# Quadratic enhancement in the reliability of collective quantum engines

Noufal Jaseem  Department of Physical Sciences, Indian Institute of Science
Education and Research Berhampur, Berhampur 760010, India.  Sai Vinjanampathy
Department of Physics, Indian Institute of Technology-Bombay, Powai, Mumbai
400076, India.  Centre for Quantum Technologies, National University of
Singapore, 3 Science Drive 2, 117543 Singapore, Singapore.  Victor Mukherjee
Department of Physical Sciences, Indian Institute of Science Education and
Research Berhampur, Berhampur 760010, India.

###### Abstract

We study fluctuations in many-body quantum heat engines operating in the
presence of collective system-bath interactions. We show that collective
effects in open quantum systems can be harnessed to develop highly consistent
many-body quantum engines. We consider quantum Otto engines, modeled by $n$
spins collectively coupled to thermal baths. Our results show that collective
effects can significantly reduce the fluctuations in the output work,
quantified by high reliability ($r$) and low thermodynamic uncertainty. In
contrast to independent engines, we demonstrate a quadratic enhancement of the
reliability $r$ for their collective counterparts. We extend our analysis to
the case of interacting spin models commonly studied in many-body physics,
such as the Lipkin-Meshkov-Glick (LMG) model, thereby broadening the regime of
applicability of collective effects in quantum thermal machines significantly.
This paves the way forward for realistic collective quantum thermal machines
in many body systems.

##  I Introduction

Modeling and control of quantum systems are crucial for the development of
quantum technologies that exhibit an advantage over their classical analogs
[1, 2, 3, 4, 5, 6, 7] such as computing, sensors and quantum thermal machines
[10, 11, 12, 14, 13, 15, 16, 17, 18, 19, 20, 21, 22, 1, 2, 3, 4, 5, 6, 7, 8,
9]. Each of these technologies is assessed by a performance benchmark, for
instance, gate complexity for computing [23, 24] and variance of an unbiased
estimator for sensors [14, 15, 16, 17, 18, 19, 20, 21, 22]. The performance of
quantum thermal machines, on the other hand, has been evaluated through three
key indicators, namely average efficiency, average output power, and more
recently reliability ($r$), quantified through the ratio of the average work
to its root-mean-square deviation [25]. In this regard, various avenues
related to quantum correlations in small quantum systems, such as non-
Markovianity [26, 27, 28, 29, 30], optimal paths [31, 32, 33, 34, 35] and non-
thermal baths [36, 37] have been shown to enhance the performance of quantum
machines. Following such theoretical investigations, several experimental
realizations [38, 39, 40, 41, 42] of microscopic thermal machines have led to
significant advancements in the fields of quantum thermodynamics and quantum
technologies.

Many-body quantum systems offer unique opportunities to engineer quantum
technologies operating in the presence of non-trivial many-body effects, such
as phase transitions [43, 44, 45], localization-delocalization transitions
[46, 47] and cooperative effects arising due to collective interactions with
the environment [48]. Interestingly, collective effects have been shown to be
highly beneficial in many quantum technologies [49, 50, 51, 52, 53, 54, 55],
and have been used to enhance efficiency [56] and work output [48, 57, 51] in
quantum heat engines, and perform high-precision quantum thermometry [51].
What remains an open question is if quantum correlations arising due to the
environment can improve the consistency of quantum thermal machines. In this
manuscript, we answer this crucial question in the affirmative and show that
cooperative quantum effects can stabilize normalized fluctuations of a many-
body engine better than their non-cooperative analogs. In general, reliability
$r$ in machines composed of $n$ subsystems scales as $\sqrt{n}$ [58]. This
leads to increased fluctuations in the microscopic regime, in machines
comprising a finite number of particles. In contrast, here we show that
collective effects can result in the reliability $r$ scaling as $n$, thus
making collective quantum heat engines highly reliable, and paving a possible
way for developing realistic and reliable quantum technologies. We begin by
reviewing a generic model of a quantum engine and discuss the metrics for
quantifying the performance of such an engine. For a quantum engine based on a
generic many-body working medium (WM), we show that a broad class of steady-
states results in a $r\sim n$ scaling of the reliability, thus emphasizing the
wide applicability of our results. We then exemplify our generic theory using
specific examples of non-interacting as well as interacting multi-spin WMs.
Finally, we summarize our main results.

##  II Reliable Collective Quantum Thermal Machines

The performance of quantum thermal machines is assessed based on a variety of
performance measures that examine the quality, quantity, and time in which a
thermodynamic task is performed. In the case of quantum engines, two well-
known metrics are output work and the efficiency of the engine. The third
metric of performance is reliability, which measures the quality of output
work considering it to be a fluctuating quantity. Reliability is defined as
$r=\langle W\rangle/\Delta W$, where $\langle
W^{p}\rangle:=\int_{-\infty}^{\infty}dWP(W)W^{p}$ is the moment of the work
distribution function and $\Delta W:=\sqrt{\langle W^{2}\rangle-\langle
W\rangle^{2}}$ is the standard deviation. Reliability hence measures the
consistency of the output work. Reliability was investigated in a series of
studies [59, 60, 61, 62, 63, 64, 65, 66]. For instance, in [59] the impacts of
diagnostic schemes to determine the performance of quantum Otto heat engines
on different figures of merit such as reliability were explored. In [60], a
two-qudit SWAP engine was studied and it was shown that for fermionic SWAP
engines the reliability is bound by
$r^{2}\leq\langle\Sigma\rangle/(2-\langle\Sigma\rangle)$, where
$\langle\Sigma\rangle$ is the entropy production. The bosonic SWAP engines, on
the other hand, obeys $r^{2}\leq\langle\Sigma\rangle/(2+\langle\Sigma\rangle)$
[61].

Figure 1:  (a) Schematic diagram of an Otto engine with an $n$-spin WM
interacting collectively with thermal baths. The energy gap of the engine
system is varied along the paths A $\to$ B and C $\to$ D, while the system
interacts collectively with baths along paths B $\to$ C and D $\to$ A. (b) The
ratio of reliability of the collective engine to that of the independent
engine, $\lambda_{r}$, is plotted against $n$ for the different Hamiltonian
given in Eqs. (1) and (2). Inset: $\lambda_{r}^{(1)}$ is plotted as a function
of $n$ and $T_{\rm h}$, for $x=1$. The advantage of collective QHEs over
independent ones becomes more pronounced for large $T_{\rm h}$ and $n$. The
green line is the contour line for $\lambda_{r}=1$. (c) The left and right-
hand sides of the TUR, in Eq.(8), are plotted as a function of the number of
spins $n$ for the collective and independent cases. For the independent case,
$\mathcal{Q}_{\rm ind}\geq 2-\langle\Sigma_{1}^{(1)}\rangle$, where
$\langle\Sigma_{1}^{(1)}\rangle$ is the single qubit entropy production (see
Appendix F). Inset: $\mathcal{Q}_{\rm col}^{(1)}$ is plotted as a function of
$n$ and $T_{\rm h}$. Low $\mathcal{Q}_{\rm col}^{(1)}$ for large $n$ and
$T_{\rm h}$ implies the collective engine is most beneficial in this regime.
The green line is the contour line for $\mathcal{Q}_{\rm col}^{(1)}=2$. For
both (a) and (b), the parameter values are fixed as $\omega_{h}=0.5$,
$\omega_{c}=0.1$, $\gamma=0.7$, and
$\Delta=\beta_{c}\omega_{c}-\beta_{h}\omega_{h}=0.005$. For the main plots,
$\beta_{h}\to 0$ and for the insets, both $T_{\rm h}$ and $T_{\rm c}$ are
varied so that $\Delta$ is kept constant. Superscripts label $x=1,2$ or LMG.

The different moments of the work distribution function are related to each
other via the fluctuation-dissipation theorem, which can be formalized in
terms of a thermodynamic uncertainty relation (TUR) [67, 66, 68, 60, 61]. A
natural question that arises is whether the reliability of the engine can be
enhanced by collective interactions. Consider $n$ non-interacting quantum
thermal machines. It is easy to see that the work and variance scale
extensively with the size of the system [58]. This implies that the
reliability scales as $\mathcal{O}(\sqrt{n})$ which can be understood as the
lower bound imposed by non-interacting systems [58]. In contrast to this, here
we consider a generic many-body quantum engine collectively coupled to thermal
baths, as depicted in Fig. 1, operating near the Carnot efficiency, $\eta_{\rm
C}$. It was previously established that for non-interacting collective quantum
heat engines the mean output work scales as the collective heat capacity
$C^{\rm col}$ in this case [51], see the Appendix A. In this manuscript, we
establish that, for a more generic setup, the variance of the output work also
scales with $C^{\rm col}$, which implies that so does $r^{2}$. Now, to
understand the collective advantage in the reliability, let the generic non-
equilibrium steady state of a $n$-body quantum engine operating near Carnot
efficiency be written as $\rho(\beta)=\exp(-\beta
n\mathcal{A})/\mbox{tr}\left[\exp(-\beta n\mathcal{A})\right]$ where $\beta$
is the inverse temperature of a bath and $\mathcal{A}$ is an operator in the
collective basis which does not scale with system size. In this case, as we
show below for appropriate $\mathcal{A}$, $C^{\rm col}\propto{n}^{2}$ and
hence the reliability scales as $n$ (see Appendices C, D, and E for details).
This is a quadratic enhancement in the reliability of generic collective
quantum thermal machines.

##  III Many Body Otto Engine and Statistics

To illustrate and formalize our general result, we consider an Otto cycle of a
many-body collective heat engine. The Otto cycle, described in Fig. 1 consists
of four strokes, namely the following. (i) (A $\to$ B): We assume the WM
starts from the steady state with respect to a thermal bath at temperature
$T_{\rm c}$. The frequency $\omega$ of the WM is increased from
$\omega=\omega_{\rm c}$ to $\omega=\omega_{\rm h}>\omega_{\rm c}$ during this
unitary stroke, thereby performing $W_{1}$ amount of work on the system. (ii)
(B $\to$ C): The WM is coupled to a hot thermal bath during this non-unitary
stroke, while the frequency is kept constant at $\omega=\omega_{\rm h}$;
$Q_{h}$ is the amount of heat flowing from the hot bath to the WM, such that
the WM reaches the steady state at the end of the stroke. (iii) (C $\to$ D):
The WM is decoupled from the bath after which the frequency is decreased from
$\omega=\omega_{\rm h}$ to $\omega=\omega_{\rm c}$ during this unitary stroke;
$W_{2}$ is the amount of work performed by the WM. (iv) (D $\to$ A): The WM is
coupled to a cold thermal bath during this non-unitary stroke, while its
frequency is kept constant at $\omega=\omega_{\rm c}$; $Q_{c}$ is the amount
of heat flowing from the WM to the cold bath, such that the WM reaches the
steady state at the end of the stroke, thereby completing the cycle.

The WM consists of multiple interacting or non-interacting identical spin $s$
particles, such as the LMG model [72, 69, 70, 71]

$H=\omega(t)\left[\frac{1}{n}(\mathcal{J}_{x}^{2}+\mathcal{J}_{y}^{2})+\gamma\mathcal{J}_{z}\right],$
(1)

or $x$-body inter-particle interactions,

$H=n\omega(t)(\mathcal{J}_{z}/n)^{x}~{}~{}~{}~{}\text{for~{}{\hbox{x =
1,2,3,...}}}.$ (2)

The collective angular momentum is defined as
$\mathcal{J}_{i}:=\sum_{k=1}^{n}J^{k}_{i}$, where $J_{i}^{k}$ is the spin
operator for the $k$th spin, along the direction $i=x,y,z$. The spins
collectively exchange energy with the thermal baths during the non-unitary
strokes. We start by focusing on the $x=1$ case, with
$H=\omega(t)\mathcal{J}_{z}$ and the WM-bath interaction Hamiltonian $H_{\rm
int}=\gamma\mathcal{J}_{x}\otimes\mathcal{B}_{\rm v}$ (${\rm v}=h,c$), where,
$\gamma$ denotes the interaction strength, while $\mathcal{B}_{c}$
($\mathcal{B}_{h}$) is the bath operator for the cold (hot) bath. This
collective system-bath interaction may result in non-zero off-diagonal terms
in the steady-state density matrix $\rho^{\rm ss}$ of the WM at the end of
non-unitary strokes in the local spin basis, signifying non-trivial steady-
state quantum coherence in the many-body system. The dynamics of the WM are
described by the master equation [48, 73, 51],

$\displaystyle\frac{d\rho}{dt}=-i[H,\rho]+\sum_{{\rm v}=c,h}\alpha_{\rm
v}\mathcal{L}_{\rm v}\rho.$ (3)

Here, $\mathcal{L}_{\rm v}\rho=\Gamma_{\rm v}(\omega_{\rm
v})\mathcal{D}(\mathcal{J}^{+})\rho+\Gamma_{\rm v}(-\omega_{\rm
v})\mathcal{D}(\mathcal{J}^{-})\rho$, $\mathcal{D}[O]\rho=(2O^{\dagger}\rho
O-OO^{\dagger}\rho-\rho OO^{\dagger})/2$,
$\mathcal{\mathcal{J}}^{\pm}:=\mathcal{\mathcal{J}}_{x}\pm
i\mathcal{\mathcal{J}}_{y}$ are the collective ladder operators of the spin
system and $\Gamma_{\rm v}(\nu)$ denotes the spectral function of the thermal
baths at frequency $\nu$, where we consider the Kubo-Martin-Schwinger
condition $\Gamma_{\rm v}(-\nu)=\exp\left(-\beta_{\rm
v}{\nu}\right)\Gamma_{\rm v}(\nu)$ [26]. We take the physical constants
$\hbar=k_{\mathrm{B}}=1$. The constant $\alpha_{\rm v}$ is set to one if the
system is coupled to the respective thermal bath, otherwise it is zero.

We consider the system to interact collectively with the baths, such that the
system steady states at A and C for the aforementioned dissipators (cf. Eq.
(3)) can be written in the collective basis $|j,m\rangle_{i}$ as [73]

$\displaystyle\rho^{ss}(\beta,\omega)$ $\displaystyle=$
$\displaystyle\sum_{j=j_{0}}^{ns}\sum_{i=1}^{l_{j}}P_{j,i}\;\rho_{j,i}^{\text{th}}(\beta,\omega),$
(4)

where
$\rho_{j,i}^{\text{th}}(\beta,\omega)=\frac{1}{Z_{j}}\sum_{m=-j}^{j}e^{-\beta
m\omega}|j,m\rangle_{i}{}_{i}\langle j,m|$ with
$Z_{j}=\sum_{m=-j}^{j}e^{-\beta m\omega}$. Here $|j,m\rangle_{i}$ are the
common eigenvectors of $\mathcal{J}_{z}$ and
$\mathcal{J}^{2}=\mathcal{J}_{x}^{2}+\mathcal{J}_{y}^{2}+\mathcal{J}_{z}^{2}$;
$-j\leq m\leq j$, $j\in[j_{0};ns]$, and $j_{0}=0$ for $s\geq 1$, while for
$s=1/2$, we have $j_{0}=1/2$ for $n$ odd and $j_{0}=0$ for $n$ even. The index
$i\in[1;l_{j}]$, where $l_{j}$ denotes the multiplicity of the eigenspaces
associated to the eigenvalue $j(j+1)$ of the $\mathcal{J}^{2}$ operator. A
detailed discussion regarding the collective angular momentum operators and
the explicit form of $l_{j}$ are given in [74] and summarized in the Appendix
D $P_{j,i}=\sum_{m=-j}^{j}\;{}_{i}\langle j,m|\rho_{0}|j,m\rangle_{i}$, with
$\rho_{0}$ being the initial state of the WM.

The dependence of $\rho^{ss}$ on the initial state through $P_{j,i}$ [cf. Eq.
(4)] implies that a choice of $P_{j,i}$ can influence the steady-state
significantly. We have $l_{j=ns}=1$, such that for $P_{j=ns}=1$ the steady
state becomes

$\rho^{ss}(\beta,\omega)=\sum_{m=-ns}^{ns}\frac{e^{-\beta
m\omega}}{Z_{ns}}|ns,m\rangle\langle ns,m|,$ (5)

which is a diagonal state in the symmetrical Dicke subspace known to produce a
collective advantage [51]. This is in contrast to the independent coupling
between the particles of the WM and the bath, in which each spin thermalizes
independently, such that the WM approaches a direct product steady state at
the end of a non-unitary stroke [26, 48, 73, 51].

We note that if the steady state is restricted to a spin $j$ subspace, then
$C^{\rm col}/(\beta\omega)^{2}\sim j(j+1)$ for $\beta\to 0$. Consequently,
even though collective effects remain present for all values of $j$ following
the scaling given above, however, as we discuss below, these effects become
most prominent when we restrict ourselves to the $j=ns$ subspace.

###  III.1 Statistics of a many-body collective quantum heat engine

Since heat and work are fluctuating quantities, a probability distribution may
be associated with obtaining a certain amount of work and heat in an engine
cycle. This probability distribution can be found using the two-point
measurement scheme, where the system is measured at the beginning and at the
end of each stroke. Let the observed system states at A, B, C, and D be
$|{\frac{n}{2},i}\rangle\equiv|{i}\rangle$, $|{j}\rangle$, $|{k}\rangle$, and
$|{l}\rangle$ respectively. Then the joint probability $P(W,Q_{h})$ of having
certain values of net work $W=W_{2}+W_{1}$ and heat $Q_{h}$ during one cycle
of the Otto engine can be readily calculated as [76, 75]

$\displaystyle P(W,Q_{h})$ $\displaystyle=$
$\displaystyle\sum_{i,j,k,l}\delta[W_{2}-(\epsilon_{l}^{0}-\epsilon_{k}^{\tau})]\;\delta[Q_{h}-(\epsilon_{k}^{\tau}-\epsilon_{j}^{\tau})]$
(6)
$\displaystyle\;\delta[W_{1}-(\epsilon_{j}^{\tau}-\epsilon_{i}^{0})]\;p_{k\to
l}^{CD}\;p_{j\to k}^{BC}\;p_{i\to j}^{AB}\;p_{i}^{A}.$

The initial occupation probability of the state $|{i}\rangle$,
$p_{i}^{A}=e^{-\beta_{c}\epsilon_{i}^{0}}/Z_{(0)}$, as the system is in a
steady state with inverse temperature $\beta_{c}$ at A. The adiabatic
transitions give $p_{i\to j}^{AB}=\delta_{i,j}$ and $p_{k\to
l}^{CD}=\delta_{k,l}$, and the transition probability $p_{j\to
k}^{BC}=e^{-\beta_{h}\epsilon_{k}^{\tau}}/Z_{(\tau)}$ since the system reaches
a steady state with inverse temperature $\beta_{h}$ at C. Here,
$\epsilon_{m}^{t=0,\tau}$ are the respective energy eigenvalues for the state
$|{m}\rangle$ at the beginning (time $t=0$) and at the end ($t=\tau$) of the
unitary stroke A $\rightarrow$ B, while the $Z_{(t=0,\tau)}$ are the
corresponding partition functions. The associated characteristic function
$\chi(\gamma_{1},\gamma_{2})$ is given by [25, 76, 75]
$\chi(\gamma_{1},\gamma_{2})=\iint_{-\infty}^{\infty}dWdQ_{h}\;e^{i\gamma_{1}W}e^{i\gamma_{2}Q_{h}}P(W,Q_{h})$,
which can then be used to compute the various moments of the heat and work
distribution function as $\langle
W^{r}Q_{h}^{s}\rangle=\frac{\partial^{r}}{\partial(i\gamma_{1})^{r}}\frac{\partial^{s}}{\partial(i\gamma_{2})^{s}}\chi(\gamma_{1},\gamma_{2})\Big{\arrowvert}_{\gamma_{1,2}=0}$.

###  III.2 $n$ qubit collective engine

Next for simplicity we focus on $s=1/2$ (see Appendix B for generic $s$) $n$
qubit engine with system Hamiltonian
$H=\omega(t)\sum_{i=1}^{n}\sigma_{z}^{i}/2$, which interacts collectively with
the thermal baths. We consider the regime close to the Carnot efficiency
bound, where the work output approaches zero and hence where work fluctuations
can be expected to be prominent. To quantify the reliability advantage of a
collective quantum heat engine, we define the ratio of reliability of the
collective engine to that of the independent engine: $\lambda_{r}=r^{2}_{\rm
col}/r^{2}_{\rm ind}$. In the limit $T_{\rm h}\to\infty$ and $\eta=-\langle
W\rangle/\langle Q_{h}\rangle\to\eta_{\rm C}=1-T_{c}/T_{h}$, we can verify
that ${\rm var}(W_{\rm col})\propto C^{\rm col}\sim n^{2}$ (see Appendix D).

A value of $\lambda_{r}>1$ implies a lower relative noise-to-signal ratio for
the collective quantum heat engine (QHE), as compared to the independent one,
signifying higher consistency in the output of collective QHEs. The parameter
$\lambda_{r}$ close to the Carnot bound is plotted in Fig. 1. The remarkable
advantage offered by the collective effects is shown by large values of
$\lambda_{r}$ in Fig. 1, specially in the limit of large $n$ and $T_{\rm h}$.
Clearly, high temperature $T_{\rm h}$ of the hot bath and large system size
$n$ allow the operation of the collective QHE with high reliability in the
work output. Interestingly, this regime is also the most beneficial for high
mean work output $\langle W_{\rm col}\rangle\sim n^{2}$, as discussed in [48]
and [51]. Further, $\lambda_{r}$ increases linearly with $n$ for $\beta_{\rm
h}\to 0$, implying that the consistency of the collective engines increases
extensively with system size. Remarkably, the quadratic advantage in
reliability persists even for the more general Hamiltonians given in Eqs. (1),
and (2) with $x>1$ [see Figs. 1, 1, and Appendices D and E for details]. The
collective advantage persists if the steady state is restricted to the $j=ns$
subspace, and is of the general form
$\rho^{ss}(\beta)=\sum_{m=-ns}^{ns}\exp[{-\beta\epsilon_{m}}]/{Z}_{ns}|ns,m\rangle\langle
ns,m|,\;{Z}_{ns}=\sum_{m=-ns}^{ns}\exp[{-\beta\epsilon_{m}}]$ where
$\epsilon_{m}$ is the respective eigen-energy of the state $|{ns,m}\rangle$.
Here the operator $\mathcal{A}$ introduced before can be identified as
$\mathcal{A}=\sum_{m=-ns}^{ns}\epsilon_{m}/n|{ns,m}\rangle\langle{ns,m}|$. We
summarize the above main results in Table 1.

Quantity | Relation with heat capacity | Scaling
---|---|---
Average work | $\langle W_{\rm col}\rangle\propto{C^{\rm col}(\theta_{h})}/{\theta_{h}^{2}}$ |  $n^{2}$ [51]
Work variance | $\text{var}(W_{\rm col})\propto{C^{\rm col}(\theta_{h})}/{\theta_{h}^{2}}$ | $n^{2}$
Reliability advantage | $\lambda_{\text{r}}\propto{C^{\rm col}(\theta_{h})}/{C^{\rm ind}(\theta_{h})}$ | $n$
Table 1:  Here we summarize the main results, which are valid in the limits
$\eta\to\eta_{\rm C}$ and $T_{h}\to\infty$ with
$\theta_{h}=\beta_{h}\omega_{h}$. The proofs are discussed in the main text
and detailed in the Appendix.

##  IV  Entropy Production and TUR

Having demonstrated the collective advantage in the reliability of a many-body
correlated heat engine, we turn to the issue of entropy production. It is
desirable to have an engine with low entropy production to enhance efficiency.
However, it is known that the entropy production and the reliability are not
independent of each other and there exists a lower bound for the product of
entropy production and the inverse of the square of the reliability in the
case of classical systems [77, 68], which has been shown to hold true in many
quantum systems as well [78, 79]. For our collective engine, we can show that
the enhanced reliability does not come at the cost of large entropy
production; i.e., their product can be made smaller than the standard TUR
bound. The efficiency $\eta$ of an engine is related to the entropy production
$\langle\Sigma\rangle$ through the relation:
$\langle\Sigma\rangle=-\frac{\langle Q_{\rm h}\rangle}{T_{\rm
h}}-\frac{\langle Q_{\rm c}\rangle}{T_{\rm c}}=\frac{\langle W\rangle}{T_{\rm
c}}\left(\frac{\eta_{\rm C}}{\eta}-1\right)$.

For the collective heat engine, the desired low fluctuating output work is
obtained at a thermodynamic cost of an increased entropy production
$\langle\Sigma\rangle$, following the inequality in the large $n$-limit [60]:

$\left(\frac{1}{r_{\rm
col}}\right)^{2}=\frac{f(\\{\beta_{i}\omega_{i}\\})}{\langle\Sigma_{\rm
col}\rangle}-1\;\geq\;\frac{2}{\langle\Sigma_{\rm col}\rangle}-1.$ (7)

where
$f(\\{\beta_{i}\omega_{i}\\})=(\beta_{c}\omega_{c}-\beta_{h}\omega_{h})(a/b)\geq
2$ (see Appendix F),
$a=\cosh(\beta_{c}\omega_{c}-\beta_{h}\omega_{h})+\cosh\beta_{c}\omega_{c}+\cosh\beta_{h}\omega_{h}-3$
and
$b={\sinh(\beta_{c}\omega_{c}-\beta_{h}\omega_{h})-\sinh\beta_{c}\omega_{c}+\sinh\beta_{h}\omega_{h}}$.
Consequently, we can arrive at a trade-off, quantified by the thermodynamic
uncertainty $\mathcal{Q}=\langle\Sigma\rangle/r^{2}$ [66, 68]. For a classical
system $\mathcal{Q}\geq 2$, this is known as the standard TUR which provides a
lower bound for $\mathcal{Q}$. For the $n$-qubit collective Otto engine, this
is given by

$\mathcal{Q}_{\rm col}\geq 2-\langle\Sigma_{\rm col}\rangle,$ (8)

which further sharpens the TUR in the quantum case. Since the entropy
production $\langle\Sigma_{\rm col}\rangle\geq 0$, the collective engine TUR,
Eq. (8), may violate the standard TUR bound, implying once again, a collective
effect induced improvement in performance. Recently, such violations were
observed for quantum-coherent and quantum-entangled systems [62, 63, 64, 65,
66, 60].

In Fig. 1 the uncertainty $\mathcal{Q}$ as well as its lower bound are plotted
for the collective and independent cases. The figure inset shows that in the
large $n$ and $T_{h}$ limit, we have $\mathcal{Q}<2$; this is the region where
the collective effects lead to a maximal advantage over the independent case
as shown in Fig. 1. As in the case of reliability, the collective advantage in
$\mathcal{Q}$ extends to WM with inter-particle interactions, such as the LMG
model [Eq. (1)], or with $x$-particle interactions [Eq. (2)].

##  V Discussion

In this manuscript, we considered QHEs modeled by generic many-body quantum
systems collectively coupled to thermal baths. We showed that for a broad
class of steady states arising due to collective system-bath coupling, we can
get a quadratic advantage in reliability, as compared to their independent
counterparts. We then explicitly showed this collective advantage in
reliability for collective QHEs with non-interacting as well as interacting
multi-spin WMs. Interestingly, this collective advantage increases with
increasing temperature $T_{h}$ of the hot bath as well as size $n$ of the
working medium. We also studied TUR in such collective QHEs; the advantage due
to collective effects persists in this case as well, in the form of low
thermodynamic uncertainty and TUR bound, as compared to the independent QHEs,
for high $T_{h}$. Furthermore, $\mathcal{Q}_{\rm col}<2$ for high $T_{h}$ and
large $n$, thereby violating the standard TUR bound in these regimes. Our
analysis not only shows a way of realizing highly reliable many-body quantum
machines but also extends the regime of validity of collective advantage to
interacting models studied in many-body physics.

Several existing platforms can be expected to be ideal for experimentally
realizing such collective QHEs. For example, experimental realization of
collective effects is well established in cavity systems, such as by using
Rydberg atoms in a cavity [80]; recently, the collective effect in the form of
superabsorption in an organic microcavity was used to charge a quantum battery
[81]. The collective phenomenon of superradiance was observed in atomic sodium
[82] as well as in quantum dots [83]. Recent advancements in techniques to
assemble atoms with great control using optical tweezers [84, 85, 86, 87, 88],
and optical lattices [89, 90, 91, 92] is also a promising development that can
be used to design desired many-body quantum systems.

Miniaturization of machines is one of the major aims of research in science
and technology. However, this is associated with the challenge of fluctuations
increasing with decreasing system size [58]. In that context, the collective
QHEs studied in this manuscript can be highly beneficial, thereby paving a
path to realizing highly reliable mesoscopic quantum thermal machines.

## ACKNOWLEDGEMENTS

V.M. acknowledges support from Science and Engineering Research Board (SERB)
through MATRICS (Project No. MTR/2021/000055) and a Seed Grant from IISER
Berhampur. S.V. acknowledges support from the Government of India DST-QUEST
Grant No. DST/ICPS/QuST/Theme-4/2019.

##  Appendix A  Work output of an Otto engine

The total work output of an Otto engine is the sum of the work done on the
system and by the system during the adiabatic unitary strokes. That is,

$\displaystyle\langle W\rangle$ $\displaystyle=$ $\displaystyle\langle
W_{1}\rangle+\langle W_{2}\rangle$ (9) $\displaystyle=$ $\displaystyle
Tr\\{[H_{h}-H_{c}]\rho_{c}\\}+Tr\\{[H_{c}-H_{h})]\rho_{h}\\}$ $\displaystyle=$
$\displaystyle(\omega_{h}-\omega_{c})Tr\\{\mathcal{G}[\rho_{h}-\rho_{c}]\\}.$

We have used the following definitions and identities here and in the
following derivations: $H_{\rm v}=\omega_{\rm v}\mathcal{G}$, $\rho_{\rm
v}=\rho^{ss}(\theta_{\rm v})$, $\theta_{\rm v}=\beta_{\rm v}\omega_{\rm
v},\;{\rm v}=\\{c,h\\}$,
$\Delta\eta=\eta_{C}-\eta=\frac{\theta_{c}-\theta_{h}}{\beta_{c}\omega_{h}}$,
$\eta_{C}$ is the Carnot efficiency, and
$\Delta\theta=\theta_{c}-\theta_{h}=\beta_{c}\omega_{h}\Delta\eta$. The
$\mathcal{G}$ could be $n(J_{z}/n)^{x}$ or
$\left[\frac{1}{n}(\mathcal{J}_{x}^{2}+\mathcal{J}_{y}^{2})+\gamma\mathcal{J}_{z}\right]$.
The collective angular momentum is defined as
$\mathcal{J}_{i}:=\sum_{k=1}^{n}J^{k}_{i}$, where $J_{i}^{k}$ is the spin
operator for the $k$th spin, along the direction $i=x,y,z$. For $s=1/2$,
$J_{i}^{k}=\sigma_{i}^{k}/2$ which is the Pauli matrix along $i$ direction
times $1/2$.

Near the Carnot limit ($\Delta\eta\approx 0$), the work can be written as

$\displaystyle\langle W\rangle$ $\displaystyle=$
$\displaystyle(\omega_{c}-\omega_{h})\beta_{c}\omega_{h}\Delta\eta\;\frac{\partial}{\partial\theta}Tr\\{\mathcal{G}\rho(\theta)\\}\arrowvert_{\theta_{h}}+O(\Delta\eta^{2}).$

One can show that,

$\displaystyle\frac{\partial}{\partial\theta}Tr\\{\mathcal{G}\rho(\theta)\\}\arrowvert_{\theta_{h}}=-\frac{C(\theta_{h})}{k_{B}\theta_{h}^{2}},$
(11)

for both independent and collective cases, where $C$ is the heat capacity of
the spin system. And also,
$\omega_{c}-\omega_{h}=(\beta_{c}\omega_{h}-\beta_{h}\omega_{h}-\beta_{c}\omega_{h}\Delta\eta)/\beta_{c}$.
Thus from Eqs. (LABEL:app_eqW) and (11), the average work is related to heat
capacity as [51],

$\displaystyle\langle
W\rangle=-\Delta\eta\omega_{h}^{2}(\beta_{c}-\beta_{h})\;\frac{C(\theta_{h})}{k_{B}\theta_{h}^{2}}+O(\Delta\eta^{2}).$
(12)

For an $n$ qubit system with $H=\omega\mathcal{J}_{z}$, inverse temperature
$\beta$, and the steady state given in Eq. (3) of the main text, the heat
capacity becomes,

$\displaystyle C^{\rm col}$ $\displaystyle=$ $\displaystyle-
k_{B}\beta^{2}\frac{\partial
E^{ss}}{\partial\beta}\;=\;-k_{B}\beta^{2}\frac{\partial}{\partial\beta}Tr(H\rho^{ss})$
$\displaystyle=$
$\displaystyle\frac{1}{4}k_{B}{\beta}^{2}{\omega}^{2}\left[\text{csch}^{2}\frac{{\beta}{\omega}}{2}-(n+1)^{2}\text{csch}^{2}\frac{(n+1){\beta\omega}}{2}\right],$

where $E^{ss}=\frac{1}{2}n\hbar\omega+Tr(H\rho^{ss})$ is the steady state
energy of the working medium (see Fig. 2 and the reference [73]). At large
temperatures, the above equation reduces to

$\displaystyle C^{\rm col}\approx\frac{1}{12}k_{B}\beta^{2}\omega^{2}n(n+2),$
(14)

which finally gives, $\langle W\rangle\propto n^{2}$. Similarly, it can be
shown that for a steady state restricted to the spin $j$ subspace, $C^{\rm
col}/(\beta\omega)^{2}\sim j(j+1)$ for $\beta\to 0$ (also see Fig.2). Equation
(14) is a special case of this with $j=n/2$.

Figure 2:  The steady state energies are plotted as a function of the inverse
temperature $\beta$ for independent and collective cases with
$H=\omega\mathcal{J}_{z}$, $\hbar=\omega=1$, $s=1/2$ and $n=10$. For the
collective case, the steady state is restricted to different $j$ subspaces and
the curves are labeled accordingly; for the independent case, the steady state
is given by the usual thermal state. Note that the heat capacity is
proportional to the slope of the steady-state energy. As shown in Table I of
the main text, all quantities of interest can be expressed in terms of heat
capacity. Therefore the different scaling advantages discussed in the main
text as well as in this appendix can be understood from the slopes of the
curves at large temperatures ($\beta\to 0$). The plot shows that for large $j$
values the collective heat capacities are larger than the independent case and
it diminishes as $j$ decreases and below a certain value of $j$, its value
becomes smaller than that of the independent case.

##  Appendix B Characteristic function for a collective $n$ spin-$s$ system
with $H=\omega\mathcal{J}_{z}$

For an engine made of $n$ spin s systems with $H=\omega\mathcal{J}_{z}$, the
characteristic function can be expressed as, $\chi=A/B$, where,

$\displaystyle A$ $\displaystyle=$
$\displaystyle\left[e^{(2ns+1)(\theta_{h}+i\gamma_{1}\omega_{h})}-e^{i(2ns+1)(\gamma_{1}\omega_{c}+\gamma_{2}\omega_{h})}\right]$
$\displaystyle\;\left[e^{i(2ns+1)\gamma_{1}\omega_{h}}-e^{(2ns+1)[\theta_{c}+i(\gamma_{1}\omega_{c}+\gamma_{2}\omega_{h})]}\right]$
$\displaystyle\;\;\sinh\frac{\theta_{c}}{2}\text{csch}\frac{(2ns+1)\theta_{c}}{2}\sinh\frac{\theta_{h}}{2}\text{csch}\frac{(2ns+1)\theta_{h}}{2}$
$\displaystyle\quad
e^{-ns\big{\\{}\theta_{c}+\theta_{h}+2i[\gamma_{1}(\omega_{c}+\omega_{h})+\gamma_{2}\omega_{h}]\big{\\}}},$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\left[e^{i\gamma_{1}\omega_{h}}-e^{\theta_{c}+i(\gamma_{1}\omega_{c}+\gamma_{2}\omega_{h})}\right]$
$\displaystyle\;\left[e^{(\theta_{h}+i\gamma_{1}\omega_{h})}-e^{i(\gamma_{1}\omega_{c}+\gamma_{2}\omega_{h})}\right],\;\theta_{c(h)}=\beta_{c(h)}\omega_{c(h)}.$

For $s=1/2$ n-qubit engine, the general spin $s$ characteristic function
simplifies to $\chi=A_{1}/B_{1}$, where,

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\sinh\frac{\theta_{c}}{2}\sinh\frac{\theta_{h}}{2}\text{csch}\frac{\theta_{c}(n+1)}{2}\text{csch}\frac{\theta_{h}(n+1)}{2}$
$\displaystyle\quad\left[e^{i\gamma_{1}(n+1)\omega_{h}}-e^{(n+1)(\theta_{c}+i(\gamma_{1}\omega_{c}+\gamma_{2}\omega_{h}))}\right]$
$\displaystyle\qquad\left[e^{(n+1)(\theta_{h}+i\gamma_{1}\omega_{h})}-e^{i(n+1)(\gamma_{1}\omega_{c}+\gamma_{2}\omega_{h})}\right]$
$\displaystyle\qquad\qquad
e^{-\frac{n}{2}\big{\\{}\theta_{c}+\theta_{h}+i2[(\gamma_{1}+\gamma_{2})\omega_{h}+\gamma_{1}\omega_{c}]\big{\\}}},$
$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle\left[e^{i\gamma_{1}\omega_{h}}-e^{\theta_{c}+i(\gamma_{1}\omega_{c}+\gamma_{2}\omega_{h})}\right]$
$\displaystyle\qquad\left[e^{(\theta_{h}+i\omega_{h}\gamma_{1})}-e^{i(\gamma_{1}\omega_{c}+\gamma_{2}\omega_{h})}\right].$

##  Appendix C  Work fluctuations

The output work fluctuations of a frictionless Otto engine originate from the
thermal energy fluctuations at the two thermalization strokes. Since the
energy fluctuations are related to the heat capacity as
$\sigma_{E}^{2}=C/(k_{B}\beta^{2})$, one would expect the work fluctuations to
be equal to the sum of the energy fluctuations at the two thermal strokes. We
can show that this is indeed true for Hamiltonians, such as the LMG model

$H=\omega(t)\left[\frac{1}{n}(\mathcal{J}_{x}^{2}+\mathcal{J}_{y}^{2})+\gamma\mathcal{J}_{z}\right],$
(15)

or with $x$-body inter-particle interactions, such as

$\displaystyle
H=n\omega(t)(\mathcal{J}_{z}/n)^{x}~{}~{}~{}~{}\text{for~{}$x\geq 1$},$ (16)

if the density matrix is given by an appropriate Gibbs’s distribution in
$j=ns$ subspace, and the variance of the work takes the form,

$\displaystyle\text{var}(W)$ $\displaystyle=$
$\displaystyle(\omega_{c}-\omega_{h})^{2}\Big{\\{}\frac{C(\theta_{h})}{k_{B}\theta_{h}^{2}}+\frac{C(\theta_{c})}{k_{B}\theta_{c}^{2}}\Big{\\}}$
$\displaystyle\approx$
$\displaystyle\frac{2(\beta_{c}-\beta_{h})^{2}}{k_{B}\beta_{c}^{2}\beta_{h}^{2}}C(\theta_{h})+O(\Delta\eta)$
$\displaystyle=$ $\displaystyle 2\omega_{h}^{2}\eta_{\rm
C}^{2}\frac{C(\theta_{h})}{k_{B}\theta_{h}^{2}}+O(\Delta\eta).$ (18)

The above equation (18) holds for both the independent and collective coupling
to the thermal baths. Again for the above cases, it can be shown that $C^{\rm
col}(\theta_{h})/\theta_{h}^{2}\propto n^{2}$. Thus the reliability for the
collective bath coupling becomes $r_{\rm col}\propto n$. Further, $r_{\rm
ind}\propto\sqrt{n}$ implies $\lambda_{r}=r_{\rm col}^{2}/r_{\rm
ind}^{2}\propto n$ which is a quadratic advantage for the collective case over
the independent analog.

##  Appendix D  $n^{2}$ scaling of work, its variance and heat capacity for
$H=n\omega{(\mathcal{J}_{z}/n)^{x}}$

Let the Hamiltonian be $H=n\omega{(\mathcal{J}_{z}/n)^{x}}$, where
$\mathcal{J}_{z}=\frac{1}{2}\sum_{i}\sigma_{z}^{i}$. Following the steps given
in the appendices A and C, we can show that

$\displaystyle\langle
W\rangle=-\Delta\eta\omega_{h}^{2}(\beta_{c}-\beta_{h})\;\frac{C(\theta_{h})}{k_{B}\theta_{h}^{2}}+O(\Delta\eta^{2})$

and

$\displaystyle\text{var}(W)\;=\;2\omega_{h}^{2}\eta_{\rm
C}^{2}\frac{C(\theta_{h})}{k_{B}\theta_{h}^{2}}+O(\Delta\eta),$

for this case as well. Let the steady state be of the form
$\rho^{ss}=e^{-\beta\omega H}/Z$ where $Z=\mbox{tr}(e^{-\beta\omega H})$. Then
heat capacity can be obtained using standard arguments as

$C=-k_{B}\beta^{2}\frac{\partial}{\partial\beta}Tr\left[H\rho\right]=k_{B}\beta^{2}{\rm
var}(H).$ (19)

If the steady state is restricted to the $j=n/2$ irreducible subspace with
Gibbs’s distribution in the common eigenbasis of $J^{2}$ and
$\mathcal{J}_{z}$, $\arrowvert j,m\rangle_{i}$ basis, then we get,

$\langle H\rangle=({n\omega/Z_{n/2}})\sum_{m}(m/n)^{x}e^{-\beta
n\omega(m/n)^{x}},$

and

$\langle H^{2}\rangle=({n^{2}\omega^{2}/Z_{n/2}})\sum_{m}(m/n)^{2x}e^{-\beta
n\omega(m/n)^{x}}.$

The index $i\in[1;l_{j}]$ in $\arrowvert j,m\rangle_{i}$, where $l_{j}$
denotes the multiplicity of the eigenspaces associated with the eigenvalue of
$\mathcal{J}^{2}$ operator. The $l_{j}$ can be explicitly expressed as [74],

$l_{j}=\frac{(2j+1)\;n!}{(\frac{n}{2}+j+1)!\;(\frac{n}{2}-j)!},$

which becomes $1$ for $j=n/2$. At large temperatures, that is inverse
temperature $\beta\to 0$, the variance simplifies to

$\displaystyle\text{var}(H)$ $\displaystyle=$ $\displaystyle\langle
H^{2}\rangle-\langle H\rangle^{2}$ $\displaystyle=$
$\displaystyle\frac{n^{2}\omega^{2}}{(n+1)}\sum_{m}\left(\frac{m}{n}\right)^{2x}-\frac{n^{2}\omega^{2}}{(n+1)^{2}}\left(\sum_{m}\left(\frac{m}{n}\right)^{x}\right)^{2}.$

For large $n$ values, the summations in Eq. (LABEL:eqvargen) can be replaced
with integration by setting $q=m/n$ giving,

$\displaystyle\text{var}(H)\approx
n^{2}\omega^{2}\left[\int_{-1/2}^{1/2}q^{2x}dq-\left(\int_{-1/2}^{1/2}q^{x}dq\right)^{2}\right].$

Here the $\text{var}(H)$ reduces to
$n^{2}\frac{\omega^{2}x^{2}}{4^{x}(2x+1)(x+1)^{2}}$ for even $x$, and
$n^{2}\frac{\omega^{2}}{4^{x}(2x+1)}$ for odd $x$. Thus both work and its
variance are found to be proportional to $n^{2}$ for $x\geq 1$.

##  Appendix E  LMG system as the WM

We consider a Lipkin-Meshkov-Glick (LMG) model given by the Hamiltonian,

$H=\omega(t)\left[\frac{1}{n}(\mathcal{J}_{x}^{2}+\mathcal{J}_{y}^{2})+\gamma\mathcal{J}_{z}\right],$
(22)

as our working medium.

Result | Scaling | Proof
---|---|---
$\langle W_{\rm col}\rangle\propto\frac{C^{\rm col}(\theta_{h})}{\theta_{h}^{2}}$ | $n^{2}$ | Ref. [51], Appendices A,
|  |  D, and E
$\text{var}(W_{\rm col})\propto\frac{C^{\rm col}(\theta_{h})}{\theta_{h}^{2}}$ | $n^{2}$ | Appendices C, D, and E
$r_{\rm col}\propto\sqrt{\frac{C^{\rm col}(\theta_{h})}{\theta_{h}^{2}}}$ | $n$ | Appendices D and E
$\lambda_{\text{r}}\propto\frac{C^{\rm col}(\theta_{h})}{C^{\rm ind}(\theta_{h})}$ | $n$ | Appendices D and E
Table 2: The above main results are tabulated which are valid in the limits
$\eta\to\eta_{\text{C}}$ and $T_{h}\to\infty$. We have defined
$\theta_{h}=\beta_{h}\omega_{h}$.

Following the same steps used to obtain the Eqs. (12) and (18), we can show
that near the Carnot efficiency, the work and its variance are proportional to
the heat capacity. That is, $W\propto{C(\theta_{h})}/{\theta_{h}^{2}}$ and
$\text{var}(W)\propto{C(\theta_{h})}/{\theta_{h}^{2}}$. Additionally, for a
steady state of the form

$\displaystyle\rho^{ss}(\beta\omega)$ $\displaystyle=$
$\displaystyle\sum_{m=-n/2}^{n/2}\frac{1}{Z_{\frac{n}{2}}}\;e^{-\beta\omega\left[\frac{1}{4n}n(n+2)-4m^{2}+m\gamma\right]}$
$\displaystyle\qquad\qquad\qquad\qquad\left|\frac{n}{2},m\right\rangle\left\langle\frac{n}{2},m\right|,$

we have $\text{var}(H)\approx
n^{2}\omega^{2}\left(\frac{1}{180}+\frac{\gamma^{2}}{12}\right)$. The
partition function is given by
$Z_{n/2}=\sum_{m=-n/2}^{n/2}e^{-\beta\omega\left[\frac{n(n+2)-4m^{2}}{4n}+m\gamma\right]}$.
The above $\text{var}(H)$ is obtained for large $n$ values by replacing the
summation with integration as we did in the Eq. (D). Thus from Eq. (19) we get
the heat capacity $C\propto n^{2}$ giving an $n$ scaling for the reliability
$r$. The main results of the above sections are tabulated in table 2.

##  Appendix F  TUR for an $n$-qubit system with $H=\omega\mathcal{J}_{z}$

Now, using the moments we get the following inequality in the large $n$-limit:

$\left(\frac{1}{r_{\rm
col}}\right)^{2}=\frac{f(\\{\beta_{i}\omega_{i}\\})}{\langle\Sigma_{\rm
col}\rangle}-1\;\geq\;\frac{2}{\langle\Sigma_{\rm col}\rangle}-1.$ (23)

where
$f(\\{\beta_{i}\omega_{i}\\})=(\beta_{c}\omega_{c}-\beta_{h}\omega_{h})(a/b)$,
$a=\cosh(\beta_{c}\omega_{c}-\beta_{h}\omega_{h})+\cosh\beta_{c}\omega_{c}+\cosh\beta_{h}\omega_{h}-3$
and $b=\sinh(\beta_{c}\omega_{c}-\beta_{h}\omega_{h})$
$-\sinh\beta_{c}\omega_{c}+\sinh\beta_{h}\omega_{h}$. Putting
$\theta_{c}=\theta_{h}+\Delta$ in the function
$f(\theta_{c},\theta_{h})=(\theta_{c}-\theta_{h})(a/b)$ gives

$\displaystyle
f(\Delta,\theta_{h})=\frac{\Delta\left[3-\cosh\Delta-\cosh(\Delta+\theta_{h})-\cosh\theta_{h}\right]}{\sinh\Delta-\sinh(\Delta+\theta_{h})+\sinh\theta_{h}}.$

For large $\theta_{h}$ limit this becomes, $f(\Delta)=\Delta\coth\Delta/2\geq
2$, and for other ranges of parameter values a simple numerical analysis
yields minima as 2. We have used this result to obtain the inequality Eq.
(23). Consequently, one can arrive at a trade-off, quantified by the
thermodynamic uncertainty $\mathcal{Q}=\langle\Sigma\rangle/r^{2}$. For the
$n$-qubits collective Otto engine with $H=\omega(t)J_{z}$, trade-off can be
found as

$\mathcal{Q}_{\rm col}^{(1)}\geq 2-\langle\Sigma_{\rm col}^{(1)}\rangle,$ (25)

which further sharpens the TUR in the collective case. Here,
$\langle\Sigma_{\rm col}^{(1)}\rangle$ and $\mathcal{Q}_{\rm col}^{(1)}$ are
the collective entropy production and the quantity $\mathcal{Q}$ for $x=1$.

For an $n$-independent qubit engine, the uncertainty

$\mathcal{Q}_{\rm ind}\;=\;\frac{\text{var}(W_{\rm ind})\langle\Sigma_{\rm
ind}\rangle}{\langle W_{\rm ind}\rangle^{2}}\;=\;\frac{\text{var}(W_{\rm
ind})\langle\Sigma_{\rm ind}\rangle}{\langle W_{\rm
ind}\rangle^{2}}{\Big{\arrowvert}}_{n=1},$

which reduces the lower bound on the thermodynamic uncertainty to a single
qubit bound, that is, $\mathcal{Q}_{\rm ind}\geq 2-\langle\Sigma_{\rm
ind}(n=1)\rangle$. Here, $\langle\Sigma_{\rm ind}(n=1)\rangle$ is the single
spin engine entropy production. For brevity we may denote the $n$ spin entropy
production $\langle\Sigma_{\rm ind}(n)\rangle$ with
$\langle\Sigma_{n}\rangle$. Also note that $\mathcal{Q}_{\rm ind}\geq
2-\langle\Sigma_{1}\rangle\geq 2-\langle\Sigma_{n}\rangle$, giving the single
qubit bound as the tighter bound than the $2-\langle\Sigma_{n}\rangle$.

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# Unleashing the Potentials of Immersive Augmented Reality for Software
Engineering

Leonel Merino 1, Mircea Lungu 2, Christoph Seidl 2  1VISUS, University of
Stuttgart, Germany  2Computer Science Department, IT University of Copenhagen,
Denmark

###### Abstract

In immersive augmented reality (IAR), users can wear a head-mounted display to
see computer-generated images superimposed to their view of the world. IAR was
shown to be beneficial across several domains, _e.g._ , automotive, medicine,
gaming and engineering, with positive impacts on, _e.g._ , collaboration and
communication. We think that IAR bears a great potential for software
engineering but, as of yet, this research area has been neglected. In this
vision paper, we elicit potentials and obstacles for the use of IAR in
software engineering. We identify possible areas that can be supported with
IAR technology by relating commonly discussed IAR improvements to typical
software engineering tasks. We further demonstrate how innovative use of IAR
technology may fundamentally improve typical activities of a software engineer
through a comprehensive series of usage scenarios outlining practical
application. Finally, we reflect on current limitations of IAR technology
based on our scenarios and sketch research activities necessary to make our
vision a reality. We consider this paper to be relevant to academia and
industry alike in guiding the steps to innovative research and applications
for IAR in software engineering.

##  I Introduction

In immersive augmented reality (IAR), a user’s view of the world is altered by
superimposing computer-generated images. Usually, IAR is realized by means of
a head-mounted display (HMD) and natural user interfaces (NUI). That is, users
can interact with the immersive environment using hand gestures as well as
head and body movements. Indeed, the use of IAR was shown to be effective
across various research fields. To name a few examples, in automotive,
researchers used IAR to enable drivers to use visualizations projected on a
car’s windshield to analyze auxiliary information while keeping their
attention on the road [1]. In medicine, researchers found that IAR allows
doctors to make more effective decisions based on computer-generated images
that represent organs of a patient during surgery [2]. Consequently, we ask
whether the benefits of employing IAR found in other fields can be transferred
to tasks involved in software engineering.

Related work. Although little research has focused on investigating the
potentials of using IAR to support software engineering, we consider these
initial works to be important contributions: For instance, IAR has been
employed to support the analysis of software architectures based on
visualization [3] and conversational interfaces [4]. IAR has also been used to
navigate 3D visualizations to support software comprehension [5] and software
performance [6] tasks. A study analyzed using a Microsoft HoloLens device to
explore UML diagrams [7]. Another study found that IAR can boost motivation of
students in software engineering education [8]. Yet another study proposed to
help create a project workspace supported by IAR [9]. In a broader context,
some studies [10, 11, 12] found that software visualizations displayed in
immersive virtual reality promote users’ engagement. Other technologies such
as multi-touch have been used to augment code reviews [13, 14]. In summary, we
think that the impact of a positive user experience of immersive technologies,
in particular IAR, on user performance warrants further examination [15].

Figure 1: A software engineer wears a Microsoft HoloLens to extend the space
of his laptops’ screens by superimposing computer-generated images in the
physical environment.

To examine what can be the role of IAR in the future of software engineering,
we follow four steps:  (i) we identify a set of dimensions that are frequently
found amongst the variables examined in evaluations of IAR approaches in other
fields,  (ii) we analyze how these dimensions could impact software
engineering tasks,  (iii) we envision usage scenarios that exemplify the use
of IAR amongst the processes of software development, and  (iv) we discuss
benefits and limitations that we foresee will be associated with the adoption
of IAR in software engineering.

In our envisioned future, software engineers wear IAR headsets as shown in
Figure 1 (_e.g._ , Microsoft HoloLens111https://www.microsoft.com/en-
us/hololens) –an optical see-through headset that corresponds to a fully
capable modern computer. That is, the device has processing power, persistent
storage, network connectivity, and I/O features such as a display, camera, and
microphone. In this vision, there is a continuum of degrees at which IAR could
be integrated into the software engineering. The most lightweight integration
we envision is using IAR as a mere companion output device that extends the
graphical user interface space offered by the computer screen and keeping
unchanged the previous display medium and interaction methods (_i.e._ ,
computer screen, keyboard and mouse). On the opposite side, we envision
adopting IAR as the main input/output device. That means to completely replace
the computer screen and mouse by exploiting the technology included in the IAR
device. For instance, software engineers could use cameras in the device to
identify elements in the physical reality (_e.g._ , text on a whiteboard,
icons on a screen, colleagues’ faces) that can be augmented with digital
representations (_e.g._ , lightweight visualizations, infographics), and
sensors that can be used to track hand gestures as well as head and eye
movements to support interactions.

##  II A Preliminary Framework

We are convinced that there is a great potential of IAR for novel applications
in software engineering that is yet untapped. We analyze the potential
benefits through the lens of a framework based on aspects mentioned frequently
in research literature on augmented reality [16] and visualization [17]. Even
though these aspects do not focus core software development artifacts
directly, we think that software engineering comprises many activities that
are inherently based on human/machine or human/human interaction. Hence, we
conjecture that all these aspects have the potential to positively impact
specific software engineering activities. Even though we present a
comprehensive list, we do not claim completeness for the identified aspects.
Instead, we aim at setting a common and relevant ground that can be used to
reflect on how IAR can support software engineering.

Collaboration. Various studies have investigated the use of augmented reality
to facilitate collaboration in teams to address tasks in multiple scenarios,
such as investigating the effects of providing real-time heart rate feedback
to remote collaborators for improving their user experience [18] or discussing
how to improve sharing and watching live 360 panorama based remote
collaborative experiences through visualizations in IAR [19]. In software
engineering, for instance, IAR could be used to help co-located collaborators
to navigate room-scaled 3D software visualizations.

Communication. Other studies analyzed the communicative value of an augmented
reality technique to goals such as teaching or presentation. E.g. the use of
augmented visual communication cues to improve users’ experience when
collaborating in a shared remote task space [20] or using eye tracking to
examine how avatar-mediated communication in shared virtual environments to
promote social presence and emotions [21]. In software engineering, IAR could,
for instance, support an operations engineer who requires orientation to find
a server in a data center.

Embodiment. The impact of the experience that comes from the sensory-motor
capacities of the body into human cognition is also frequently present across
several studies in augmented reality research. For example, a study [22]
examined how visual embodiment and social behavior influence perception of
intelligent virtual agents using augmented reality. Another study [23]
analyzed users’ behavior when interacting with each other through a remote
embodied space based on a high fidelity immersive environment that presents
full scale and real-time images of co-participants. In software engineering,
embodiment promoted by IAR could help to establish a more dynamic working
environment and overcome sedentary behavior.

Mediated reality. A frequent topic of interest in augmented reality research
is mediated reality, which relates to changing the appearance of physical
objects and scenes. For instance, a study [24] investigated the effectiveness
of interactive exploration of 3D visualizations in immersive tangible
augmented reality. Another study [25] presented an approach in augmented
reality that alters the perception of the appearance of a physical object by
controlling visual aspects of its surrounding space. Software engineers could
use IAR with objects of daily use by adding to them digital enhancements that
blend with reality.

Mobility. The mobile capabilities of augmented reality approaches also are a
frequent subject of study. For example, a study [26] presented an approach
that enables a dynamic 3D reconstruction of indoor and outdoor scenes of
everyday environments based on cameras worn by a user in IAR. Another study
[27] analyzed awareness of surroundings, empowerment, positive surprise,
amazement and fascination, immersion, and social connectivity of early stage
mobile augmented reality applications. In software engineering, for instance,
IAR could boost the productiveness of developers waiting in an airport, by
enabling them to see multiple monitors as if they were at their office.

Multi-device. A frequent topic is also the study of how we can interact with
multiple devices of various characteristics. Some devices can have a screen
(of various sizes), whereas some others project images on surfaces or involve
see-through displays. For instance, a study [28] presented a metaphor for
multi-device interaction with handhelds and large displays based on position,
size, and orientation control of a handheld optical projector. Another study
[29] investigated the use of IAR to extend the available screen of smartphones
beyond their physical limits with a co-planar virtual surface. Software
engineers could use IAR to integrate such interfaces displayed in multiple
devices.

Pervasiveness.  In IAR, users can have a continuous experience based on
mobility and multi-device interaction. In consequence, some studies
investigated the impact of such pervasiveness. For instance, a study [30]
discussed pervasive augmented reality of scenarios that provide a continuous
and multi-purpose user experience by adapting the system based on the changing
requirements and constraints of users’ current context. Another study [6]
presented a pervasive visualization approach that uses IAR to enhance software
performance awareness. Indeed, software engineers could benefit, for example,
from omnipresent views of multiple information, _e.g._ , status of a
continuous integration system, communication within a team.

Privacy. An important topic of interest is how the technologies involved in
augmented reality can be used to promote privacy protection. For instance, a
study [14] presented a virtual elastic ”tether” for mobile devices that allows
groups to have quick, simple, and privacy-preserving pedestrian meetups.
Another study [31] presented a method that hides people from image sequences
taken with a hand-held camera for privacy protection. In software engineering,
IAR could be beneficial for projects that involve confidential information by
preventing access to unauthorized viewers.

We observe that the framework defined by these eight aspects can be used to
identify research gaps by analyzing the applicability of IAR to particular
software engineering tasks. We elaborate on such analysis by means of
visionary usage scenarios.

##  III Usage Scenarios

We now present usage scenarios for an envisioned future life of software
engineers using IAR. Each scenario highlights specific examples that stress
the support of IAR. Even though state-of-the-art software development
processes are diverse, they contain common fundamental activities, _e.g._ ,
requirements engineering, software architecture or software design,
implementation. We structure our usage scenarios along those activities
highlighting improvements through IAR for each one of them. In Section IV, we
relate aspects of our conceptual framework from Section II to each one of
these scenarios, for which we introduce markers in this section, _e.g._ ,
(MOB1) for the first mobility example.

Requirements Engineering. Grace is a requirements engineer. Her work consists
of understanding business needs to produce a specification of the requirements
of a software system. She wears an IAR headset that supports her on various
requirements engineering tasks. One such task is _augmented requirements
elicitation._ While interviewing a stakeholder  _(MOB1)_ (i) she can obtain an
interactive view of the domain model, which reacts by highlighting nodes as
the customer is talking  _(PER1)_ , and  (ii) she is able to glimpse at the
domain model concepts, which helps her formulate more comprehensive questions
_(COL1)_.  Another task is _augmented requirements analysis._ Grace likes
taking notes on paper and drawing diagrams using a whiteboard in her office.
The camera of her IAR headset allows her to digitize hand written notes to
boost their analysis  _(MED1)_. Using IAR, she can categorize and organize
notes by spatially rearranging them as well as decomposing notes into more
atomic ones  _(EMB1)_. Then, notes can be linked to produce a view that
supports Grace to trace requirements to their source.

Software Design. Pete, a software designer, wears his IAR headset to virtually
project a UML class diagram of a software system on a wall at his office to
use the ample surface for a diagram overview  _(MED2)_. He needs to identify
what classes can be reused and how the system should be extended. He stands in
front of the wall  _(EMB2)_ and adds classes to the diagram by drawing boxes
and links from the boxes to their corresponding hierarchies with a marker. The
marker does not contain ink but virtually traces movements and gives haptic
feedback  _(MED3)_ , which makes his drawing more precise. He also uses the
microphone in the IAR device and speech recognition to enter the name of the
newly added classes. Once finished, he uses a hand gesture swiping the diagram
onto his computer screen to create a source code skeleton of the new classes
in the IDE  _(MUL1)_.

Implementation. Alice, a software developer, is ready to start the
implementation. On her desk, there is only a keyboard. Although the keyboard
exists in physical reality, labels on the keys are displayed in augmented
reality  _(MED4)_. The IAR headset functions as _augmented navigation_
_(MED5)_ that enables her to efficiently navigate through the code while
helping her create a mental map of the source code [32]  _(PER2)_. To the left
of her keyboard she projects the most recently visited files as well as most
likely files to which she might want to navigate based on a degree-of-interest
model [33]. To the right of her keyboard, she keeps a series of visualizations
of the source code visible only through her IAR headset. Interacting with
these virtual navigation affordances is faster and more natural than using the
mouse to ”peck” tabs from history and allows her to build a spatial
representation of the source code. On the wall behind her screen, she displays
context-aware documentation that is related to the code at hand  _(MOB2)_.
This is where she sometimes brings up the traceability matrix for the project,
or the project backlog. During her long commute by train Alice is using her
IAR as an the main output device: besides the fact that she can continue her
work seamlessly from the office, she also does not have to worry about the
privacy of her data being compromised by passers by looking at her screen
_(PRI1)_.

Table I: A summary of aspects of IAR with potential benefits for software engineering. | Collaboration | Communication | Embodiment | Mediated Reality | Mobility | Multi-device | Pervasiveness | Privacy
---|---|---|---|---|---|---|---|---
Requirements Engineering | COL1 | – | EMB1 | MED1 | MOB1 | – | PER1 | –
Software Design | – | – | EMB2 | MED2–3 | – | MUL1 | – | –
Implementation | – | – | – | MED4–5 | MOB2 | – | PER2 | PRI1
DevOps | – | COM1 | — | MED6–8 | MOB3–4 | – | – | –
Software Testing | COL2 | – | EMB3–4 | MED9–10 | – | – | PER3 | –
Software Maintenance | COL3 | – | EMB5 | MED11–12 | MOB5 | – | – | –

DevOps. Don is a DevOps engineer working on a desktop setup with mouse and
keyboard but augmented with an IAR headset. On the desk, he keeps a set of
_augmented commands_ –a series of extra programmable input areas represented
as virtual buttons visible only through his IAR headset  _(MED6)_. They
capture repetitive actions he needs frequently, such as building and
redeploying an application on the test server222This is inspired by the
original setup that Douglas Engelbart envisioned for his Mother of All Demos
presentation but it is not limited to only five keys and the user does not
have to memorize the meaning of the keys.
https://www.youtube.com/watch?v=yJDv-zdhzMY. A message arrives on his IAR
headset  _(COM1)_ and Don is warned that the application deployed in
production is having performance issues. The components of the application are
deployed in multiple servers in the data center. An alarm warns Don that one
of the servers is shut down, and he needs to configure the load balancer on
site  _(MOB3)_. Don goes to the data center  _(MOB4)_ and uses his IAR headset
to retrieve a visualization of the traffic overlaid on the physical servers.
He filters the traffic of applications of other customers to isolate the
traffic between the components of the application of his client. In the
visualization, a glyph rendered on top of physical racks shows information of
deployed components of the application in a server  _(MED7)_. Colored thick
edges placed on the floor represent metrics of the traffic in the network
amongst servers (_e.g._ , bandwidth, latency)  _(MED8)_. Don spots the cause
of the issue. He uses a hand gesture to deploy an instance of the involved
component in a less demanded server and reroute network traffic.

Software Testing. Craig, is a quality assurance engineer. He keeps a view of
the debugger window on the wall on his right. With a slight head movement, he
can move from writing tests to running them  _(EMB3)_. To obtain a pervasive
view of the health of the collaborative project  _(COL2)_ , he uses a city
visualization that he places virtually on his desk close to the keyboard. In
the city, each building represents a test class. Buildings close to each other
in a district represent test suites. Empty districts represent source code
still not covered by tests. In the dynamic visualization (that is connected to
the continuous integration server), buildings become green whenever a test
passed and red otherwise  _(PER3)_. Craig is writing new tests. He likes to
use the whiteboard. The IAR device is able to capture the written code. When
he finishes writing a new test and the code does not compile, his hand written
code is highlighted (in IAR) to point to a syntax error  _(MED9)_. IAR also
supports Craig by displaying a list of keywords (_e.g._ , existing interface
methods) simulating an auto-complete feature. When the code finally compiles,
a red button appears (in IAR) on the whiteboard next the the code  _(MED10)_.
He now can push the button to run the test. He feels good of adopting this
embodied approach to write tests  _(EMB4)_.

Software Maintenance. Marcus and Bob are software maintainers discussing a
strategy to package the software system for distribution  _(COL3)_. They stand
at the whiteboard to write snippets of a build configuration. Working at the
whiteboard makes them feel more creative and open to engage in a conversation
_(EMB5)_. They wear IAR headsets and share a collaborative application that
uses their headsets’ front camera to capture and parse the configuration
description. The application can display in augmented reality extended
information related to the configuration that Marcus is writing  _(MED11)_ ,
_e.g._ , transitive library versions. The application also informs them in
real time when the package name they write is sufficiently unique, so they can
save time by not having to write it fully on the whiteboard. The application
also enables Marcus to compile the system using his configuration. Suddenly,
Marcus and Bob want to relocate to the cafeteria. As the IAR application
captured the specified configuration, they are able to continue their
discussion anywhere  _(MOB5)_. For instance, they can use a wall in the
cafeteria or they can even display the specification floating in the air
_(MED12)_. They could choose between using a marker and writing on a glass
wall in the cafeteria to completely immerse in augmented reality.

##  IV Discussion

A summary of the aspects of IAR that could benefit software engineering is
presented in Table I. We observe that embodiment and mediated reality crosscut
almost all software engineering processes. That is, IAR could boost cognitive
abilities of software engineers by involving natural user interfaces such as
hand gestures and head movements. Software engineers could also benefit from
manipulating visual perception by involving other senses, _e.g._ , haptic.

We describe usage scenarios that exemplify multiple benefits of adopting IAR
in software engineering. However, we observe that IAR adoption implies several
challenges as well. There are challenges for researchers who will need to
investigate new domain-specific interaction techniques for the domains of
software modeling and code writing. Although some of the applications we have
envisioned in this paper are specific to working with code (_e.g._ , augmented
auto-complete on the whiteboard), we think applications for IAR in software
engineering are not limited to it. We reflect that an implication of using IAR
in software engineering will be investigating to which extent immersion brings
benefits to particular tasks, and in which cases immersion could represent a
drawback. There are also challenges for software engineers, _e.g._ , to modify
their behavior and adopt natural user interfaces such as head and eye
movements to interact with IAR interfaces. Finally, there are challenges for
society that will have to deal with software engineers wearing IAR devices in
public spaces. An implication of it could be requiring to define protocols for
using IAR devices in public spaces, _e.g._ , to warn when they are recording.

##  V Conclusion

We describe our vision of how immersive augmented reality (IAR) can support
tasks involved in software engineering. To this end, we define a preliminary
framework of eight relevant aspects to software engineering, which are
frequently found in evaluations of IAR research. We elaborate on envisioned
usage scenarios that span various processes involved in software engineering.
Finally, we discuss the implications of using IAR on a daily basis. In the
future, we plan to further our investigation and realize such blending of IAR
into software engineering.

## Acknowledgments

Merino acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – Project-ID 251654672 – TRR 161.

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