusions

In this paper we focus on the problem of quantifying state activation in
scenarios where the quantum switch (QS) is applied. It is well known that
passive state in quantum thermodynamics cannot be activated by any unitary
operations [38, 39, 40, 41, 42], needing extra resources (e.g. coherences in
the state) to be able to be used in thermodynamical tasks. With the rapid
advancement of indefinite causal order (ICO) research in its quantum switch
(QS) form [6, 7, 8, 9, 10, 11, 12, 22, 14, 15, 16, 17, 18, 19, 20, 21, 13, 23,
24, 25, 26, 27, 28, 29, 30, 31], especially in the context of communication,
computation and metrology, the matter of its resourcefulness in performing
thermodynamic tasks is still open, apart from a few works on this topic [32,
33, 34, 35]. We then put forward the result that, the QS by itself _does not_
ensure state activation, needing resources outside of itself to enable energy
extraction from passive states (Sec. III). In order to activate the composite
state of system plus control, non-diagonal elements (in the computational
basis defined by the action of QS) on both the state of the control and the
Hamiltonian of the control are necessary (but not sufficient) conditions. The
first corresponds to _coherences_ and the second is related to the inner
transitions between the two states of the control. Moreover, when the latter
does not exist, it is still possible to activate the state of the system alone
by measuring the state of the control. The measurement that ensures energy
extraction is case dependent (what is the kind of system considered, what are
the unitaries, etc.). In examples (Sec. IV) we then present different
scenarios where state activation happens or does not happen, always taking as
a reference point for the state of the system the Gibbs thermal state. These
results point to the fact that, when considering thermodynamical tasks
involving the QS, one must always consider a very specific setup, where all
parameters are well controlled, otherwise, it is difficult to predict whether
the QS is in fact bringing anything new to what is already known in quantum
thermodynamics.

This work points to a few future possible studies. First, it is still
necessary to quantify the energetic cost of measuring the control qubit. From
_Landauer’s principle_ we know that to every measurement corresponds some
finite dissipation of heat. Thus, it is imperative to compare this energetic
cost to what is gained from state activation by means of measurement of the
control qubit. This comparison will present what are the situations where a
net energetic gain is obtained. Second, the QS is not the only example of
indefinite causal order. Using process matrices [3], one is able to devise
situations where causal separability does not hold and causal inequalities are
violated [25]. Still, little is known about these scenarios and what are their
possible use cases. An interesting open problem for the near future is thus to
understand how they act on passive states and whether they are capable of
activating such states without the need of extra resources.

## Acknowledgements

We acknowledge support by the Foundation for Polish Science (IRAP project,
ICTQT, contract no. 2018/MAB/5, co-financed by EU within Smart Growth
Operational Programme).

## References

  * Aharonov  _et al._ [1990] Y. Aharonov, J. Anandan, S. Popescu, and L. Vaidman, Superpositions of time evolutions of a quantum system and a quantum time-translation machine, Phys. Rev. Lett. 64, 2965 (1990).
  * Chiribella  _et al._ [2013] G. Chiribella, G. M. D’Ariano, P. Perinotti, and B. Valiron, Quantum computations without definite causal structure, Phys. Rev. A 88, 022318 (2013).
  * Oreshkov  _et al._ [2012] O. Oreshkov, F. Costa, and Č. Brukner, Quantum correlations with no causal order, Nature Communications 3, 1092 (2012).
  * Hardy [2005] L. Hardy, Probability theories with dynamic causal structure: A new framework for quantum gravity (2005), arXiv:gr-qc/0509120 [gr-qc] .
  * Wechs  _et al._ [2021] J. Wechs, H. Dourdent, A. A. Abbott, and C. Branciard, Quantum circuits with classical versus quantum control of causal order, PRX Quantum 2, 030335 (2021).
  * Araújo  _et al._ [2014] M. Araújo, F. Costa, and i. c. v. Brukner, Computational advantage from quantum-controlled ordering of gates, Phys. Rev. Lett. 113, 250402 (2014).
  * Quintino  _et al._ [2019] M. T. Quintino, Q. Dong, A. Shimbo, A. Soeda, and M. Murao, Reversing unknown quantum transformations: Universal quantum circuit for inverting general unitary operations, Phys. Rev. Lett. 123, 210502 (2019).
  * Renner and Brukner [2022] M. J. Renner and i. c. v. Brukner, Computational advantage from a quantum superposition of qubit gate orders, Phys. Rev. Lett. 128, 230503 (2022).
  * Guérin  _et al._ [2016] P. A. Guérin, A. Feix, M. Araújo, and i. c. v. Brukner, Exponential communication complexity advantage from quantum superposition of the direction of communication, Phys. Rev. Lett. 117, 100502 (2016).
  * Ebler  _et al._ [2018] D. Ebler, S. Salek, and G. Chiribella, Enhanced communication with the assistance of indefinite causal order, Phys. Rev. Lett. 120, 120502 (2018).
  * Wei  _et al._ [2019] K. Wei, N. Tischler, S.-R. Zhao, Y.-H. Li, J. M. Arrazola, Y. Liu, W. Zhang, H. Li, L. You, Z. Wang, Y.-A. Chen, B. C. Sanders, Q. Zhang, G. J. Pryde, F. Xu, and J.-W. Pan, Experimental quantum switching for exponentially superior quantum communication complexity, Phys. Rev. Lett. 122, 120504 (2019).
  * Zhao  _et al._ [2020] X. Zhao, Y. Yang, and G. Chiribella, Quantum metrology with indefinite causal order, Phys. Rev. Lett. 124, 190503 (2020).
  * Chapeau-Blondeau [2021a] F. m. c. Chapeau-Blondeau, Quantum parameter estimation on coherently superposed noisy channels, Phys. Rev. A 104, 032214 (2021a).
  * Xie  _et al._ [2021] D. Xie, C. Xu, and A. Wang, Quantum metrology with coherent superposition of two different coded channels, Chinese Physics B 30, 090304 (2021).
  * Chapeau-Blondeau [2022] F. Chapeau-Blondeau, Indefinite causal order for quantum metrology with quantum thermal noise, Physics Letters A 447, 128300 (2022).
  * Yin  _et al._ [2023] P. Yin, X. Zhao, Y. Yang, Y. Guo, W.-H. Zhang, G.-C. Li, Y.-J. Han, B.-H. Liu, J.-S. Xu, G. Chiribella, G. Chen, C.-F. Li, and G.-C. Guo, Experimental super-heisenberg quantum metrology with indefinite gate order (2023), arXiv:2303.17223 [quant-ph] .
  * Chiribella  _et al._ [2021] G. Chiribella, M. Banik, S. S. Bhattacharya, T. Guha, M. Alimuddin, A. Roy, S. Saha, S. Agrawal, and G. Kar, Indefinite causal order enables perfect quantum communication with zero capacity channels, New Journal of Physics 23, 033039 (2021).
  * Abbott  _et al._ [2020] A. A. Abbott, J. Wechs, D. Horsman, M. Mhalla, and C. Branciard, Communication through coherent control of quantum channels, Quantum 4, 333 (2020).
  * Guérin  _et al._ [2019] P. A. Guérin, G. Rubino, and i. c. v. Brukner, Communication through quantum-controlled noise, Phys. Rev. A 99, 062317 (2019).
  * Jia and Costa [2019] D. Jia and F. Costa, Causal order as a resource for quantum communication, Phys. Rev. A 100, 052319 (2019).
  * Liu  _et al._ [2023] Q. Liu, Z. Hu, H. Yuan, and Y. Yang, Optimal strategies of quantum metrology with a strict hierarchy, Phys. Rev. Lett. 130, 070803 (2023).
  * Chapeau-Blondeau [2021b] F. m. c. Chapeau-Blondeau, Noisy quantum metrology with the assistance of indefinite causal order, Phys. Rev. A 103, 032615 (2021b).
  * Procopio  _et al._ [2015] L. M. Procopio, A. Moqanaki, M. Araújo, F. Costa, I. Alonso Calafell, E. G. Dowd, D. R. Hamel, L. A. Rozema, Č. Brukner, and P. Walther, Experimental superposition of orders of quantum gates, Nature Communications 6, 7913 (2015).
  * Rubino  _et al._ [2017] G. Rubino, L. A. Rozema, A. Feix, M. Araújo, J. M. Zeuner, L. M. Procopio, Časlav Brukner, and P. Walther, Experimental verification of an indefinite causal order, Science Advances 3, e1602589 (2017), https://www.science.org/doi/pdf/10.1126/sciadv.1602589 .
  * Araújo  _et al._ [2015] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Časlav Brukner, Witnessing causal nonseparability, New Journal of Physics 17, 102001 (2015).
  * Goswami  _et al._ [2018] K. Goswami, C. Giarmatzi, M. Kewming, F. Costa, C. Branciard, J. Romero, and A. G. White, Indefinite causal order in a quantum switch, Phys. Rev. Lett. 121, 090503 (2018).
  * Goswami and Romero [2020] K. Goswami and J. Romero, Experiments on quantum causality, AVS Quantum Science 2, 037101 (2020).
  * Guo  _et al._ [2020] Y. Guo, X.-M. Hu, Z.-B. Hou, H. Cao, J.-M. Cui, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, and G. Chiribella, Experimental transmission of quantum information using a superposition of causal orders, Phys. Rev. Lett. 124, 030502 (2020).
  * Rubino  _et al._ [2021] G. Rubino, L. A. Rozema, D. Ebler, H. Kristjánsson, S. Salek, P. Allard Guérin, A. A. Abbott, C. Branciard, i. c. v. Brukner, G. Chiribella, and P. Walther, Experimental quantum communication enhancement by superposing trajectories, Phys. Rev. Res. 3, 013093 (2021).
  * Taddei  _et al._ [2021] M. M. Taddei, J. Cariñe, D. Martínez, T. García, N. Guerrero, A. A. Abbott, M. Araújo, C. Branciard, E. S. Gómez, S. P. Walborn, L. Aolita, and G. Lima, Computational advantage from the quantum superposition of multiple temporal orders of photonic gates, PRX Quantum 2, 010320 (2021).
  * Caleffi and Cacciapuoti [2020] M. Caleffi and A. S. Cacciapuoti, Quantum switch for the quantum internet: Noiseless communications through noisy channels, IEEE Journal on Selected Areas in Communications 38, 575 (2020).
  * Felce and Vedral [2020] D. Felce and V. Vedral, Quantum refrigeration with indefinite causal order, Phys. Rev. Lett. 125, 070603 (2020).
  * Guha  _et al._ [2020] T. Guha, M. Alimuddin, and P. Parashar, Thermodynamic advancement in the causally inseparable occurrence of thermal maps, Phys. Rev. A 102, 032215 (2020).
  * Simonov  _et al._ [2022] K. Simonov, G. Francica, G. Guarnieri, and M. Paternostro, Work extraction from coherently activated maps via quantum switch, Phys. Rev. A 105, 032217 (2022).
  * Dieguez  _et al._ [2023] P. R. Dieguez, V. F. Lisboa, and R. M. Serra, Thermal devices powered by generalized measurements with indefinite causal order, Phys. Rev. A 107, 012423 (2023).
  * Giacomini  _et al._ [2016] F. Giacomini, E. Castro-Ruiz, and Časlav Brukner, Indefinite causal structures for continuous-variable systems, New Journal of Physics 18, 113026 (2016).
  * Alicki and Kosloff [2018] R. Alicki and R. Kosloff, Introduction to quantum thermodynamics: History and prospects, in  _Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions_, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer International Publishing, Cham, 2018) pp. 1–33.
  * Haag  _et al._ [1974] R. Haag, D. Kastler, and E. B. Trych-Pohlmeyer, Stability and equilibrium states, Communications in Mathematical Physics 38, 173 (1974).
  * Lenard [1978] A. Lenard, Thermodynamical proof of the Gibbs formula for elementary quantum systems, Journal of Statistical Physics 19, 575 (1978).
  * Pusz and Woronowicz [1978] W. Pusz and S. L. Woronowicz, Passive states and KMS states for general quantum systems, Communications in Mathematical Physics 58, 273 (1978).
  * Allahverdyan  _et al._ [2004] A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, Maximal work extraction from finite quantum systems, Europhysics Letters 67, 565 (2004).
  * Šafránek  _et al._ [2023] D. Šafránek, D. Rosa, and F. C. Binder, Work extraction from unknown quantum sources, Phys. Rev. Lett. 130, 210401 (2023).
  * Alicki and Fannes [2013] R. Alicki and M. Fannes, Entanglement boost for extractable work from ensembles of quantum batteries, Phys. Rev. E 87, 042123 (2013).
  * Nielsen and Chuang [2010] M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information: 10th Anniversary Edition_ (Cambridge University Press, 2010).
  * Agarwal [2012] G. S. Agarwal,  _Quantum Optics_ (Cambridge University Press, 2012).
  * Nieto and Truax [1997] M. M. Nieto and D. R. Truax, Holstein-primakoff/bogoliubov transformations and the multiboson system, Fortschritte der Physik/Progress of Physics 45, 145 (1997).

##  Appendix A Displacement operators

The joint energy variation of system plus control after applying the quantum
switch of two displacement operators is equal to ($\bra{1}H_{C}\ket{0}=t^{*}$,
$\bra{0}\rho_{C}\ket{1}=e^{i\varphi_{C}}\sin(\theta_{C}/2)\cos(\theta_{C}/2)$):

$\displaystyle\Delta_{\text{QS}}$ $\displaystyle=$
$\displaystyle\Delta_{S}+2\real\\{\bra{0}\rho_{C}\outerproduct{1}{1}H_{C}\ket{0}(\chi-1)\\}$
(70) $\displaystyle=$
$\displaystyle\Delta_{S}+\sin\theta_{C}\real\\{e^{i\varphi_{C}}t^{*}(\chi-1)\\}$
(71)

where

$\Delta_{S}=\cos^{2}\left(\frac{\theta_{C}}{2}\right)E_{12}+\sin^{2}\left(\frac{\theta_{C}}{2}\right)E_{21}-E_{S}$
(72)

with

$E_{12}=\tr\\{D(\alpha_{2})D(\alpha_{1})\rho_{S}D^{\dagger}(\alpha_{1})D^{\dagger}(\alpha_{2})H_{S}\\},$
(73)
$E_{21}=\tr\\{D(\alpha_{1})D(\alpha_{2})\rho_{S}D^{\dagger}(\alpha_{2})D^{\dagger}(\alpha_{1})H_{S}\\},$
(74) $\displaystyle E_{S}$ $\displaystyle=$
$\displaystyle\tr\\{\rho_{S}H_{S}\\}$ (75) $\displaystyle=$
$\displaystyle\omega(1-e^{-\beta\omega})\sum_{n}e^{-\beta\omega
n}\bra{n}\left(a^{\dagger}a+\frac{\mathds{1}}{2}\right)\ket{n}$ (76)
$\displaystyle=$ $\displaystyle\omega\left(\langle
n\rangle_{\text{th}}+\frac{1}{2}\right).$ (77)

where

$\langle n\rangle_{\text{th}}=\frac{1}{e^{\beta\omega}-1}$

is the thermal boson occupation number (or _Bose-Einstein distribution_ with
zero chemical potential) and $\chi$ is connected to the cross-map unitary
(check main text for definition). Moreover,

$\displaystyle E_{21}$ $\displaystyle=$
$\displaystyle\tr\\{D(\alpha_{1})D(\alpha_{2})\rho_{S}D^{\dagger}(\alpha_{2})D^{\dagger}(\alpha_{1})H_{S}\\}$
(78) $\displaystyle=$
$\displaystyle\frac{\omega}{2}+\omega\tr\\{\rho_{S}D^{\dagger}(\alpha_{2})D^{\dagger}(\alpha_{1})a^{\dagger}aD(\alpha_{1})D(\alpha_{2})\\}$
(79) $\displaystyle=$ $\displaystyle E_{S}+\omega|\alpha^{\prime}|^{2}$ (80)

where $\alpha^{\prime}:=\alpha_{1}+\alpha_{2}$ and we used the fact that
$D^{\dagger}(\alpha)a^{\dagger}D(\alpha)=a^{\dagger}+\alpha^{*}$,
$D^{\dagger}(\alpha)aD(\alpha)=a+\alpha$ [45]. Similarly,

$\displaystyle E_{12}$ $\displaystyle=$
$\displaystyle\tr\\{D(\alpha_{2})D(\alpha_{1})\rho_{S}D^{\dagger}(\alpha_{1})D^{\dagger}(\alpha_{2})H_{S}\\}$
(81) $\displaystyle=$ $\displaystyle E_{S}+\omega|\alpha^{\prime}|^{2}.$ (82)

Moreover,

$\displaystyle\chi$ $\displaystyle=$
$\displaystyle\tr\\{D(\alpha_{2})D(\alpha_{1})\rho_{S}D^{\dagger}(\alpha_{2})D^{\dagger}(\alpha_{1})\\}$
(83) $\displaystyle=$
$\displaystyle\tr\\{e^{\alpha_{1}^{*}\alpha_{2}-\alpha_{1}\alpha_{2}^{*}}D(\alpha_{1})D(\alpha_{2})\rho_{S}D^{\dagger}(\alpha_{2})D^{\dagger}(\alpha_{1})\\}$
(84) $\displaystyle=$ $\displaystyle
e^{\alpha_{1}^{*}\alpha_{2}-\alpha_{1}\alpha_{2}^{*}}$ (85)

in which we use the following relation [45]:

$D(\alpha_{2})D(\alpha_{1})=e^{\alpha_{1}^{*}\alpha_{2}-\alpha_{1}\alpha_{2}^{*}}D(\alpha_{1})D(\alpha_{2}).$

Hence

$\Delta_{\text{QS}}=\omega|\alpha^{\prime}|^{2}+|t|(\text{cos}(\theta-\varphi_{C}+2|\alpha_{1}||\alpha_{2}|\sin(\phi_{1}-\phi_{2}))-\cos(\theta-\varphi_{C}))\sin\theta_{C}$
(86)

After measuring the control qubit, the final state of the system contains
coherence terms which contribute to the final energy of the system. The final
energy difference of the system post-measurement is written as

$\Delta_{S,M}=\frac{1}{N_{M}}\bigg{(}\cos^{2}\left(\frac{\theta_{C}}{2}\right)\cos^{2}\left(\frac{\theta_{M}}{2}\right)\Delta_{12}+\sin^{2}\left(\frac{\theta_{C}}{2}\right)\sin^{2}\left(\frac{\theta_{M}}{2}\right)\Delta_{21}+\frac{1}{2}\sin\theta_{C}\sin\theta_{M}\real\\{\Delta_{F}e^{i(\varphi_{C}+\varphi_{M})}\\}\bigg{)}$
(87)

where

$N_{M}=\frac{1}{2}(1+\cos\theta_{C}\cos\theta_{M}+\sin\theta_{C}\sin\theta_{M}\real\\{\chi
e^{i(\varphi_{C}+\varphi_{M})})\\}$ (88)

is a normalization constant and

$\Delta_{12}:=\tr\\{D(\alpha_{2})D(\alpha_{1})\rho_{S}D^{\dagger}(\alpha_{1})D^{\dagger}(\alpha_{2})\\}-E_{S}$
(89)
$\Delta_{21}:=\tr\\{D(\alpha_{1})D(\alpha_{2})\rho_{S}D^{\dagger}(\alpha_{2})D^{\dagger}(\alpha_{1})\\}-E_{S}$
(90) $\Delta_{F}:=F_{S}-\chi E_{S}$ (91)

being $F_{S}$ defined by

$F_{S}:=\tr\\{D(\alpha_{2})D(\alpha_{1})\rho_{S}D^{\dagger}(\alpha_{2})D^{\dagger}(\alpha_{1})H_{S}\\}.$
(92)

As a first step to calculate $\Delta_{S,M}$, we find that

$\displaystyle F_{S}$ $\displaystyle=$
$\displaystyle\tr\\{D(\alpha_{2})D(\alpha_{1})\rho_{S}D^{\dagger}(\alpha_{2})D^{\dagger}(\alpha_{1})H_{S}\\}$
(93) $\displaystyle=$
$\displaystyle\chi\tr\\{D(\alpha_{1})D(\alpha_{2})\rho_{S}D^{\dagger}(\alpha_{2})D^{\dagger}(\alpha_{1})H_{S}\\}$
(94) $\displaystyle=$ $\displaystyle\chi E_{21}$ (95) $\displaystyle=$
$\displaystyle\chi(E_{S}+\omega|\alpha^{\prime}|^{2})$ (96)

and then

$\Delta_{F}=\chi\omega|\alpha^{\prime}|^{2}$ (97)

The values of $E_{12}$ and $E_{21}$ are the same as before, hence

$\Delta_{12}=\Delta_{21}=\omega|\alpha^{\prime}|^{2}.$ (98)

Finally, putting everything together:

$\Delta_{S,M}=\omega|\alpha^{\prime}|^{2}\geq 0$ (99)

and then it is impossible to activate the state of the system by measuring the
state of the control, for whatever choice of measurement state. This result is
a consequence of the fact that two displacement operators have a very specific
commutation relation,

$[D(\alpha_{1}),D(\alpha_{2})]=(1-e^{\alpha_{1}^{*}\alpha_{2}-\alpha_{1}\alpha_{2}^{*}})D(\alpha_{1})D(\alpha_{2}).$
(100)

The physical implication of this property is discussed in the main text.

##  Appendix B Displacement operator and squeeze operator

In the case that one of the unitaries being the displacement operator and the
squeeze operator the final energy difference is

$\Delta_{\text{QS}}=\Delta_{S}+\sin\theta_{C}\real\\{e^{i\varphi_{C}}t^{*}(\chi-1)\\}$
(101)

where

$\Delta_{S}=\cos^{2}\left(\frac{\theta_{C}}{2}\right)E_{12}+\sin^{2}\left(\frac{\theta_{C}}{2}\right)E_{21}-E_{S}$
(102)

and

$E_{12}=\tr\\{S(z)D(\alpha)\rho_{S}D^{\dagger}(\alpha)S^{\dagger}(z)H_{S}\\},$
(103)
$E_{21}=\tr\\{D(\alpha)S(z)\rho_{S}S^{\dagger}(z)D^{\dagger}(\alpha)H_{S}\\},$
(104) $\chi=\tr\\{S(z)D(\alpha)\rho_{S}S^{\dagger}(z)D^{\dagger}(\alpha)\\}.$
(105)

Applying the relations
$D^{\dagger}(\alpha)a^{\dagger}D(\alpha)=a^{\dagger}+\alpha^{*}$,
$D^{\dagger}(\alpha)aD(\alpha)=a+\alpha$,
$S^{\dagger}(z)a^{\dagger}S(z)=a^{\dagger}\cosh|z|+ae^{-i\xi}\sinh|z|$ and
$S^{\dagger}(z)aS(z)=a\cosh|z|+a^{\dagger}e^{i\xi}\sinh|z|$ [45], one finds
that

$E_{21}=\omega|\alpha|^{2}+\frac{\omega}{2}\left(2\langle
n\rangle_{\text{th}}+1\right)\cosh(2|z|)$ (106)
$E_{12}=E_{21}+\omega|\alpha|^{2}\cos(\xi-2\phi)\sinh(2|z|).$ (107)

Therefore,

$\Delta_{S}=-\frac{\omega}{2}+\omega|\alpha|^{2}+\frac{\omega\cosh(2|z|)}{2}+2\omega\langle
n\rangle_{\text{th}}\sinh^{2}|z|+\omega|\alpha|^{2}\cos^{2}\left(\frac{\theta_{C}}{2}\right)\cos(\xi-2\phi)\sinh(2|z|).$
(108)

In order to calculate $\chi$, we use the braiding relation
$D(\alpha)S(z)=S(z)D(\gamma)$, where
$\gamma=|\alpha|e^{i\phi}\cosh|z|-|\alpha|e^{i(\xi-\phi)}\sinh|z|$ [46]. By
applying the _P-representation_ of the thermal state [45],

$\rho_{S}=\int P_{T}(\eta)\outerproduct{\eta}{\eta}\differential^{2}\eta$

where $\ket{\eta}$ is a coherent state and

$P_{T}(\eta)=\frac{1}{\pi\langle n\rangle_{\text{th}}}e^{-|\eta|^{2}/\langle
n\rangle_{\text{th}}}$ (109)

then

$\chi=\int
P_{T}(\eta)\bra{\eta}D^{\dagger}(\gamma)D(\alpha)\ket{\eta}\differential^{2}\eta=\int
P_{T}(\eta)e^{\frac{\eta}{2}(\gamma^{*}-\alpha^{*})}e^{\frac{\eta^{*}}{2}(\alpha-\gamma)}\innerproduct{\eta+\gamma}{\eta+\alpha}\differential^{2}\eta$
(110)

hence, by evaluating the integral:

$\chi=\innerproduct{\gamma}{\alpha}e^{-\langle
n\rangle_{\text{th}}|\alpha-\gamma|^{2}}$ (111)

in which we used the fact that for two coherent states [45]:

$\innerproduct{\alpha}{\beta}=e^{\alpha^{*}\beta-|\alpha|^{2}/2-|\beta|^{2}/2}.$

Finally:

$\Delta_{\text{QS}}=\frac{\omega}{2}+\omega|\alpha|^{2}+\frac{\omega\cosh(2|z|)}{2}+2\omega\langle
n\rangle_{\text{th}}\sinh^{2}|z|+\omega|\alpha|^{2}\cos^{2}\left(\frac{\theta_{C}}{2}\right)\cos(\xi-2\phi)\sinh(2|z|)+|t|\sin\theta_{C}\real\\{e^{i(\varphi_{C}-\theta)}(\chi-1)\\}.$
(112)

Once again, after measuring the control qubit, the final state of the system
has extra terms add to the final energy of the system. The final energy of the
system post-measurement is written as

$\Delta_{S,M}=\frac{1}{N_{M}}\bigg{(}\cos^{2}\left(\frac{\theta_{C}}{2}\right)\cos^{2}\left(\frac{\theta_{M}}{2}\right)\Delta_{12}+\sin^{2}\left(\frac{\theta_{C}}{2}\right)\sin^{2}\left(\frac{\theta_{M}}{2}\right)\Delta_{21}+\frac{1}{2}\sin\theta_{C}\sin\theta_{M}\real\\{\Delta_{F}e^{i(\varphi_{C}+\varphi_{M})}\\}\bigg{)}$
(113)

where

$N_{M}=\frac{1}{2}(1+\cos\theta_{C}\cos\theta_{M}+\sin\theta_{C}\sin\theta_{M}\real\\{\chi
e^{i(\varphi_{C}+\varphi_{M})})\\}$ (114)
$\Delta_{12}:=\tr\\{S(z)D(\alpha)\rho_{S}D^{\dagger}(\alpha)S^{\dagger}(z)\\}-E_{S}$
(115)
$\Delta_{21}:=\tr\\{D(\alpha)S(z)\rho_{S}S^{\dagger}(z)D^{\dagger}(\alpha)\\}-E_{S}$
(116)

and

$\Delta_{F}:=F_{S}-\chi E_{S}$ (117)

being $F_{S}$ defined by

$\displaystyle F_{S}$ $\displaystyle:=$
$\displaystyle\tr\\{S(z)D(\alpha)\rho_{S}S^{\dagger}(z)D^{\dagger}(\alpha)H_{S}\\}$
(118) $\displaystyle=$
$\displaystyle\omega\frac{\chi}{2}+\tr\\{S(z)D(\alpha)\rho_{S}S^{\dagger}(z)D^{\dagger}(\alpha)a^{\dagger}a\\}$
(119)

which, by using the previous braiding relation is equal to

$F_{S}=\omega\frac{\chi}{2}+\int
P_{T}(\eta)\bra{\eta}D^{\dagger}(\gamma)a^{\dagger}aD(\alpha)\ket{\eta}\differential^{2}\eta$
(120)

with

$\bra{\eta}D^{\dagger}(\gamma)a^{\dagger}aD(\alpha)\ket{\eta}=e^{\frac{\eta}{2}(\gamma^{*}-\alpha^{*})}e^{\frac{\eta^{*}}{2}(\alpha-\gamma)}(\eta^{*}+\gamma^{*})(\eta+\alpha)\innerproduct{\eta+\gamma}{\eta+\alpha}$

since coherent states are eigenvectors of the annihilation operator,

$a\ket{\alpha}=\alpha\ket{\alpha}.$

Then, after computing the integral in the complex plane, one has

$F_{S}=\omega\chi\Bigg{(}\frac{1}{2}+\gamma^{*}\alpha+(1+2\gamma^{*}\alpha-|\alpha|^{2}-|\gamma|^{2})\langle
n\rangle_{\text{th}}-|\alpha-\gamma|^{2}\langle
n\rangle^{2}_{\text{th}}\Bigg{)}$ (121)

and

$\Delta_{F}=\omega\chi(\gamma^{*}\alpha+(2\gamma^{*}\alpha-|\gamma|^{2}-|\alpha|^{2})\langle
n\rangle_{\text{th}}-|\alpha-\gamma|^{2}\langle n\rangle^{2}_{\text{th}}).$
(122)

Moreover, using the values of $E_{12}$ and $E_{21}$, one obtains

$\Delta_{21}=\omega|\alpha|^{2}+\omega\langle
n\rangle_{\text{th}}(\cosh(2|z|)-1)+\omega\sinh^{2}|z|$ (123)
$\Delta_{12}=\Delta_{21}+\omega|\alpha|^{2}\cos(\xi-2\phi)\sinh(2|z|)$ (124)

and finally

$\displaystyle\Delta_{S,M}$
$\displaystyle=\frac{1}{N_{M}}\Bigg{[}\frac{\omega}{4}(1+\cos\theta_{C}\cos\theta_{M})(2|\alpha|^{2}+(2\langle
n\rangle_{\text{th}}+1)(\cosh(2|z|)-1))$ (125)
$\displaystyle+\omega|\alpha|^{2}\cos^{2}\left(\frac{\theta_{C}}{2}\right)\cos^{2}\left(\frac{\theta_{M}}{2}\right)\cos(\xi-2\phi)\sinh(2|z|)+\frac{1}{2}\sin\theta_{C}\sin\theta_{M}\real\\{\Delta_{F}e^{i(\varphi_{C}+\varphi_{M})}\\}\Bigg{]}$

which taking the cases (i) $\xi-2\phi=0$:

$\displaystyle\Delta_{S,M}^{0}$
$\displaystyle=\frac{1}{N_{M}^{0}}\Bigg{[}\frac{\omega}{4}(1+\cos\theta_{C}\cos\theta_{M})(2|\alpha|^{2}+(2\langle
n\rangle_{\text{th}}+1)(\cosh(2|z|)-1))+\omega|\alpha|^{2}\cos^{2}\left(\frac{\theta_{C}}{2}\right)\cos^{2}\left(\frac{\theta_{M}}{2}\right)\sinh(2|z|)$
(126)
$\displaystyle-\frac{\omega|\alpha|^{2}}{2}\sin\theta_{C}\sin\theta_{M}e^{-2|z|-|\alpha|^{2}e^{-|z|}(2\langle
n\rangle_{\text{th}}+1)(\cosh|z|-1)}\bigg{(}\langle
n\rangle^{2}_{\text{th}}(e^{2|z|}-2e^{|z|}+1)+\langle
n\rangle_{\text{th}}(e^{2|z|}-2e^{|z|}+|\alpha|^{2}e^{-2|z|})-e^{|z|}\bigg{)}\cos(\varphi_{C}+\varphi_{M})\Bigg{]}$

where

$N_{M}^{0}=\frac{1}{2}\Bigg{(}1+\cos\theta_{C}\cos\theta_{M}+\sin\theta_{C}\sin\theta_{M}e^{-2|\alpha|^{2}\sinh^{2}(|z|/2)(\cosh|z|-\sinh|z|)}\cos(\varphi_{C}+\varphi_{M})\Bigg{)}$
(127)

and (ii) $\xi-2\phi=\pi$:

$\displaystyle\Delta_{S,M}^{\pi}$
$\displaystyle=\frac{1}{N_{M}^{\pi}}\Bigg{[}\frac{\omega}{4}(1+\cos\theta_{C}\cos\theta_{M})(2|\alpha|^{2}+(2\langle
n\rangle_{\text{th}}+1)(\cosh(2|z|)-1))-\omega|\alpha|^{2}\cos^{2}\left(\frac{\theta_{C}}{2}\right)\cos^{2}\left(\frac{\theta_{M}}{2}\right)\sinh(2|z|)$
(128)
$\displaystyle-\frac{\omega|\alpha|^{2}}{2}\sin\theta_{C}\sin\theta_{M}e^{-\frac{|\alpha|^{2}}{2}(e^{|z|}-1)^{2}(2\langle
n\rangle_{\text{th}}+1)}\bigg{(}\langle
n\rangle^{2}_{\text{th}}(e^{2|z|}-2e^{|z|}+1)+\langle
n\rangle_{\text{th}}(|\alpha|^{2}e^{4|z|}-2e^{|z|}+1)-e^{|z|}\bigg{)}\cos(\varphi_{C}+\varphi_{M})\Bigg{]}$

with

$N_{M}^{\pi}=\frac{1}{2}\Bigg{(}1+\cos\theta_{C}\cos\theta_{M}+\sin\theta_{C}\sin\theta_{M}e^{-\frac{|\alpha|^{2}}{2}(e^{|z|}-1)^{2}(2\langle
n\rangle_{\text{th}}+1)}\cos(\varphi_{C}+\varphi_{M})\Bigg{)}.$ (129)