# Selective weak measurement reveals super ergotropy

M. A. Balkanlu, E. Faizi and B. Ahansaz
Physics Department, Azarbaijan Shahid Madani University, Tabriz, Iran
<EMAIL_ADDRESS>
###### Abstract

The concept of ergotropy was previously introduced as the maximum extractable
work from a quantum state. Its enhancement, which is induced by quantum
correlation via projective measurement, was formulated as the daemonic
ergotropy. In this work, we investigate the ergotropy in the presence of
quantum correlation via weak measurement because of its elegant effects on the
measured system. By considering a bipartite correlated quantum system
consisting of main and ancillary systems, we demonstrate that the extractable
work by the non-selective weak measurement on the ancilla is always equal to
the situation captured by the strong measurement. However, the selective weak
measurement interestingly reveals more work than the daemonic ergotropy and
the ergotropy of the total system is greater than or equal to the daemonic
ergotropy. Moreover, it is shown that for Bell diagonal states, at the cost of
losing quantum correlation, the total extractable and thus non-local
extractable works can be increased by using measurement. Also, we find that
there is no direct relationship between quantum correlation and non-local
extractable work for these cases.

Keywords: Quantum thermodynamics, Ergotropy, Weak measurement, Quantum
correlation, Non-local extractable work.

##  I. Introduction

In quantum thermodynamics, concepts such as heat, work, thermal equilibrium,
temperature, entropy and heat transfer at the atomic scale, need to be
redefined and re-examined. Quantum thermodynamic concepts of heat and work
were first defined by R. Alicki [1], where the changes in the local
Hamiltonian and the state of a system are associated with work and heat
respectively. Recently, new definitions have been proposed as the change in
the energy which is accompanied by a change in the entropy is identified as
heat, while any change in the energy which does not lead to a change in the
entropy is known as work [2]. For the case of an individual system it has been
published that, similar to classical thermodynamics, the optimal extractable
work is equal to its free energy change [3, 4]. In general, work can only be
extracted from a system whose initial state is non-passive. Thermodynamic
passivity expresses that the quantum state’s energy cannot be reduced by any
unitary process that acts on the system. These states are so-called passive
states [5, 6, 7]. However, several copies of a passive state can become non-
passive which allows work to be extracted from these states otherwise, the
state is called completely passive [8]. It is worth mentioning that, only the
completely passive states are thermal states [9]. The maximal extractable work
from a quantum non-passive state using cyclic unitary transformation is
defined as ergotropy in which unitary transformation is governed by time-
dependent external potentials as work sources [10]. There are also some
ancilla-assisted protocols to work extraction processes based on quantum
correlations [11, 12, 13, 14]. In a bipartite quantum system, the non-local
effect on one party due to the measurement on the other is usually called
quantum correlation. The non-local properties of quantum correlation make it a
key element for extracting work from a global state consisting of local
thermal states by using nonlocal extraction operations. Nonlocal extractable
work is defined as the difference between the total and the local extractable
works. G. Francica et al research [11] shows that, in a bipartite closed
quantum system which is composed of the main and the ancillary systems, thanks
to quantum correlation, it is possible to reach more work than the ergotropy.
In this scenario, a non-interacting ancillary system is joined to the main
system. Then a measurement by a set of orthogonal projectors is implemented on
the ancilla which is followed by ergotropic transformation of all conditional
states. This enhanced extractable work which is daemonic ergotropy is the
average ergotropy of conditional states of the main system over all the
possible outcomes of the measurement on the ancillary system.

The general belief that ”the more information is obtained from a quantum
system, the more its state is disturbed by measurement” has created a great
deal of interest in finding a balance between the two [15, 16, 17, 18, 19,
20]. Measuring the system that is necessary to observe it, will disturb the
state of the system and thus its coherence. For instance, strong measurement,
with complete destructing of coherence, irreversibly collapses the system
state to the eigenstates of its projectors [21]. While protecting the unknown
state of a quantum system from decoherence is a key requirement for quantum
information processing. Recovering the initial state after a measurement is
one kind of state protection. If the interaction between the system and
measuring device is weak, then the system will partially be perturbed and its
coherence will not be completely annihilated. So it is possible to retrieve
the initial state by undoing the effect of the measurement [22]. Such an
observation of the quantum systems is known as a weak measurement [23, 24, 25,
26, 27, 28, 29]. It has recently been shown that weak measurement also helps
protect quantum entanglement against decoherence based on its reversibility
[30, 31]. In a bipartite quantum system, the weak measurement performed on one
of the subsystems can reveal greater quantum correlation than strong
measurement [32].

The measurement on the ancillary system is the critical instrument to extract
the enhanced work from the system in ancilla-assisted protocols. The
reversibility of the system state under weak measurement motivated us to
investigate whether applying the weak measurement would lead to a further gain
in work extraction compared by projective measurement? On the hand, the
relationship between geometrically defined correlation and non-local
extractable work for the two-qubit pure and Bell diagonal states has been
confirmed [33]. Therefore, we were interested in investigating the effect of
projective measurement on total and consequently non-local extractable works
and reexamining the relationship between quantum correlation and non-local
extractable work in Bell diagonal states.

The rest of this paper is organized as follows. Section II briefly reviewed
the definitions of ergotropy and daemonic ergotropy. In Section III, we
performed non-selective weak measurement instead of projective measurement on
the ancillary system and showed that the maximum extractable work is always
equal to the daemonic ergotropy. But as discussed in Section IV, we
demonstrated that using the selective weak measurement reveals more work than
the daemonic ergotropy which we named super ergotropy. Moreover, Section V
contains some geometric discussion of the local, non-local, and general
extractable works for the correlated ancilla-assisted two-qubit systems. The
manuscript is finished with some concluding remarks in Section VI.

##  II. Ergotropy and daemonic ergotropy

In this section, we are going to review the definitions of ergotropy and
daemonic ergotropy. According to Ref. [10], we assume a thermally isolated
quantum system $S$ prepared in a non-passive state
$\hat{\rho}(0)=\sum_{j}r_{j}|r_{j}\rangle\langle r_{j}|(r_{j}\geq r_{j+1}$
$j=1,2,...)$ with respect to its Hamiltonian
$\hat{H}=\sum_{k}\varepsilon_{k}|\varepsilon_{k}\rangle\langle\varepsilon_{k}|(\varepsilon_{k}\leq\varepsilon_{k+1}$
$k=1,2,...)$ which exchange work with an external macroscopic source. To
extract the ergotropy, any arbitrary Hamiltonian perturbation $\hat{V}(t)$ is
cyclically imposed on the system over a fixed period $[0,\tau]$. So under the
action of this potential, the system undergoes a unitary transformation as
$\hat{U}=\sum_{i}|\varepsilon_{i}\rangle\langle r_{i}|$ from $\hat{\rho}(0)$
to the final state $\hat{\rho}(\tau)$ with the lowest final energy. Then the
system exchanges released energy as work with the external source. The
ergotropy which is defined as the maximal extractable work from the state
$\hat{\rho}(0)$ can be expressed as

$\displaystyle
W=Tr[\hat{\rho}(0)\hat{H}_{s}]-Tr[\hat{\rho}(\tau)\hat{H}_{s}]=\sum_{j,k}\varepsilon_{j}r_{k}(|\langle\varepsilon_{j}|r_{k}\rangle|^{2}-\delta_{jk}).$
(1)

The ergotropy is enhanced by using the concept of quantum correlation which is
obtained in Ref. [11]. The procedure is that the system $S$ and a non-
interacting ancillary system $A$ are prepared in a joint state
$\hat{\rho}_{SA}$. Then the local measurement is done on subsystem $A$ and
finally, ergotropic transformation is performed for all conditional density
matrices $\hat{\rho}_{S|A}$. The optimal extracted work from the state of the
system $S$ is as follows

$\displaystyle
W_{\\{\hat{\Pi}^{A}_{a}\\}}=Tr[\hat{\rho}_{s}\hat{H}_{s}]-\sum_{a}p_{a}Tr[\hat{U_{a}}\hat{\rho}_{S|a}\hat{U}^{\dagger}_{a}\hat{H}_{s}],$
(2)

where $p_{a}=Tr[\hat{\rho}_{SA}\hat{\Pi}^{A}_{a}]$ is the probability of
finding the system $S$ in the conditional state
$\hat{\rho}_{S|a}=Tr_{A}[\hat{\Pi}^{A}_{a}\hat{\rho}_{SA}\hat{\Pi}^{A}_{a}]/p_{a}$,
$\hat{U_{a}}$ is the cyclic unitary process that transforms $\hat{\rho}_{S|a}$
in order to minimize its internal energy and ${\\{\hat{\Pi}^{A}_{a}\\}}$ is a
set of orthogonal projectors of measurement. The extracted work
$W_{\\{\hat{\Pi}^{A}_{a}\\}}$ which is known as Daemonic Ergotropy is
calculated as

$\displaystyle
W_{\\{\hat{\Pi}^{A}_{a}\\}}=Tr[\hat{\rho}_{s}\hat{H}_{s}]-\sum_{a}p_{a}\sum_{k}r^{a}_{k}\varepsilon_{j},$
(3)

where $r^{a}_{k}$ are the eigenvalues of $\hat{\rho}_{S|a}$. The state
$\hat{\rho}_{S}$ can be written as
$\hat{\rho}_{S}=\sum_{a}p_{a}\hat{\rho}_{S|a}=\sum_{a}p_{a}\sum_{k}r^{a}_{k}|r^{a}_{k}\rangle\langle
r^{a}_{k}|$ which implies $r_{k}=\sum_{a}p_{a}\sum_{j}r^{a}_{j}|\langle
r_{k}|r^{a}_{j}\rangle|^{2}$and finally the difference
$W_{\\{\hat{\Pi}^{A}_{a}\\}}-W$ reads as

$\displaystyle
W_{\\{\hat{\Pi}^{A}_{a}\\}}-W=\sum_{k}\varepsilon_{k}(r_{k}-\sum_{a}p_{a}r^{a}_{k})=\sum_{a}p_{a}\sum_{k,j}r^{a}_{j}\varepsilon_{k}(|\langle
r_{k}|r^{a}_{j}\rangle|^{2}-\delta_{jk}).$ (4)

This is the average of all the conditional states ergotropy relative to
Hamiltonian $\hat{H}=\sum_{k}\varepsilon_{k}|r_{k}\rangle\langle r_{k}|$ which
are non-negative by definition. From the positivity of Eq. (4), we have
$W_{\\{\hat{\Pi}^{A}_{a}\\}}\geq W$. When the subsystems are initially
uncorrelated as $\hat{\rho}_{SA}=\hat{\rho}_{S}\otimes\hat{\rho}_{A}$, we have
$\hat{\rho}_{S|a}=\hat{\rho}_{S}$ for any set $\\{\hat{\Pi}^{A}_{a}\\}$ and
outcome $a$, which means there would be no gain in work extraction and
$W_{\\{\hat{\Pi}^{A}_{a}\\}}=W$.

##  III. Non-selective weak measurement and daemonic ergotropy

The notion of weak measurement with post-selection was first introduced in
Ref. [23], which leads to the definition of weak values of the measured
observable. Weak values can be calculated by using the pre and post-selected
quantum systems and can be beyond the permissible values for observable. We
will not be considering the measurements of this type. Throughout this paper,
we will use another expression for weak measurement in terms of measurement
operators which is given by Oreshkov and Brun for the weak measurement in Ref.
[28]. In this section, we have shown that by applying the mentioned
restrictions on the system dimension and projectors, the same results as in
the previous section are obtained by non-selective weak measurement on the
ancillary system in a joint state $\hat{\rho}_{SA}$. Nevertheless, its
advantage is to maintain the correlation between subsystems. Following the
second formalism of the weak measurement, the operators of weak measurement
have the form as

$\displaystyle\hat{P}(x)=\sqrt{b_{0}(x)}\hat{\Pi}_{0}+\sqrt{b_{1}(x)}\hat{\Pi}_{1},$
(5)
$\displaystyle\hat{P}(-x)=\sqrt{b_{0}(-x)}\hat{\Pi}_{0}+\sqrt{b_{1}(-x)}\hat{\Pi}_{1},$
(6)

and the POVM elements $\hat{E(x)}$, $\hat{E(-x)}$ for this measurement is
given as

$\displaystyle\hat{E}(x)=\hat{p}^{\dagger}(x)\hat{p}(x)=b_{0}(x)\hat{\Pi}_{0}+b_{1}(x)\hat{\Pi}_{1},$
(7)
$\displaystyle\hat{E}(-x)=\hat{p}^{\dagger}(-x)\hat{p}(-x)=b_{0}(-x)\hat{\Pi}_{0}+b_{1}(-x)\hat{\Pi}_{1},$
(8)

where $b_{0}(x)=(1-\mathrm{tanh}x)/2,b_{1}(x)=(1+\mathrm{tanh}x)/2$ and
$x\in[0,\infty]$ is the parameter to control the strength of measurement.
$\hat{\Pi}_{0}$ and $\hat{\Pi}_{1}$ are two orthogonal projectors with
$\hat{\Pi}_{0}+\hat{\Pi}_{1}=I$. Note that in the limit of $x\rightarrow 0$,
the operators $\hat{P}(x)$ and $\hat{P}(-x)$ are proportional to the identity
which means no measurement is done. In the limit of strong measurement, we
have $\mathrm{lim}_{x\rightarrow\infty}\hat{P}(-x)=\hat{\Pi}_{0}$ and
$\mathrm{lim}_{x\rightarrow\infty}\hat{P}(x)=\hat{\Pi}_{1}$. These operators
also satisfy $\hat{P}(x)\hat{P}(y)\propto\hat{P}(x+y)$ and
$[\hat{P}(x),\hat{P}(y)]=I$. We can write the extractable work from the state
of systems average ergotropies of two possible outcomes of measurement

$\displaystyle W_{\\{\hat{P}^{A}_{\pm
x}\\}}=Tr[\hat{\rho}_{s}\hat{H}_{s}]-\sum_{y=x,-x}p_{y}Tr[\hat{U_{y}}\hat{\rho}_{S|y}\hat{U}^{\dagger}_{y}\hat{H}_{s}].$
(9)

$\bf{Theorem\ 1.}$ For any system $S$ and ancillary system $A$ prepared in a
state $\hat{\rho}_{SA}$, by performing weak measurement on the ancillary
system, we always have $W_{\\{\hat{P}^{A}_{\pm
x}\\}}=W_{\\{\hat{\Pi}^{A}_{a}\\}}$.

$\bf{Proof.\ }$ We can write probabilities of measurement as below

$\displaystyle p_{x}$
$\displaystyle=Tr[(\hat{I}\otimes\hat{E}(x))\hat{\rho}_{SA}]$ (10)
$\displaystyle=Tr\\{[\hat{I}\otimes(b_{0}(x)\hat{\Pi}_{0}+b_{1}(x)\hat{\Pi}_{1})]\hat{\rho}_{SA}\\}=b_{0}(x)p_{0}+b_{1}(x)p_{1},$

and $\hat{\rho}_{S|x}$ which is related to $\hat{\rho}_{S|a}$ is given by

$\displaystyle\hat{\rho}_{S|x}=\frac{Tr_{A}[(\hat{I}\otimes\hat{E}(x))\hat{\rho}_{SA}]}{p(x)}=\frac{\sum_{a}b_{a}(x)p_{a}\hat{\rho}_{S|a}}{p(x)}.$
(11)

By considering equations (10) and (11), Eq. (9) reduces as follows

$\displaystyle W_{\\{\hat{P}^{A}_{\pm
x}\\}}=Tr[\hat{\rho}_{s}\hat{H}_{s}]-\sum_{a}\sum_{y=x,-x}b_{a}(y)p_{a}Tr[\hat{U_{a}}\hat{\rho}_{S|a}\hat{U}^{\dagger}_{a}\hat{H}_{s}].$
(12)

Similar to Eq. (4), the difference $W_{\\{\hat{P}^{A}_{\pm x}\\}}-W$ is given
by

$\displaystyle W_{\\{\hat{P}^{A}_{\pm
x}\\}}-W=\sum_{k}\varepsilon_{k}(r_{k}-\sum_{a}\sum_{y=x,-x}b_{a}(y)p_{a}r^{a}_{k}).$
(13)

By using the fact $\sum_{y=x,-x}b_{a}(y)=1$, we have $W_{\\{\hat{P}^{A}_{\pm
x}\\}}=W_{\\{\hat{\Pi}^{A}_{a=0,1}\\}}$. Which shows the equality of maximum
extractable works for the two sets of measurement operators.

##  IV. Selective weak measurement and super ergotropy

The daemonic ergotropy can be obtained via weak measurement on the ancillary
system to avoid quantum correlation loss. This fact motivated the authors to
optimize the extractable work which led to the investigation of selective weak
measurement i.e. by applying one of its operators. We proceed with the work
extraction by measuring the ancillary system with one of the weak measurement
operators. So the conditional density matrix of the system $S$ given the
measurement operator $\hat{P}{(\pm x)}$ on the system $A$ can be written as

$\displaystyle\hat{\rho}_{S|\pm x}=\frac{\sum_{a}b_{a}(\pm
x)p_{a}\hat{\rho}_{S|a}}{p_{(\pm x)}}$ (14)

where

$\displaystyle p_{\pm x}=b_{0}(\pm x)p_{0}+b_{1}(\pm x)p_{1}.$ (15)

The extractable work from the state of system $S$ reads as

$\displaystyle W_{\hat{P}^{A}_{\pm
x}}=Tr[\hat{\rho}_{s}\hat{H}_{s}]-\frac{1}{p_{(\pm x)}}\sum_{a=0}^{1}b_{a}(\pm
x)p_{a}Tr[\hat{U_{a}}\hat{\rho}_{S|a}\hat{U}^{\dagger}_{a}\hat{H}_{s}].$ (16)

So the difference between the maximal extractable work by using non-selective
(projective measurement) and selective (weak measurement) on the subsystem $A$
can be obtained as

$\displaystyle\delta_{W(\pm x)}=W_{\hat{P}^{A}_{\pm
x}}-W_{\\{\hat{\Pi}^{A}_{a}\\}}$
$\displaystyle=\sum_{a=0}^{1}(1-\frac{b_{a}(\pm x)}{p_{(\pm
x)}})Tr[\hat{U_{a}}\hat{\rho}_{S|a}\hat{U}^{\dagger}_{a}\hat{H}_{s}]$ (17)
$\displaystyle=\pm
C_{\pm}(Tr[\hat{U_{0}}\hat{\rho}_{S|0}\hat{U}^{\dagger}_{0}\hat{H}_{s}]-Tr[\hat{U_{1}}\hat{\rho}_{S|1}\hat{U}^{\dagger}_{1}\hat{H}_{s}]),$

where
$C_{\pm}=\frac{2p_{0}p_{1}\mathrm{tanh}x}{1\pm(p_{1}-p_{0})\mathrm{tanh}x}$
and signs $+$ and $-$ are related to $\hat{P}(+x)$ and $\hat{P}(-x)$
respectively. These coefficients are positive and depend on the probability of
measurement outcomes and its intensity.

$\bf{Theorem\ 2.}$ For any system $S$ and ancillary system $A$ prepared in a
state $\hat{\rho}_{SA}$, by performing the selective weak measurement on the
ancillary system we always have $W_{\hat{P}^{A}_{\pm x}}\geq
W_{\\{\hat{\Pi}^{A}_{a=0,1}\\}}$.

$\bf{Proof.\ }$ Considering Eq. (17), we select measurement operator $P(x)$
when $Tr[\hat{U_{0}}\hat{\rho}_{S|0}\hat{U}^{\dagger}_{0}\hat{H}_{s}]$ is
greater than $Tr[\hat{U_{1}}\hat{\rho}_{S|1}\hat{U}^{\dagger}_{1}\hat{H}_{s}]$
and we chose measurement operator $P(-x)$ when
$Tr[\hat{U_{1}}\hat{\rho}_{S|1}\hat{U}^{\dagger}_{1}\hat{H}_{s}]$ is greater
than $Tr[\hat{U_{0}}\hat{\rho}_{S|0}\hat{U}^{\dagger}_{0}\hat{H}_{s}]$ in
order to $\delta_{W(\pm x)}$ be positive. If two traces are equal, the
extractable work by both measurements is the same.

Therefore, depending on the final energy of conditional states
$\hat{\rho}_{S|a}$, extractable work greater than or equal to the demonic
ergotropy can be achieved. We called extractable work $W_{\hat{P}^{A}_{\pm
x}}$ as super ergotropy.

###  1\. Example

In the following, we introduce some examples to make the theorem more clear.
If we consider the pure states $\hat{\rho}_{1}=|\psi\rangle_{SA}\langle\psi|$
with
$|\psi\rangle_{SA}=\mathrm{cos}(\theta)|00\rangle+\mathrm{sin}(\theta)|00\rangle$
and $\hat{\rho}_{2}=|\phi\rangle_{SA}\langle\phi|$ with
$|\phi\rangle_{SA}=\frac{1}{\sqrt{3}}(|11\rangle+|10\rangle+|01\rangle)$ the
maximum work which can be extracted by using the selective weak measurement is
the same as with the demonic ergotropy. Here we will show the ability of the
selective weak measurement to gain more work than demonic ergotropy. As
another example, we consider the mixed state
$\hat{\rho}_{SA}=\frac{1}{3}(|11\rangle\langle 11|+|10\rangle\langle
10|+|01\rangle\langle 01|)$ which its daemonic ergotropy is calculated from
Eq. (2) as
$W_{\\{\hat{\Pi}^{A}_{a=0,1}\\}}=\frac{\varepsilon_{1}-\varepsilon_{0}}{3}=\frac{\Delta\varepsilon}{3}$,
where $\varepsilon_{1}$ and $\varepsilon_{0}$ are the energy of levels $1$ and
$0$ of the system $S$, respectively. By measuring the ancillary system via
operator $P(x)$ followed by the ergotropic transformation, the maximal
extractable work reads as

$\displaystyle
W_{\hat{P}^{A}_{+x}}=\frac{3+\mathrm{tanh}x}{3-\mathrm{tanh}x}(\Delta\varepsilon/3).$
(18)

The Fig. 1 shows $W_{\hat{P}^{A}_{+x}}/W_{\\{\hat{\Pi}^{A}_{a}\\}}$ as a
function of measurement strength $x$ which is greater than one. The more
strongly one perturbs the subsystem $A$, the more work can be revealed from
system $S$. So for sufficiently large values of $x$, we have
$W_{\hat{P}^{A}_{+x}}=2W_{\\{\hat{\Pi}^{A}_{a}\\}}$. In this case, selective
strong measurement with projector $\Pi^{A}_{1}$ emerges instead of $P(+x)$.
However, when we set $x=0$, the extractable works $W_{\hat{P}^{A}_{+x}}$ and
$W_{\\{\hat{\Pi}^{A}_{a}\\}}$ became equal.

##  V. Measurement effect on total and non-local extractable works

Non-local extractable work of a joint system is defined as the difference
between the total and the local extractable works [33]. We are also interested
in measuring the total system by a set of projective operators
$\\{\hat{\Pi}_{ij}\\}$. We define a classical non-local extractable work as
the difference between the average ergotropy of projective measurement on the
general system and the local extractable work as

$\displaystyle W^{\\{\hat{\Pi}_{ij}\\}}_{Non-
loc}=W^{\\{\hat{\Pi}_{ij}\\}}_{tot}-W_{loc}.$ (19)

The definition above enables us to compare the extractable work from quantum
and classically correlated states and examine the role of measurement in this
scenario.

In this section, we consider a general two-qubit system with computational
bases $\\{|11\rangle,|10\rangle,\\\ |01\rangle,|00\rangle\\}$ which consists
of non-interacting main and ancillary subsystems. The density matrix of such a
system can be written in geometrically Bloch representation as

$\displaystyle\hat{\rho}_{SA}=\frac{1}{4}(\hat{I}+\sum_{i=1}^{3}s_{i}\hat{\sigma}_{i}\otimes\hat{I}+\sum_{j=1}^{3}r_{j}\hat{I}\otimes\hat{\sigma}_{j}+\sum_{i,j=1}^{3}t_{ij}\hat{\sigma}_{i}\otimes\hat{\sigma}_{j}),$
(20)

where $\hat{I}$ is the identity matrix,
$s_{i}=Tr[\hat{\rho}_{SA}(\hat{\sigma}_{i}\otimes\hat{I})]$,
$r_{j}=Tr[\hat{\rho}_{SA}(\hat{I}\otimes\hat{\sigma}_{j})]$ and
$t_{ij}=Tr[\hat{\rho}_{SA}(\hat{\sigma}_{i}\otimes\hat{\sigma}_{j})]$ is the
$3\times 3$ correlation matrix and $\hat{\sigma}_{i}(\hat{\sigma}_{j})$ are
the Pauli matrices. Now, we perform measurement on the ancillary system with
the set of orthogonal projectors $\hat{\Pi}_{1}$ and $\hat{\Pi}_{0}$
respectively. The conditional density matrix of the system $S$ given projector
$\hat{\Pi}_{1}$ on the system $A$ reads as

$\displaystyle\hat{\rho}_{{S|1}}=\frac{1}{4}[(1+r_{3})\hat{I}+\sum_{i}(s_{i}+t_{i3})\hat{\sigma}_{i}]=\lambda_{+}|\lambda_{+}\rangle\langle\lambda_{+}|+\lambda_{-}|\lambda_{-}\rangle\langle\lambda_{-}|,$
(21)

with

$\displaystyle\lambda_{\pm}=\frac{1+r_{3}\pm|\vec{s}+\vec{t}_{i3}|}{4},$ (22)

where
$|\vec{s}+\vec{t}_{i3}|=\sqrt{(s_{1}+t_{13})^{2}+(s_{2}+t_{23})^{2}+(s_{3}+t_{33})^{2}}$.
The conditional density matrix of the system $S$ given projector
$\hat{\Pi}_{0}$ on the system $A$ reads as

$\displaystyle\hat{\rho}_{{S|0}}=\frac{1}{4}[(1-r_{3})\hat{I}+\sum_{i}(s_{i}-t_{i3})\hat{\sigma}_{i}]=\lambda_{+}^{\prime}|\lambda_{+}^{\prime}\rangle\langle\lambda_{+}^{\prime}|+\lambda_{-}^{\prime}|\lambda_{-}^{\prime}\rangle\langle\lambda_{-}^{\prime}|,$
(23)

with

$\displaystyle\lambda_{\pm}^{\prime}=\frac{1+r_{3}\pm|\vec{s}-\vec{t}_{i3}|}{4},$
(24)

where
$|\vec{s}-\vec{t}_{i3}|=\sqrt{(s_{1}-t_{13})^{2}+(s_{2}-t_{23})^{2}+(s_{3}-t_{33})^{2}}$.
The unitary operators of the ergotropic transformation are
$\hat{U}_{1}=|1\rangle\langle\lambda_{+}|+|0\rangle\langle\lambda_{-}|$ and
$\hat{U}_{0}=|1\rangle\langle\lambda_{+}^{\prime}|+|0\rangle\langle\lambda_{-}^{\prime}|$
which gives the daemonic ergotropy as

$\displaystyle
W_{\\{\hat{\Pi}^{A}_{a}\\}}=(2s_{3}+|\vec{s}+\vec{t}_{i3}|+|\vec{s}-\vec{t}_{i3}|)\frac{\Delta\varepsilon}{4}.$
(25)

On the other hand, the reduced density matrix of the system $S$ is given by
$\hat{\rho}_{S}=Tr_{A}(\hat{\rho}_{SA})=\frac{1}{2}(\hat{I}+\sum_{i}s_{i}\hat{\sigma}_{i})$.
Due to the zero Hamiltonian of the ancillary system, the local extractable
work will be only the ergotropy of the main system as
$W_{loc}=(s_{3}+s)\frac{\Delta\varepsilon}{2}$ [33]. As mentioned in section
II, we have $W_{\\{\hat{\Pi}^{A}_{a}\\}}\geq W_{loc}$, which is compatible
with this geometric interpretation which states for two arbitrary vectors $A$
and $B$, $|\vec{A}+\vec{B}|+|\vec{A}-\vec{B}|\geq 2|\vec{A}|$. Classical non-
local extractable work from Eq. (19) obtains as
$W^{\\{\hat{\Pi}_{ij}\\}}_{Non-loc}=(1-s)\frac{\Delta\varepsilon}{2}$
(Appendix A). As an interesting result, when the pre-measurement state of the
main system is pure, $W^{\\{\hat{\Pi}_{ij}\\}}_{Non-loc}$ is always equal to
zero, but for the mixed pre-measurement state of this system, it will be
greater than or equal to zero.

Up to local unitary equivalence, a general two-qubit state is always reducible
to a state at the Bloch normal form as [34]

$\displaystyle\hat{\rho}_{SA}=\frac{1}{4}(\hat{I}+\vec{s}\hat{\sigma}\otimes\hat{I}+\hat{I}\otimes\vec{r}\hat{\sigma}_{j}+\sum_{i=1}^{3}c_{i}\hat{\sigma}_{i}\otimes\hat{\sigma}_{i}),$
(26)

where $\vec{s}$ and $\vec{r}$ are the Bloch vectors of the reduced density
matrices of the systems $S$ and $A$ respectively and $c_{i}$ are the real
numbers such that $|c_{i}|\leq 1$. We assume that $\vec{s}=\vec{r}=0$ to
reduce it to the Bell-diagonal states

$\displaystyle\hat{\rho}_{SA}$
$\displaystyle=\frac{1}{4}(\hat{I}+\sum_{i=1}^{3}c_{i}\hat{\sigma}_{i}\otimes\hat{\sigma}_{i})=\sum_{i=1}^{3}\lambda_{i}|\lambda_{i}\rangle\langle\lambda_{i}|,$
(27)

with
$\lambda_{0}=1-c_{1}-c_{2}-c_{3},\lambda_{1}=1-c_{1}+c_{2}+c_{3},\lambda_{2}=1+c_{1}-c_{2}+c_{3}$
and $\lambda_{3}=1+c_{1}+c_{2}-c_{3}$. The reduced density matrices are
maximally mixed states and cannot be transformed by unitary operations, so the
local extractable work of $\hat{\rho}_{SA}$ will be zero $W_{loc}=0$. The
conditional density matrices of the system given the projective measurement
operators $\hat{\Pi}_{i}(i=0,1)$ on the ancillary system can be obtained as

$\displaystyle\hat{\rho}_{S|1}=\frac{1}{2}[\hat{I}+c_{3}\hat{\sigma}_{3}]=\frac{1}{2}[(1+c_{3})|1\rangle\langle
1|+(1-c_{3})|0\rangle\langle 0|]\text{ with}\hskip
2.84526ptp_{1}=\frac{1}{2},$ (28)
$\displaystyle\hat{\rho}_{S|0}=\frac{1}{2}[\hat{I}-c_{3}\hat{\sigma}_{3}]=\frac{1}{2}[(1-c_{3})|1\rangle\langle
1|+(1+c_{3})|0\rangle\langle 0|]\text{ with}\hskip
2.84526ptp_{0}=\frac{1}{2}.$ (29)

So the daemonic ergotropy of the system $S$ is given by

$\displaystyle
W_{\\{\hat{\Pi}^{A}_{a}\\}}=|c_{3}|\frac{\Delta\varepsilon}{2}.$ (30)

On the other hand, the total extractable work is obtained by the ergotropic
transformation $\hat{U}_{SA}$ on the state $\hat{\rho}_{SA}$. Depending on
which $|c_{i}|$ is greatest than the others, the corresponding cyclic unitary
transformation is required to extract work from the joint system state. The
ergotropy of the total system is given by

$\displaystyle W_{tot}=\begin{cases}|c_{1}|\Delta\varepsilon/2&\text{
with}\hskip
8.53581pt\hat{U}_{1}=|11\rangle\langle\lambda_{0}|+|10\rangle\langle\lambda_{1}|+|01\rangle\langle\lambda_{2}|+|00\rangle\langle\lambda_{3}|,\\\
|c_{2}|\Delta\varepsilon/2&\text{ with}\hskip
8.53581pt\hat{U}_{2}=|11\rangle\langle\lambda_{0}|+|10\rangle\langle\lambda_{2}|+|01\rangle\langle\lambda_{1}|+|00\rangle\langle\lambda_{3}|,\\\
|c_{3}|\Delta\varepsilon/2&\text{ with}\hskip
8.53581pt\hat{U}_{3}=|11\rangle\langle\lambda_{0}|+|10\rangle\langle\lambda_{3}|+|01\rangle\langle\lambda_{1}|+|00\rangle\langle\lambda_{2}|,\par\end{cases}$
(31)

which is greater than or equal to the daemonic ergotropy
$W_{\\{\hat{\Pi}^{A}_{a}\\}}$. Now, we perform the projective measurement on
the total system by a set of $\hat{\Pi}_{ij}$ and calculate the conditional
density matrices as

$\displaystyle\begin{array}[]{c}\hat{\rho}^{SA}_{\hat{\Pi}_{11}}=\frac{Tr(\hat{\Pi}_{11}\hat{\rho}_{SA})}{p_{11}}=|11\rangle\langle
11|\hskip 8.53581pt\text{ with}\hskip 8.53581ptp_{11}=\frac{1}{4}(1+c_{3}),\\\
\hat{\rho}^{SA}_{\hat{\Pi}_{10}}=\frac{Tr(\hat{\Pi}_{10}\hat{\rho}_{SA})}{p_{10}}=|10\rangle\langle
10|\hskip 8.53581pt\text{ with}\hskip 8.53581ptp_{10}=\frac{1}{4}(1-c_{3}),\\\
\hat{\rho}^{SA}_{\hat{\Pi}_{01}}=\frac{Tr(\hat{\Pi}_{01}\hat{\rho}_{SA})}{p_{01}}=|01\rangle\langle
01|\hskip 8.53581pt\text{ with}\hskip 8.53581ptp_{01}=\frac{1}{4}(1-c_{3}),\\\
\hat{\rho}^{SA}_{\hat{\Pi}_{00}}=\frac{Tr(\hat{\Pi}_{00}\hat{\rho}_{SA})}{p_{00}}=|00\rangle\langle
00|\hskip 8.53581pt\text{ with}\hskip 8.53581ptp_{00}=\frac{1}{4}(1+c_{3}).\\\
\end{array}$ (36)

The energy of each conditional state is reduced to $\varepsilon_{0}$ by using
the cyclic unitary operators, so the average ergotropy of all the conditional
states reads as follows

$\displaystyle
W^{\\{\hat{\Pi}_{ij}\\}}_{tot}=Tr[\hat{\rho}_{SA}\hat{H}_{SA}]-\sum_{i,j}p_{ij}Tr[\hat{U_{ij}}\hat{\rho}_{SA|ij}\hat{U}^{\dagger}_{ij}\hat{H}_{SA}]=\Delta\varepsilon/2,$
(37)

which is greater than or equal to the total extractable work of the joint
system state. It means we can extract more work by performing the projective
measurement on the state $\hat{\rho}_{SA}$ than both the total extractable
work and the daemonic ergotropy. On the other hand, since $W_{loc}=0$
therefore $W^{\\{\hat{\Pi}_{ij}\\}}_{Non-loc}=W^{\\{\hat{\Pi}_{ij}\\}}_{tot}$
. Thus for the post-measurement state
$\hat{\rho}_{SA}=\sum_{i,j}p_{ij}\hat{\rho}_{SA|ij}$, the non-local
extractable work is maximized, however, the quantum correlation is completely
lost. Before such measurement, by using a measure which is introduced in Ref.
[35], one can get the quantum correlation as

$\displaystyle
C(\hat{\rho})=\sqrt{2-\sqrt{4-[c^{2}_{1}c^{2}_{2}+c^{2}_{1}c^{2}_{3}+c^{2}_{2}c^{2}_{3}}}].$
(38)

As we see from Eq. (34), the quantum correlation depends on all $c_{i}$s
$(i=1,2,3)$, however non-local extractable work $W_{non-loc}$ is only
proportional to one of them. In general, there is not an explicit relationship
between these two quantities. The Fig. 2 shows the quantum correlation
$C(\hat{\rho})$ as the function of $c_{1}/c_{2}$ and $c_{3}/c_{2}$. As one can
see this quantity is maximal when all $c_{i}$s tend to $1$.

###  1\. Example

As an example of Bell-diagonal states, we ploted
$W^{\\{\hat{\Pi}_{ij}\\}}_{Non-loc}$ and $W_{Non-loc}$ in Fig. 3 for
$c_{1}=\frac{1}{2}$, $c_{2}=-\frac{1}{2}$ and $c_{3}=\mathrm{sin}(\theta)$.
For post measurement classical correlated state
$\hat{\rho}_{SA}=\frac{1}{4}\sum_{j,k}(1+(-1)^{j+k}c_{3})|jk\rangle\langle
jk|(j,k=0,1)$, the Fig. 3 shows that $W^{\\{\hat{\Pi}_{ij}\\}}_{Non-loc}$ is
greater than $W_{Non-loc}$ except in $\theta=\frac{\pi}{2}$ which are equal.
In Fig. 4, we ploted the quantum correlation in terms of $\theta$. It shows
the correlation of the initial state before the work extraction process which
is calculated as $\frac{1}{2}\sqrt{8-\sqrt{63-8sin^{2}(\theta)}}$. As one can
see the quantum correlation is maximized at $\theta=\frac{\pi}{2}$, which
specifies the extremum point for two types of non-local extractable work in
Fig. 3.

##  VI. Conclusions

In this paper, we studied the concept of gain in work extraction in closed
quantum systems in the presence of quantum correlation through the performance
of the weak measurement. We considered a bipartite quantum system and showed
that the extractable work induced by the weak measurement on the ancillary
subsystem was always equal to the daemonic ergotropy captured by the
projective measurement independently of the strength of the weak measurement.
As an interesting result, we extracted more work by using the selective weak
measurements on the ancillary system which we called super ergotropy. It
depends on the final internal energy of the conditional principal system
states to specify which of the weak measurement operators to be imposed. We
also provided a geometric confirmation for daemonic ergotropy in the case of
general two-qubit states. In this regard, we defined a non-local extractable
work for bipartite systems which are classically correlated and showed that if
the state of the main subsystem is pure, regardless of whether the general
two-qubit state after the perfect projective measurement on the total system
has a classical correlation or not, the defined classical non-local work is
equal to zero, and if its state is mixed, this work can be non-zero. Moreover,
we investigated total system ergotropy in the case of Bell diagonal states and
showed that in the effect of perfect projective measurement on the total
system, the average ergotropy of conditional states was greater than total
extractable work in the cost of losing quantum correlation between subsystems.
It can be generalized daemonic ergotropy. Our research showed no explicit
relationship between non-local extractable work and the quantum correlation of
subsystems in the pre-measurement state of the last-mentioned total system.

## Data availability

The present study’s data are available from the corresponding author upon a
reasonable request.

## Acknowledgments

The authors would like to thank Abbas Ektesabi for very useful comments and
advice.

## Appendix A

The Hamiltonian of the two-qubit state is

$\displaystyle\hat{H}=\hat{H}_{A}\otimes\hat{I}.$ (39)

So initial energy of the total system is given by

$\displaystyle Tr[\hat{\rho}_{SA}\hat{H}_{SA}]$
$\displaystyle=\frac{1}{4}Tr[(\hat{I}+\sum_{i=1}^{3}s_{i}\hat{\sigma}_{i}\otimes\hat{I}+\sum_{j=1}^{3}r_{j}\hat{I}\otimes\hat{\sigma}_{j}+\sum_{i,j=1}^{3}t_{ij}\hat{\sigma}_{i}\otimes\hat{\sigma}_{j})(\hat{H}_{A}\otimes\hat{I})]$
(40)
$\displaystyle=\frac{1}{2}(\varepsilon_{0}+\varepsilon_{1}+s_{3}\Delta\varepsilon).$

By measuring the two-qubit system via a set of projective operators
$\hat{\Pi}_{ij}(i,j=0,1)$ and calculating the average ergotropy overall
outcomes of measurement can be obtained as

$\displaystyle\begin{array}[]{c}\hat{\rho}^{SA}_{\hat{\Pi}_{11}}=\frac{Tr(\hat{\Pi}_{11}\hat{\rho}_{SA})}{p_{11}}=|11\rangle\langle
11|\hskip 8.53581pt\text{ with}\hskip
8.53581ptp_{11}=\frac{1}{4}(1+s_{3}+r_{3}+t_{33}),\\\
\hat{\rho}^{SA}_{\hat{\Pi}_{10}}=\frac{Tr(\hat{\Pi}_{10}\hat{\rho}_{SA})}{p_{10}}=|10\rangle\langle
10|\hskip 8.53581pt\text{ with}\hskip
8.53581ptp_{10}=\frac{1}{4}(1+s_{3}-r_{3}-t_{33}),\\\
\hat{\rho}^{SA}_{\hat{\Pi}_{01}}=\frac{Tr(\hat{\Pi}_{01}\hat{\rho}_{SA})}{p_{01}}=|01\rangle\langle
01|\hskip 8.53581pt\text{ with}\hskip
8.53581ptp_{01}=\frac{1}{4}(1-s_{3}+r_{3}-t_{33}),\\\
\hat{\rho}^{SA}_{\hat{\Pi}_{00}}=\frac{Tr(\hat{\Pi}_{00}\hat{\rho}_{SA})}{p_{00}}=|00\rangle\langle
00|\hskip 8.53581pt\text{ with}\hskip
8.53581ptp_{00}=\frac{1}{4}(1-s_{3}-r_{3}+t_{33}).\\\ \end{array}$ (45)

The cyclic unitary operators which minimize the energy of the conditional
state are given by

$\displaystyle\begin{array}[]{c}\hat{U}_{11}=|01\rangle\langle
11|+|11\rangle\langle 01|\text{or}\hskip
2.84526pt\hat{U}_{11}=|00\rangle\langle 11|+|11\rangle\langle 00|,\\\
\hat{U}_{10}=|01\rangle\langle 10|+|10\rangle\langle 01|\hskip
2.84526pt\text{or}\hskip 2.84526pt\hat{U}_{10}=|00\rangle\langle
10|+|10\rangle\langle 00|,\\\ \hat{U}_{01}=\hat{I},\\\
\hat{U}_{00}=\hat{I}.\end{array}$ (50)

All the cyclic unitary operators lead the energy of the total system to the
lowest eigenstate of the Hamiltonian. So the total extractable work is

$\displaystyle W^{\\{\hat{\Pi}_{ij}\\}}_{tot}$
$\displaystyle=Tr[\hat{\rho}_{SA}\hat{H}_{SA}]-\sum_{i,j}p_{i,j}Tr[\hat{U_{ij}}\hat{\rho}_{SA|ij}\hat{U}^{\dagger}_{ij}\hat{H}_{SA}]$
(51) $\displaystyle=(1+s_{3})\Delta\varepsilon/2.$

So the classical non-local extractable work abstains as

$\displaystyle W^{\\{\hat{\Pi}_{ij}\\}}_{Non-
loc}=W^{\\{\hat{\Pi}_{ij}\\}}_{tot}-W=(1-s)\Delta\varepsilon/2.$ (52)

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Fig. 1. $\frac{W_{\hat{P}^{A}_{+x}}}{W_{\\{\hat{\Pi}^{A}_{a}\\}}}$ as a
function of measurement strength $x$.

Figure 1:

Fig. 2. The quantum correlation of Bell-diagonal state as the function of
$c_{1}/c_{2}$ and $c_{3}/c_{2}$.

Figure 2:

Fig. 3. The Non-local extractable works $W_{Non-loc}$ (solid line) and
$W^{\\{\hat{\Pi}_{i,j}\\}}_{Non-loc}$ (dashed line) divided by
$\Delta\varepsilon$ of Bell-diagonal state for the case
$c_{1}=\frac{1}{2},c_{2}=-\frac{1}{2}$ and $c_{3}=\mathrm{sin}(\theta)$.

Figure 3:

Fig. 4. The quantum correlation of Bell-diagonal state for the case
$c_{1}=\frac{1}{2},c_{2}=-\frac{1}{2}$ and $c_{3}=\mathrm{sin}(\theta)$.

Figure 4:


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##  Appendix A Appendix

###  A.1 Language Codes

Code | Language | Code | Language
---|---|---|---
af | Afrikaans | ja | Japanese
ar | Arabic | ko | Korean
be | Belarusian | lt | Lithuanian
bg | Bulgarian | lv | Latvian
ca | Catalan | mr | Marathi
cs | Czech | nl | Dutch
da | Danish | no | Norwegian
de | German | pl | Polish
el | Greek | pt | Portuguese
en | English | ro | Romanian
es | Spanish | ru | Russian
et | Estonian | sk | Slovak
fa | Persian | sl | Slovenian
fi | Finnish | sr | Serbian
fr | French | sv | Swedish
ga | Irish | sw | Swahili
gl | Galician | ta | Tamil
he | Hebrew | th | Thai
hi | Hindi | tr | Turkish
hr | Croatian | uk | Ukranian
hu | Hungarian | ur | Urdu
hy | Armenian | vi | Vietnamese
id | Indonesian | wo | Wolof
it | Italian | zh | Chinese
Table 8: ISO 2-Letter language codes of the languages included in the
challenge set

###  A.2 Permitted Unit Conversions

We allow the following unit conversions for the challenge set that covers such
errors:

Distance:

  * •

miles $\rightarrow$ metres

  * •

kilometres $\rightarrow$ miles

  * •

kilometres $\rightarrow$ metres

  * •

metres $\rightarrow$ feet

  * •

metres $\rightarrow$ yards

  * •

feet $\rightarrow$ metres

  * •

feet $\rightarrow$ yards

  * •

centimetres $\rightarrow$ inches

  * •

centimetres $\rightarrow$ millimetres

  * •

inches $\rightarrow$ centimetres

  * •

inches $\rightarrow$ millimetres

  * •

millimetres $\rightarrow$ centimetres

  * •

millimetres $\rightarrow$ inches

  * •

millimetres $\rightarrow$ inches

Speed:

  * •

miles per hour $\rightarrow$ kilometres per hour

  * •

kilometres per hour $\rightarrow$ miles per hour

  * •

kilometres per second $\rightarrow$ miles per second

  * •

miles per second $\rightarrow$ kilometres per second

Time:

  * •

hours $\rightarrow$ minutes

  * •

minutes $\rightarrow$ seconds

  * •

seconds $\rightarrow$ minutes

  * •

days $\rightarrow$ hours

  * •

months $\rightarrow$ weeks

  * •

weeks $\rightarrow$ days

Volume:

  * •

barrels $\rightarrow$ gallons

  * •

barrels $\rightarrow$ litres

  * •

gallons $\rightarrow$ barrels

  * •

gallons $\rightarrow$ litres

Weight:

  * •

kilograms $\rightarrow$ grams

  * •

kilograms $\rightarrow$ pounds

  * •

grams $\rightarrow$ ounces

  * •

ounces $\rightarrow$ grams

Area:

  * •

square kilometres $\rightarrow$ square miles

###  A.3 Zero Shot Performance Scores

| antonym-replacement |  | real-world knowledge
---
-commonsense
nonsense
| WMT |  | Non-
---
WMT
WMT |  | Non-
---
WMT
WMT |  | Non-
---
WMT
BERTScore | -0.376 | -0.408 |  - 0.007 |  - 0.060 |  - 0.790 | -0.678
BLEURT-20 |  - 0.024 | -0.008 |  - 0.396 |  - 0.195 | -0.273 | -0.622
COMET-20 |  - 0.152 |  - 0.104 |  - 0.087 |  - 0.020 |  - 0.706 | -0.315
COMET-QE |  - 0.616 |  - 0.664 |  - 0.168 |  - 0.356 |  - 0.245 |  - 0.538
COMET-22 |  - 0.744 |  - 0.664 |  - 0.584 |  - 0.557 |  - 0.706 |  - 0.175
metricx_xl_DA_2019 |  - 0.728 |  - 0.760 |  - 0.570 |  - 0.624 |  - 0.790 |  - 0.357
metricx_xl_MQM_2020 |  - 0.888 |  - 0.936 |  - 0.517 |  - 0.611 |  - 0.944 |  - 0.762
metricx_xxl_DA_2019 |  - 0.312 |  - 0.296 |  - 0.718 |  - 0.758 |  - 0.706 |  - 0.441
metricx_xxl_MQM_2020 |  - 0.696 |  - 0.632 |  - 0.691 |  - 0.758 |  - 0.930 |  - 0.734
UniTE-ref |  - 0.664 |  - 0.696 |  - 0.409 |  - 0.396 |  - 0.091 | -0.147
UniTE |  - 0.632 |  - 0.552 |  - 0.409 |  - 0.409 |  - 0.441 | -0.203
COMETKiwi |  - 0.744 |  - 0.696 |  - 0.745 |  - 0.772 |  - 0.510 |  - 0.469
Cross-QE |  - 0.680 |  - 0.616 |  - 0.638 |  - 0.450 |  - 0.720 |  - 0.538
HWTSC-Teacher-Sim |  - 0.504 |  - 0.296 |  - 0.248 |  - 0.168 |  - 0.930 |  - 0.580
UniTE-src |  - 0.776 |  - 0.680 |  - 0.651 |  - 0.490 |  - 0.524 |  - 0.552
Table 9: Zero-shot performance of neural metrics on three phenomena to measure
the ability of metrics to generalise to new language pairs. WMT language pairs
consist of a subset of languages seen during training of the metrics, while
non-WMT language pairs are unseen. Results show that the metrics are able to
generalise to unseen languages.

Table 9 contains the Kendall tau-like correlation scores for neural metrics on
WMT language pairs (a subset of those seen during training) and non-WMT
language pairs (unseen), for three phenomena: antonym replacement, real-world
knowledge commonsense, and nonsense. The table contains the complete set of
scores, and complements Table 7, which reports only the difference between the
non-WMT and WMT correlation scores. See Section 8.3.1 on zero-shot
performance. We shall now list the language pairs across the different
phenomena:

  * Antonym Replacement
WMT: de-en
non-WMT: ko-en, es-en

  * Real-world Knowledge - Commonsense
WMT: de-en, ru-en, en-ru, en-de
non-WMT: ru-de, fr-ru, ru-fr, de-ru

  * Nonsense
WMT: de-en
non-WMT: fr-ja, ko-ja, en-ko, ko-en

Note that the subset of examples used in this analysis only consists of
mid/high resource language pairs; investigation into the performance on low-
resource languages is left for future work.

###  A.4 Distribution of Examples Across Language Pairs

|  | tgt $\rightarrow$
---|---|---
|  | af | ar | be | bg | ca | cs | da | de | el | en | es | et | fa | fi | fr | ga | gl | he | hi | hr | hu | hy | id | it | ja | ko | lt | lv | mr | nl | no | pl | pt | ro | ru | sk | sl | sr | sv | sw | ta | th | tr | uk | ur | vi | wo | zh
src | af |  |  |  |  |  |  |  |  |  | 96 |  |  | 25 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
$\downarrow$ | ar |  |  |  |  |  |  |  |  |  | 361 |  |  |  |  | 102 |  |  |  | 17 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| be |  |  |  |  |  |  |  |  |  | 67 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| bg |  |  |  |  |  |  |  |  |  | 393 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 40 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| ca |  |  |  |  |  |  |  |  |  | 79 | 175 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| cs |  |  |  |  |  |  |  |  |  | 85 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| da |  |  |  |  |  |  |  |  |  | 83 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| de |  |  |  |  |  |  |  |  |  | 4163 | 84 |  |  |  | 394 |  |  |  |  |  |  |  |  |  | 113 | 63 |  |  |  |  |  |  |  |  | 104 |  |  |  |  |  |  |  |  |  |  |  |  | 75
| el |  |  |  |  |  |  |  |  |  | 387 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| en | 25 | 5 | 6 | 15 | 347 | 368 | 46 | 6964 | 21 |  | 725 | 25 | 20 | 12 | 800 |  | 16 | 18 | 343 | 27 | 44 | 3 | 31 | 10 | 430 | 545 | 17 | 19 | 52 | 50 | 53 | 349 | 44 | 46 | 698 | 27 | 45 | 15 | 39 |  | 1 |  | 10 | 16 | 10 | 25 |  | 333
| es |  |  |  |  | 88 |  |  | 64 |  | 1263 |  |  |  |  | 125 |  |  |  |  |  |  |  |  |  | 117 | 74 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 67
| et |  |  |  |  |  |  |  |  |  | 70 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| fa | 16 |  |  |  |  |  |  |  |  | 85 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| fi |  |  |  |  |  |  |  |  |  | 79 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| fr |  |  |  |  |  |  |  | 683 |  | 2868 | 78 |  |  |  |  |  |  |  |  |  |  |  |  |  | 403 | 59 |  |  | 344 |  |  |  |  |  | 46 |  |  |  |  |  |  |  |  |  |  |  |  | 61
| ga |  |  |  |  |  |  |  |  |  | 17 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| gl |  |  |  |  |  |  |  |  |  | 70 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| he |  |  |  |  |  |  |  |  |  | 59 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 51 |  |  |  |  |  |  |  |  |
| hi |  | 8 |  |  |  |  |  |  |  | 367 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| hr |  |  |  |  |  |  |  |  |  | 81 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 29 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| hu |  |  |  |  |  |  |  |  |  | 53 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| hy |  |  |  |  |  |  |  |  |  | 48 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 13 |  |
| id |  |  |  |  |  |  |  |  |  | 63 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| it |  |  |  |  |  |  |  |  |  | 801 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| ja |  |  |  |  |  |  |  | 60 |  | 912 | 67 |  |  |  | 122 |  |  |  |  |  |  |  |  |  |  | 163 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 74
| ko |  |  |  |  |  |  |  | 70 |  | 1004 | 72 |  |  |  | 110 |  |  |  |  |  |  |  |  |  | 358 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 73
| lt |  |  |  | 28 |  |  |  |  |  | 68 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| lv |  |  |  |  |  |  |  |  |  | 61 |  |  |  |  |  |  |  |  |  | 24 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| mr |  |  |  |  |  |  |  |  |  | 63 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| nl |  |  |  |  |  |  |  |  |  | 73 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| no |  |  |  |  |  |  |  |  |  | 53 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| pl |  |  |  |  |  |  |  |  |  | 65 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 111 |  |  |  |  |  |  | 58 |  |  |  |  |  |  |  |  |  |  |  |
| pt |  |  |  |  |  |  |  |  |  | 89 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 40 |  |  |  |  |  |  |  |  |  |
| ro |  |  |  |  |  |  |  |  |  | 91 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| ru |  |  |  |  |  |  |  | 106 |  | 472 | 87 |  |  |  | 42 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| sk |  |  |  |  |  |  |  |  |  | 54 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 17 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| sl |  |  |  |  |  |  |  |  |  | 69 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| sr |  |  |  |  |  |  |  |  |  | 64 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 54 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| sv |  |  |  |  |  |  |  |  |  | 79 |  |  |  |  |  |  |  | 28 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| sw |  |  |  |  |  |  |  |  |  | 327 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| ta |  |  |  |  |  |  |  |  |  | 39 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| th |  |  |  |  |  |  |  |  |  | 299 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| tr |  |  |  |  |  |  |  |  |  | 386 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| uk |  |  |  |  |  |  |  |  |  | 77 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| ur |  |  |  |  |  |  |  |  |  | 372 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| vi |  |  |  |  |  |  |  |  |  | 391 |  |  |  |  |  |  |  |  |  |  |  | 3 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| wo |  |  |  |  |  |  |  |  |  | 11 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| zh |  |  |  |  |  |  |  | 150 |  | 1209 | 59 |  |  |  | 113 |  |  |  |  |  |  |  |  |  | 128 | 80 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Table 10: Number of examples per language pair. Rows: source language;
Columns: target language

Table 10 contains the total number of examples per language pair in the
challenge set. As can be seen in the table, the distribution of examples is
variable across language pairs. The dominant language pairs are: en-de, de-en,
and fr-en.

###  A.5 Distribution of Language Pairs Across Phenomena

phenomena |  language pairs |  phenomena |  language pair
---|---|---|---
ambiguous-translation-wrong- discourse-connective-causal hallucination-unit-conversion-unit-matches-ref |  fr-en, de-en |  hallucination-real-data-vs-ref-word |  en-de, de-en, fr-de
ambiguous-translation-wrong-discourse-connective-while-contrast |  fr-en |  hallucination-real-data-vs-synonym |  en-mr, de-en, en-de, fr-de
ambiguous-translation-wrong-discourse-connective-while-temporal |  fr-en |  untranslated-vs-ref-word |  en-de, de-en, fr-de
ambiguous-translation-wrong-gender-female-anti |  fr-en, de-en, it-en |  untranslated-vs-synonym |  en-de, de-en, fr-de
ambiguous-translation-wrong-gender-male-anti |  fr-en, de-en, it-en |  modal_verb:deletion |  de-en
ambiguous-translation-wrong-gender-male-pro |  fr-en, de-en, it-en |  modal_verb:substitution |  de-en
ambiguous-translation-wrong-sense-frequent |  en-de, en-ru |  nonsense |  ko-en, ko-ja, en-ko, fr-ja, de-en
ambiguous-translation-wrong-sense-infrequent |  en-de, en-ru |  ordering-mismatch |  en-de, de-en, fr-de
anaphoric_group_it-they:deletion |  en-de |  overly-literal-vs-correct-idiom |  en-de, de-en
anaphoric_group_it-they:substitution |  en-de |  overly-literal-vs-explanation |  en-de, de-en
anaphoric_intra_non-subject_it:deletion |  en-de |  overly-literal-vs-ref-word |  en-de, de-en, fr-de
anaphoric_intra_non-subject_it:substitution |  en-de |  overly-literal-vs-synonym |  en-mr, de-en, en-de, fr-de
anaphoric_intra_subject_it:deletion |  en-de |  pleonastic_it:deletion |  en-de
anaphoric_intra_subject_it:substitution |  en-de |  pleonastic_it:substitution_pro_trans_different_to_ref |  en-de
anaphoric_intra_they:deletion |  en-de |  punctuation:deletion_all |  en-de
anaphoric_intra_they:substitution |  en-de |  punctuation:deletion_commas |  en-de
anaphoric_singular_they:deletion |  en-de |  punctuation:deletion_quotes |  en-de
anaphoric_singular_they:substitution |  en-de |  punctuation:statement-to-question |  en-de
antonym-replacement |  fr-en, ko-en, ja-en, es-en, zh-en, de-en |  real-world-knowledge-entailment |  en-de, de-en
similar-language-high |  en-hi, en-cs, en-es |  real-world-knowledge-hypernym-vs-distractor |  en-de, de-en
similar-language-low |  fr-mr, en-pl, en-ca |  real-world-knowledge-hypernym-vs-hyponym |  en-de, de-en
coreference-based- on-commonsense |  en-de, en-ru, en-fr |  real-world-knowledge-synonym-vs-antonym |  en-de, de-en
hallucination-named-entity-level-1 hallucination-named-entity-level-2 hallucination-named-entity-level-3 hallucination-number-level-1 hallucination-number-level-2 hallucination-number-level-3 |  en-de, ja-de, en-ko, de-zh, ja-en, es-de, fr-en, es-ko, ko-ja, es-ja, de-ja, zh-es, fr-zh, fr-ja, es-en, fr-ko, zh-en, ko-de, ko-es, de-ko, ko-en, fr-es, ja-es, ja-ko, zh-fr, en-es, de-en, ja-fr, ko-zh, en-fr, de-fr, ko-fr, es-fr, zh-ko, fr-de, ja-zh, de-es, es-zh, en-ja, zh-de, en-zh, zh-ja |  hyponym-replacement hypernym-replacement |  fr-en, ko-en, ja-en, es-en, zh-en, de-en
lexical-overlap |  fr-en, en-fr, de-fr, ko-en, es-ja, ja-en, ko-fr, es-fr, ko-ja, de-ja, zh-en, ja-fr, zh-fr, en-ja, es-en, fr-ja, de-en, zh-ja |  xnli-addition-contradiction xnli-addition-neutral xnli-omission-contradiction xnli-omission-neutral |  fr-en, vi-en, sw-en, tr-en, zh-en, ru-en, bg-en, el-en, th-en, es-en, hi-en, de-en, ar-en, ur-en
hallucination-unit-conversion-amount-matches-ref hallucination-unit-conversion-unit-matches-ref |  et-en, wo-en, da-en, no-en, uk-en, ta-en, fi-en, pl-en, ja-en, hy-en, ur-en, hr-en, fr-en, lt-en, tr-en, he-en, bg-en, ro-en, sv-en, ru-en, es-en, nl-en, zh-en, hu-en, be-en, lv-en, ko-en, ga-en, sk-en, af-en, sl-en, sr-en, ca-en, de-en, mr-en, id-en, vi-en, gl-en, pt-en, fa-en, hi-en, el-en, ar-en, it-en, cs-en |  hallucination-date-time |  en-de, et-en, ca-es, en-et, hr-lv, da-en, no-en, uk-en, fi-en, en-da, ta-en, pl-en, ja-en, en-hr, hy-en, ur-en, fr-en, hr-en, lt-en, sr-pt, en-sv, tr-en, en-no, en-sl, he-en, pl-sk, ru-en, ro-en, sv-en, en-lt, es-en, en-nl, nl-en, bg-en, he-sv, zh-en, hu-en, be-en, lv-hr, lv-en, bg-lt, en-ro, sk-pl, ko-en, ga-en, sk-en, af-en, sl-en, en-hu, sr-en, en-es, ca-en, en-sk, de-en, mr-en, id-en, vi-en, gl-en, en-fr, de-fr, pt-en, fr-de, en-pt, fa-en, hi-en, el-en, ar-en, it-en, en-pl, cs-en
commonsense-only-ref-ambiguous commonsense-src-and-ref-ambiguous |  en-de, fr-en, ru-fr, en-fr, de-fr, ru-de, fr-de, ru-en, en-ru, fr-ru, de-ru, de-en |  copy-source |  ar-fr, ru-es, ur-en, fr-en, tr-en, zh-de, bg-en, ru-en, es-en, zh-en, sw-en, ja-ko, th-en, de-en, pl-mr, vi-en, hi-en, el-en, ar-en
addition omission |  en-ca, en-el, en-et, en-ta, pl-en, hr-en, he-en, pl-sk, en-ar, ru-en, en-fi, zh-en, hu-en, be-en, lv-hr, en-he, ko-en, en-fa, sl-en, ca-en, en-gl, en-tr, en-sk, de-en, en-sr, fa-af, fa-en, ar-en, cs-en, en-de, en-hy, ar-hi, no-en, uk-en, fi-en, en-be, sr-pt, en-ru, sv-en, nl-en, sk-pl, en-hi, en-hu, mr-en, hi-ar, id-en, gl-en, en-fr, en-lv, fr-de, ca-es, en-uk, |  addition omission |  en-ur, en-hr, ur-en, en-no, en-sl, ro-en, en-vi, en-lt, es-en, en-nl, he-sv, en-it, en-ro, af-fa, en-id, lt-bg, en-af, af-en, es-ca, vi-en, sv-he, de-fr, pt-en, en-pl, et-en, hr-lv, wo-en, da-en, en-ko, en-da, ja-en, hy-en, pt-sr, hy-vi, fr-en, en-cs, lt-en, en-sv, tr-en, bg-en, lv-en, bg-lt, sr-en, en-es, en-bg, en-pt, hi-en, el-en, it-en
Table 11: Collection of list of languages per phenomena

Table 11 contains the list of language pairs per phenomena in the challenge
set. As can be seen in the table, the distribution of language pairs is
variable across phenomena. Addition and omission have the highest variety of
language pairs. en-de is the most frequent language pair across all phenomena.