#  Correlation-boosted quantum engine: A proof-of-principle demonstration

Marcela Herrera  Electronic Address<EMAIL_ADDRESS>Centre for
Bioinformatics and Photonics—CIBioFi, Edificio E20 No. 1069, Universidad del
Valle, 760032 Cali, Colombia  Departamento de Física, Universidad del Valle,
760032 Cali, Colombia  Departamento de Ciencias Naturales, Universidad
Autónoma de Occidente, Cali, Colombia  John H. Reina  Electronic Address:
<EMAIL_ADDRESS>Centre for Bioinformatics and
Photonics—CIBioFi, Edificio E20 No. 1069, Universidad del Valle, 760032 Cali,
Colombia  Departamento de Física, Universidad del Valle, 760032 Cali, Colombia
Irene D’Amico  Electronic Address<EMAIL_ADDRESS>Department of
Physics, University of York, York YO10 5DD, United Kingdom  Roberto M. Serra
Electronic Address<EMAIL_ADDRESS>Centro de Ciências Naturais e Humanas,
Universidade Federal do ABC, Avenida dos Estados 5001, 09210-580 Santo André,
São Paulo, Brazil  Department of Physics, Zhejiang Normal University, Jinhua
321004, China

###### Abstract

Employing currently available quantum technology, we design and implement a
non-classically correlated SWAP heat engine that allows to achieve an
efficiency above the standard Carnot limit. Such an engine also boosts the
amount of extractable work, in a wider parameter window, with respect to
engine’s cycle in the absence of initial quantum correlations in the working
substance. The boosted efficiency arises from a trade-off between the entropy
production and the consumption of quantum correlations during the full
thermodynamic cycle. We derive a generalized second-law limit for the
correlated cycle and implement a proof-of-principle demonstration of the
engine efficiency enhancement by effectively tailoring the thermal engine on a
cloud quantum processor.

##  I Introduction

Recent years have witnessed the rise of quantum thermodynamics (QTD), which
has rapidly become an arena to test and debate fundamental concepts such as
the laws of thermodynamics and the related fluctuation theorems at the quantum
scale [1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 12]. In parallel, QTD is
facilitating the extension of practical concepts such as heat and work towards
the design [13, 14, 15, 16, 17, 18, 19, 20] and, more recently, the
implementation [2, 26, 21, 22, 23, 24, 25, 27] of thermal machines based on a
handful of quantum degrees of freedom. Particular attention is being given to
the concept of entropy production for systems out of equilibrium [28, 29, 30,
31, 32] and to the – related – second law of thermodynamics [6, 33, 34], as
well as to the role of quantum correlations in both fundamental concepts [35,
36, 37, 38, 39] and in the functioning of thermal devices [40, 41, 42, 43, 44,
45]. Quantum features can be considered as extra resources. Squeezed states,
for instance, can enhance the performance of a microscopic engine above its
classical limits [17, 46]. Coherence with a dynamical interference [48, 49,
47], quantum measurements [50, 51, 52, 53, 54, 55, 56], and quantum operations
causal order [57, 58, 59, 60, 61, 62] also play nontrivial roles in the
performance of thermodynamic tasks [40, 41].

Through the design and proof-of-principle implementation of a two-qubit
thermal machine (on a quantum processor), here, we demonstrate that the use of
quantum correlations as an extra resource can indeed lead to a generalized
second law of thermodynamics encompassing regimes with efficiency larger than
the standard Carnot limit. As an added bonus, quantum correlations increase
the amount of extractable work, as well as extend the parameter region
corresponding to useful work extraction for the proposed cycle.

This paper is organized as follows. Taking advantage of initial non-classical
correlations in a two-qubit working substance, in Sec. II we design a
correlated quantum heat-engine based on the concept of a partial SWAP
operation. The engine’s nonequilibrium-thermodynamics quantifiers are
introduced and analytically computed in Sec. III. In Sec. IV, we introduce a
quantum processor implementation of the quantum heat engine setup and
experimentally demonstrate the performance enhancement of the proposed
correlated SWAP heat engine. In Sec. V, we derive an analytical expression for
the SWAP engine efficiency and provide a criterion for performance over the
classical limit (above Carnot) to occur. In Sec. V.1, we prove that such an
engine can exceed the conventional classical limits, with an out-performance
that is well described by an information-to-energy trade-off relation for the
cycle efficiency, which can be seen as a quantum generalized efficiency limit
for a two-stroke cycle in the presence of non-classical correlations. In Sec.
V.2 and V.3, based on the ancilla-assisted two-point-measurement method
developed in Ref. [26], we use a cloud quantum processor [63] to implement an
experimental proof-of-principle of our design. Summary of results and
concluding remarks are given in Sec. VI.

##  II Design and functioning of a correlated quantum heat engine

The quantum engine is composed by two qubits with energy gaps
$\varepsilon_{A}$ and $\varepsilon_{B}$. Each of them is initially coupled to
an effective heat environment, as schematized in Fig. 1(a), stroke 1.1, which
lead to the product of Gibbs states,
$\tilde{\rho}_{A}^{0}\otimes\tilde{\rho}_{B}^{0}$. After such a complete
thermalization, the two qubits get correlated through stroke 1.2 (Fig. 1(a))
in a state given by

$\rho_{AB}^{0}=\rho_{A}^{0}\otimes\rho_{B}^{0}+\chi_{AB},$ (1)

where the reduced local state for each qubit
$\rho_{i}^{0}=\text{exp}\left(-\beta_{i}\mathcal{H}_{i}\right)/\mathcal{Z}_{i}$
is a thermal equilibrium state at the inverse temperature
$\beta_{i}=\left(k_{B}T_{i}\right)^{-1}$, $i=A,B$;
$\mathcal{Z}_{i}=\text{Tr}_{i}\,\text{exp}\left(-\beta_{i}\mathcal{H}_{i}\right)$
is the corresponding partition function, and
$\chi_{AB}=\alpha\left|01\right\rangle\left\langle
10\right|+\alpha^{*}\left|10\right\rangle\left\langle 01\right|$ gives the
relevant correlation term, with $\text{Tr}_{i}\chi_{AB}=0$. Here, the states
$\left|0\right\rangle$ and $\left|1\right\rangle$ denote the ground and
excited eigenstates of the qubit Hamiltonian
$\mathcal{H}_{i}=-\frac{1}{2}\varepsilon_{i}\sigma_{Z}^{(i)}$; the total two-
qubit Hamiltonian $\mathcal{H}_{AB}=\mathcal{H}_{A}+\mathcal{H}_{B}$. Qubit
$A$ is assumed to be hotter than the qubit $B$, that is $\beta_{A}<\beta_{B}$.
We remark that the correlation term $\chi_{AB}$ does not contribute to the
energy of qubits $A$ and $B$. The populations of states $\tilde{\rho}_{A}^{0}$
and $\tilde{\rho}_{B}^{0}$ are chosen in a way to obtain the desired
correlated state in Eq. (1) (see Appendix A). The energy exchange between the
two qubits is determined by means of a partial SWAP operation, as illustrated
in Fig 1(a), stroke 2, where work can be extracted from or performed on the
quantum system. Physically, this can be implemented by an effective unitary
Heisenberg exchange Hamiltonian evolution ($j=X,Y,Z$)

$\mathcal{U}_{t}=\exp\left(\frac{-iJt}{2}\sum_{j}\sigma^{(A)}_{j}\otimes\sigma^{(B)}_{j}\right).$
(2)

At $t=0$ there is no interaction at all; a complete SWAP operation [64, 65]
takes place at $t=\pi/(2J)$.

Figure 1: (a) Thermodynamic cycle of a correlated SWAP quantum heat engine.
The two-qubit ($AB$) system acts as the working substance. $Q_{A}$ ($Q_{B}$)
denotes the heat exchanged with the hot (cold) environment. (b) Quantum-
circuit to test the correlated heat engine concept. The input to the circuit
is the ground state $\left|0000\right\rangle$. The qubits $q_{0}$ and $q_{3}$
are employed as ancillae to prepare the engine in the initial thermal state
$\rho_{1}^{0}\otimes\rho_{2}^{0}\equiv\tilde{\rho}_{A}^{0}\otimes\tilde{\rho}_{B}^{0}$
(box 1); the rotation angles $\theta_{i}$ depend on $\alpha$ and the qubits
effective temperature. The $AB$ correlations are generated in boxes 2 and 3
via $U_{-}$, and the final partial SWAP implementation, via $U_{+}$, in box 4.

Although the initial state Eq. (1) seems to be similar to the one considered
in Ref. [41], we stress that the off-diagonal parameter has a different phase
and the evolution is driven by a different Hamiltonian, which makes all the
difference to exploit non-convectional energy flows to thermal tasks.
Moreover, the correlation parameter used in [41] would not allow for an
advantage in the quantum heat engine here implemented.

Next, we consider a proof-of-principle implementation on a quantum processor.
Figure 1(b) shows a quantum circuit that implements a partial SWAP engine in
the presence of initial correlations. It is based on the use of the
(controlled) gate
$U_{\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}}(x)\equiv\mathds{1}\sqrt{1-x}\pm
i\sigma_{Y}\sqrt{x}$, as follows. Qubits $q_{1}$ and $q_{2}$ constitute the
working substance, while qubits $q_{0}$ and $q_{3}$ are employed as ancillae
to prepare the engine in the initial thermal state
$\rho_{1}^{0}\otimes\rho_{2}^{0}$, and emulate the actions of the hot and cold
environments, respectively. This initial state is prepared by means of a
rotation gate $R_{x}(\theta_{i})$ in the ancillae and a CNOT gate between the
ancillae and the qubits $i=A,B$ ($q_{1},q_{2}$ in box 1, Fig. 1(b)). The
rotation angles are associated with the effective temperature of each qubit
and with the correlation parameter $\alpha$ as follows:

$\displaystyle\theta_{A}$ $\displaystyle=$
$\displaystyle\arccos{\sqrt{p_{-}}},\;\;\;\theta_{B}=\arccos{\sqrt{p_{+}}},$
(3) $\displaystyle p_{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(p_{A}+p_{B}\pm\sqrt{(p_{B}-p_{A})^{2}+4\alpha^{2}}\right),$
(4) $\displaystyle\frac{\varepsilon_{B}}{\varepsilon_{A}}$ $\displaystyle=$
$\displaystyle\frac{\beta_{A}\ln\left(p_{B}\mathcal{Z}_{B}\right)}{\beta_{B}\ln\left(p_{A}\mathcal{Z}_{A}\right)},$
(5)

where $p_{i}=\exp(-\beta_{i}\varepsilon_{i})/\mathcal{Z}_{i}$ and
$\varepsilon_{B}/\varepsilon_{A}$ gives the qubits energy-gap ratio.

In the present implementation, the initial correlation in the working
substance ($q_{1}$ and $q_{2}$) is achieved by a CNOT gate framing one
controlled $U_{-}(x)$ gate, $x=x(\alpha)=(p_{A}-p_{B})/(p_{+}-p_{-})+1/2$, as
shown in Fig. 1(b) (box 3) [66], and $\alpha$ should satisfy $\alpha\leq
1/\left(\mathcal{Z}_{A}\mathcal{Z}_{B}\right)$ to fulfill that the system’s
density operator is a positive semi-definite. We also produced the same
reduced correlated state of the working substance in the ancillary qubits
$q_{0}$ and $q_{3}$, through box 2 of Fig. 1(b), in order to have a copy of
the initial correlated state at the circuit end, thus having an abridged
measurement strategy. After boxes 2 and 3 (Fig. 1(b)), the working substance
is in the state given by Eq. (1). Two distinct scenarios are identified: i)
$\alpha=0$, the qubits are initially uncorrelated, and ii) $\alpha\neq 0$, the
bipartite state is initially correlated. The partial SWAP operation, stroke 2,
can be effectively implemented through a quantum circuit composed by two CNOT
gates framing one controlled-gate $U_{+}(\lambda)$ (Fig. 1(b), box 4). For
$\lambda=1$, we have a full SWAP between the qubit states, while for
$\lambda=0$ there is no operation at all between such qubits. The partial SWAP
takes place at a time $\tau$, with $0<\tau<\pi/(2J)$. We finally measure all
qubits in their respective ($\sigma_{Z}^{(i)}$) energy basis.

Figure 2: Parameters phase diagram for engine operation mode: (a) initially
uncorrelated qubits, work extraction is only allowed for parameter ratios
$\\{\beta_{B}/\beta_{A},\varepsilon_{B}/\varepsilon_{A}\\}$ above the dashed
curve (blue region); (b) initially correlated qubits, work extraction is
allowed for all $0\leq\varepsilon_{B}/\varepsilon_{A}<1$,
$\beta_{A}<\beta_{B}$. Experimental results for the re-scaled average work
($\left\langle W\right\rangle/\varepsilon_{A}$), and heat from the hot
($\left\langle Q_{A}\right\rangle/\varepsilon_{A}$) and cold ($\left\langle
Q_{B}\right\rangle/\varepsilon_{A}$) reservoirs for (c) initially uncorrelated
qubits, and (d) initially correlated qubits; in all experimental runs in the
quantum processor, we set: $\beta_{B}=2\beta_{A}$, $\lambda=0.6$, and
$\alpha_{max}=1/\left(\mathcal{Z}_{A}\mathcal{Z}_{B}\right)$. The error bars
were estimated using the standard deviation of the measured data. The solid
curves correspond to the theoretical prediction.  Figure 3:  Quantum-
processor-implementation results for the average work, and heat from the hot
($\left\langle Q_{A}\right\rangle$) and cold ($\left\langle
Q_{B}\right\rangle$) reservoirs for (a) initially uncorrelated qubits
($\lambda=0$), and initially correlated qubits: (b) $\lambda=0.8$, (c)
$\lambda=0.6$, and (d) $\lambda=0.2$. In all runs in the quantum processor,
$\beta_{B}=2\beta_{A}$ and
$\alpha_{max}=1/\left(\mathcal{Z}_{A}\mathcal{Z}_{B}\right)$. The error bars
were estimated using the standard deviation of the measured data. The solid
curves are obtained from our theoretical predictions from Eqs. (6) to (8) and
numerical simulations.

##  III Quantum-engine nonequilibrium-thermodynamics quantifiers

For the aforementioned cycle, we calculate the mean energies involved in the
whole process: the average values for the work, $\langle
W\rangle=\text{Tr}[(\rho_{AB}^{f}-\rho_{AB}^{0})\mathcal{H}_{AB}]$, (that
takes place in stroke 2) and the heat contributions from the hot
($\left\langle Q_{A}\right\rangle$) and cold ($\left\langle
Q_{B}\right\rangle$) environments, $\left\langle
Q_{i}\right\rangle=-\text{Tr}[(\rho_{i}^{f}-\rho_{i}^{0})\mathcal{H}_{i}]$,
where
$\rho_{A(B)}^{f}=\text{Tr}_{B(A)}\left(\rho_{AB}^{f}\right)=\text{Tr}_{B(A)}\left(\mathcal{U}_{\tau}\rho_{AB}^{0}\mathcal{U}^{\dagger}_{\tau}\right)$
is the final out-of-equilibrium reduced state for qubit $A$ ($B$) after stroke
2. We obtain:

$\displaystyle\left\langle W\right\rangle$ $\displaystyle=$ $\displaystyle
2\left(\varepsilon_{B}-\varepsilon_{A}\right)f(\Delta\nu,\lambda,\alpha),$ (6)
$\displaystyle\left\langle Q_{A}\right\rangle$ $\displaystyle=$ $\displaystyle
2\varepsilon_{A}f(\Delta\nu,\lambda,\alpha),$ (7) $\displaystyle\left\langle
Q_{B}\right\rangle$ $\displaystyle=$
$\displaystyle-2\varepsilon_{B}f(\Delta\nu,\lambda,\alpha),$ (8)

with
$f(\Delta\nu,\lambda,\alpha)=\frac{\sinh(\Delta\nu)}{\mathcal{Z}_{A}\mathcal{Z}_{B}}\lambda+2\alpha\sqrt{\lambda(1-\lambda)}$,
where
$\Delta$$\nu=$$\left(\varepsilon_{B}\beta_{B}-\varepsilon_{A}\beta_{A}\right)$/2.
Since the total energy is conserved, Eq. (6) to (8) fulfill energy
conservation, i.e., $\left\langle W\right\rangle=-(\left\langle
Q_{A}\right\rangle+\left\langle Q_{B}\right\rangle)$. These results for the
correlated case ($\alpha\neq 0$) are plotted in Fig. 2(b) and as the
continuous curves in Fig. 2(d) (see also Fig. 3). For comparison, in Fig. 2(a)
and (c) we plot the corresponding energies in the absence of initial
correlations ($\alpha=0$).

We characterized the performance of the quantum engine by varying the energy
gap ratio $\varepsilon_{B}/\varepsilon_{A}$, setting $\beta_{B}=2\beta_{A}$,
and fixing the SWAP parameter to $\lambda=0.6$ (other values are considered in
Fig. 3). For the correlated initial state we considered $\alpha$’s maximum
value, $\alpha_{max}=1/\left(\mathcal{Z}_{A}\mathcal{Z}_{B}\right)$.

In Fig. 2(a) and (b) we plot the parameters’ diagram for temperature and
energy gap ratios $\\{\beta_{B}/\beta_{A},\varepsilon_{B}/\varepsilon_{A}\\}$
required for work extraction for both, initially uncorrelated, and correlated
scenarios. The dashed curve in the diagram for the case without initial
correlations, Fig. 2(a), separates the regions for which the system can work
as a refrigerator (work injection) or as a heat engine (work extraction). In
contrast, for initially correlated qubits (Fig. 2(b)), work injection gets
suppressed in the whole gap ratio $\varepsilon_{B}/\varepsilon_{A}<1$, for all
$\beta_{A}<\beta_{B}$, and work extraction becomes much higher; the maximal
extracted work can be seen, e.g., for $\beta_{B}\gtrsim 3\beta_{A}$ and
$0.2\lesssim\varepsilon_{B}/\varepsilon_{A}\lesssim 0.5$ (Fig. 2(b)).

In Fig. 2(c) and (d) we plot the mean energy (re-scaled) quantities, work
($\left\langle W\right\rangle$, black dots) and heat from the hot
($\left\langle Q_{A}\right\rangle$, red dots) and cold ($\left\langle
Q_{B}\right\rangle$, blue dots) environments obtained from the experimental
runs in the quantum processor. The error bars were estimated using the
standard deviation of the measured data in the quantum processor. Three
operational regimes [65] appear in the initially uncorrelated qubits scenario
(Fig. 2(c)): refrigerator ($0<\varepsilon_{B}/\varepsilon_{A}<1/2$), heat
engine ($1/2<\varepsilon_{B}/\varepsilon_{A}<1$), and heat accelerator
($\varepsilon_{B}/\varepsilon_{A}>1$). However, when the qubits are initially
correlated, the partial SWAP engine only exhibits two operational modes: heat
engine ($0<\varepsilon_{B}/\varepsilon_{A}<1$) and heat accelerator
($\varepsilon_{B}/\varepsilon_{A}>1$) (Fig. 2(d)). In the engine operation
mode, quantum correlations boost the amount of work that can be extracted,
making it at its maximum about an order of magnitude larger than the one
obtained in the absence of initial qubit correlations. As already explained
above, this result has also been verified for other $\lambda$ values (see Fig.
3). In Figs. 2(c), (d) and Fig. 3 we find a very good agreement between the
quantum-processor-implementation results and the corresponding theoretical
prediction from Eqs. (6) to (8) (solid curves). Furthermore, quantum
correlations enlarge the $\varepsilon_{B}/\varepsilon_{A}$ values’ window
where work extraction is possible.

In Fig. 3 the average work and heat from the hot and cold environments, for
different $\lambda$ values are plotted. As established above, Fig. 3(a) shows
the three different modes of operation of the SWAP quantum engine that are
available if we consider initially uncorrelated qubits ($\lambda=0$). The
engine mode occurs for $0.5<\varepsilon_{B}/\varepsilon_{A}<1$. Otherwise, for
initially correlated qubits, the engine mode expands its range to
$0.0<\varepsilon_{B}/\varepsilon_{A}<1$, as it is shown in Fig. 3(b)-(d),
$\lambda=0.8,0.6,0.2$, respectively. The lower the $\lambda$ parameter the
greater the amount of extracted work. In fact, executing a full SWAP
($\lambda=1$) in the engine would erase the advantage due to its quantum
correlations hence obtaining the same result as in the absence of initial
correlations.

From Eqs. (6) and (7) it is straightforward to obtain the SWAP engine
efficiency,

$\eta=-\frac{\left\langle W\right\rangle}{\left\langle
Q_{A}\right\rangle}=1-\frac{\varepsilon_{B}}{\varepsilon_{A}}.$ (9)

For qubit energies such that
$\frac{\varepsilon_{B}}{\varepsilon_{A}}=\frac{\beta_{A}}{\beta_{B}}$, the
quantum engine achieves the standard Carnot limit,

$\eta_{Carnot}\equiv 1-\frac{\text{{\hbox{\beta_{A}}}}}{\beta_{B}}.$ (10)

Figure 4:  Schematic representation of the transpiled circuit implemented by 3
basis gates: $R_{z}$ rotations, $\sqrt{X}$, and CNOT. The circuit depth or
“layers” of quantum gates are executed in parallel according to ibmq_manila
topology to achieve the computation indicated in the original circuit of Fig.
1(b). The $q[i]$ input labeling indicates the physical qubits used by the
5-qubit processor and the circuit output $c[i]$ represents the classical data
where the measurement result is recorded.

##  IV QUANTUM HEAT ENGINE EXPERIMENTAL SETUP

Here we give further details about the quantum processor implementation of a
quantum heat engine and corresponding data analysis. For the implementation
and characterization of the quantum heat engine we ran several experiments on
the 5-qubit ibmq_manila quantum processor [63]. Below we experimentally
demonstrate the performance enhancement of a correlated SWAP heat engine and
confirm our theoretical predictions.

For each run in the quantum processor, we implemented the quantum circuit
depicted in Fig. 1(b) and collected the qubits statistics over a sample of
size 20000. We performed the experiment for different values of the partial
SWAP parameter $\lambda$. Additionally, for each $\lambda$, we ran 10
experiments with the aforementioned sample size in order to see the
fluctuations of the experimental setup. From these, we estimated the error
propagation, using the standard deviation. Each circuit run is determined by
the correlation ($\alpha$) and thermalization ($\lambda$) parameters.

Engine implementation experimental parameters
---
|  |  |  | $\lambda=0.2$ | $\lambda=0.6$ | $\lambda=0.8$
$\varepsilon_{B}/\varepsilon_{A}$ | $\theta_{A}(\text{rad})$ | $\theta_{B}(\text{rad})$ | $\phi_{A}(\text{rad})$ | $\phi_{B}(\text{rad})$ | $\phi_{B}(\text{rad})$ | $\phi_{B}(\text{rad})$
0.05 | 2.41 | 1.34 | 2.57 | 0.10 | -0.32 | -0.54
0.20 | 2.47 | 1.45 | 2.50 | 0.17 | -0.25 | -0.47
0.40 | 2.55 | 1.56 | 2.41 | 0.27 | -0.15 | -0.37
0.60 | 2.64 | 1.69 | 2.31 | 0.37 | -0.05 | -0.27
0.70 | 2.67 | 1.74 | 2.26 | 0.42 | -0.001 | -0.22
0.80 | 2.71 | 1.79 | 2.21 | 0.467 | 0.05 | -0.17
1.00 | 2.78 | 1.86 | 2.12 | 0.56 | 0.14 | -0.08
1.20 | 2.84 | 1.92 | 2.03 | 0.65 | 0.22 | 0.002
1.40 | 2.89 | 1.96 | 1.96 | 0.72 | 0.30 | 0.08
1.50 | 2.91 | 1.97 | 1.92 | 0.75 | 0.33 | 0.11
Table 1: Parameters used in the transpiled circuit displayed in Fig. 4 for the
engine implementation in a quantum processor.

In Figure 4, we outline the transpiled circuit that has been implemented
following the topology and optimization of ibmq_manila. This corresponds to
the engine thermodynamic cycle portrayed in the circuit of Fig. 1(b). The
correlated SWAP engine implementation parameters (see Table 1) have energy-gap
ratio $\varepsilon_{B}/\varepsilon_{A}$, and angles $\theta_{A}$ and
$\theta_{B}$, with $p_{\pm}$ and $p_{i}$ as given in Eqs. (3) to (5). Here,

$\displaystyle\phi_{A}$ $\displaystyle=$
$\displaystyle\arcsin{\sqrt{x(\alpha)}},$ (11) $\displaystyle\phi_{B}$
$\displaystyle=$ $\displaystyle\phi_{A}-\arcsin{\sqrt{\lambda}},$ (12)
$\displaystyle x(\alpha)$ $\displaystyle=$
$\displaystyle(p_{A}-p_{B})/(p_{+}-p_{-})+1/2,$ (13)

where $x(\alpha)$ and $\lambda$ give the correlation and partial swap
parameters, respectively. In Table 1 we display the experimental angles used
in all the runs in the quantum processor here reported.

ibmq_manila calibration data
---
Qubit | $T_{1}(\mu\text{s})$ | $T_{2}(\mu\text{s})$ | Gate time (ns)
$q_{0}$ | 177.13 | 78.73 | 0_1: 277.33
$q_{1}$ | 186.02 | 75.55 | 1_2: 469.33
|  |  | 1_0: 312.89
$q_{2}$ | 136.19 | 22.30 | 2_3: 355.56
|  |  | 2_1: 504.88
$q_{3}$ | 184.82 | 46.64 | 3_4: 334.22
|  |  | 3_2: 391.11
$q_{4}$ | 122.91 | 43.53 | 4_3: 298.67
Table 2: Calibration data for the qubits involved in the generation of the
heat engine at the 5-qubit ibmq_manila quantum processor (Fig. 4).

In the transpiled circuit (Fig. 4), three basic gates are used; $R_{z}$
rotations, $\sqrt{X}$ and CNOT, in order to parallel implement the original
quantum circuit of Fig. 1(b). We resort to the gating and decoherence times
for ibmq_manila [63] as figures of merit to check that the coherence
properties required in the implementation of the quantum thermodynamic cycle
are guaranteed. Clearly, relaxion ($T_{1}$) and decoherence ($T_{2}$) times
are about three orders of magnitude longer than the qubits’ gating times.
These times are reported in Table 2.

The last column gives the execution times of the ‘$i\\_j$’ two-qubit gates
according to the ibmq_manila topology ($i,j=0,1,...,4$). By assuming that each
gate takes its longest possible execution time ($\sim 500$ ns), we can
overestimate the runtime of the full circuit to approximately
$11\;\mu\text{s}$, which is shorter than the shortest $T_{2}$ time reported in
Table 2.

In Fig. 5 we plot the average work and efficiency for (a) and (c) initially
uncorrelated qubits, (b) and (d) initially correlated qubits. The blue dots
give the results of 10 experiments for each energy gap ratio value shown in
the figure. Each dot is obtained by running the circuit Fig. 4 with 20000
shots, using the parameters given in Table 1. Figures 5 (a) and (b)
demonstrate that the extractable work (negative work) under initially
correlated qubits is larger (about an order of magnitude larger at its
maximum) than the one obtained in the absence of initial qubit correlations.
In Fig. 5 (c) and (d) we show the efficiency obtained for the experimental
implementation of the thermal machine. This is in agreement with the criterion
given below (see Eqs. (16) and (17)): efficiency for uncorrelated initial
qubits remain below the standard Carnot limit, while initially correlated
qubits may allow for a boost in efficiency, with values well above the Carnot
limit.

Figure 5:  Quantum-processor-implementation results for the average work and
efficiency, considering (a) and (c) initially uncorrelated qubits; (b) and (d)
initially correlated qubits. The blue dots indicate the realization of 10
different experiments for each $\varepsilon_{B}/\varepsilon_{A}$ value, and
each one performed with 20000 shots. The full lines in (c) and (d) correspond
to the theoretical prediction. In all the figures, $\lambda=0.6$,
$\beta_{B}=2\beta_{A}$, and
$\alpha_{max}=1/\left(\mathcal{Z}_{A}\mathcal{Z}_{B}\right)$.  Figure 6:
Experimental results for the correlated SWAP quantum heat engine
implementation; figures of merit: (a) efficiency and booster, (b) entropy
production, variation of mutual information and of quantum discord, and (c)
test of the generalized second law limit for efficiency, which must satisfy
$\eta+\mathcal{B_{E}}=\eta_{Carnot}=1/2$. The black and blue dashed lines in
(a) correspond to the theoretical expectation for the correlated SWAP engine
efficiency and efficiency booster, respectively. The shaded areas in (b)
denote the theoretical results but calculated with the actual initial
experimental states (instead of the ideal ones) of the various experimental
runs. In all experiments, we set: $\beta_{B}=2\beta_{A}$, $\lambda=0.6$, and
$\alpha_{max}=1/\left(\mathcal{Z}_{A}\mathcal{Z}_{B}\right)$. The error bars
were estimated using the standard deviation of the measured data.

##  V Boosting quantum engine efficiency by quantum correlations

The mutual information $I(A:B)=S_{A}+S_{B}-S_{AB}$ gives a measure of the
total correlations between systems $A$ and $B$, where
$S_{i}=-\text{Tr}_{i}\left(\rho_{i}\ln\rho_{i}\right)$ is the von Neumann
entropy of state $\rho_{i}$. We next derive an analytical expression for the
SWAP engine efficiency. Such efficiency involves an information-to-energy
trade-off relation written in terms of the single-cycle variation of the
mutual information between qubits $A$ and $B$, $\Delta I(A:B)=\Delta
S_{A}+\Delta S_{B}$ (where $\Delta S_{i}=S(\rho_{i}^{f})-S(\rho_{i}^{0})$) and
of the entropy production cycle,
$\Sigma_{\text{eng}}=D[\rho_{A}^{f}||\rho_{A}^{0}]+D[\rho_{B}^{f}||\rho_{B}^{0}]$.
Here,
$D\left[\rho||\sigma\right]=\text{Tr}\left[\rho\left(\ln\rho-\ln\sigma\right)\right]$
is the Kullback–Leibler divergence [1]. For the cycle, we arrive at the
following generalized efficiency (see Appendix B):

$\eta=\eta_{Carnot}-\frac{\Sigma_{\text{eng}}+\Delta
I(A:B)}{\beta_{B}\left\langle Q_{A}\right\rangle}.$ (14)

Equation (14) can be applied to all cycles based on bipartite working
substance (of any dimension) when work and heat exchanges are performed in
different strokes. Let us now introduce an efficiency booster quantifier
$\mathcal{B_{E}}$,

$\mathcal{B_{E}}\equiv\frac{\Sigma_{\text{eng}}+\Delta
I}{\beta_{B}\left\langle Q_{A}\right\rangle},$ (15)

which tracks the direct competition between entropy production and
correlations consumption.

Figure 7: Entropy and correlation quantities during the thermodynamic cycle:
(a) $\lambda=0.8$, (b) $\lambda=0.6$, and (c) $\lambda=0.2$. Experimental data
in black, blue, and red correspond to the entropy production, variation of
quantum discord and variation of mutual information, respectively. In all runs
in the quantum processor, the temperature relation $\beta_{B}=2\beta_{A}$ and
the correlation factor
$\alpha_{max}=1/\left(\mathcal{Z}_{A}\mathcal{Z}_{B}\right)$. The shaded areas
denote the theoretical results calculated with the actual initial experimental
states (instead of the ideal one), see e.g. Fig. 8. The error bars were
estimated using the standard deviation of the measured data.

###  V.1 Engine efficiency criterion

Equation (14) implies that there may exist efficiencies above Carnot,
$\eta>\eta_{Carnot}$, depending on the sign of $\Delta I$, with the following
engine efficiency criterion arising

$\displaystyle\mathcal{\mathcal{B_{E}}}<0,\;\eta>\eta_{Carnot},$ (16)
$\displaystyle\mathcal{\mathcal{B_{E}}}\geq 0,\;\eta\leq\eta_{Carnot}.$ (17)

Equation (14) can be seen as a quantum generalization of the second law
efficiency and expression (16) is the condition for performance over the
classical limit to occur, and indeed it is satisfied in Fig. 6(a). This result
arises since the variation of mutual information $\Delta I$ is always
negative, and there is a trade-off with the always positive entropy production
$\Sigma_{\text{eng}}$ (see Fig. 6(b)). The variation of $\Delta
I=\Delta\mathcal{D_{Q}}+\Delta\mathcal{C}$, where $\mathcal{D_{Q}}$ quantifies
purely quantum correlations (here given by the quantum discord [67]), and
$\mathcal{C}$ represents classical correlations, demonstrates that there is a
consumption of quantum correlations during the thermodynamic cycle. For
$\varepsilon_{B}/\varepsilon_{A}<1/2$, this makes $\Sigma_{eng}<\lvert\Delta
I(A:B)\lvert$ and hence $\mathcal{\mathcal{B_{E}}}<0$, which in turn implies
$\eta>\eta_{Carnot}$ in Eq. (14).

For the calculation of the quantum correlations we have used the geometrical
quantum discord $\mathcal{D_{Q}}$ for a two-qubit $X$ state [69, 70], which
can be written in terms of the elements of the density matrix as:

$\mathcal{D_{Q}}=\left[k_{1}-2k_{2}+4z^{2}-max\left(k_{1}-2k_{2},2z^{2}\right)\right],$
(18)

where $k_{1}=a^{2}+b^{2}+c^{2}+d^{2}$ and $k_{2}=ac+bd$.

The emergence of condition (16) depends on a proper choice of initially
correlated states and of the driving Hamiltonian in the stroke 2. It only
arises if $\mathcal{B_{E}}<0$. Otherwise, $\eta\leq\eta_{Carnot}$ (Eq. (17));
$\mathcal{B_{E}}\geq 0$ also describes engine operation for initially
uncorrelated qubits (Fig. 2(a)): entropy production is always greater or equal
than variation of mutual information. An experimental verification of the
engine efficiency criterion Eqs. (16) and (17) (see Figs. 5, 6 and 7) is
provided below.

###  V.2 Experimental demonstration of performance boosting in the correlated
SWAP heat engine

In Fig. 6, we plot the quantum-processor-implementation results for the
efficiency, the entropy and the generalized second law related quantities as a
function of the energy gap ratio. These quantifiers have been obtained by
using quantum state tomography (QST), as detailed below. The efficiency $\eta$
(black dots) and the efficiency booster $\mathcal{\mathcal{B_{E}}}$ (blue
dots) are displayed in Fig. 6(a). The plotted error bars were estimated as in
Fig. 2. The booster $\mathcal{\mathcal{B_{E}}}$ reaches negative values hence
the efficiencies go above the Carnot limit, $\eta_{Carnot}=0.5$, in agreement
with our theoretical findings (solid lines). In Fig. 6(b), we plot the entropy
production (black dots), the variation of mutual information (red dots), and
of quantum discord (blue dots) obtained in the quantum processor
implementation. For $\varepsilon_{B}/\varepsilon_{A}<0.5$, we obtain
$\lvert\Delta I(A:B)\lvert>\Sigma_{\text{eng}}$ and since $\Delta I$ is always
negative, $\mathcal{\mathcal{B_{E}}}<0$ and the SWAP engine efficiency
surpasses the standard Carnot limit $\eta>\eta_{Carnot}$, which is in perfect
agreement with Fig. 6(a) and with the criterion Eq. (16).

We make explicit the role of the measured correlations (both quantum and
classical), by plotting $\Delta I=\Delta\mathcal{D_{Q}}+\Delta\mathcal{C}$ and
$\Delta\mathcal{D_{Q}}$, during the thermodynamic cycle (see the two lower
curves of Fig. 6(b)). Here, the variation of quantum discord closely follows
that of mutual information and both are always negative. This means that while
entropy production increases, both variations in classical and quantum
correlations are consumed during the cycle. The larger correlations
consumption is of purely quantum origin, and come from the discord. These
results have been further verified and plotted for $\Sigma_{\text{eng}}$,
$\Delta I$ and $\Delta\mathcal{D_{Q}}$ in Fig. 7, for other values of
$\lambda$. The shaded areas in Fig. 6(b) represent the theoretical results
calculated with the actual initial states in the quantum processor (instead of
the ideal one) of the various runs. The theoretical predictions are in very
good agreement with the implementation results.

Figure 6(c) shows the quantum-processor-implementation results for a
verification of the generalized efficiency formula as a function of the
qubits’ energy-gap ratio. The black dots give the results for the sum of the
engine efficiency and the booster $\eta+\mathcal{\mathcal{B_{E}}}$, as
averaged measurements following the qubits statistics from collected data from
20000 runs. We find that the generalized second law limit measured with the
ibmq_manila quantum processor is at most within two standard deviations from
the analytical result Eq. (14), thus verifying, with Fig. 6(b), the quantum
origin of the working thermodynamical principle for enhancing the efficiency
of the correlated SWAP quantum heat engine.

###  V.3 Entropy related quantities and quantum state tomography

In Fig 7 we plot the variation of entropic and correlation quantities as
function of $\varepsilon_{B}/\varepsilon_{A}$, namely the variation of the
engine entropy production $\Sigma_{\text{eng}}$, of the mutual information
$\Delta I$ and of the quantum discord $\mathcal{D_{Q}}$, for (a)
$\lambda=0.8$, (b) $\lambda=0.6$, and (c) $\lambda=0.2$. Figure 7 further
demonstrates that the variation of mutual information is mostly due to the
consumption of quantum correlations between the qubits. It also confirms that
for initially correlated systems $|\Delta I|$ may be greater than
$\Sigma_{\text{eng}}$, leading to a correlation boosting of the engine
efficiency.

Figure 8: Quantum state tomography of the two-bit working substance for the
initially correlated state (upper row) and corresponding final state (lower
row) for $\lambda=0.6$ and for the following qubits energy gap ratios
$\varepsilon_{B}/\varepsilon_{A}$: (a) and (f) 0.10; (b) and (g) 0.30; (c) and
(h) 0.55; (d) and (i) 0.70; (e) and (j) 0.90. Data shown represent the result
over 20000 shots for each circuit used in one of the QST implementations.
Re$(\rho)$ denotes the real values of the density matrix (the corresponding
imaginary part entries are of the order of $10^{-3}$, not shown).

For the quantum state tomography (QST) implementation required in this
analysis we have used a module in the qiskit-ignis library [68]. For a
complete QST of a two-qubits state, nine circuits are needed. The result for
each circuit is averaged over 20000 shots and, additionally, in order to
average over the system fluctuations, we repeat the process five times for the
initial state and five times for the final one. In Figure 8(a)-(e) we show the
QST result for the initially correlated state $\rho^{0}_{AB}$, considering
$\varepsilon_{B}/\varepsilon_{A}=0.10,0.30,0.55,0.70,0.90$, respectively. In
the same way, in Fig. 8(f)-(j) we give the QST result for the final state,
with $\lambda=0.6$ and
$\varepsilon_{B}/\varepsilon_{A}=0.10,0.30,0.55,0.70,0.90$, respectively.
Similar results were obtained for $\lambda=0.2,0.8$ (not shown). In the
computational basis, the qubits density matrix can then be approximated as

$\rho=\left(\begin{array}[]{cccc}a&0&0&0\\\ 0&b&z&0\\\ 0&z&c&0\\\
0&0&0&d\end{array}\right).$ (19)

##  VI Summary

In summary, the limits posed by the second law of thermodynamics may be
affected by the presence of initial quantum correlations in the working fluid
of a thermal machine, leading to efficiency higher than the Carnot standard
limit and to a boost in the extractable work in each cycle. A criterion for
the construction of such enhanced thermodynamic feature is given in terms of a
trade-off between entropy production and quantum correlations consumption
during the implemented thermal machine’s cycle. The design of thermal machines
that use extra resources based on quantum correlations highlight the need for
a revision of the standard thermodynamics limits. In this framework, the
energetic cost of building initial correlations should not be included in the
efficiency definition. This is in line with the practice of not including
costs related to the production of (hot) heat sources in classical internal-
combustion engines (e.g., fuel production/refining, etc.). Our results with an
IBM quantum processor clearly demonstrate that the effect we propose allows to
obtain advantages in thermal tasks using available quantum technology.

###### Acknowledgements.

M.H. and J.H.R acknowledge the financial support from MinCiencias (Colombia)
through a Postdoctoral Fellowship Award (Grant No. 270-2021/CI 71295) and the
Norwegian Ministry of Education and Research, through the QTECNOS consortium
(NORPART 2021-10436/CI 71331). R.M.S. also acknowledges CNPq, FAPESP (Brazil)
and Ministry of Science and Technology (China), through the High-End Foreign
Expert Program (Grant No. G2021016021L). I.D.A. acknowledges the kind
hospitality of the Instituto de Física de São Carlos, University of São Paulo,
São Carlos (Brazil). We also acknowledge the use of ibmq_manila quantum
processor of IBM Quantum services [63].

##  Appendix A ABOUT ENERGY EXCHANGE, CORRELATIONS PRODUCTION AND EFFICIENCY

Here we show the effect of taking into account the initial-state correlation
production in the thermodynamic cycle. To do so, we start from the state in
stroke 1.1 (Fig. 1(a)). Such a state,
$\tilde{\rho}_{A}^{0}\otimes\tilde{\rho}_{B}^{0}$, explicitly reads

$\left(\begin{array}[]{cccc}p_{+}p_{-}&0&0&0\\\ 0&p_{+}(1-p_{-})&0&0\\\
0&0&p_{-}(1-p_{+})&0\\\ 0&0&0&(1-p_{+})(1-p_{-})\end{array}\right).$ (20)

The populations of states $\tilde{\rho}_{A}^{0}$ and $\tilde{\rho}_{B}^{0}$
are chosen as
$p_{\pm}=\left(p_{A}+p_{B}\pm\sqrt{(p_{B}-p_{A})^{2}+4\alpha^{2}}\right)/2$ to
obtain the desired correlated state
$\rho_{AB}^{0}=\rho_{A}^{0}\otimes\rho_{B}^{0}+\chi_{AB}$, where $p_{-}$
($p_{+}$) denotes the ground state population of $\tilde{\rho}_{A}^{0}$
($\tilde{\rho}_{B}^{0}$). Hence, $\rho_{AB}^{0}$ becomes

$\left(\begin{array}[]{cccc}p_{A}p_{B}-\alpha^{2}&0&0&0\\\
0&p_{A}(1-p_{B})+\alpha^{2}&\alpha&0\\\
0&\alpha&p_{B}(1-p_{A})+\alpha^{2}&0\\\
0&0&0&(1-p_{A})(1-p_{B})-\alpha^{2}\end{array}\right),$ (21)

and the von Neumann entropy of $\rho_{A}^{0}\otimes\rho_{B}^{0}+\chi_{AB}$ is
the same than that of $\tilde{\rho}_{A}^{0}\otimes\tilde{\rho}_{B}^{0}$. In
fact, using the expressions for $p_{\pm}$ and $p_{i}$, we obtain the same
eigenvalues for both density operators, $\rho_{AB}^{0}$ and
$\tilde{\rho}_{A}^{0}\otimes\tilde{\rho}_{B}^{0}$:

$\begin{split}\nu_{1}&=\frac{1}{2}\left[p_{A}+p_{B}-2p_{A}p_{B}+2\alpha^{2}-\sqrt{\left(p_{B}-p_{A}\right)^{2}+4\alpha^{2}}\right],\\\
\nu_{2}&=\left(1-p_{A}\right)\left(1-p_{B}\right)-\alpha^{2},\\\
\nu_{3}&=p_{A}p_{B}-\alpha^{2},\\\
\nu_{4}&=\frac{1}{2}\left[p_{A}+p_{B}-2p_{A}p_{B}+2\alpha^{2}+\sqrt{\left(p_{B}-p_{A}\right)^{2}+4\alpha^{2}}\right].\end{split}$
(22)

From the initial state $\tilde{\rho}_{A}^{0}\otimes\tilde{\rho}_{B}^{0}$, we
calculate the heat and work as
$\left\langle\tilde{Q}_{i}\right\rangle=-\text{Tr}\left[\left(\rho_{i}^{f}-\tilde{\rho}_{i}^{0}\right)\mathcal{H}_{i}\right]$
and
$\left\langle\tilde{W}\right\rangle=\text{Tr}\left[\left(\rho_{AB}^{f}-\tilde{\rho}_{A}^{0}\otimes\tilde{\rho}_{B}^{0}\right)\mathcal{H}_{AB}\right]$,
respectively. In Fig. 9(a), the dashed lines correspond to the heat generation
($\langle\tilde{Q}_{A}\rangle/\epsilon_{A}$, pink area) and work production
($\langle\tilde{W}_{A}\rangle/\epsilon_{A}$, yellow area) when considering
$\tilde{\rho}_{A}^{0}\otimes\tilde{\rho}_{B}^{0}$ as the initial state. We see
that $\langle\tilde{Q}_{A}\rangle$ and $\langle\tilde{W}\rangle$ are greater
than $\langle Q_{A}\rangle$ and $\langle W\rangle$ (in absolute value), but
the cycle efficiency when considering the correlation production
$\tilde{\eta}=\langle\tilde{W}\rangle/\langle\tilde{Q}_{A}\rangle$ corresponds
to the same one computed from the correlated state $\eta=\langle
W\rangle/\langle Q_{A}\rangle$, as it is shown in Fig. 9(b). This is so
because there is a compensation in the extra amounts obtained for the
corresponding heat and work, as illustrated in the shaded areas of Fig. 9(a).
These show the extra heat generation (pink area) and the correlation
production cost (yellow area) due to the generation of correlations in the
initial state.

Figure 9: (a) Average work
$\left\langle\tilde{W}\right\rangle/\varepsilon_{A}$, heat from the hot
reservoir $\left\langle\tilde{Q}_{A}\right\rangle/\varepsilon_{A}$, and (b)
efficiency $\tilde{\eta}$, including in the cycle the initial-state
correlation production. In the same graphs, we plot the corresponding
quantities but now without including the initial-state correlation production
step: $\left\langle W\right\rangle/\varepsilon_{A}$, $\left\langle
Q_{A}\right\rangle/\varepsilon_{A}$, and $\eta$. We set
$\beta_{B}=2\beta_{A}$, $\lambda=0.6$ and
$\alpha_{max}=1/\left(\mathcal{Z}_{A}\mathcal{Z}_{B}\right)$.

##  Appendix B DEMONSTRATION OF THE GENERALIZED SECOND LAW LIMIT

Here we give a demonstration for the efficiency result Eq. (14). The entropy
production of the correlated SWAP quantum engine can be expressed as the sum
of two relative entropies that reads

$\begin{split}D\left[\rho_{A}^{f}||\rho_{A}^{0}\right]+D\left[\rho_{B}^{f}||\rho_{B}^{0}\right]=&-\Delta
S_{A}-\Delta S_{B}\\\
&+\beta_{A}\text{Tr}\left[\left(\rho_{A}^{f}-\rho_{A}^{0}\right)\mathcal{H}_{A}\right]\\\
&+\beta_{B}\text{Tr}\left[\left(\rho_{B}^{f}-\rho_{B}^{0}\right)\mathcal{H}_{B}\right],\end{split}$
(23)

where $\rho_{i}^{f}$ is the final out-of-equilibrium state for the qubit
$i=A,B$. We simplify this equation by introducing $\left\langle
Q_{i}\right\rangle=-\text{Tr}\left[\left(\rho_{i}^{f}-\rho_{i}^{0}\right)\mathcal{H}_{i}\right]$,
hence

$\begin{split}D\left[\rho_{A}^{f}||\rho_{A}^{0}\right]+D\left[\rho_{B}^{f}||\rho_{B}^{0}\right]=&-\Delta
S_{A}-\Delta S_{B}\\\ &-\beta_{A}\left\langle
Q_{A}\right\rangle-\beta_{B}\left\langle Q_{B}\right\rangle.\end{split}$ (24)

We use the fact that $\Delta I(A:B)=\Delta S_{A}+\Delta S_{B}$ to rewrite Eq.
(S4) as
$D\left[\rho_{A}^{f}||\rho_{A}^{0}\right]+D\left[\rho_{B}^{f}||\rho_{B}^{0}\right]=-\Delta
I(A:B)-\beta_{A}\left\langle Q_{A}\right\rangle-\beta_{B}\left\langle
Q_{B}\right\rangle$. This is equivalent to

$\frac{D\left[\rho_{A}^{f}||\rho_{A}^{0}\right]+D\left[\rho_{B}^{f}||\rho_{B}^{0}\right]+\Delta
I(A:B)}{\beta_{B}\left\langle
Q_{A}\right\rangle}=-\frac{\beta_{A}}{\beta_{B}}-\frac{\left\langle
Q_{B}\right\rangle}{\left\langle Q_{A}\right\rangle}.$ (25)

Energy conservation implies that the average values for the heat and work,
$\left\langle Q_{A}\right\rangle+\left\langle Q_{B}\right\rangle+\left\langle
W\right\rangle=0$. The extracted work $\left\langle
W_{ext}\right\rangle=-\left\langle W\right\rangle=\left\langle
Q_{A}\right\rangle+\left\langle Q_{B}\right\rangle$ and the quantum heat
engine efficiency $\eta=\left\langle W_{ext}\right\rangle/\left\langle
Q_{A}\right\rangle$ reads

$\eta=1-\frac{\beta_{A}}{\beta_{B}}-\frac{D\left[\rho_{A}^{f}||\rho_{A}^{0}\right]+D\left[\rho_{B}^{f}||\rho_{B}^{0}\right]+\Delta
I(A:B)}{\beta_{B}\left\langle Q_{A}\right\rangle},$ (26)

where $1-\beta_{A}/\beta_{B}$ is the standard Carnot limit. Equation (26)
defines a generalized second law limit for bipartite quantum engine in the
presence of initial correlations. As discussed in the main text, the
efficiency booster $\mathcal{B_{E}}\equiv(\Sigma_{\text{eng}}+\Delta
I)/\beta_{B}\left\langle Q_{A}\right\rangle$ sets a criterion (Eq. (16)) for
the enhancement of the engine’s efficiency and extractable work (see Figs. 3,
5 and 7).

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# Strategies and payoffs in quantum minority games

Puya Sharif and Hoshang Heydari
Physics Department, Stockholm university 10691 Stockholm Sweden

Email<EMAIL_ADDRESS>
###### Abstract

Game theory is the mathematical framework for analyzing strategic interactions
in conflict and competition situations. In recent years quantum game theory
has earned the attention of physicists, and has emerged as a branch of quantum
information theory [1]. With the aid of entanglement and linear superposition
of strategies, quantum games are shown to yield significant advantage over
their classical counterparts. In this paper we explore optimal and equilibrium
solutions to quantum minority games. Initial states with different level of
entanglement are investigated. Focus will be on 4 and 6 player games with some
$N$-player generalizations.

##  1 Introduction

Game theory is the systematic study of decision-making in strategic
situations. Its models are widely used in economics, political science,
biology and computer science to capture the behavior of individual
participants in conflict and competition situations. The field attempts to
describe how decision makers do and should interact within a well-defined
system of rules to maximize their payoff. The kind of games we will be
considering here is called minority games and arises in situations when a
group of non communicating agents has to independently choose between two
different choices $\left|0\right\rangle$ and $\left|1\right\rangle$. Payoff $
of one unit goes to those agents that makes the minority choice. If agents are
evenly distributed between the two choices, everybody loses. In the classical
case, the game has a mixed-strategy solution, where agents chooses randomly
between $\left|0\right\rangle$ and $\left|1\right\rangle$. This yields an
expected payoff $<\$>$ which is given by the number of combinations that
results in some player being in the minority group divided by the total number
of possible combinations. For a four player game there are 16 possible
combinations, with two minority states for each player. This gives an expected
payoff ¡$¿ of 1/8 to each.

Generally a game is defined as a set
$\Gamma=\Gamma(N,\,\\{s_{i}\\},\,\\{\$_{i}\\})$, where $N$ denotes the number
of players, $\\{s_{i}\\}$ the set of available strategies of player $i$, and
$\\{\$_{i}\\}$ the payoffs of different game outcomes. For quantum games, we
add the associated Hilbert space $\mathcal{H}$, generally of dim $2^{N}$, and
the initial state $\rho$. In a quantum game, the choice of strategy $s_{i}$
translates to choosing a unitary operator $M_{i}$, which is applied locally on
the qubit held by the player. The games will be analyzed with regard to two of
the most important solution concepts in game theory is the Nash equilibrium
and Pareto optimality. Nash equilibrium is defined as the combination of
strategies $s_{i}$ for which no player gains by unilaterally changing their
strategy. Pareto optimality occurs when no player can rise its payoff without
lowering the payoff of others.

##  2 Quantum minority games

Following the scheme presented in [2], in the quantum version of the minority
game, each player is provided with a qubit from an entangled set. Strategy
$s_{i}$ of player $i$ is played by doing a local unitary operation on the
players own qubit, by applying its strategy operator $M\in$ SU(2). $M$ will be
parameterized in the following way:

$M\left(\theta,\alpha,\beta\right)=\left(\begin{array}[]{cc}e^{i\alpha}\cos\left(\theta/2\right)&ie^{i\beta}\sin\left(\theta/2\right)\\\
ie^{-i\beta}\sin\left(\theta/2\right)&e^{-i\alpha}\cos\left(\theta/2\right)\end{array}\right),$
(2.0.1)

with $\theta\in[0,\pi]$ and $\alpha,\beta\in[-\pi,\pi]$. The game starts out
in an entangled initial state $\rho_{in}$.

$\rho_{in}=\left|\psi\right\rangle\langle\psi|,$ (2.0.2)

where$\left|\psi\right\rangle$ is usually taken to be a $N$ qubit GHZ-state,
from which each player is provided with one qubit [2][3]. The final state
$\rho_{\mathrm{fin}}$ of the game becomes

$\rho_{\mathrm{fin}}=(\bigotimes_{i=1}^{N}M_{i})\rho_{\mathrm{in}}(\bigotimes_{i=1}^{N}M_{i}){}^{\dagger}.$
(2.0.3)

To calculate the expected payoff of player $i$ we take the trace of the final
state $\rho_{\mathrm{fin}}$ multiplied with the projection operator $P_{i}$ of
the player. The projection operator projects the final state onto the desired
states of player $i$.

$P_{i}=\sum_{j=1}^{k}\left|\xi_{i}^{j}\right\rangle\left\langle\xi_{i}^{j}\right|.$
(2.0.4)

The sum is over all the $k$ different states $\left|\xi_{i}^{j}\right\rangle$,
for which player $i$ is in the minority. For $N=4$, we have the following
projection operator $P_{1}$ for player 1.
$P_{1}=\left|1000\right\rangle\left\langle
1000\right|+\left|0111\right\rangle\left\langle 0111\right|$. In the 6-player
game, each player has a sum of 12 such states. The expected payoff $<\$>$ is
finally given by:

$\mathrm{<\$_{i}>=Tr[}\rho_{\mathrm{fin}}P_{i}].$ (2.0.5)

The local unitary operations of the players eliminates the possibility for the
system to end up in most states where nobody wins, and therefore yields higher
than classical payoff.

###  2.1 Solutions with different initial states

As a generalization of the broadly used GHZ-state as the initial state
$\left|\psi_{in}\right\rangle$ we consider a superposition with products of
symmetric bell pairs. A four qubit version of this state was used in a
experimental implementation of a quantum minority game by C. Schmid and A.P.
Flitney in [3].

$\left|\Psi(x)\right\rangle=\frac{x}{\sqrt{2}}\left|\mathrm{GHZ}_{N}\right\rangle+\sqrt{\frac{1-x^{2}}{2^{N/2}}}(\left|01\right\rangle+\left|10\right\rangle)^{\otimes
N/2}.$ (2.1.1)

The parameter $x\in[0,1]$ denotes the level of mixture. $x=1$ just gives back
the GHZ-state and $x=0$ product of the Bell-pairs. To account for loss in
fidelity in the creation of the initial state, we form a density matrix
$\rho_{in}$ out of $\mid\Psi_{in}\rangle$ and add noise that can be controlled
by the parameter $f$. We get:

$\rho_{in}=f\mid\Psi_{in}\rangle\langle\Psi_{in}\mid+\frac{1-f}{64}\mathbb{I_{\mathrm{64}}},$
(2.1.2)

where $\mathbb{I_{\mathrm{64}}}$ is the $64\times 64$ identity matrix. By
adjusting $f\rightarrow 0$, the initial state gets mixed with an even
distribution of all basis states in $\mathcal{H}=(\mathbb{C}^{2})^{\otimes 6}$

For the GHZ-state alone i.e $x=1$ and $f=1$ it has been shown that the Nash
equilibrium solution $s_{NE}=M(\theta,\alpha,-\alpha)$ for the 4-player game
is $M(\frac{\pi}{2},-\frac{\pi}{8},\frac{\pi}{8})$, and for the 6-player game,
$M(\frac{\pi}{2},-\frac{\pi}{12},\frac{\pi}{12})$. For the state above, Schmid
and Flitney showed that when starting with only the product of Bell-pairs i.e
$x=0$, no advantage is achieved over the classical counterpart. For
$x\leq\sqrt{\frac{2}{3}}$, a new set of Nash equilibria occurs, where the
payoff is a function of $x$. This Bell-dominated region has a new Pareto
optimal strategy: $M(\frac{\pi}{4},0,0)$ compared to
$M(\frac{\pi}{2},-\frac{\pi}{8},\frac{\pi}{8})$ in the GHZ-dominated region
$x>\sqrt{\frac{2}{3}}$.

Figure 1: Payoffs for $\hat{M}(\theta,\alpha,-\alpha).$ When $N=4$. Top left:
$x=1$. Top right: $x=\sqrt{2/3}+0.1$, Bottom left: $x=\sqrt{2/3}$. Bottom
right: $x=\sqrt{2/3}-0.1$

The case is different for $N=6$, here the equilibrium strategy remains the
same throughout any change of $x$. For $f=1$, the payoff is given by

$<\$>=\frac{1}{4}+\frac{x^{2}}{16}.$ (2.1.3)

For the pure GHZ-state this gives an equilibrium payoff of 5/16. When
$x\rightarrow 0$ the payoff approaches 1/4, which is still better than the
classical payoff of 3/16. This shows that even the initial state containing
only the products of Bell-pairs yields an advantage compared to the classical
expected payoff. This is not the case for general $N$. When noise is taken
into account the payoff function becomes

$<\$>=\frac{1}{16}(3+f+fx^{2}).$ (2.1.4)

When the noise reaches maximum: $f\rightarrow 0$, the classical payoff of 3/16
returns. It can be demonstrated that
$M_{NE}=M(\frac{\pi}{2},-\frac{\pi}{12},\frac{\pi}{12})$ is a Nash equilibrium
solution for all $x\in[0,1]$, by letting one player deviate from the NE
solution, by playing $M_{D}(\theta^{*},-\alpha^{*},\alpha^{*})$. The following
inequality holds for a Nash equilibrium:

$\$_{i}(M_{NE}^{\otimes 6})\geq\$_{i}(M_{D}\otimes M_{NE}^{\otimes 5}).$
(2.1.5)

Figure 2: Payoff of a player that plays
$M_{D}(\theta^{*},-\frac{\pi}{12},\frac{\pi}{12})$, when the rest plays
$M_{NE}$

.

Figure 3: Payoff of a player that plays
$M_{D}(\frac{\pi}{2},-\alpha^{*},\alpha^{*})$, when the rest plays $M_{NE}$

.

####  2.1.1 exponential entangler

A GHZ-state can be created by acting with an entanglement operator $J(\gamma)$
on a product state
$\left|0\right\rangle\otimes\left|0\right\rangle\otimes\left|0\right\rangle\otimes\cdots\otimes\left|0\right\rangle$
, where

$J(\gamma)=\mathrm{exp\left(i\frac{\gamma}{2}\sigma_{x}^{\otimes\mathit{N}}\right)}.$
(2.1.6)

We then have

$\left|\Psi(\gamma)\right\rangle=J\left|00\cdots 0\right\rangle,$ (2.1.7)

where $\gamma\in[0,\frac{\pi}{2}]$ is a parameter that controls the level of
entanglement. This gives an output state of the following form

$\left|\Psi(\gamma)\right\rangle=\cos(\frac{\gamma}{2})\left|00\cdots
0\right\rangle+i\sin(\frac{\gamma}{2})\left|11\cdots 1\right\rangle.$ (2.1.8)

Maximum is reached for $\gamma=\frac{\pi}{2}.$ If
$\left|\Psi(\gamma)\right\rangle$ is used as initial state for a quantum
minority game, the Nash equilibrium payoffs will depend on the parameter
$\gamma$. For $\gamma=0$ the classical payoffs are obtained, since the game
starts out in an unentangled initial state [6]. A $N$-player generalization
has been conjectured:

$<\$>_{N}=\left(<\$>_{C}-\frac{1}{2}<\$>_{Q}\right)\left(\cos\frac{\gamma}{2}-\sin\frac{\gamma}{2}\right)^{2}+\frac{1}{2}<\$>_{Q}\left(\cos\frac{\gamma}{2}+\sin\frac{\gamma}{2}\right)^{2},$
(2.1.9)

where $<\$>_{C}$ is the classically obtainable payoffs for for classical NE
strategies, and $<\$>_{C}$ for the quantum versions [5].

####  2.1.2 Products of W-states

A six player game could use a product of two three qubit W-states as its
initial state $\left|\psi_{in}\right\rangle$.

$\left|\psi_{in}\right\rangle=\left|W_{3}\right\rangle\otimes\left|W_{3}\right\rangle,$
(2.1.10)

where

$\left|W_{3}\right\rangle=\frac{1}{\sqrt{3}}(\left|001\right\rangle+\left|010\right\rangle+\left|100\right\rangle).$
(2.1.11)

$\left|\psi_{in}\right\rangle$ is a symmetric superposition of nine states
with four qubits in the $\left|0\right\rangle$ -state and two in the
$\left|1\right\rangle$ -state, compactly written as
$\left|\psi_{in}\right\rangle=\left|4,2\right\rangle$. This state therefore
has tree minority combinations for each player, and no undesired states! The
game simply starts out in the best possible configuration, and the only thing
the players should do is to apply the identity operator $I$, to obtain an
expected payoff 1/3, the theoretical maximum for a six-player game. This
solution Pareto optimal, compared to the six-player game starting with an GHZ-
state, which is not.

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