# Role of quantum correlations in daemonic expected utility

Gianluca Francica and Luca Dell’Anna  Dipartimento di Fisica e Astronomia e
Sezione INFN, Università di Padova, via Marzolo 8, 35131 Padova, Italy

###### Abstract

Fluctuations can challenge the possibility of improving work extraction from
quantum correlations. This uncertainty in the work extraction process can be
addressed resorting to the expected utility hypothesis which can provide an
optimal method for work extraction. We study a bipartite quantum system and
examine the role of quantum correlations in a daemonic work extraction
performed by certain local operations and classical communication.
Specifically, we demonstrate and explain how, depending on the so-called
absolute risk aversion, a non-neutral risk agent, influenced by fluctuations,
views quantum correlations differently from a neutral risk agent who is
affected solely by the average work.

##  I Introduction

The role of correlations among the parties of a quantum system in
thermodynamics has been recently investigated focusing on work extraction from
finite systems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. In particular, the daemonic
ergotropy [4], which can be defined as the maximum average work locally
extractable from a bipartite quantum system by performing certain local
operations and classical communication, gives a gain in work extraction if
there are quantum correlations. Beyond its standard definition, generalized
measurements and multipartite extensions have been also discussed [5]. Among
its different uses, e.g., in continuously-monitored open quantum batteries
[11], daemonic ergotropy also found applications in order to investigate the
role of indefinite causal order structures in thermodynamics. In particular,
when the communication is not classical and it is achieved with a process
matrix [12], the role of indefinite causal order has been investigated with
the help of daemonic ergotropy [13]. Furthermore, by considering a quantum
switch [14], the activation of states by applying maps in an indefinite causal
order [15, 16] and the effects of non-Markovianity [17] have been also
investigated. Here, we wonder whether and when there are advantages in using
the daemonic ergotropy protocol of Ref. [4] if we take in account fluctuations
by means of a utility function, i.e., by using the expected utility
hypothesis, first formalized by von Neumann and Morgenstern within the theory
of games and economic behaviour eighty years ago [18]. Recently, expected
utility has been also related to fluctuation theorems [19, 20]. In particular,
in Ref. [20] we investigated a possible relation with an entropy coming from a
fluctuation theorem, which in certain cases can be a guideline for making a
choice. Concerning the work fluctuations, when the initial state is not
incoherent with respect to the energy basis, there may not be a probability
distribution for the work done, as proven by a no-go theorem [21] due to the
existence of quantum contextuality [22]. Thus, we can adopt the
quasiprobability distribution of work introduced in Ref. [23], which is
selected if some fundamental conditions need to be satisfied [24]. In work
extraction from thermally isolated quantum system the optimal expected utility
has been introduced in Ref. [25] in order to optimize the work extraction by
taking into account also the fluctuations, which can be dominant in finite
systems. Here, we study the gain achieved from quantum correlations when
fluctuations are taken into account. After a brief introduction to some
preliminary notions in Sec. II, we examine the expected utility for the work
extraction protocol in Sec. III. In particular, we aim to clarify the role of
quantum correlations when the fluctuations are taken in account with a utility
function. Finally, in Sec. IV we summarize and further discuss the obtained
results.

##  II Preliminaries

We start our discussion by introducing some preliminary notions, which are the
protocol of work extraction (see Sec. II.1), some rudiments about quantum
correlations, i.e. the quantum discord and the entanglement (see Sec. II.2),
and the expected utility hypothesis (see Sec. II.3).

###  II.1 Work extraction

We consider a bipartite system having Hilbert space
$\mathcal{H}=\mathcal{H}_{S}\otimes\mathcal{H}_{A}$, where $\mathcal{H}_{S}$
is the Hilbert space of a system $S$, where we can perform unitary
transformations, and $\mathcal{H}_{A}$ is the one of a system (a so-called
ancilla) $A$, where we can perform projective measurements. The two subsystems
$S$ and $A$ are not interacting but they are prepared in a state $\rho_{SA}$
that can show correlations among the parties. A ‘daemonic’ protocol can be
realized through local operations and classical communication. In this
section, we are interested to extract the optimal work locally from $S$ by
neglecting the fluctuations and focusing only on the average extracted work,
following Ref. [4]. We consider an Hilbert $\mathcal{H}_{S}$ with dimension
$d_{S}$ and the Hamiltonian of the system $S$ is

$H_{S}=\sum_{k}\epsilon_{k}|\epsilon_{k}\rangle\langle\epsilon_{k}|\,,$ (1)

with $\epsilon_{k}<\epsilon_{k+1}$. The reduced state of the system $S$ is
given by

$\rho_{S}=\text{Tr}_{A}\left\\{\rho_{SA}\right\\}=\sum_{k}r_{k}|r_{k}\rangle\langle
r_{k}|\,,$ (2)

with $r_{k}\geq r_{k+1}$. The system $S$ is thermally isolated, and an amount
of average work is locally extracted by cyclically changing some Hamiltonian
parameters of $S$, so that at the end of the cycle the final Hamiltonian is
equal to the initial one, and they are both equal to $H_{S}$ in Eq. (1). It
results a unitary cycle with a local unitary time-evolution operator
$U_{S}=\mathcal{T}e^{-i\int_{0}^{\tau}H_{S}(t)dt}$ for $S$, generated by the
time-dependent Hamiltonian $H_{S}(t)$ such that $H_{S}(0)=H_{S}(\tau)=H_{S}$,
where $t=0$ and $t=\tau$ are the initial and final time, and $\mathcal{T}$ is
the time ordering operator. The final reduced state of $S$ is
$U_{S}\rho_{S}U_{S}^{\dagger}$ and the average work is minus the change of
average energy and reads

$W(\rho_{S},U_{S})=E(\rho_{S})-E(U_{S}\rho_{S}U_{S}^{\dagger})\,,$ (3)

where the average energy of the initial and final state is calculated with
respect to the Hamiltonian $H_{S}$, and we have defined
$E(\rho_{S})=\text{Tr}\left\\{H_{S}\rho_{S}\right\\}$. The optimal work
extraction is achieved by maximizing the average work in Eq. (3) over all the
unitary cycles, i.e.,

$\mathcal{E}(\rho_{S})=\max_{U_{S}}W(\rho_{S},U_{S})\geq 0\,.$ (4)

The optimal value $\mathcal{E}(\rho_{S})$ is called ergotropy [26], and it is
achieved by performing an optimal unitary cycle with unitary operator
$U_{S}=\sum_{k=1}^{d}e^{i\phi_{k}}|\epsilon_{k}\rangle\langle r_{k}|$, so that
the ergotropy reads

$\mathcal{E}(\rho_{S})=\sum_{k,j}r_{j}\left(\left|\langle\epsilon_{k}|r_{j}\rangle\right|^{2}-\delta_{j,k}\right)\epsilon_{k}\,.$
(5)

We note that the ergotropy $\mathcal{E}(\rho_{S})$ is zero if and only if the
initial state $\rho_{S}$ is passive, i.e., commutates with the Hamiltonian,
$[\rho_{S},H_{S}]=0$, and the populations with respect to the energy basis are
sorted in decreasing order,
$r_{k}=\langle\epsilon_{k}|\rho_{S}|\epsilon_{k}\rangle$.

A larger amount of work can be extracted through the following daemonic
protocol, which exploits the information about the state of $S$ that is
obtained by performing projective measurements on $A$. These measurements are
described by the set of projectors $\\{\Pi^{A}_{a}=|a\rangle\langle a|\\}$
with $\langle a|a^{\prime}\rangle=\delta_{a,a^{\prime}}$, where
$a=1,\ldots,d_{A}$ and $d_{A}$ is the dimension of the Hilbert space
$\mathcal{H}_{A}$. By performing a projective measurement with projector
$\Pi^{A}_{a}$ on the state of the ancilla $A$, the system $S$ collapses into
the state

$\rho_{S|a}=\frac{\text{Tr}_{A}\left\\{I^{S}\otimes\Pi^{A}_{a}\rho_{SA}I^{S}\otimes\Pi^{A}_{a}\right\\}}{p_{a}}\,,$
(6)

with probability
$p_{a}=\text{Tr}\left\\{I^{S}\otimes\Pi^{A}_{a}\rho_{SA}\right\\}$. Through
classical communication, we can perform unitary cycles conditioned by the
outcomes $a$ of the measurements, extracting the maximum average work
$\mathcal{E}_{\\{\Pi^{A}_{a}\\}}(\rho_{SA})$ that is called daemonic ergotropy
[4] and reads

$\mathcal{E}_{\\{\Pi^{A}_{a}\\}}(\rho_{SA})=\sum_{a}p_{a}\mathcal{E}(\rho_{S|a})\,.$
(7)

We note that if the information gained from the measurements on $A$ is not
exploited, the cycles are not conditioned by $a$ and the maximum average work
extracted remains equal to the ergotropy $\mathcal{E}(\rho_{S})$. Thus, the
maximum gain obtained by exploiting the information acquired reads

$\delta\mathcal{E}(\rho_{SA})=\max_{\\{\Pi^{A}_{a}\\}}\mathcal{E}_{\\{\Pi^{A}_{a}\\}}(\rho_{SA})-\mathcal{E}(\rho_{S})\,,$
(8)

and it is related to the presence of quantum correlations [4].

###  II.2 Quantum correlations

Quantum correlations will play a crucial role in our discussion. For our
purposes, to identify them we define the set of classical-quantum states

$\mathcal{C}_{S}=\left\\{\rho_{SA}\,|\,\rho_{SA}=\sum_{k}p_{k}P^{S}_{k}\otimes\rho^{A}_{k}\right\\}$
(9)

and the set of separable states

$\mathcal{S}=\left\\{\rho_{SA}\,|\,\rho_{SA}=\sum_{k}p_{k}\rho^{S}_{k}\otimes\rho^{A}_{k}\right\\}\,,$
(10)

so that $\mathcal{C}_{S}\subseteq\mathcal{S}$, where $p_{k}\geq 0$ and sum to
one, $P^{S}_{k}$’s are rank one projectors,
$P^{S}_{k}P^{S}_{j}=\delta_{k,j}P^{S}_{k}$ and $\sum_{k}P^{S}_{k}=I^{S}$, and
$\rho^{S}_{k}$ and $\rho^{A}_{k}$ are density matrices. We recall that all the
separable states $\rho_{SA}\in\mathcal{S}$ can be prepared from a product
state $\rho_{S}\otimes\rho_{A}$ by performing local operations and classical
communication. On the other hand, states $\rho_{SA}\notin\mathcal{S}$ are
called entangled (see, e.g., Ref. [27] for a review). Correlations can be
quantified through distance based measures (see, e.g., Ref. [28]). For
instance, the entanglement in a state $\rho_{SA}$ can be quantified through
the relative entropy of entanglement [29],
$E_{re}(\rho_{SA})=\min_{\sigma_{SA}\in\mathcal{S}}S(\rho_{SA}||\sigma_{SA})$,
where we have defined the quantum relative entropy
$S(\rho||\eta)=\text{Tr}\left\\{\rho(\log_{2}\rho-\log_{2}\eta)\right\\}$.
Although a state $\rho_{SA}$ is separable, it can still show quantum features
(quantum discord), if $\rho_{SA}\notin\mathcal{C}_{S}$. Originally, quantum
discord has been introduced as the difference between the mutual information
and the maximum one way classical information [30, 31]. A measure based on
quantum relative entropy reads
$D_{re}^{A|S}(\rho_{SA})=\min_{\chi^{c-q}_{SA}\in\mathcal{C}_{S}}S(\rho_{SA}||\chi^{c-q}_{SA})$.
In particular, given a classical-quantum state
$\chi^{c-q}_{SA}\in\mathcal{C}_{S}$ there exists a set $\\{P^{S}_{k}\\}$ such
that the state remains unperturbed if we perform these projective
measurements, i.e., $\sum_{k}P^{S}_{k}\otimes
I^{A}\chi^{c-q}_{SA}P^{S}_{k}\otimes I^{A}=\chi^{c-q}_{SA}$.

###  II.3 Expected utility hypothesis

Here, we aim to investigate the work extraction performed by an agent non-
neutral to risk, which takes in account also the fluctuations and not only the
average work. For our purposes, we focus on an agent who must choose between
two procedures that yield two different values of extracted work represented
by the random variables $w_{1}$ and $w_{2}$. To give an example, we consider
an agent who must choose between extracting a certain work $w_{det}=50$ or
flipping a coin and extracting a work $w_{head}=100$ if heads or nothing
otherwise. If the agent is risk neutral, he is indifferent to the choice,
since if he flips the coin he will extract the average work $w_{det}$. An
agent non-neutral to risk will choose the certain work $w_{det}$ or to flip
the coin depending on his risk aversion (e.g., an agent that is averse to
risk, tends to choose the deterministic work extraction of the amount
$w_{det}$, preferring situations with small fluctuations). The risk aversion
of the agent can be fully characterized by using a utility function $u(w)$,
which quantifies the satisfaction gained from a choice, so that the agent will
choose the procedure yielding the work $w_{1}$ instead of $w_{2}$ if

$\langle u(w_{1})\rangle>\langle u(w_{2})\rangle\,.$ (11)

It is easy to see that the inequality in Eq. (11) remains unchanged if we
perform an affine transformation on the utility function, i.e., the
transformation $u(w)\mapsto au(w)+b$, where $a$ is a positive variable. This
means that the utility function is defined up to affine transformations, since
two utility functions related by such transformation gives the same preference
ordering given by Eq. (11). From Eq. (11), any linear utility function
$u(w)=aw+b$, with $a>0$, gives the condition $\langle w_{1}\rangle>\langle
w_{2}\rangle$, thus this utility function characterizes an agent neutral to
risk. To characterize the risk aversion we can focus on a strictly increasing
utility function that is concave. In our example, from Jensen’s inequality,
for arbitrary values of $w_{det}$ and $w_{head}$, the agent chooses the
certain work $w_{det}$ instead to flip the coin if $w_{det}>w_{head}/2$. Thus,
this suggests that the risk aversion is related to the concavity of the
utility function, and we recall that in general it can be measured with the
Arrow-Pratt coefficient of absolute risk aversion defined as

$r_{A}(w)=-\frac{u^{\prime\prime}(w)}{u^{\prime}(w)}\,,$ (12)

which is non-negative for a utility function that is concave and strictly
increasing. We note that, while $u^{\prime\prime}(w)$ quantifies the
concavity, dividing by $u^{\prime}(w)$ guarantees that the measure in Eq. (12)
is invariant under affine transformations. More details can be found, e.g., in
Refs. [32, 33].

##  III Daemonic expected utility

Given a utility function $u(w)$, we can define the optimal expected utility of
an arbitrary state $\rho_{S}$ as [25]

$\mathcal{U}(\rho_{S})=\max_{U_{S}}\langle u(w)\rangle\,,$ (13)

where the average is calculated as

$\langle u(w)\rangle=\int u(w)p_{q}(w,\rho_{S},U_{S})dw$ (14)

and $p_{q}(w,\rho_{S},U_{S})$ is the quasiprobability distribution of work
defined as [23, 24]

$\displaystyle p_{q}(w,\rho_{S},U_{S})$ $\displaystyle=$
$\displaystyle\sum_{k,j,i}\text{Re}\langle\epsilon_{i}|\rho_{S}|\epsilon_{j}\rangle\langle\epsilon_{j}|U_{S}^{\dagger}|\epsilon_{k}\rangle\langle\epsilon_{k}|U_{S}|\epsilon_{i}\rangle$
(15)
$\displaystyle\times\delta(w-q\epsilon_{i}-(1-q)\epsilon_{j}+\epsilon_{k})\,.$

We note that for a risk neutral agent, so that $u(w)=w$, the optimal expected
utility is equal to the ergotropy,
$\mathcal{U}(\rho_{S})=\mathcal{E}(\rho_{S})$. In particular, we will focus on
the symmetric quasiprobability representation with $q=1/2$, giving a minimum
of $\mathcal{U}(\rho_{S})$ in the function of $q$ [25]. We recall that
$p_{q}(w,\rho_{S},U_{S})$ reduces to the quasiprobability distribution of Ref.
[34] for $q=0,1$ and the one of Ref. [35] for $q=1/2$. Thus, we consider a
daemonic protocol for the work extraction, with certain optimal unitary cycles
not necessarily equal to the daemonic ergotropy ones. This daemonic protocol
gives an optimal expected utility which is equal to the ‘daemonic’ expected
utility defined as

$\mathcal{U}_{\\{\Pi^{A}_{a}\\}}(\rho_{SA})=\sum_{a}p_{a}\mathcal{U}(\rho_{S|a})\,.$
(16)

We note that, for any convex combination $\rho_{S}=\sum_{a}p_{a}\rho_{a}$ of
density matrices $\rho_{a}$, we get
$p_{q}(w,\rho_{S},U_{S})=\sum_{a}p_{a}p_{q}(w,\rho_{a},U_{S})$ from which it
is easy to see that
$\mathcal{U}(\rho_{S})\leq\sum_{a}p_{a}\mathcal{U}(\rho_{a})$. Thus, by noting
that $\rho_{S}=\sum_{a}p_{a}\rho_{S|a}$, we get
$\mathcal{U}_{\\{\Pi^{A}_{a}\\}}(\rho_{SA})\geq\mathcal{U}(\rho_{S})$. We
define the maximum gain

$\delta\mathcal{U}(\rho_{SA})=\max_{\\{\Pi^{A}_{a}\\}}\mathcal{U}_{\\{\Pi^{A}_{a}\\}}(\rho_{SA})-\mathcal{U}(\rho_{S})\geq
0\,.$ (17)

Thus, the agent prefers the optimal daemonic protocol instead of a local
unitary cycle in order to extract work from $S$ if the gain is positive,
$\delta\mathcal{U}(\rho_{SA})>0$, otherwise the agent is indifferent to the
choice. The situation is schematically represented in Fig. 1.

Figure 1: The system is made of two parties, $S$ and $A$, represented by a
circle and a square, respectively. The two parties are connected by a line
representing the initial correlations. Local measurements and unitary
operations are performed on the square and the circle, respectively. An agent
must choose between extracting work from $S$ using the daemonic protocol or
conventionally using a local unitary cycle without performing measurements on
$A$ or communicating the outcomes $a$.

###  III.1 Constant absolute risk aversion

We start to investigate the case of a constant absolute risk aversion
$r_{A}(w)=\text{const}$. In particular, we focus on the exponential utility

$u(w)=\frac{1}{r}(1-e^{-rw})$ (18)

for $r\neq 0$, and $u(w)=w$ for $r=0$, which is a strictly increasing function
having absolute risk aversion $r_{A}(w)=r$. To perform our calculations, it is
useful to consider the spectral decompositions
$e^{-rH_{S}/2}\rho_{S}e^{-rH_{S}/2}=\sum_{k}u_{k}|u_{k}\rangle\langle u_{k}|$
with $u_{k}\geq u_{k+1}$ and
$e^{-rH_{S}/2}\rho_{S|a}e^{-rH_{S}/2}=\sum_{k}u^{a}_{k}|u^{a}_{k}\rangle\langle
u^{a}_{k}|$ with $u^{a}_{k}\geq u^{a}_{k+1}$ for all $a$. For any state
$\rho_{S}$, from Eq. (13) we get [25]

$\mathcal{U}(\rho_{S})=\frac{1}{r}\left(1-\sum_{k}u_{k}e^{r\epsilon_{k}}\right)\,,$
(19)

from which the daemonic expected utility trivially reads

$\mathcal{U}_{\\{\Pi^{A}_{a}\\}}(\rho_{SA})=\sum_{a}p_{a}\frac{1}{r}\left(1-\sum_{k}u^{a}_{k}e^{r\epsilon_{k}}\right)\,.$
(20)

With the aim to study the states $\rho_{SA}$ such that
$\delta\mathcal{U}(\rho_{SA})=0$ in the case of a constant absolute risk
aversion we introduce some lemmas. By considering the identity

$u_{k}=\sum_{a}p_{a}\sum_{j}u^{a}_{j}\left|\langle
u_{k}|u^{a}_{j}\rangle\right|^{2}\,,$ (21)

coming from $\rho_{S}=\sum_{a}p_{a}\rho_{S|a}$, we get that:

######  Lemma 1.

For a given set $\\{\Pi^{A}_{a}\\}$, the daemonic gain
$\mathcal{U}_{\\{\Pi^{A}_{a}\\}}(\rho_{SA})-\mathcal{U}(\rho_{S})$ can be
expressed as

$\mathcal{U}_{\\{\Pi^{A}_{a}\\}}(\rho_{SA})-\mathcal{U}(\rho_{S})=\sum_{a}p_{a}\tilde{\mathcal{E}}(\tilde{\rho}_{S|a})\geq
0\,,$ (22)

where $\tilde{\mathcal{E}}(\tilde{\rho}_{S|a})$ is the ergotropy of the non-
normalized state $\tilde{\rho}_{S|a}=e^{-rH_{S}/2}\rho_{S|a}e^{-rH_{S}/2}$
with respect to the Hamiltonian
$\tilde{H}_{S}=\sum_{k}y_{k}|u_{k}\rangle\langle u_{k}|$ with energies
$y_{k}=\frac{e^{r\epsilon_{k}}}{r}$ so that $y_{k}<y_{k+1}$, which explicitly
reads

$\tilde{\mathcal{E}}(\tilde{\rho}_{S|a})=\sum_{k,j}u^{a}_{j}\left(\left|\langle
u_{k}|u^{a}_{j}\rangle\right|^{2}-\delta_{j,k}\right)y_{k}\,.$ (23)

Thus, $\mathcal{U}_{\\{\Pi^{A}_{a}\\}}(\rho_{SA})=\mathcal{U}(\rho_{S})$ if
and only if all the ergotropies $\tilde{\mathcal{E}}(\tilde{\rho}_{S|a})$ are
zero, i.e., for all the outcomes $a$, $\tilde{\rho}_{S|a}$ is passive with
respect to the Hamiltonian $\tilde{H}_{S}$. This means that, for all the
outcomes $a$, $[\tilde{\rho}_{S|a},\tilde{H}_{S}]=0$, i.e., all the
conditioned non-normalized states $\tilde{\rho}_{S|a}$ are diagonal with
respect to the basis $\\{|u_{k}\rangle\\}$, and $\langle
u_{k}|\tilde{\rho}_{S|a}|u_{k}\rangle$ are in decreasing order, i.e.,$\langle
u_{k}|\tilde{\rho}_{S|a}|u_{k}\rangle\geq\langle
u_{k+1}|\tilde{\rho}_{S|a}|u_{k+1}\rangle$. Furthermore, we get that:

######  Lemma 2.

Given a state $\rho_{SA}$, if all the conditional states $\rho_{S|a}$ are
diagonal with respect to the same basis $\\{|r_{k}\rangle\\}$ for any set
$\\{\Pi^{A}_{a}\\}$, then

$\rho_{SA}=\sum_{k}|r_{k}\rangle\langle r_{k}|\otimes C^{A}_{k}\,,$ (24)

where $C^{A}_{k}$ are positive semidefinite matrices.

We note that the state in Eq. (24) is a classical-quantum state,
$\rho_{SA}\in\mathcal{C}_{S}$.

###### Proof.

To prove Eq. (24), we note that an arbitrary state $\rho_{SA}$ can be written
as

$\rho_{SA}=\sum_{a,a^{\prime},k,k^{\prime}}C^{aa^{\prime}}_{kk^{\prime}}|r_{k}\rangle\langle
r_{k^{\prime}}|\otimes|a\rangle\langle a^{\prime}|\,,$ (25)

where $\\{|a\rangle\\}$ is a basis of $\mathcal{H}_{A}$. By assumption, we
have that $\rho_{S|a}=\sum_{k}r^{a}_{k}|r_{k}\rangle\langle r_{k}|$ for all
$a$, thus $C^{aa}_{kk^{\prime}}=p_{a}r^{a}_{k}\delta_{kk^{\prime}}$, where
$p_{a}$ is the probability corresponding to the outcome $a$. We expand the sum
in Eq. (25) getting

$\displaystyle\rho_{SA}$ $\displaystyle=$
$\displaystyle\sum_{a,k}p_{a}r^{a}_{k}|r_{k}\rangle\langle
r_{k}|\otimes|a\rangle\langle a|+\sum_{a\neq
a^{\prime},k}C^{aa^{\prime}}_{kk}|r_{k}\rangle\langle
r_{k}|\otimes|a\rangle\langle a^{\prime}|$ (26)
$\displaystyle+\sum_{a>a^{\prime},k\neq k^{\prime}}|r_{k}\rangle\langle
r_{k^{\prime}}|\otimes(C^{aa^{\prime}}_{kk^{\prime}}|a\rangle\langle
a^{\prime}|+C^{a^{\prime}a}_{kk^{\prime}}|a^{\prime}\rangle\langle a|)\,.$

Thus, if the last term is zero, we get Eq. (24) with the matrices $C^{A}_{k}$
having entries $C^{aa^{\prime}}_{kk}$ with respect to the basis
$\\{|a\rangle\\}$. To show that it is zero, we note that if there exist at
least $\bar{a}$ and $\bar{a}^{\prime}\neq a$ such that
$C^{\bar{a}\bar{a}^{\prime}}_{kk^{\prime}}\neq 0$ and
$C^{\bar{a}\bar{a}^{\prime}}_{kk^{\prime}}\neq 0$ for some $k\neq k^{\prime}$
and the last term is non-zero, we can perform measurements with respect to
some basis different from $\\{|a\rangle\\}$, e.g., containing the states
$(|\bar{a}\rangle\pm|\bar{a}^{\prime}\rangle)/\sqrt{2}$, getting
$\rho_{S|\bar{a}}$ non-diagonal with respect to the basis
$\\{|r_{k}\rangle\\}$, which is absurd since $\rho_{S|a}$ is diagonal for any
set $\\{\Pi_{a}\\}$, which completes the proof. ∎

In general we define a dephasing map $\Delta$ as
$\Delta(X)=\sum_{k}\Pi_{k}X\Pi_{k}$, where $\Pi_{k}=|k\rangle\langle k|$ such
that $\Pi_{k}\Pi_{j}=\delta_{k,j}\Pi_{k}$ and $\sum_{k}\Pi_{k}=I$. By
recalling that $A\geq B$ means that the matrix $A-B$ is positive semidefinite,
we get the following lemma:

######  Lemma 3.

Given two matrices $A$ and $B$, $\Delta(A)\geq\Delta(B)$ for any dephasing map
$\Delta$ if and only if $A\geq B$.

###### Proof.

To prove it, we note that the dephasing map is a linear map, thus
$\Delta(A)\geq\Delta(B)$ is equivalent to $\Delta(A-B)\geq 0$. Thus,
$\Delta(A)\geq\Delta(B)$ for any dephasing map $\Delta$ if and only if the
diagonal elements of the matrix $A-B$ are non-negative with respect to any
basis. This is equivalent to $\langle\psi|A-B|\psi\rangle\geq 0$ for any state
$|\psi\rangle$, i.e., $A-B$ is positive semidefinite, $A-B\geq 0$, which
completes the proof. ∎

Thus, with the help of these three lemmas, we get the theorem:

######  Theorem 1.

We have a zero maximum gain $\delta\mathcal{U}(\rho_{SA})=0$ if and only if
$\rho_{SA}\in\mathcal{S}$ and has the form

$\rho_{SA}=\sum_{k}e^{rH_{S}/2}|u_{k}\rangle\langle u_{k}|e^{rH_{S}/2}\otimes
C^{A}_{k}\,,$ (27)

where the positive semidefinite matrices $C^{A}_{k}\geq 0$ are such that
$C^{A}_{k}\geq C^{A}_{k+1}$ for all $k=1,\ldots,d_{S}-1$.

###### Proof.

From Lemma 1, we get that $\delta\mathcal{U}(\rho_{SA})=0$ if and only if
$\tilde{\rho}_{S|a}$ are passive with respect to the Hamiltonian
$\tilde{H}_{S}$ for all the outcomes $a$ for any set $\\{\Pi^{A}_{a}\\}$.
Thus, by considering $\tilde{\rho}_{SA}=e^{-rH_{S}/2}\otimes
I^{A}\rho_{SA}e^{-rH_{S}/2}\otimes I^{A}$, since $\tilde{\rho}_{S|a}$ is
diagonal with respect to the basis $\\{|u_{k}\rangle\\}$ for any set
$\\{\Pi^{A}_{a}\\}$, from Lemma 2 we get
$\tilde{\rho}_{SA}=\sum_{k}|u_{k}\rangle\langle u_{k}|\otimes C^{A}_{k}$, from
which it results Eq. (27). Furthermore, $\langle
u_{k}|\tilde{\rho}_{S|a}|u_{k}\rangle$ are in decreasing order for any set
$\\{\Pi^{A}_{a}\\}$, by noting that $\langle
u_{k}|\tilde{\rho}_{S|a}|u_{k}\rangle=\langle a|C^{A}_{k}|a\rangle$ where we
have considered $\Pi^{A}_{a}=|a\rangle\langle a|$, this is equivalent to
$\Delta(C^{A}_{k})\geq\Delta(C^{A}_{k+1})$ for any dephasing map $\Delta$.
Thus, from Lemma 3 we get $C^{A}_{k}\geq C^{A}_{k+1}$. ∎

We note that the set of the states $\rho_{SA}$ with zero gain
$\delta\mathcal{U}(\rho_{SA})=0$ has measure zero since these states are
obtained from classical-quantum states $\chi^{c-q}_{SA}$ by performing the
transformation $\rho_{SA}=e^{rH_{S}/2}\otimes
I^{A}\chi^{c-q}_{SA}e^{rH_{S}/2}\otimes
I^{A}/\text{Tr}\left\\{\chi^{c-q}_{SA}e^{rH_{S}}\otimes I^{A}\right\\}$, which
form a set of measure zero [36]. Thus, almost all states $\rho_{SA}$ give a
positive gain $\delta\mathcal{U}(\rho_{SA})>0$. Furthermore, from Theorem 1 it
follows that:

######  Corollary 1.

We get $r=0$ or $[\rho_{S},H_{S}]=0$ if and only if
$\delta\mathcal{U}(\rho_{SA})=0\Rightarrow\rho_{SA}\in\mathcal{C}_{S}$.

###### Proof.

For $r=0$ or if $[\rho_{S},H_{S}]=0$, the states $e^{rH_{S}/2}|u_{k}\rangle$
are mutually orthogonal so that $\rho_{SA}$ in Eq. (27) is a classical-quantum
state, $\rho_{SA}\in\mathcal{C}_{S}$. Vice versa, if $\rho_{SA}$ in Eq. (27)
is a classical-quantum state, $e^{rH_{S}}$ is diagonal with respect to the
basis $\\{|u_{k}\rangle\\}$, from which we get $r=0$ or $[\rho_{S},H_{S}]=0$.
∎

In particular, for $r=0$ we get the ergotropy gain
$\delta\mathcal{U}(\rho_{SA})=\delta\mathcal{E}(\rho_{SA})$ in Eq. (8). This
shows how the presence of initial quantum coherence with respect to the energy
basis, so that $[\rho_{S},H_{S}]\neq 0$, can make
$\delta\mathcal{U}(\rho_{SA})=0$ for separable states
$\rho_{SA}\notin\mathcal{C}_{S}$ with $\delta\mathcal{E}(\rho_{SA})>0$.

###  III.2 Non-constant absolute risk aversion

In general, the agent can be characterized with a non-exponential utility
function, so that the absolute risk aversion is not constant. To study the
case of an arbitrary utility function $u(w)$, we focus on $d_{S}=2$. We
consider the unitary operator

$U_{S}=\left(\begin{array}[]{cc}\alpha&-\beta^{*}\\\ \beta&\alpha^{*}\\\
\end{array}\right)\,,$ (28)

with $\left|\alpha\right|^{2}+\left|\beta\right|^{2}=1$, the state and the
Hamiltonian

$\rho_{S}=\left(\begin{array}[]{cc}p&c\\\ c^{*}&1-p\\\
\end{array}\right)\,,\quad H_{S}=\left(\begin{array}[]{cc}\epsilon_{1}&0\\\
0&\epsilon_{2}\\\ \end{array}\right)\,,$ (29)

where $\left|c\right|\leq\sqrt{p(1-p)}$ so that $\rho_{S}\geq 0$. Since the
utility is defined up to an additive constant, without loss of generality we
focus on a function $u(w)$ such that $u(0)=0$. For $q=1/2$, from Eq. (14) we
get the general expression for the expected utility

$\langle
u(w)\rangle=\left|\beta\right|^{2}X-p\left|\beta\right|^{2}Y-2\text{Re}(c\alpha\beta)Z\,,$
(30)

where we have defined $X=u(\epsilon_{2}-\epsilon_{1})$,
$Y=2u_{o}(\epsilon_{2}-\epsilon_{1})$ and
$Z=2u_{o}(\frac{\epsilon_{2}-\epsilon_{1}}{2})$, where
$u_{o}(w)=(u(w)-u(-w))/2$ is the odd part of the utility function $u(w)$. We
note that arbitrary $X$, $Y$ and $Z$ can be obtained from a cubic utility, for
instance for $\epsilon_{2}-\epsilon_{1}=1$, we can consider

$u(w)=\frac{8Z-Y}{6}w+\frac{2X-Y}{2}w^{2}+\frac{2(Y-2Z)}{3}w^{3}\,.$ (31)

In detail, $Y\geq X>0$ and $Z>0$ if $u(w)$ is a strictly increasing function.
To obtain the optimal expected utility $\mathcal{U}(\rho_{S})$, we must to
maximize Eq. (30) over all the complex $\alpha=ae^{i\theta}$ and
$\beta=be^{i\phi}$ such that $a^{2}+b^{2}=1$. We search the stationary point
of the Lagrangian $L=\langle u(w)\rangle-\lambda(a^{2}+b^{2})$, where we have
introduced the Lagrange multiplier $\lambda$ such that $a^{2}+b^{2}=1$. For
$c$ real, we get

$\displaystyle\cos(\theta+\phi)=\pm 1\,,$ (32) $\displaystyle
a=\mp\frac{cZb}{\lambda}\,,$ (33) $\displaystyle
b=\frac{1}{\sqrt{1+\frac{c^{2}Z^{2}}{\lambda^{2}}}}\,,$ (34)
$\displaystyle\lambda^{2}-(X-pY)\lambda-c^{2}Z^{2}=0\,,$ (35)

where the sign $\cos(\theta+\phi)$ is chosen such that $a\geq 0$, and

$\mathcal{U}(\rho_{S})=\frac{X-pY+\frac{2c^{2}Z^{2}}{\lambda}}{1+\frac{c^{2}Z^{2}}{\lambda^{2}}}=\frac{\lambda}{1+\frac{c^{2}Z^{2}}{\lambda^{2}}}\,.$
(36)

Thus we deduce that $\lambda\geq 0$ is the largest solution of Eq. (35), which
explicitly reads

$\lambda=\frac{X-pY}{2}+\sqrt{\left(\frac{X-pY}{2}\right)^{2}+c^{2}Z^{2}}\,,$
(37)

so that $\mathcal{U}(\rho_{S})\geq 0$. We note that the expression in Eq. (36)
can be easily generalized for a complex coherence $c$ by replacing $c^{2}$
with $|c|^{2}$. As $|c|Z\to 0$, we get the expansion

$\mathcal{U}(\rho_{S})=\frac{1}{2}(X-pY+|X-pY|)+\frac{|c|^{4}Z^{4}}{|X-pY|^{3}}+O(|c|^{6}Z^{6})\,,$
(38)

so that, for an incoherent state, i.e., for $c=0$, or for a utility function
such that $Z=0$, we get $\mathcal{U}(\rho_{S})=0$ if $X-pY\leq 0$, otherwise
$\mathcal{U}(\rho_{S})=X-pY>0$. We aim to characterize the states $\rho_{SA}$
such that $\delta\mathcal{U}(\rho_{SA})=0$. Although for an exponential
utility function they are separable, for an arbitrary utility in principle
they can be not. In particular, we focus on $d_{A}=2$ and we start to consider
the states

$\rho_{SA}=p|00\rangle\langle 00|+(1-p)|11\rangle\langle
11|+\frac{C}{2}(|00\rangle\langle 11|+|11\rangle\langle 00|)\,,$ (39)

where $C$, such that $0\leq C\leq\sqrt{p(1-p)}$, is the quantum concurrence
and quantifies the entanglement between $S$ and $A$, e.g., for $C>0$ the state
$\rho_{SA}$ is entangled. By randomly generating the states $\rho_{SA}$ in Eq.
(39) and the values of $X$,$Y$ and $Z$ such that $X-pY\leq 0$, we find that
there are entangled states $\rho_{SA}$ showing a practically zero gain
$\delta\mathcal{U}(\rho_{SA})\approx 0$ (see Fig. 2).

Figure 2:  The gain $\delta\mathcal{U}(\rho_{SA})$ versus the concurrence $C$
for $10^{4}$ random states $\rho_{SA}$ of Eq. (39). We randomly generate $X$,
$Y$ and $Z$ in the interval $[-1,1]$, such that $X-pY\leq 0$.

This suggests that there are zero-gain entangled states depending on the
utility function. However, it results that all these zero-gain states are
obtained when $X-qY\leq 0$ for any $q\in[0,1]$, as $Z\to 0$, since in this
case $\mathcal{U}(\rho_{S|a})\sim Z^{4}\to 0$ for any state (see Eq. (38)).
Thus, $\delta\mathcal{U}(\rho_{SA})=0$ for all the states $\rho_{SA}$ when
$Z=0$ if $X\leq 0$ and $Y\geq 0$ or if $X\leq Y<0$. In particular, in this
case the utility function is not a strictly increasing function. Thus, for
instance we have zero-gain for $X<0$ and $Y=Z=0$, i.e., for the quadratic
utility $u(w)=-w^{2}$, or for $Y>0$ and $X=Z=0$, i.e., for
$u(w)=-w-3w^{2}+4w^{3}$. Furthermore, we note that
$\delta\mathcal{U}(\rho_{SA})=0$ for all the states $\rho_{SA}$ when $Z=0$ if
$X>Y\geq 0$ or if $X>0$ and $Y<0$, since in this case $X-qY>0$ for any
$q\in[0,1]$ and by using Eq. (38) we get
$\mathcal{U}_{\\{\Pi^{A}_{a}\\}}(\rho_{SA})=\mathcal{U}(\rho_{S})=X-pY$ for
any set $\\{\Pi^{A}_{a}\\}$. We note that for utility functions $u(w)$ such
that $Z=0$, the optimal expected utility is equal to

$\mathcal{U}(\rho_{S})=\mathcal{U}(\Delta(\rho_{S}))$ (40)

for any state $\rho_{S}$ and for the dephasing map with energy projectors
$\Delta(\rho_{S})=\sum_{k}|\epsilon_{k}\rangle\langle\epsilon_{k}|\rho_{S}|\epsilon_{k}\rangle\langle\epsilon_{k}|$.
Thus, the initial quantum coherence does not give any contribution for these
utility functions, which we call ‘incoherent’ utility functions. In detail, we
define the incoherent utility functions as the utility functions $u(w)$ such
that $\mathcal{U}_{c}(\rho_{S})=0$ for all $\rho_{S}$, where in general the
coherent contribution is defined as
$\mathcal{U}_{c}(\rho_{S})=\mathcal{U}(\rho_{S})-\mathcal{U}(\Delta(\rho_{S}))\geq
0$ [25]. For $d_{S}=2$, the incoherent utility functions are the functions
$u(w)$ such that $Z=0$. For an arbitrary incoherent utility function, the gain
can be expressed as

$\delta\mathcal{U}(\rho_{SA})=\max_{\\{\Pi^{A}_{a}\\}}\sum_{a}p_{a}\mathcal{U}(\Delta(\rho_{S|a}))-\mathcal{U}(\Delta(\rho_{S}))$
(41)

and, for instance, it is zero for $d_{S}=2$ for any state if $X\leq 0$ and
$Y\geq 0$, if $X\leq Y<0$, if $X>Y\geq 0$ or if $X>0$ and $Y<0$. For a Werner
state $\rho_{SA}=\frac{1-z}{4}I+z|\psi\rangle\langle\psi|$, where
$|\psi\rangle=(|00\rangle+|11\rangle)/\sqrt{2}$, which is entangled for
$z>1/3$, given an incoherent utility function, from Eq. (41) we get

$\delta\mathcal{U}(\rho_{SA})=\max_{q\in[0,1]}\frac{1}{4}\left|\tilde{X}+\tilde{Y}_{q}\right|+\frac{1}{4}\left|\tilde{X}-\tilde{Y}_{q}\right|-\frac{1}{2}\left|\tilde{X}\right|\,,$
(42)

where $\tilde{X}=X-Y/2$ and $\tilde{Y}_{q}=\left(q-\frac{1}{2}\right)zY$. If
$2X>Y\geq 0$, from Eq. (42) we get $\delta\mathcal{U}(\rho_{SA})=0$ for $z\leq
z_{0}=2X/Y-1$, whereas $\delta\mathcal{U}(\rho_{SA})>0$ for $z>z_{0}$. Thus,
for $z_{0}>1/3$ there are Werner states with zero-gain although they are
entangled states. The results can be summarized as in Table 1.

Table 1: The role of quantum correlations can be summarized as follows. risk aversion | utility gain
---|---
$r_{A}=0$ | $\delta\mathcal{U}=\delta\mathcal{E}=0\Rightarrow\rho_{SA}\in\mathcal{C}_{S}$
$r_{A}=\text{const}\neq 0$ | $\delta\mathcal{U}=0\Rightarrow\rho_{SA}\in\mathcal{S}$
$r_{A}\neq\text{const}$ | $\delta\mathcal{U}=0\nRightarrow\rho_{SA}\in\mathcal{S}$

In contrast, in the case of arbitrary non-incoherent utility functions, if Eq.
(41) is zero, the gain is given by the coherent contribution and reads
$\delta\mathcal{U}(\rho_{SA})=\max_{\\{\Pi^{A}_{a}\\}}\sum_{a}p_{a}\mathcal{U}_{c}(\rho_{S|a})-\mathcal{U}_{c}(\rho_{S})$.
In this case, we find that the gain $\delta\mathcal{U}(\rho_{SA})$ is positive
if $\rho_{SA}$ is entangled at least for $d_{S}=2$. Furthermore, we note that
the condition $\Delta(\rho_{S|a})=\rho_{S|a}$ for all $a$ and any set
$\\{\Pi^{A}_{a}\\}$, from which $\rho_{S}=\Delta(\rho_{S})$ and Eq. (41)
follows, is not a sufficient condition in order to can obtain
$\delta\mathcal{U}(\rho_{SA})=0$ for $\rho_{SA}\notin\mathcal{S}$. To prove
it, it is enough to note that if this condition is satisfied, by using Lemma
2, we get $\rho_{SA}\in\mathcal{C}_{S}$ having the form
$\rho_{SA}=\sum_{k}|\epsilon_{k}\rangle\langle\epsilon_{k}|\otimes C^{A}_{k}$.
Then, in this case there are no states $\rho_{SA}\notin\mathcal{S}$ such that
$\delta\mathcal{U}(\rho_{SA})=0$, for any utility function $u(w)$. This
suggests that only for incoherent utility functions, which we recall to be
defined such that Eq. (40) is satisfied for any $\rho_{S}$, there can be
states $\rho_{SA}\notin\mathcal{S}$ such that
$\delta\mathcal{U}(\rho_{SA})=0$.

##  IV Conclusions

We considered a bipartite quantum system and took into account an agent that
can extract work locally through a daemonic protocol introduced in Ref. [4],
which is affected by work fluctuations when the agent is non-neutral to risk.
We introduced the daemonic expected utility, so that the agent can select the
optimal work extraction protocol by looking on the utility gain defined from
this quantity. We completely characterized the role of correlations among the
two parties of the system, showing how quantum correlations can influence the
selection done by the agent depending on the absolute risk aversion.
Furthermore, our results clarify the role of initial quantum coherence with
respect to the energy basis, which is tricky. We showed how the presence of an
initial quantum coherence can give zero utility gain although the states have
positive ergotropy gain. Furthermore, in general, if the contribution of this
initial quantum coherence is always absent due to the particular form of the
utility function, then there are entangled state with zero utility gain. In
conclusion, we believe our results represent a substantial step forward in
understanding work fluctuations, showing how these strongly alter the
optimization of the work extraction daemonic protocol (when they are taken in
account by a utility function), resulting in a sensible deviation of a risk
non-neutral agent’s decision-making behavior from a risk neutral one.

### Acknowledgements

The authors acknowledge financial support from the project BIRD 2021
”Correlations, dynamics and topology in long-range quantum systems” of the
Department of Physics and Astronomy, University of Padova, from the European
Union - Next Generation EU within the National Center for HPC, Big Data and
Quantum Computing (Project No. CN00000013, CN1 - Spoke 10 Quantum Computing)
and from the Project ”Frontiere Quantistiche” (Dipartimenti di Eccellenza) of
the Italian Ministry for Universities and Research.

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