# Nonlocal Locking of Observable Quantities: A Faithful Signature of
Nonclassical Correlations

Mir Alimuddin  ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute
of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels
(Barcelona), Spain.  Snehasish Roy Chowdhury  Physics and Applied Mathematics
Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700108  Ram Krishna
Patra  Department of Physics of Complex Systems, S. N. Bose National Center
for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700106, India.
Subhendu B. Ghosh  Physics and Applied Mathematics Unit, Indian Statistical
Institute, 203 B.T. Road, Kolkata 700108  Tommaso Tufarelli  School of
Mathematical Sciences, University of Nottingham, University Park, Nottingham,
NG7 2RD, United Kingdom.  Gerardo Adesso  School of Mathematical Sciences,
University of Nottingham, University Park, Nottingham, NG7 2RD, United
Kingdom.  Manik Banik  Department of Physics of Complex Systems, S. N. Bose
National Center for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata
700106, India.

###### Abstract

Nonclassicality in composite quantum systems depicts several puzzling
manifestations, with Einstein-Podolsky-Rosen entanglement, Schrödinger
steering, and Bell nonlocality being the most celebrated ones. In addition to
those, an unentangled quantum state can also exhibit nonclassicality, as
evidenced from notions such as quantum discord and work deficit. Here, we
propose a general framework to investigate nonclassical correlations in
multipartite quantum states. The distinct signatures left on observable
quantities, depending on whether the sub-parts of a composite system are
probed separately or jointly, provide an operational avenue to construct
different quantifiers that faithfully capture signatures of nonclassicality in
quantum states. Along the line we unveil an intriguing phenomenon referred to
as ‘nonlocal locking of observable quantities’, where the value of an
observable quantity gets locked in the correlation of a nonclassical state.
Our approach reduces the experimental demand for verification of
nonclassicality in composite systems and can find applications for enhaced
energy storage in quantum thermodynamical devices.

Introduction.– Correlations stand at the forefront of scientific inquiry
across various fields of study. Their systematic investigation plays a crucial
role in enhancing our understanding of the physical world [1]. This is
especially evident while studying quantum systems consisting of multiple sub-
parts. For instance, the puzzling features of quantum correlations as
exemplified by the phenomena of entanglement and steering have been recognized
since the early days of quantum theory [2, 3, 4, *Schrdinger1936]. Later J. S.
Bell, in his seminal work, built on the notion of quantum entanglement to
establish the physical phenomenon of quantum nonlocality [6, *Bell1966] (see
also [8]). Along with capturing fundamentally distinct notions of
nonclassicality [9, 10, 11], quantum entanglement, steering, and nonlocality
also find a number of utilities across a spectrum of applications at the heart
of modern quantum technologies [12, 13, 14, 15]. More recently, it has been
identified that quantum discord encapsulates nonclassicality even in an
unentangled quantum state [16, 17, 18] (see also [19, 20]). The study of
nonclassical correlations, with an emphasis on their identification,
quantification, and classification, thus stands as a pivotal objective in
quantum information research during the last few decades.

In this paper, we adopt an operationally-driven approach to explore
nonclassical correlations in multipartite quantum systems. A multipartite
quantum system can be probed in different ways depending on the availability
of resources among the spatially separated sub-parts. For instance, global
probing requires quantum communication channels among the sub-parts, and
without such channels only local probing is possible, where distant parties
address their respective sub-parts individually. On the other hand, allowing
only classical communication among the sub-parts confines a probing mechanism
intermediate to the global and local probings (see Fig. 1). A correlated
quantum state can lead to distinct observable values depending on how the
system is probed and this observation provides operational pathways for
exploring nonclassical correlations within it. In this work, we pursue this
approach to introduce quantifiers of general nonclassical correlations beyond
entanglement. While some quantifiers are derived from resource theories and
hinge on entropic quantities, the generality of our framework allows for
constructing alternative quantifiers leveraging simple observables on
composite systems, thus mitigating experimental complexity. For bipartite
systems, such quantifiers are proven to be faithful, meaning that they vanish
on and only on states lacking nonclassical correlations. Our analysis unveils
an intriguing phenomenon hereby termed ‘nonlocal locking of observable
quantities,’ wherein observable values get locked within the nonclassical
correlation inherent in multipartite quantum states. Furthermore, we provide
exact quantitative results focusing on the simplest composite quantum system
comprising two qubits. Finally, we discuss on the potential implications of
our findings in designing efficient energy storage devices that utilize
nonclassically correlated states of many-body quantum systems.

Preliminaries.– The state of an individual quantum system ($A$) is described
by a density operator $\rho\in\mathcal{D}(\mathcal{H}_{A})$;
$\mathcal{D}(\mathcal{X})$ be the set of positive trace-one operators acting
on the Hilbert space $\mathcal{X}$. For a finite-dimensional system, which we
assume throughout this work, $\mathcal{H}_{A}$ is isomorphic to some complex
Euclidean space $\mathbb{C}^{d_{A}}$, where $d_{A}$ denotes the dimension of
the Hilbert space. The state of a composite system ${\bf A}\equiv A_{1}\cdots
A_{n}$ is called separable if it is a statistical mixture of independently
prepared constituent sub-parts, i.e., $\rho_{\bf
A}=\sum_{i}p_{i}(\otimes_{j=1}^{n}\psi^{i}_{j})$, where $\vec{p}:=\\{p_{i}\\}$
is a probability vector, and $\psi:=\ket{\psi}\bra{\psi}$ with
$\ket{\psi^{i}}_{j}\in\mathbb{C}^{d_{j}}$. States that are not separable are
called entangled, and they capture the most celebrated aspect of nonclassical
correlation in composite quantum systems [12, 13]. Importantly, a separable
state can also accommodate another distinct nonclassical aspect of correlation
recognized as quantum discord [16, 17, 18]. A multipartite state is called
classically correlated if it allows a decomposition of the form $\rho_{\bf
A}=\sum p_{i_{1}\cdots
i_{n}}\psi^{i_{1}}_{1}\otimes\cdots\otimes\psi^{i_{n}}_{n}$, where
$\\{\ket{\psi^{i_{j}}}_{j}\\}$ forms an orthonormal basis (ONB) for the
$j^{th}$ Hilbert space, and $\\{p_{i_{1}\cdots i_{n}}\\}$ denotes a
probability vector. A function
$\mathbb{F}:\mathcal{D}(\otimes_{j=1}^{n}\mathbb{C}^{d_{j}}_{A_{j}})\mapsto\mathbb{R}$
is said to capture the signature of nonclassical correlation faithfully if it
takes value zero for all classically correlated states and nonzero otherwise.
For the bipartite case, the original entropic measure of quantum discord,
defined in terms of the difference between two different expressions for the
mutual information [16, 17, 18], is an example of a faithful quantifier [20].
The notion of quantum discord has been generalized to quantum systems
involving more than two sub-parts, albeit with caveats in its operational
interpretation [21, 22, 23]. Notably, this notion of nonclassicality is party
asymmetric, as a state can be a statistical mixture of product states where
$\\{\ket{\psi^{i_{j}}}\\}$ form ONBs only for some sub-parts, but not for all.
In the following, we develop a general operational framework for studying
nonclassical correlations in multipartite quantum states.

Framework.– Any physical operation acting on a quantum system is described by
a completely positive trace-preserving (CPTP) map [24]. Given a quantum system
$A$, we start by defining a generic function
$f_{A}:\mathcal{D}(\mathbb{C}^{d_{A}})\mapsto\mathbb{R}$. For a CPTP map
$\phi_{A}:\mathcal{D}(\mathbb{C}^{d_{A}})\mapsto\mathcal{D}(\mathbb{C}^{d_{A}})$,
we define the variation of the function $f_{A}$ on some state $\rho_{A}$ as
$\Delta f_{A}^{\phi_{A}}(\rho_{A}):=f_{A}(\rho_{A})-f_{A}(\phi(\rho_{A}))$.
For a set of CPTP maps $\Phi_{A}$, the variation is given by $\Delta
f_{A}^{\Phi_{A}}(\rho_{A}):=\max_{\phi_{A}\in\Phi_{A}}\Delta
f_{A}^{\phi_{A}}(\rho_{A})$, which quantifies the maximum extracted value of
the function $f_{A}$ on the given state $\rho_{A}$ under the set of operations
$\Phi_{A}$. For a multipartite quantum system ${\bf A}(=A_{1}\cdots A_{n})$,
fixing $(f_{j},\Phi_{j})$ pair for the $j^{th}$ party
$j\in\\{1,\cdots,n\\}\equiv[n]$, and fixing $(f_{\bf A},\Phi_{\bf A})$ pair
for the joint system, we define another function $\mathbb{F}_{GL}(\rho_{\bf
A}):=\Delta f^{\Phi_{\bf A}}_{\bf A}(\rho_{\bf A})-\sum_{j}\Delta
f^{\Phi_{j}}_{j}(\rho_{j})$, where
$\rho_{j}:=\operatorname{Tr}_{A_{j}^{c}}(\rho_{\bf A})$ denotes the marginal
state of the $j^{th}$ sub-part. The function $\mathbb{F}_{GL}$ quantifies the
difference between the extracted value of the global function and that of the
local functions evaluated on the global state and the local sub-parts under
the fixed sets of operations. Interestingly, as we will see, for suitable
choices of $(f_{\gamma},\Phi_{\gamma})$’s, with $\gamma\in\\{[n]\cup{\bf
A}\\}$, the function $\mathbb{F}_{GL}$ can capture the signature of different
forms of correlation within the multipartite state. As an example, for
bipartite systems, the following observation is noteworthy (see Appendix-I for
more details).

Figure 1: Resource Extraction from a bipartite quantum state $\rho_{\bf
A}\in\mathcal{D}(\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}})$; here ${\bf
A}=A_{1}A_{2}$. (a) $\Delta f^{\Phi_{12}}_{12}(\rho_{\bf A})$ quantifies the
amount of resource extracted under the set of global operations $\Phi_{12}$.
(b) $\sum_{j}\Delta f^{\Phi_{j}}_{j}(\rho_{j})$ is the amount of resource
extracted under local operations $(\Phi_{1},\Phi_{2})$. (c) $\Delta
f^{\\{\Phi_{\gamma}\\}}_{\mathcal{N}^{0}_{C}}(\rho_{\bf A})$ is the amount of
resource extracted under $(\Phi_{1},\Phi_{2},\Phi_{12})$ when from the sub-
part $A_{1}$ to $A_{2}$ free classical channel $\mathcal{N}^{0}_{C}$ are
available. The quantity $\mathbb{F}_{G\mathcal{N}^{0}_{C}}(\rho_{\bf
A}):=\Delta f^{\Phi_{12}}_{12}(\rho_{\bf A})-\Delta
f^{\\{\Phi_{\gamma}\\}}_{\mathcal{N}^{0}_{C}}(\rho_{\bf A})$ amounts to
’nonlocal locking of the resource’, and can capture the signature of
nonclassical correlation in state $\rho_{\bf A}$.

######  Observation 1.

The choices
$(f_{\gamma},\Phi_{\gamma})\equiv(f_{\gamma}^{pe},\Phi_{\gamma}^{no}),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ \gamma\in\\{[2]\cup{\bf A}\\}$ lead to
$\mathbb{F}^{pe}_{GL}(\rho_{\bf A})=S(\rho_{1})+S(\rho_{2})-S(\rho_{\bf
A}):=I(A_{1}:A_{2})$; where
$S(\rho_{\gamma}):=-\mathrm{tr}(\rho_{\gamma}\log\rho_{\gamma})$,
$f^{pe}_{\gamma}(\rho_{\gamma}):=\log d_{\gamma}-S(\rho_{\gamma})$, and
$\Phi^{no}_{\gamma}$ is the set of noisy operations.

Notably, the function $f^{pe}_{\gamma}(\rho_{\gamma})$ denotes the asymptotic
rate of purity extraction from the state $\rho_{\gamma}$ under noisy
operations [25]. Accordingly, $\mathbb{F}^{pe}_{GL}$ in Observation 1,
quantifies the difference between asymptotic rates of purity extraction under
global and local noisy operations [26], which interestingly finds an exact
match with the total correlation of the state $\rho_{\bf A}$, quantified
through the quantum mutual information $I(A_{1}:A_{2})$ [27].

At this point it is worth noting how the function $\mathbb{F}_{GL}$ be
evaluated operationally for a multipartite state (see Fig. 1). Firstly, each
of the local parties can evaluate the corresponding $\Delta f^{\Phi_{j}}_{j}$
locally. On the other hand, for evaluating $\Delta f^{\Phi_{\bf A}}_{\bf A}$,
the spatially separated parties can send their respective sub-parts to a
common party (say $j^{\star}$). This however demands a perfect quantum
communication channel from all the other parties to the designated party
$j^{\star}$. At this point, one can consider intermediate scenarios where only
a subset of channels $\mathcal{N}$ are available from the remaining parties to
$j^{\star}$. In such cases one can define the variation $\Delta
f^{\\{\Phi_{\gamma}\\}}_{\mathcal{N}}(\rho_{\bf
A}):=\max_{\\{\phi_{\gamma}\in\Phi_{\gamma},\leavevmode\nobreak\
N\in\mathcal{N}\\}}[\sum_{j\in[n]}\Delta f^{\phi_{j}}_{j}(\rho_{j})+\Delta
f^{\phi_{\bf A}}_{\bf A}(N(\sigma_{\bf A}))-\mathcal{C}(N)]$, where
$\sigma_{\bf A}:=(\otimes_{j}\phi_{j})(\rho_{\bf A})$, $N(\sigma_{\bf A})$ is
the composite state received by $j^{\star}$, and $\mathcal{C}(N)$ is a
suitably defined cost function. This function quantifies the value extraction
from a given $\rho_{\bf A}$ over the maps $\\{\Phi_{\bf A}\\}$’s when
collaboration from the remaining parties to $j^{\star}$ is limited by the set
of maps $\mathcal{N}$. To say more elaborately, each of the parties extracts
$\Delta f^{\phi_{j}}_{j}$ from their respective parts under their local
operations resulting the state $\rho_{\bf A}$ to evolve into a new state
$\sigma_{\bf A}$. Accordingly, the $j^{\star}$ party receives the state
$N(\sigma_{\bf A})$ and extracts $\Delta f^{\phi_{\bf A}}_{\bf A}$. Clearly,
in absence of $\mathcal{N}$, the value of $\Delta
f^{\\{\Phi_{\gamma}\\}}_{\mathcal{N}}$ should exactly match with
$(\sum_{j\in[n]}\Delta f^{\Phi_{j}}_{j})$, whereas with perfect quantum
communication channel it should match with $\Delta f^{\Phi_{\bf A}}_{\bf A}$.
The cost function $\mathcal{C}(N)$ is defined accordingly to satisfy these
demands [28, 29]. At this point, we define a new function
$\mathbb{F}_{G\mathcal{N}}(\rho_{\bf A}):=\Delta f^{\Phi_{\bf A}}_{\bf
A}(\rho_{\bf A})-\Delta f^{\\{\Phi_{\gamma}\\}}_{\mathcal{N}}(\rho_{\bf A})$
quantifying the difference between optimal global extracted value and the
$\mathcal{N}$ channel assisted optimal extracted value, which will be shown to
be useful to explore nonclassical correlation in multipartite states.

Resource theoretic quantifiers.– To see application of the aforesaid general
framework we start by analyzing how the function $\mathbb{F}_{G\mathcal{N}}$
can capture the notion of quantum discord in bipartite systems (${\bf
A}=A_{1}A_{2}$) for suitable choices of the tuple
$(f_{\gamma},\Phi_{\gamma},\mathcal{N})$. For that once again we fix
$(f_{\gamma},\Phi_{\gamma})\equiv(f^{pe}_{\gamma},\Phi_{\gamma}^{no}),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ \gamma\in\\{1,2,{\bf A}\\}$, and fix $\mathcal{N}$
to be classical communication channel between $A_{1}$ and $A_{2}$. Recall that
quantum discord is a party asymmetric concept: a bipartite state $\rho_{\bf
A}\in\mathcal{D}(\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}})$ having the
classical-quantum (CQ) form $\rho_{\bf A}=\sum
p_{i_{1}i_{2}}\psi^{i_{1}}_{1}\otimes\psi^{i_{2}}_{2}$, with
$\langle\psi^{i_{1}}_{1}|\psi^{i^{\prime}_{1}}_{1}\rangle=\delta_{i_{i}i^{\prime}_{1}}$,
has discord zero from $A_{1}\to A_{2}$ [16, 17, 18]. A generic classical
communication channel $N\in\mathcal{N}_{C}$ sending the $A_{1}$ part to the
$A_{2}$ can be thought as a measure-and-prepare channel [30, 31], with
preparations being restricted to rank-1 orthogonal states (see Appendix III).
Such a channel could map completely mixed state to a resourceful state, and
therefore is costly from the perspective of purity resource theory. One can
bypass this cost association by further limiting these channels to be cost
free. As we argue in Appendix III, a cost-free classical channel in this case
takes the form
$N(\rho_{A_{1}}):=\sum_{l\in[d_{1}]}\operatorname{Tr}(\pi_{l}\rho_{1})\ket{\alpha^{l}_{1}}\bra{\alpha^{l}_{1}}$,
where
$\langle\alpha^{l}_{1}|\alpha^{l^{\prime}}_{1}\rangle=\delta_{ll^{\prime}}$,
and $\\{\pi_{l}\\}$ is a positive-operator-value-measurement (POVM) with
$\operatorname{Tr}(\pi_{l})=1,\leavevmode\nobreak\ \forall\leavevmode\nobreak\
l\in[d_{1}]$. By denoting the set of all such zero cost classical channels as
$\mathcal{N}^{0}_{C}$, we obtain the following result.

######  Theorem 1.

For
$(f_{\gamma},\Phi_{\gamma},\mathcal{N})\equiv(f_{\gamma}^{pe},\Phi_{\gamma}^{no},\mathcal{N}^{0}_{C})$
the function $\mathbb{F}^{pe}_{G\mathcal{N}^{0}_{C}}$ faithfully captures
nonclassical correlation from $A_{1}\to A_{2}$ in any bipartite state
$\rho_{\bf A}\in\mathcal{D}(\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}})$
having maximally mixed marginal on $A_{1}$ sub-part.

Proof is provided in Appendix III. During the process we also prove the
following Proposition which is interesting on its own right (see Appendix II).

######  Proposition 1.

For $\Lambda:\mathcal{D}(\mathbb{C}^{d})\to\mathcal{D}(\mathbb{C}^{d})$ being
a CPTP unital map, the states $\rho$ and $\Lambda(\rho)$ have the same
spectrum whenever $S(\Lambda(\rho))=S(\rho)$.

At this point we would like to point out that a special case of our Theorem 1
reproduces one of Zurek’s results [32]. While the set of free channels there
consists of measure-and-orthogonal preparation with the measurement being
rank-1 projective measurements, in our case we consider more general kind of
measurements. Notably, similar analysis as of Theorem 1 is worth exploring by
fixing
$(f_{\gamma},\Phi_{\gamma},\mathcal{N})\equiv(f_{\gamma}^{fe},\Phi_{\gamma}^{to},\mathcal{N}_{tc})$,
where $f_{\gamma}^{fe}$and $\Phi_{\gamma}^{to}$ respectively denote
thermodynamic free energy and thermal operations for a given inverse
temperature $\beta$, and $\mathcal{N}_{tc}$ are thermal classical channels
compatible to this thermodynamic resource theory. The resulting study will go
in line with the works in Refs. [33, 34, 35, 36].

Generic nonclassicality quantifiers.– We now proceed to explore the generality
of our framework to construct quantifiers for nonclassical correlations that
need not to be motivated from a resource theory. To this aim, we start by
noting that the function $f:\mathcal{D}(\mathbb{C}^{d})\mapsto\mathbb{R}$ can
simply be the expectation value of some observable
$\mathcal{O}\in\mbox{Herm}(\mathbb{C}^{d})$, i.e.,
$\mathcal{O}:\rho\mapsto\langle\mathcal{O}\rangle_{\rho}:=\operatorname{Tr}(\mathcal{O}\rho)$.
Unlike the earlier case, here the variation is evaluated over the set of
reversible operations, resulting $\Phi$ to be the set of unitary operations
$\mathrm{U}$. In this case the variation
$\Delta\mathcal{O}^{\mathrm{U}}(\rho):=\max_{u\in\mathrm{U}}[\operatorname{Tr}(\mathcal{O}\rho)-\operatorname{Tr}(\mathcal{O}u\rho
u^{\dagger})]$ quantifies the optimal value of the observable $\mathcal{O}$
extracted from the state $\rho$ when the system is evolved under closed system
dynamics. While studying multipartite systems, generality of our framework
allows choosing different observables for different sub-parts. Although the
choice of the global observable can also be arbitrary, here we consider
$\mathcal{O}_{\bf
A}\equiv\sum_{j\in[n]}\tilde{\mathcal{O}}_{j}\leavevmode\nobreak\ $, where
$\tilde{\mathcal{O}}_{j}:=\mathcal{O}_{j}\otimes\mathbf{I}_{rest}$. If
system’s Hamiltonian $h$ is taken to be the observable, the variation $\Delta
h^{\mathrm{U}}(\rho)$, known as the ergotropy of the state, which has a long
history in literature [37, 38, 39]. The corresponding function
$\mathbb{H}_{GL}(\rho_{\bf A})$ quantifies the ergotropic gap – the difference
between the extractable work from a state $\rho_{\bf A}$ under global and
local unitary operations, respectively. Interestingly, this function and its
variants can capture different forms of nonclassical correlations,
particularly entanglement in bipartite states and genuine entanglement in
multipartite quantum states [40, 41, 42, 43, 44].

Figure 2: Nonlocal energy locking in two qubit Werner states
$\mathcal{W}^{\alpha}_{\bf A}$. Red dashed curve denotes the original quantum
discord $\mathcal{Q}(\mathcal{W}^{\alpha}_{\bf
A})=\frac{1-\alpha}{4}\log(1-\alpha)-\frac{1+\alpha}{2}\log(1+\alpha)+\frac{1+3\alpha}{4}\log(1+3\alpha)$
establishing the states are nonclassical for $\alpha\in(0,1]$ [17] (see also
[45]). Blue curve amounts to energy
$\mathbb{H}_{G\mathcal{N}^{0}_{C}}(\mathcal{W}^{\alpha}_{\bf
A})\leavevmode\nobreak\ [=\alpha\epsilon_{2}/2]$ locked into the nonclassical
correlation of the state $\mathcal{W}^{\alpha}_{\bf A}$. Here the plot is
given by fixing $\epsilon_{2}=2$.

While studying the quantity $\mathbb{O}_{G\mathcal{N}^{0}_{C}}$, which is one
of our core interests in the present work, we first need to identify the free
classical channels $\mathcal{N}^{0}_{C}$. As it can be argued a classical
channel $N\in\mathcal{N}_{C}$ communicating $A_{j}$ sub-part to the
$j^{\star}$ party will be considered to be free if
$\operatorname{Tr}(\mathcal{O}_{j}N(\rho_{j}))\leq\operatorname{Tr}(\mathcal{O}_{j}\rho_{j})\leavevmode\nobreak\
\&\leavevmode\nobreak\
\Delta\mathcal{O}^{U_{j}}_{j}N(\rho_{j})\leq\Delta\mathcal{O}^{U_{j}}_{j}(\rho_{j}),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ \rho_{j}\in\mathcal{D}(\mathbb{C}^{d_{j}})$. A
more elaborate justification of this demand as well as a complete
characterization of free classical channels $\mathcal{N}^{0}_{C}$ are provided
in Appendix IV. With this we now present a general result for bipartite
systems (proof is deferred to Appendix IV).

######  Theorem 2.

The function $\mathbb{O}_{G\mathcal{N}^{0}_{C}}(\rho_{\bf A})$ faithfully
captures nonclassical correlation from $A_{1}\to A_{2}$ in any bipartite state
$\rho_{\bf A}$, whenever $\mathcal{O}_{\bf A}$ has non degenerate spectrum.

Notably unlike Theorem 1, the function
$\mathbb{O}_{G\mathcal{N}^{0}_{C}}(\rho_{\bf A})$ faithfully captures
signature of nonclassical correlation in any bipartite state without assuming
any further constraint on the state. Theorem 2 has quite interesting
implication. A non zero value of $\mathbb{O}_{G\mathcal{N}^{0}_{C}}(\rho_{\bf
A})$ implies that extractable value of the observable $\mathcal{O}_{\bf A}$
under global operations is strictly greater than the value extractable under
local operations assisted with classical communication from $A_{1}$ to
$A_{2}$. In other words, the value of the observable $\mathcal{O}_{\bf A}$
gets locked in the nonclassical correlation of the state. This phenomenon we
call the nonlocal locking of observable quantities in nonclassical
correlation. The motivation of this name arises from the pioneering
‘nonlocality without entanglement’ phenomenon, where distinguishability of a
set of multipartite product states under local operation and classical
communication is suboptimal than the global distinguishability [46, 47] (see
also [48, 49, 50, 51]), and the works [52, 53, 54] that report several locking
phenomena utilizing quantum resources). The following crucial property (proof
provided in Appendix V) of the quantity discussed in Theorem 2 is worth
noting.

######  Proposition 2.

$\mathbb{O}_{G\mathcal{N}^{0}_{C}}(\rho_{\bf
A})=\mathbb{O}_{G\mathcal{N}^{0}_{C}}(\sigma_{\bf A})$, where $\sigma_{\bf
A}=u_{1}\otimes u_{2}(\rho_{\bf A})u^{\dagger}_{1}\otimes u^{\dagger}_{2}$ for
arbitrary $u_{i}$’s on the respective parts.

A simplified from of $\mathbb{O}_{G\mathcal{N}^{0}_{C}}(\rho_{\bf A})$ can be
obtained for generic two-qubit state $\rho_{\bf
A}=\frac{1}{4}(\mathbf{I}+\bm{\sigma}^{\intercal}_{1}{\bf r}_{1}+{\bf
r}^{\intercal}_{2}\bm{\sigma}_{2}+\bm{\sigma}^{\intercal}_{1}{\bf
T}\bm{\sigma}_{2})$, which is stated in the following theorem by considering
$\mathbb{O}$ to be the Hamiltonian of the system.

######  Theorem 3.

$\mathbb{H}_{G\mathcal{N}^{0}_{C}}(\rho_{\bf A})=\min_{\bf
m}\sum_{k=1}^{4}E_{k}\left(q_{k}({\bf m})-p_{k}\right)$; where $\\{E_{k}\\}$
are the non-degenerate increasing eigenvalues of $\mathbb{H}$, $\\{p_{k}\\}$
are the decreasing spectral values of $\rho_{\bf A}$, and $\\{q_{k}({\bf
m})\\}$’s take values from the set
$\\{{\footnotesize\frac{1}{4}\left(1+(-1)^{i}{\bf m}^{\intercal}{\bf
r}_{1}+(-1)^{j}\left|{\bf r}^{\intercal}_{2}+(-1)^{i}{\bf m}^{\intercal}{\bf
T}\right|\right)}\\}_{i,j=0}^{1}$ in decreasing order, and ${\bf m}$ is an
arbitrary Bloch vector with $|{\bf m}|=1$.

Proof is provided in Appendix VI. There we also obtain a closed letter
expression of this quantity for the states having completely mixed marginals
which is stated in the following Corollary.

######  Corollary 1.

For a two-qubit state having completely mixed marginals,
$\mathbb{H}_{G\mathcal{N}^{0}_{C}}(\rho_{\bf
A})=\omega(E_{1}+E_{2})/2+(1-\omega)(E_{3}+E_{4})/2-\sum_{k}p_{k}E_{k}$, where
$2\omega:=1+\sqrt{\lambda}$, with $\lambda$ being the maximum eigenvalue of
the matrix ${\bf TT}^{\intercal}$.

The plot depicted in Fig.1 exhibits the phenomena of energy locking in two-
qubit Werner class of states. Notably, this phenomenon promises nontrivial
usages of nonclassical correlation in devising efficient quantum batteries
[55, 56, 57]. In multi-cell quantum batteries, stored energy can be lost due
to various factors such as decoherence, global and local interactions, etc.
For an isolated system the evolution is restricted to unitary processes only.
Additionally, if interactions between individual systems are not allowed,
energy drainage can occur only due to local interactions. Our findings
indicate that energy storage is not possible in classically correlated states
when local interactions and classical channels are available between pairs of
isolated quantum batteries. In contrast, our Theorem 2 shows that for a system
with a non-degenerate Hamiltonian, non-classical correlation in the form of
discord can efficiently store energy in such a scenario.

Discussions.– We start by pointing out the recent work of Ref. [58]. There
also the authors explored the role of nonclassical correlation in ergotropic
work extraction. As established in their Theorem 1, the zero value of the
ergotropic quantity implies the bipartite state to be classically correlated.
But, the other way implication does not hold. Subsequently, this study is
extended by considering generalised measurements [59], albeit the question of
both way implication remains unresolved. In contrast, our Theorem 2 presents
faithful quantifiers of nonclassical correlations for arbitrary bipartite
systems. An interesting question is whether these quantities serve as valid
measures of nonclassicality. Recall that, a correlation measure ($M$) is
expected to fulfill the following three criteria [19]: (i) faithfulness,
meaning $M(\rho_{\bf A})=0\Leftrightarrow\rho_{\bf A}$ is a CQ state, (ii)
unitary invariance, indicating $M((u_{1}\otimes u_{2})(\rho_{\bf
A})(u_{1}^{\dagger}\otimes u_{2}^{\dagger}))=M(\rho_{\bf
A}),\leavevmode\nobreak\ \forall\leavevmode\nobreak\
u_{1}\in\mathrm{U}_{1}\leavevmode\nobreak\ \&\leavevmode\nobreak\
u_{2}\in\mathrm{U}_{2}$, and (iii) monotonicity, where
$M((\mathcal{I}\otimes\Lambda)(\rho_{\bf A}))\leq M(\rho_{\bf A})$ holds for
any CPTP map $\Lambda$ acting on the $A_{2}$ subsystem. While we have
established that the first two properties are satisfied, the last one requires
further study. Another intriguing question, which we leave for further
exploration, is whether these nonclassical quantifiers are independent of the
original quantum discord measure. Two quantifiers $M_{1}$ and $M_{2}$ are
considered independent if there exist nonclassical states $\rho_{\bf A}$ and
$\sigma_{\bf A}$ such that $M_{1}(\rho_{\bf A})>M_{1}(\sigma_{\bf A})$, while
$M_{2}(\rho_{\bf A})<M_{2}(\sigma_{\bf A})$. Furthermore, although our current
investigation primarily focuses on bipartite systems, the generality of our
framework suggests potential for extension to multipartite systems.

###### Acknowledgements.

Acknowledgements: MA acknowledges the funding supported by the European Union
(Project QURES- GA No. 101153001). SRC acknowledges support from University
Grants Commission, India (reference no. 211610113404). MB acknowledges funding
from the National Mission in Interdisciplinary Cyber-Physical systems from the
Department of Science and Technology through the I-HUB Quantum Technology
Foundation (Grant no: I-HUB/PDF/2021-22/008).

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##  I Appendix I: More On Observation 1

A noisy operation
$\phi_{\gamma}^{no}:\mathcal{D}(\mathbb{C}^{d_{\gamma}})\mapsto\mathcal{D}(\mathbb{C}^{d_{\gamma}})$
is defined as

$\displaystyle\phi_{\gamma}^{no}(\rho_{\gamma}):=\operatorname{Tr}_{z}\left[U_{kz}\left(\rho_{\gamma}\otimes\frac{\mathbf{I}_{z}}{d_{z}}\right)U^{\dagger}_{kz}\right].$
(1)

Consider now the function $f_{\gamma}(\rho_{\gamma})\equiv
f_{\gamma}^{pe}(\rho_{\gamma}):=\log d_{\gamma}-S(\rho_{\gamma})$ which
satisfies the following properties [25]:

  * (p1)

$0\leq f^{pe}(\rho_{\gamma})\leq\log d_{\gamma}$ for a state
$\rho_{\gamma}\in\mathcal{D}(\mathbb{C}^{d_{\gamma}})$;

  * (p2)

$f^{pe}$ remains invariant under unitaries, i.e., $f^{pe}\left(u_{\gamma}\rho
u_{\gamma}^{\dagger}\right)=f^{pe}(\rho_{\gamma})$;

  * (p3)

$f^{pe}$ is non-increasing under noisy operations, i.e.,
$f^{pe}\left(\phi_{\gamma}(\rho)\right)\leq
f^{pe}(\rho_{\gamma}),\leavevmode\nobreak\ \forall\leavevmode\nobreak\
\phi_{\gamma}\in\Phi^{no}_{\gamma}$.

The variation of the function $f^{pe}$ over the set of noisy operations
$\Phi^{no}$ thus becomes

$\displaystyle\Delta(f^{pe})_{\gamma}^{\Phi^{no}_{\gamma}}(\rho_{\gamma})$
$\displaystyle:=\underset{\phi_{\gamma}\in\Phi^{no}_{\gamma}}{\mbox{max}}\left[f^{pe}_{\gamma}\left(\rho_{\gamma}\right)-f^{pe}_{\gamma}\left(\phi_{\gamma}\left(\rho_{\gamma}\right)\right)\right]$
$\displaystyle=f^{pe}_{\gamma}(\rho_{\gamma})-\underset{\phi_{\gamma}\in\Phi^{no}_{\gamma}}{\mbox{min}}\left[f^{pe}_{\gamma}\left(\phi_{\gamma}\left(\rho_{\gamma}\right)\right)\right]$
$\displaystyle=f^{pe}_{\gamma}(\rho_{\gamma}).$ (2)

The last line follows from the fact that under noisy operation the minimum
value of $f^{pe}_{\gamma}\left(\phi_{\gamma}\left(\rho_{\gamma}\right)\right)$
can be achieved to zero by evolving the state $\rho_{\gamma}$ to the
completely mixed state $\mathbf{I}_{\gamma}/{d_{\gamma}}$. Now, for the
bipartite case $({\bf A}\equiv A_{1}A_{2})$ we have

$\displaystyle\mathbb{F}_{GL}^{pe}(\rho_{\bf A})$ $\displaystyle=f^{pe}_{\bf
A}\left(\rho_{\bf A}\right)-f^{pe}(\rho_{1})-f^{pe}_{2}(\rho_{2})$
$\displaystyle=\log(d_{1}d_{2})-S(\rho_{\bf A})-\log d_{1}+S(\rho_{1})-\log
d_{2}+S(\rho_{2})$ $\displaystyle=S(\rho_{1})+S(\rho_{2})-S(\rho_{\bf
A})=I(A_{1}:A_{2}).$ (3)

This establishes the claim in Observation 1.

##  II Appendix II:Proof of Proposition 1

###### Proof.

Let $\vec{p}=(p_{1},p_{2},\cdots p_{d})$ and $\vec{q}=(q_{1},q_{2},\cdots
q_{d})$ are two probability vectors with their elements arranged in increasing
order. The vector $\vec{p}$ majorizes $\vec{q}$, denoted as
$\vec{p}\succ\vec{q}$, if and only if [60]

$\displaystyle\sum_{i\in[l]}\left(p_{i}-q_{i}\right)\leq
0,\leavevmode\nobreak\ \forall\leavevmode\nobreak\
l\in[d-1],\leavevmode\nobreak\ \leavevmode\nobreak\ \&\leavevmode\nobreak\
\leavevmode\nobreak\
\sum_{i\in[d]}\left(p_{i}-q_{i}\right)=0\leavevmode\nobreak\ .$ (4)

Consider a density operator $\rho\in\mathcal{D}(\mathbb{C}^{d})$ and a unital
CPTP map
$\Lambda:\mathcal{D}(\mathbb{C}^{d})\mapsto\mathcal{D}(\mathbb{C}^{d})$, with
$\vec{p}$ and $\vec{q}$ denoting eigenvalues (arranged in increasing order) of
$\rho$ and $\Lambda(\rho)$, respectively. Clearly, the vector $\vec{p}$
majorizes $\vec{q}$ [61]. Whenever $\vec{p}\succ\vec{q}$, the majorization
conditions ensure that $-\infty<p_{i}\log q_{i}\leq 0,\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in[d]$. Thus we have,

$\displaystyle\sum_{i\in[d]}(q_{i}-p_{i})\log q_{i}$
$\displaystyle=\sum_{i=1}^{d}(q_{i}-p_{i})\log q_{i}$
$\displaystyle=\left(q_{1}-p_{1}\right)\log q_{1}+\left(q_{2}-p_{2}\right)\log
q_{2}+\cdots+\left(q_{d}-p_{d}\right)\log q_{d}$
$\displaystyle=\left(q_{1}-p_{1}\right)\log
q_{1}+\Big{(}(q_{1}+q_{2})-(p_{1}+p_{2})\Big{)}\log
q_{2}+\cdots+\left(\sum_{i\in[d]}q_{i}-\sum_{i\in[d]}p_{i}\right)\log q_{d}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
-\left\\{(q_{1}-p_{1})\log q_{2}+\Big{(}(q_{1}+q_{2})-(p_{1}+p_{2})\Big{)}\log
q_{3}+\cdots+\left(\sum_{i\in[d-1]}q_{i}-\sum_{i\in[d-1]}p_{i}\right)\log
q_{d}\right\\}$ $\displaystyle=\left(q_{1}-p_{1}\right)\left(\log q_{1}-\log
q_{2}\right)+\Big{(}(q_{1}+q_{2})-(p_{1}+p_{2})\Big{)}\left(\log q_{2}-\log
q_{3}\right)+\cdots$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\left(\sum_{i\in[d-1]}q_{i}-\sum_{i\in[d-1]}p_{i}\right)\left(\log
q_{d-1}-\log
q_{d}\right)+\left(\sum_{i\in[d]}q_{i}-\sum_{i\in[d]}p_{i}\right)\log q_{d}$
$\displaystyle=\sum_{l=2}^{d}\left\\{\sum_{i\in[l-1]}\left(q_{i}-p_{i}\right)\right\\}\Big{(}\log
q_{l-1}-\log q_{l}\Big{)}+\left(\sum_{i\in[d]}(q_{i}-p_{i})\right)\log q_{d}.$
(5)

Now using Eq.(4) and the fact that $\log$ is a monotonically increasing
function, from Eq.(II) we have,

$\displaystyle\sum_{i\in[d]}^{d}(q_{i}-p_{i})\log q_{i}\leq 0.$ (6)

Now, suppose that $S(\Lambda(\rho))=S(\rho)$, i.e., $\sum_{i\in[d]}q_{i}\log
q_{i}=\sum_{i\in[d]}p_{i}\log p_{i}$. Then again using the fact
$-\infty<p_{i}\log q_{i}\leq 0,\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in[d]$, we have,

$\displaystyle\sum_{i\in[d]}q_{i}\log q_{i}-\sum_{i\in[d]}p_{i}\log q_{i}$
$\displaystyle=\sum_{i\in[d]}p_{i}\log p_{i}-\sum_{i\in[d]}p_{i}\log q_{i},$
$\displaystyle\Rightarrow\sum_{i\in[d]}\left(q_{i}-p_{i}\right)\log q_{i}$
$\displaystyle=\sum_{i\in[d]}p_{i}\left(\log p_{i}-\log q_{i}\right),$
$\displaystyle\Rightarrow\sum_{i\in[d]}\left(q_{i}-p_{i}\right)\log q_{i}$
$\displaystyle=H\left(\vec{p}||\vec{q}\right)\geq 0.$ (7)

The last inequality follows from the non negativity of relative entropy
$H(\vec{p}||\vec{q})$. Now comparing Eq.(6) and Eq.(II), we have

$\displaystyle H\left(\vec{p}||\vec{q}\right)=0\Rightarrow\vec{p}=\vec{q}.$
(8)

This completes the proof. ∎

##  III Appendix III: Proof of Theorem 1

Before proving Theorem 1, in the following we first characterize the the set
of free channel $\mathcal{N}^{0}_{C}$ through which $A_{1}$ sub-part is send
to $A_{2}$.

A quantum channel
$N:\mathcal{D}(\mathbb{C}^{d})\mapsto\mathcal{D}(\mathbb{C}^{d})$ is called a
measure-and-prepare channel if it is of the form

$\displaystyle
N(\rho)=\sum_{i\in[n]}\operatorname{Tr}(\pi_{i}\rho)\ket{\psi_{i}}\bra{\psi_{i}},$
(9)

where the $\pi_{i}\geq 0\leavevmode\nobreak\ \&\leavevmode\nobreak\
\sum_{i\in[n]}\pi_{i}=\mathbf{I}_{d}$, where $n$ can take any integer value.
Notably, such a channel breaks entanglement and hence has zero quantum
capacity [62]. Even though, theses channels can not send any quantum
information we will not consider them to be classical channels. A channel $N$
is called classical channel if it carries information of distinct classical
symbols only. Set of such channels $\mathcal{N}_{C}$ are proper subset of
measure-and-prepare channel and takes the form

$\displaystyle
N(\rho)=\sum_{i\in[n]}\operatorname{Tr}(\pi_{i}\rho)\ket{\psi_{i}}\bra{\psi_{i}},$
(10)

where $n\leq d$ and
$\langle\psi_{i}|\psi_{i^{\prime}}\rangle=\delta_{ii^{\prime}}$. However, for
the choices $(f^{pe}_{\gamma},\Phi^{no}_{\gamma})$ not all $\mathcal{N}_{C}$
are free. In the following lemma we characterize the set of free channel
$\mathcal{N}^{0}_{C}\subset\mathcal{N}_{C}$.

######  Lemma 1.

In resource theory of purity a free classical channel
$N\in\mathcal{N}^{0}_{C}$ takes the form

$\displaystyle
N(\rho)=\sum_{i\in[d]}\operatorname{Tr}(\pi_{i}\rho)\ket{\psi_{i}}\bra{\psi_{i}},$
(11)

where $\langle\psi_{i}|\psi_{i^{\prime}}\rangle=\delta_{ii^{\prime}}$ and
$\operatorname{Tr}(\pi_{i})=1,\leavevmode\nobreak\ \forall\leavevmode\nobreak\
i\in[d]$.

###### Proof.

Recall that $S(\Lambda^{u}(\rho))\geq S(\rho)$ for a unital CPTP map
$\Lambda^{u}$, and hence $f^{pe}(\Lambda^{u}(\rho))\leq f^{pe}(\rho)$. On the
other hand, for any non-unital CPTP map $\Lambda^{nu}$, we have
$f^{pe}(\Lambda^{nu}(\mathbf{I}/d))>f^{pe}(\mathbf{I}/d)=0$, and therefore
they are not free. Thus any free classical channel $N\in\mathcal{N}^{0}_{C}$
must satisfy $N(\mathbf{I}/d)=\mathbf{I}/d$, which yields the form of $N$ as
in Lemma. ∎

###  III.1 Proof of Theorem 1

###### Proof.

(Theorem 1)
if part: Here we will prove that for every CQ state

$\displaystyle\rho_{\bf A}=\sum
p_{i_{1}i_{2}}\psi^{i_{1}}_{1}\otimes\psi^{i_{2}}_{2},\leavevmode\nobreak\
\mbox{with}\leavevmode\nobreak\
\langle\psi^{i_{1}}_{1}|\psi^{i^{\prime}_{1}}_{1}\rangle=\delta_{i_{i}i^{\prime}_{1}},$
(12)

we have $\mathbb{F}^{pe}_{G\mathcal{N}^{0}_{C}}(\rho_{\bf A})=0$.

As have already discussed in the main manuscript

$\displaystyle\Delta
f^{\\{\Phi^{no}_{\gamma}\\}}_{\mathcal{N}_{C}^{0}}(\rho_{\bf A})\leq\Delta
f^{\Phi^{no}_{\bf A}}_{\bf A}(\rho_{\bf A}),\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\ \rho_{\bf
A}\in\mathcal{D}(\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}).$ (13)

Also recall that, evaluating $\Delta
f^{\\{\Phi^{no}_{\gamma}\\}}_{\mathcal{N}_{C}^{0}}(\rho_{\bf A})$ essentially
contains two key steps: first applying $\phi_{1}\in\Phi^{no}_{1}$ in $A_{1}$’s
local lab, and then sending the $A_{1}$ sub-part to the $A_{2}$ party through
the channel $N\in\mathcal{N}_{C}^{0}$. The channel $N$ has the following form
(follows from Lemma 1)

$\displaystyle
N(\rho_{1})=\sum_{l\in[d_{1}]}\operatorname{Tr}(\pi_{l}\rho_{1})\ket{\alpha^{l}}_{1}\bra{\alpha^{l}},\leavevmode\nobreak\
\mbox{with}\leavevmode\nobreak\
\langle\alpha^{l}|\alpha^{l^{\prime}}\rangle=\delta_{ll^{\prime}},$ (14)
$\displaystyle\mbox{and}\leavevmode\nobreak\ \\{\pi_{l}\\}\leavevmode\nobreak\
\mbox{is a POVM with}\leavevmode\nobreak\
\operatorname{Tr}(\pi_{l})=1,\leavevmode\nobreak\ \forall\leavevmode\nobreak\
l\in[d_{1}].$

Given a CQ state $\rho_{\bf A}$ of the form of Eq.(12), let no $\phi_{1}$ is
applied on the $A_{1}$ part, rather part is sent through the free channel

$\displaystyle
N^{\star}(\rho_{1})=\sum_{l\in[d_{1}]}\operatorname{Tr}(\psi^{l}\rho_{1})\ket{\psi^{l}}_{1}\bra{\psi^{l}},$
(15)

where $\\{\ket{\psi^{l}}_{1}\\}_{l=1}^{d_{1}}$ are the states appearing in the
$A_{1}$ part of expression Eq.(12). Since
$N^{\star}\otimes\mathcal{I}(\rho_{\bf A})=\rho_{\bf A}$, therefore the CQ
state $\rho_{\bf A}$ will reach in $A_{2}$’s lab completely unchanged; here
$\mathcal{I}$ denotes the identity map. Thus, the extracted value of $\Delta
f^{\\{\Phi^{no}_{\gamma}\\}}_{\mathcal{N}_{C}^{0}}(\rho_{\bf A})$ achieves it
maximum value $\Delta f^{\Phi^{no}_{\bf A}}_{\bf A}(\rho_{\bf A})$, implying
$\mathbb{F}^{pe}_{G\mathcal{N}^{0}_{C}}(\rho_{\bf A})=0$. This completes the
if part of the proof.

only if part: Starting with an arbitrary state $\chi_{\bf A}$ we will now show
that

$\displaystyle\mathbb{F}^{pe}_{G\mathcal{N}^{0}_{C}}(\chi_{\bf
A}):=\underbrace{\Delta\left(f^{pe}\right)_{\bf A}^{\Phi_{\bf A}}(\chi_{\bf
A})}_{1^{st}\leavevmode\nobreak\
term}-\underbrace{\Delta\left(f^{pe}\right)^{\\{\Phi_{\gamma}\\}}_{\mathcal{N}_{C}^{0}}(\chi_{\bf
A})}_{2^{nd}\leavevmode\nobreak\ term}=0\leavevmode\nobreak\
\implies\leavevmode\nobreak\ \chi_{\bf A}\leavevmode\nobreak\ \mbox{is a CQ
state}.$ (16)

Recall that a channel $\Lambda:\mathcal{D}(C^{d})\to\mathcal{D}(C^{d})$ is
called unital if it maps completely mixed state to itself, i.e.,
$\Lambda\left(\mathbf{I}/d\right)=\mathbf{I}/d$. Set of all such channels is
denoted by $\mathcal{U}(d)$. Since for a $\Lambda\in\mathcal{U}(d)$ we have
$f^{pe}(\Lambda(\rho))\leq f^{pe}(\rho),\leavevmode\nobreak\
\forall\rho\in\mathcal{D}(C^{d})$, theses channels are considered to be free.
On the other hand, as shown in Lemma 1, a free classical channel
$N\in\mathcal{N}^{0}_{C}$ reads as

$\displaystyle
N(\rho_{1}):=\sum_{l\in[d_{1}]}\operatorname{Tr}(\pi_{l}\rho_{1})\ket{\alpha^{l}_{1}}\bra{\alpha^{l}_{1}},\leavevmode\nobreak\
\mbox{where}\leavevmode\nobreak\
\langle\alpha^{l}_{1}|\alpha^{l^{\prime}}_{1}\rangle=\delta_{ll^{\prime}},$
(17) $\displaystyle\mbox{and}\leavevmode\nobreak\
\\{\pi_{l}\\}\leavevmode\nobreak\ \mbox{is a POVM with}\leavevmode\nobreak\
\operatorname{Tr}(\pi_{l})=1,\leavevmode\nobreak\ \forall\leavevmode\nobreak\
l\in[d_{1}].$

Manifestly it follows that
$\mathcal{N}^{0}_{C}\subset\mathcal{U}(\mathbb{C}^{d_{1}})$. Now, the
"$1^{st}\leavevmode\nobreak\ term$" in Eq.(16) simply boils down to

$\displaystyle\Delta\left(f^{pe}\right)_{\bf A}^{\Phi_{\bf A}}(\chi_{\bf
A})=\log(d_{1}d_{2})-S(\chi_{\bf A}).$ (18)

For the "$2^{nd}\leavevmode\nobreak\ term$" let us recall its expression:

$\displaystyle\Delta\left(f^{pe}\right)^{\\{\Phi_{\gamma}\\}}_{\mathcal{N}_{C}^{0}}(\chi_{\bf
A})=\max_{\left\\{\phi_{1}\in\Phi^{no}_{1},\leavevmode\nobreak\
\phi_{2}\in\Phi^{no}_{2},\leavevmode\nobreak\ \phi_{\bf A}\in\Phi^{no}_{\bf
A},\leavevmode\nobreak\ \leavevmode\nobreak\
N\in\mathcal{N}^{0}_{C}\right\\}}\left[\Delta
f^{\phi_{1}}_{1}(\chi_{1})+\Delta f^{\phi_{2}}_{2}(\chi_{2})+\Delta
f^{\phi_{\bf A}}_{\bf A}(N(\sigma_{\bf A}))-\mathcal{C}(N)\right],$ (19)

where $\sigma_{\bf A}=(\phi_{1}\otimes\phi_{2})(\chi_{\bf A})$. Clearly,
$C(N)=0$ as $N\in\mathcal{N}^{0}_{C}$. Note that the marginal $\chi_{1}$ being
completely mixed, makes the optimization over $\Phi^{no}_{1}$ irrelevant.
Furthermore, the action of $\phi_{2}$ can be absorbed in $\phi_{\bf A}$, which
in turn reduces the optimization over $\Phi^{no}_{2}$. Accordingly, Eq.(19)
modified as

$\displaystyle\Delta\left(f^{pe}\right)^{\\{\Phi_{\gamma}\\}}_{\mathcal{N}_{C}^{0}}(\chi_{\bf
A})=\max_{\left\\{\phi_{\bf A}\in\Phi^{no}_{\bf A},\leavevmode\nobreak\
\leavevmode\nobreak\ N\in\mathcal{N}^{0}_{C}\right\\}}\left[\Delta
f^{\phi_{\bf A}}_{\bf A}(N(\chi_{\bf A}))\right].$ (20)

From Eqs.(16), (18), and (20), we therefore have

$\displaystyle\max_{\left\\{\phi_{\bf A}\in\Phi^{no}_{\bf
A},\leavevmode\nobreak\ \leavevmode\nobreak\
N\in\mathcal{N}^{0}_{C}\right\\}}\left[\Delta f^{\phi_{\bf A}}_{\bf
A}(N(\chi_{\bf A}))\right]=\log(d_{1}d_{2})-S(\chi_{\bf A}).$ (21)

Suppose the optimization over $\mathcal{N}^{0}_{C}$ is achieved for some
$N^{\star}$. Then we have

$\displaystyle\max_{\left\\{\phi_{\bf A}\in\Phi^{no}_{\bf
A}\right\\}}\left[\Delta f^{\phi_{\bf A}}_{\bf A}(N^{\star}(\chi_{\bf
A}))\right]=\log(d_{1}d_{2})-S(\chi_{\bf A})$ $\displaystyle\Rightarrow
S((N_{1}^{\star}\otimes\mathcal{I}_{2})(\chi_{\bf A}))=S(\chi_{\bf A}).$ (22)

Using Proposition 1, we thus have

$\displaystyle\chi_{\bf A}=(N_{1}^{\star}\otimes\mathcal{I}_{2})(\chi_{\bf
A})=\sum_{l\in[d_{1}]}\operatorname{Tr}((\pi^{\star}_{l}\otimes\mathbf{I})\chi_{\bf
A})\ket{\alpha^{l}}_{1}\bra{\alpha^{l}}\otimes\chi_{A_{2}|l}.$ (23)

Here,
$\chi_{A_{2}|l}:=\operatorname{Tr}_{1}\left[\left(\pi^{\star}_{l}\otimes\mathbf{I}\right)\chi_{\bf
A}\right]/\operatorname{Tr}\left[\left((\pi^{\star}_{l}\otimes\mathbf{I})\right)\chi_{\bf
A}\right]$ represents the normalized state of subpart $A_{2}$, conditioned on
the positive operator $\\{\pi^{\star}_{l}\\}$ acting on part $A_{1}$. This
ensures that $\chi_{\bf A}$ conforms to the CQ form. This completes only if
part of the proof, and hence the Theorem 1 is proved. ∎

##  IV Appendix IV: Proof of Theorem 2

We first present few Lemmas that will be needed for proving Theorem 2. We
start by characterizing the relevant set of zero-cost classical communication
channels $\mathcal{N}^{0}_{C}$. Let us recall that a free classical channel
$N$ in resource theory of purity satisfies two conditions: (i)
$f^{pe}(N(\rho))\leq f^{pe}(\rho),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ \rho$, and (ii) $\Delta
f^{{\Phi^{no}}}(N(\rho))\leq\Delta f^{{\Phi^{no}}}(\rho),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ \rho$. It is worth noting that both of these
conditions solely depend on the entropy of the given state, and moreover one
implies the other. However, within the observable approach, the relationship
between the expectation value of an observable
$\langle\mathcal{O}\rangle_{(\rho)}$ and its variation
$\Delta\mathcal{O}^{U}(\rho)$ exhibit more complex behaviour. Hence, we demand
a free classical channel $N$ to fulfill both the following two criteria:

$\displaystyle\mbox{\bf(C1)}\leavevmode\nobreak\
\operatorname{Tr}(\mathcal{O}N(\rho))\leq\operatorname{Tr}(\mathcal{O}\rho),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ \rho\in\mathcal{D}(\mathbb{C}^{d});$
$\displaystyle\mbox{\bf(C2)}\leavevmode\nobreak\
\Delta\mathcal{O}^{U}(N(\rho))\leq\Delta\mathcal{O}^{U}(\rho),\leavevmode\nobreak\
\forall\rho\in\mathcal{D}(\mathbb{C}^{d}).$

In the following lemma we completely characterize the set of free channels
that satisfies these two criteria.

######  Lemma 2.

Given a non-degenerate $\mathcal{O}\in$ Herm$(\mathbb{C}^{d})$, a classical
channel $N$ satisfying the criteria (C1) and (C2) reads as

$\displaystyle either\leavevmode\nobreak\ \leavevmode\nobreak\
N\left(\rho\right)$ $\displaystyle=\ket{o_{1}}\bra{o_{1}},$ (24a)
$\displaystyle or\leavevmode\nobreak\ \leavevmode\nobreak\ N\left(\rho\right)$
$\displaystyle=\sum_{l\in[d]}\operatorname{Tr}\left(\ket{o_{l}}\bra{o_{l}}\rho\right)\ket{o_{l}}\bra{o_{l}},$
(24b)

where, $\\{\ket{o_{l}}\\}_{l\in[d]}$ are the eigenstates of $\mathcal{O}$ and
forms a complete basis of $\mathbb{C}^{d}$.

###### Proof.

Recalling from Eq.(10), a classical channel reads as

$\displaystyle
N(\rho)=\sum_{l\in[n]}\operatorname{Tr}(\pi_{l}\rho)\ket{\psi_{l}}\bra{\psi_{l}},$
(25)

where $\langle\psi_{l}|\psi_{l^{\prime}}\rangle=\delta_{ll^{\prime}}$,
$\pi_{l}\geq 0\leavevmode\nobreak\ \&\leavevmode\nobreak\
\sum_{l\in[n]}\pi_{l}=\mathbf{I}_{d}$ with $n\leq d$. Let
$\mathcal{O}:=\sum_{l\in[d]}\lambda_{l}|o_{l}\rangle\langle o_{l}|$ be the
spectral representation of $\mathcal{O}$ with $\lambda_{l}$’s being arranged
in increasing order, i.e,
$\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{d}$.

Consider a classical channel as of Eq.(25) with $n=1$ which represents the pin
map $N(\rho)=\ket{\psi_{1}}\bra{\psi_{1}}$ with $\pi_{1}=\mathbf{I}_{d}$. When
the channel is fed with the state $\ket{o_{1}}$, the eigenstate of
$\mathcal{O}$ corresponding to the lowest eigenvalue, the condition (C1) will
be satisfied only if $\ket{\psi_{1}}=\ket{o_{1}}$. Moreover, this choice also
turns out to be compatible with the requirement of condition (C2).
Accordingly, the free channel takes the form of Eq.(24a).

Consider, now the case $1<n\leq d$. Feeding the state
$\rho=\ket{o_{1}}\bra{o_{1}}$ into the channel and imposing (C1) we obtain the
following conclusions: $\pi_{1}=\ket{o_{1}}\bra{o_{1}}$,
$\ket{\psi_{1}}=\ket{o_{1}}$, and
$\operatorname{Tr}(\pi_{l}\pi_{1})=\delta_{l1},\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\in[n]$. Accordingly, the channel takes the form:

$\displaystyle
N(\rho)=\operatorname{Tr}(\ket{o_{1}}\bra{o_{1}}\rho)\ket{o_{1}}\bra{o_{1}}+\sum_{l=2}^{n}\operatorname{Tr}(\pi_{l}\rho)\ket{\psi_{l}}\bra{\psi_{l}}.$
(26)

Moving forward, by setting $\rho=\ket{o_{2}}\bra{o_{2}}$ and imposing
condition (C1), we obtain $\pi_{2}=\ket{o_{2}}\bra{o_{2}}$,
$\ket{\psi_{2}}=\ket{o_{2}}$, and
$\operatorname{Tr}(\pi_{l}\pi_{2})=\delta_{l2},\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\in[n]$, which thus lead to

$\displaystyle
N(\rho)=\operatorname{Tr}(\ket{o_{1}}\bra{o_{1}}\rho)\ket{o_{1}}\bra{o_{1}}+\operatorname{Tr}(\ket{o_{2}}\bra{o_{2}}\rho)\ket{o_{2}}\bra{o_{2}}+\sum_{l=3}^{n}\operatorname{Tr}(\pi_{l}\rho)\ket{\psi_{l}}\bra{\psi_{l}}.$
(27)

Continuing this we extend the argument for $l\in[n-1]$ and obtain

$\displaystyle
N(\rho)=\sum_{l\in[n-1]}\operatorname{Tr}(\ket{o_{l}}\bra{o_{l}}\rho)\ket{o_{l}}\bra{o_{l}}+\operatorname{Tr}(\pi_{n}\rho)\ket{\psi_{n}}\bra{\psi_{n}}.$
(28)

Consider now the completely mixed state $\rho=\mathbf{I}/d$ for which the
variation $\Delta\mathcal{O}^{U}(\mathbf{I}/d)=0$. The action of the channel
(28) on this state yields

$\displaystyle
N(\mathbf{I}/d)=\frac{1}{d}\sum_{l\in[n-1]}\ket{o_{l}}\bra{o_{l}}+\left(1-\frac{n-1}{d}\right)\ket{\psi_{n}}\bra{\psi_{n}}.$
(29)

$\displaystyle\left(1-\frac{n-1}{d}\right)$
$\displaystyle\leq\frac{1}{d}\Rightarrow d\leq n,$
$\displaystyle\&\leavevmode\nobreak\ \ket{\psi_{n}}\bra{\psi_{n}}$
$\displaystyle=\ket{o_{n}}\bra{o_{n}}.$

Since for the classical in Eq.(25) we have $n\leq d$, thus we conclude $n=d$.
Accordingly a free classical channel $N\in\mathcal{N}^{0}_{C}$ takes the form
of Eq.(24b). This concludes the proof. ∎

Figure 3: A measure-and-prepare channel
$N_{MP}:\mathcal{D}(\mathbb{C}^{d})\mapsto\mathcal{D}(\mathbb{C}^{d})$ reads
as
$N_{MP}(\rho)=\sum_{i\in[n]}\operatorname{Tr}(\pi_{i}\rho)\ket{\psi_{i}}\bra{\psi_{i}}$,
where $\pi_{i}\geq 0\leavevmode\nobreak\ \&\leavevmode\nobreak\
\sum_{i\in[n]}\pi_{i}=\mathbf{I}_{d}$, with $n$ taking integer values. Let
$\mathcal{N}_{MP}$ denotes the set of all such channels. A classical channel
$N_{C}\in\mathcal{N}_{C}$ takes the form
$N_{C}(\rho)=\sum_{i\in[n]}\operatorname{Tr}(\pi_{i}\rho)\ket{\psi_{i}}\bra{\psi_{i}}$,
where $n\leq d$ and
$\langle\psi_{i}|\psi_{i^{\prime}}\rangle=\delta_{ii^{\prime}}$. A free
classical channel $N^{0(1)}_{C}\in\mathcal{N}^{0(1)}_{C}$ appearing in Theorem
1 reads as
$N^{0(1)}_{C}(\rho)=\sum_{i\in[d]}\operatorname{Tr}(\pi_{i}\rho)\ket{\psi_{i}}\bra{\psi_{i}}$,
where $\langle\psi_{i}|\psi_{i^{\prime}}\rangle=\delta_{ii^{\prime}}$ and
$\operatorname{Tr}(\pi_{i})=1,\leavevmode\nobreak\ \forall\leavevmode\nobreak\
i\in[d]$ (see Lemma 1). For a non-degenerate $\mathcal{O}\in$
Herm$(\mathbb{C}^{d})$, a free classical channel
$N^{0(2)}_{C}\in\mathcal{N}^{0(2)}_{C}$ appearing in Theorem 2 takes the form
either $N^{0(2)}_{C}\left(\rho\right)=\ket{o_{1}}\bra{o_{1}}$ or
$N^{0(2)}_{C}\left(\rho\right)=\sum_{l\in[d]}\operatorname{Tr}\left(\ket{o_{l}}\bra{o_{l}}\rho\right)\ket{o_{l}}\bra{o_{l}}$,
where $\\{\ket{o_{l}}\\}_{l\in[d]}$ are the eigenstates of $\mathcal{O}$ (see
Lemma 2). The above figure depicts the set inclusion relations,
$\mathcal{N}^{0(2)}_{C}\subsetneq\mathcal{N}^{0(1)}_{C}\subsetneq\mathcal{N}_{C}\subsetneq\mathcal{N}_{MP}$.

Given a quantum state $\rho$ and an observable $\mathcal{O}$, we are
interested in the variation
$\Delta\mathcal{O}^{\mathrm{U}}(\rho):=\max_{u\in\mathrm{U}}[\operatorname{Tr}(\mathcal{O}\rho)-\operatorname{Tr}(\mathcal{O}u\rho
u^{\dagger})]$. The state having same entropy as of the state $\rho$ and
yielding minimum expectation value for the observable $\mathcal{O}$ is thus
relevant in this regard. We call this state the $\mathcal{O}$-passive state.

######  Definition 1.

Given an observable $\mathcal{O}\in$Herm$(\mathbb{C}^{d})$ state
$\rho_{p}^{o}\in\mathcal{D}(\mathbb{C}^{d})$ is said to be the
$\mathcal{O}$-passive state, if

$\displaystyle\operatorname{Tr}(\mathcal{O}\rho_{p}^{o})\leq\operatorname{Tr}(\mathcal{O}u\rho_{p}^{o}u^{\dagger}),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ u\in\mathrm{U}.$ (30)

######  Theorem 4.

[Schur-Horn theorem [63, 64, 60]] Let $\mathcal{Z}\in$Herm$(\mathbb{C}^{d})$
has spectral
$\mathbb{R}^{d}\ni\vec{\lambda}\equiv(\lambda_{1},\cdots,\lambda_{d})$ and let
$\mathbb{R}^{d}\ni\vec{\mu}\equiv(\mu_{1},\cdots,\mu_{d})$ be the diagonal
elements of $\mathcal{Z}$ represented in some other basis, with both the
vectors arranged in increasing order. Then $\vec{\lambda}\succ\vec{\mu}$.

Using this Theorem we provide a sufficient form for the $\mathcal{O}$-passive
state.

######  Lemma 3.

For the observable $\mathcal{O}=\sum_{l\in[d]}\lambda_{l}|o_{l}\rangle\langle
o_{l}|$, with $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{d}$, the state
$\rho_{p}^{o}\in\mathcal{D}(\mathbb{C}^{d})$ of the following form is always
an $\mathcal{O}$-passive state:

$\displaystyle\rho_{p}^{o}=\sum_{l\in[d]}r_{l}|o_{l}\rangle\langle
o_{l}|,\mbox{ where},\leavevmode\nobreak\ r_{1}\leq r_{2}\leq\cdots r_{d}.$
(31)

###### Proof.

Let, $\rho_{u}:=u\rho_{p}^{o}u^{\dagger}=\sum_{l\in[d]}r_{l}|l\rangle\langle
l|$, for an arbitrary $u\in\mathrm{U}$. The state $\rho_{u}$ can also be
written in $\\{|o_{l}\rangle\langle o_{l^{\prime}}|\\}$ basis as
$\rho_{u}=\sum_{l,l^{\prime}\in[d]}s_{ll^{\prime}}|o_{l}\rangle\langle
o_{l^{\prime}}|$. The diagonal entries $s_{ll}$ are denoted as $s_{l}$ for the
sake of brevity. Note that the $\vec{s}_{u}\equiv(s_{1},\cdots,s_{d})$ is not
arranged in any particular order. However, we can obtain
$\rho_{uv}:=v\rho_{u}v^{\dagger}$ by apply another unitary $v$ such that
diagonal entries
$\vec{s}_{uv}\equiv(s_{1}^{\uparrow},\cdots,s_{d}^{\uparrow})$ are arranged in
increasing order. Manifestly it follows that

$\displaystyle\operatorname{Tr}(\mathcal{O}\rho_{uv})\leq\operatorname{Tr}(\mathcal{O}\rho_{u}).$
(32)

Applying Schur-Horn theorem between $\rho_{p}^{o}$ and $\rho_{uv}$ we can
conclude $\vec{r}\succ\vec{s}_{uv}$. Now we have,

$\displaystyle\operatorname{Tr}(\mathcal{O}\rho_{p}^{o})-\operatorname{Tr}(\mathcal{O}\rho_{uv})$
$\displaystyle=\sum_{l\in[d]}\lambda_{l}\left(r_{l}-s_{l}^{\uparrow}\right)$
$\displaystyle=\left(r_{1}-s_{1}^{\uparrow}\right)\lambda_{1}+\Big{(}(r_{1}+r_{2})-(s_{1}^{\uparrow}+s_{2}^{\uparrow})\Big{)}\lambda_{2}+\cdots+\left(\sum_{l\in[d]}r_{l}-\sum_{l\in[d]}s_{l}^{\uparrow}\right)\lambda_{d}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
-\left\\{(r_{1}-s_{1}^{\uparrow})\lambda_{2}+\Big{(}(r_{1}+r_{2})-(s_{1}^{\uparrow}+s_{2}^{\uparrow})\Big{)}\lambda_{3}+\cdots+\left(\sum_{l\in[d-1]}r_{l}-\sum_{l\in[d-1]}s_{l}^{\uparrow}\right)\lambda_{d}\right\\}$
$\displaystyle=\left(r_{1}-s_{1}^{\uparrow}\right)\left(\lambda_{1}-\lambda_{2}\right)+\Big{(}(r_{1}+r_{2})-(s_{1}^{\uparrow}+s_{2}^{\uparrow})\Big{)}\left(\lambda_{2}-\lambda_{3}\right)+\cdots$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+\left(\sum_{l\in[d-1]}r_{l}-\sum_{l\in[d-1]}s_{l}^{\uparrow}\right)\left(\lambda_{d-1}-\lambda_{d}\right)+\left(\sum_{l\in[d]}r_{l}-\sum_{l\in[d]}s_{l}^{\uparrow}\right)\lambda_{d}$
$\displaystyle=\sum_{m=2}^{d}\left\\{\sum_{l\in[l-1]}\left(r_{l}-s_{l}^{\uparrow}\right)\right\\}\Big{(}\lambda_{m-1}-\lambda_{m}\Big{)}+\left(\sum_{l\in[d]}(r_{l}-s_{l}^{\uparrow})\right)\lambda_{d}\leq
0.$ (33)

The inequality in last line follows from the fact that
$\vec{r}\succ\vec{s}_{uv}$ and the fact that $\vec{\lambda}$ is arranged in
increasing order. Now, Eq.(32) and Eq.(IV) together imply
$\operatorname{Tr}(\mathcal{O}\rho_{p}^{o})\leq\operatorname{Tr}(\mathcal{O}u\rho_{p}^{o}u^{\dagger})$.
This completes the proof.

Furthermore, it is important to notice that, when $\mathcal{O}$ is non-
degenerate, the form of the state in in Eq.(31) is the only allowed form of
the $\mathcal{O}$-passive state. ∎

######  Lemma 4.

Given an observable $\mathcal{O}\in$ Herm$(\mathbb{C}^{d})$ and a state
$\rho\in\mathcal{D}(\mathbb{C}^{d})$, we have
$\Delta\mathcal{O}^{\mathrm{U}}(\rho)=\langle\mathcal{O}\rangle_{\rho}-\mathcal{O}^{p}(\rho)$,
where $\mathcal{O}^{p}:\mathcal{D}(\mathbb{C}^{d})\mapsto\mathbb{R}$ is
defined as:
$\mathcal{O}^{p}(\rho):=\operatorname{Tr}(\mathcal{O}\rho_{p}^{o})$, with
$\rho_{p}^{o}$ being a $\mathcal{O}$-passive state having the same spectrum as
$\rho$.

###### Proof.

Recalling the definition of $\Delta\mathcal{O}^{\mathrm{U}}(\rho)$, we have

$\displaystyle\Delta\mathcal{O}^{\mathrm{U}}(\rho)$
$\displaystyle=\max_{u\in\mathrm{U}}[\operatorname{Tr}(\mathcal{O}\rho)-\operatorname{Tr}(\mathcal{O}u\rho
u^{\dagger})]$
$\displaystyle=\operatorname{Tr}(\mathcal{O}\rho)-\min_{u\in\mathrm{U}}\operatorname{Tr}(\mathcal{O}u\rho
u^{\dagger})$
$\displaystyle=\operatorname{Tr}(\mathcal{O}\rho)-\operatorname{Tr}(\mathcal{O}\rho_{p}^{o}).$
(34)

The last line directly follows from the Definition 1 of $\mathcal{O}$-passive
state. ∎

######  Lemma 5.

Given a non-degenerate $\mathcal{O}\in$ Herm$(\mathbb{C}^{d})$ and a channel
$N$ of the form Eq.(24b), if $\mathcal{O}^{p}(\rho)=\mathcal{O}^{p}(N(\rho))$,
then $\rho$ is a fixed point of $N$, i.e., $N(\rho)=\rho$.

###### Proof.

Let, $\vec{p}=(p_{1},p_{2},\cdots,p_{d})$ be the eigenvalues of $\rho$
arranged in increasing order. Further assume,
$\vec{\lambda}=(\lambda_{1},\lambda_{2},\cdots,\lambda_{d})$ be the
eigenvalues of the Hermitian operator $\mathcal{O}$ arranged in decreasing
order. It follows from Lemma 3

$\displaystyle\mathcal{O}^{p}(\rho)=\vec{\lambda}\cdot\vec{p}=\sum_{i\in[d]}\lambda_{i}p_{i}.$
(35)

Consider now, $\vec{q}=(q_{1},q_{2},\cdots,q_{d})$ to be the eigenvalues of
$N(\rho)$ arranged in increasing order. Therefore, we have

$\displaystyle\mathcal{O}^{p}(N(\rho))-\mathcal{O}^{p}(\rho)$
$\displaystyle=\sum_{i\in[d]}\lambda_{i}(q_{i}-p_{i})$
$\displaystyle=\left(q_{1}-p_{1}\right)\lambda_{1}+\Big{(}(q_{1}+q_{2})-(p_{1}+p_{2})\Big{)}\lambda_{2}+\cdots+\left(\sum_{i\in[d]}q_{i}-\sum_{i\in[d]}p_{i}\right)\lambda_{d}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
-\left\\{(q_{1}-p_{1})\lambda_{2}+\Big{(}(q_{1}+q_{2})-(p_{1}+p_{2})\Big{)}\lambda_{3}+\cdots+\left(\sum_{i\in[d-1]}q_{i}-\sum_{i\in[d-1]}p_{i}\right)\lambda_{d}\right\\}$
$\displaystyle=\left(q_{1}-p_{1}\right)\left(\lambda_{1}-\lambda_{2}\right)+\Big{(}(q_{1}+q_{2})-(p_{1}+p_{2})\Big{)}\left(\lambda_{2}-\lambda_{3}\right)+\cdots$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+\left(\sum_{i\in[d-1]}q_{i}-\sum_{i\in[d-1]}p_{i}\right)\left(\lambda_{d-1}-\lambda_{d}\right)+\left(\sum_{i\in[d]}q_{i}-\sum_{i\in[d]}p_{i}\right)\lambda_{d}$
$\displaystyle=\sum_{l=2}^{d}\left\\{\sum_{i\in[l-1]}\left(q_{i}-p_{i}\right)\right\\}\Big{(}\lambda_{l-1}-\lambda_{l}\Big{)}+\left(\sum_{i\in[d]}(q_{i}-p_{i})\right)\lambda_{d}\leq
0.$ (36)

Inequality in last line follows from the fact that $N$ is a unital and hence
$\vec{p}$ majorizes $\vec{q}$, and the fact that $\vec{\lambda}$ is arranged
in decreasing order.

Now, $\mathcal{O}$ being non-degenerate we have
$\lambda_{l-1}-\lambda_{l}>0,\leavevmode\nobreak\ \forall\leavevmode\nobreak\
l$. Therefore, $\mathcal{O}^{p}(\rho)=\mathcal{O}^{p}(N(\rho))$ implies
$\vec{p}=\vec{q}$. This concludes the proof. ∎

###  IV.1 Proof of Theorem 2

###### Proof.

(Theorem 2).
if part: We first show that for every CQ state $\rho_{\bf A}$ of Eq.(12),
$\mathbb{O}_{G\mathcal{N}_{C}^{0}}(\rho_{\bf A})=0$. Notice, for every such
$\rho_{\bf A}$, one can always find a unitary
$u_{1}^{\star}\in\mathrm{U}_{1}$, such that:

$\displaystyle\sigma_{\bf
A}=\Big{(}u_{1}^{\star}\otimes\mathbf{I}_{2}\Big{)}\rho_{\bf
A}\Big{(}\left(u_{1}^{\star}\right)^{\dagger}\otimes\mathbf{I}_{2}\Big{)}=\sum
p_{i_{1}i_{2}}\ket{o^{i_{1}}}_{1}\bra{o^{i_{1}}}\otimes\psi^{i_{2}}_{2}.$ (37)

Clearly, after applying $u_{1}^{\star}$ if the channel $N$ as defined in
Eq.(24b) is used to send the $A_{1}$ sub-part to $A_{2}$, then the state
remains unchanged, i.e., $(N\otimes\mathcal{I})\sigma_{\bf A}=\sigma_{\bf A}$.
The states $\sigma_{\bf A}$ and $\rho_{\bf A}$ having same spectral yield
$\mathcal{O}_{\bf A}^{p}(\sigma_{\bf A})=\mathcal{O}_{\bf A}^{p}(\rho_{\bf
A})$. Therefore, under the choice
$\\{\mathrm{U}_{1},\mathrm{U}_{2},\mathrm{U}_{12}\\}=\\{u_{1}^{\star},\mathbf{I}_{2},u_{12}^{\star}\\}$,
with $u_{12}^{\star}$ mapping $\sigma_{\bf A}$ to the passive state having the
same spectrum, one have

$\displaystyle\Delta\mathcal{O}^{\\{u_{1}^{\star},\mathbf{I}_{2},u_{12}^{\star}\\}}_{N}(\rho_{\bf
A})$
$\displaystyle=\Delta\mathcal{O}_{1}^{u_{1}^{\star}}(\rho_{1})+\Delta\mathcal{O}_{2}^{\mathbf{I}_{2}}(\rho_{2})+\Delta\mathcal{O}_{\bf
A}^{u_{12}^{\star}}(\sigma_{\bf A})$
$\displaystyle=\left(\langle\mathcal{O}_{1}\rangle_{\rho_{1}}-\langle\mathcal{O}_{1}\rangle_{\sigma_{1}}\right)+\left(\langle\mathcal{O}_{2}\rangle_{\rho_{2}}-\langle\mathcal{O}_{2}\rangle_{\sigma_{2}}\right)+\Delta\mathcal{O}_{\bf
A}^{u_{12}^{\star}}(\sigma_{\bf A})\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{[Since $\rho_{2}=\sigma_{2}$]}$
$\displaystyle=\left(\langle\mathcal{O}_{1}\rangle_{\rho_{1}}+\langle\mathcal{O}_{2}\rangle_{\rho_{2}}\right)-\left(\langle\mathcal{O}_{1}\rangle_{\sigma_{1}}+\langle\mathcal{O}_{2}\rangle_{\sigma_{2}}\right)+\Delta\mathcal{O}_{\bf
A}^{u_{12}^{\star}}(\sigma_{\bf A})$
$\displaystyle=\Big{(}\langle\mathcal{O}_{\bf A}\rangle_{\rho_{\bf
A}}-\langle\mathcal{O}_{\bf A}\rangle_{\sigma_{\bf
A}}\Big{)}+\left(\langle\mathcal{O}_{\bf A}\rangle_{\sigma_{\bf
A}}-\mathcal{O}_{\bf A}^{p}\left(\rho_{\bf
A}\right)\right)\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{[using Lemma
\ref{lem4}]}$ $\displaystyle=\langle\mathcal{O}_{\bf A}\rangle_{\rho_{\bf
A}}-\mathcal{O}_{\bf A}^{p}\left(\rho_{\bf A}\right)=\Delta\mathcal{O}_{\bf
A}^{\mathrm{U}_{\bf A}}(\rho_{\bf A})\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{[using Lemma \ref{lem4}]}.$

Hence, the choice $\\{u_{1}^{\star},\mathbf{I}_{2},u_{12}^{\star}\\}$ being
optimal yields $\mathbb{O}_{G\mathcal{N}^{0}_{C}}(\rho_{\bf A})=0$.

only if part: Given a state $\chi_{\bf
A}\in\mathcal{D}(\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}})$, let
$\\{v_{1},v_{2},v_{12}\\}$ be the optimal choice for evaluating
$\mathbb{O}_{G\mathcal{N}^{0}_{C}}(\chi_{\bf A})$. Furthermore, let us assume,
$\eta_{\bf A}=(v_{1}\otimes v_{2})\chi_{\bf A}(v_{1}^{\dagger}\otimes
v_{2}^{\dagger})$. Thereafter when $A_{1}$ sub-part is send through the
channel $N$ of the form Eq.(24b), the state becomes
$(N\otimes\mathcal{I})\eta_{\bf A}=\tilde{\eta}_{\bf A}$ (say). Thus we have,

$\displaystyle\Delta\mathcal{O}^{\\{\mathrm{U}_{1},\mathrm{U}_{2},\mathrm{U}_{12}\\}}_{\mathcal{N}^{0}_{C}}(\chi_{\bf
A})$ $\displaystyle=\Delta\mathcal{O}^{\\{v_{1},v_{2},v_{12}\\}}_{N}(\chi_{\bf
A})$ $\displaystyle=\Big{(}\langle\mathcal{O}_{\bf A}\rangle_{\chi_{\bf
A}}-\langle\mathcal{O}_{\bf A}\rangle_{\eta_{\bf
A}}\Big{)}+\left(\langle\mathcal{O}_{\bf A}\rangle_{\tilde{\eta}_{\bf
A}}-\mathcal{O}_{\bf A}^{p}\left(\tilde{\eta}_{\bf A}\right)\right)$
$\displaystyle=\langle\mathcal{O}_{\bf A}\rangle_{\chi_{\bf
A}}-\mathcal{O}_{\bf A}^{p}\left(\tilde{\eta}_{\bf A}\right).$ (38)

In the last line we use the fact that $\mathcal{O}_{\bf A}(\zeta_{\bf
A})=\mathcal{O}_{\bf A}((N\otimes\mathcal{I})\zeta_{\bf
A}),\leavevmode\nobreak\ \forall\leavevmode\nobreak\ \zeta_{\bf
A}\in\mathcal{D}(\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}})$. Now, using
Eq.(38) we obtain

$\displaystyle\mathbb{O}_{G\mathcal{N}^{0}_{C}}(\chi_{\bf A})$
$\displaystyle=\Delta\mathcal{O_{\bf
A}}(\chi_{A})-\Delta\mathcal{O}^{\\{\mathrm{U}_{1},\mathrm{U}_{2},\mathrm{U}_{12}\\}}_{\mathcal{N}^{0}_{C}}(\chi_{\bf
A})$ $\displaystyle=\left(\mathcal{O}_{\bf A}\left(\chi_{\bf
A}\right)-\mathcal{O}^{p}_{\bf A}\left(\chi_{\bf
A}\right)\right)-\left(\mathcal{O}_{\bf A}\left(\chi_{\bf
A}\right)-\mathcal{O}_{\bf A}^{p}\left(\tilde{\eta}_{\bf A}\right)\right)$
$\displaystyle=\mathcal{O}_{\bf A}^{p}\left(\tilde{\eta}_{\bf
A}\right)-\mathcal{O}_{\bf A}^{p}\left(\chi_{\bf A}\right)$
$\displaystyle=\mathcal{O}_{\bf A}^{p}\left(\tilde{\eta}_{\bf
A}\right)-\mathcal{O}_{\bf A}^{p}\left(\eta_{\bf A}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{[Since $\mathcal{O}^{p}(\rho)=\mathcal{O}^{p}(u\rho
u^{\dagger})$]}.$ (39)

Therefore, $\mathbb{O}_{G\mathcal{N}^{0}_{C}}(\chi_{\bf A})=0$ implies
$\mathcal{O}_{\bf A}^{p}\left(\tilde{\eta}_{\bf A}\right)=\mathcal{O}_{\bf
A}^{p}\left(\eta_{\bf A}\right)$. Using Lemma 5, we can further claim that,
$\eta_{\bf A}$ is a fixed point of $N$, i.e,

$\displaystyle\eta_{\bf A}$
$\displaystyle=\left(N\otimes\mathcal{I}\right)\eta_{\bf A}$
$\displaystyle=\sum_{l\in[d]}\operatorname{Tr}\left(\left(\ket{o^{l}}_{1}\bra{o^{l}}\right)\eta_{\bf
A}\right)\ket{o^{l}}_{1}\bra{o^{l}}\otimes\eta_{A_{2}|l}$
$\displaystyle\Rightarrow(v_{1}\otimes v_{2})\chi_{\bf
A}(v_{1}^{\dagger}\otimes v_{2}^{\dagger})$
$\displaystyle=\sum_{l\in[d]}\operatorname{Tr}\left(\left(\ket{o^{l}}_{1}\bra{o^{l}}\right)\eta_{\bf
A}\right)\ket{o^{l}}_{1}\bra{o^{l}}\otimes\eta_{A_{2}|l}$
$\displaystyle\Rightarrow\chi_{\bf A}$ $\displaystyle=(v_{1}^{\dagger}\otimes
v_{2}^{\dagger})\underbrace{\left(\sum_{l\in[d]}\operatorname{Tr}\left(\left(\ket{o^{l}}_{1}\bra{o^{l}}\right)\eta_{\bf
A}\right)\ket{o^{l}}_{1}\bra{o^{l}}\otimes\eta_{A_{2}|l}\right)}_{\mbox{CQ
state}}(v_{1}\otimes v_{2}),$ (40)

where
$\eta_{A_{2}|l}:=\operatorname{Tr}_{1}\left[\left(\ket{o^{l}}_{1}\bra{o^{l}}\right)\eta_{\bf
A}\right]/\operatorname{Tr}\left[\left(\ket{o^{l}}_{1}\bra{o^{l}}\right)\eta_{\bf
A}\right]$ represents the normalized state of subpart $A_{2}$, conditioned
that the projector $\ket{o^{l}}_{1}\bra{o^{l}}$ acts on part $A_{1}$. Since
the set of CQ states is invariant under local unitary operations and
$\eta_{\bf A}$ is a CQ state, therefore $\chi_{\bf A}$ is also a CQ state.
This completes the proof. ∎

##  V Appendix V: Proof of Proposition 2

###### Proof.

While the Proposition holds for any observable $\mathbb{O}$, here we provide
the proof by considering it to be the system’s Hamiltonian. Recall the
expression

$\displaystyle\mathbb{H}_{GN^{0}_{c}}(\rho_{{\bf
A}})=\min_{U_{1}}E(\widetilde{\chi}_{\bf A}^{p})-E(\rho_{\bf A}^{p}),$

where $\rho^{p}_{\bf A}$ and $\widetilde{\chi}^{p}_{\bf A}$ respectively
denotes the passive state corresponding to $\rho_{\bf A}$ and $N_{c}^{o}\circ
U_{1}(\rho_{\bf A})$. Let $U_{1}^{\star}$ be the optimizing unitary operation.
Then, the expression simplifies to

$\displaystyle\mathbb{H}_{GN^{0}_{c}}(\rho_{\bf A})=E(\chi_{\bf
A}^{p})-E(\rho_{\bf A}^{p}),$ (41)

with $\chi^{p}_{\bf A}$ denoting the passive state corresponding to
$N_{c}^{o}\circ U^{\star}_{1}(\rho_{\bf A})$. The evolution of the state is
thus given by

$\displaystyle\rho_{\bf A}\xrightarrow{U^{\star}_{1}}\tau_{\bf
A}=(U^{\star}_{1})\rho_{\bf
A}(U^{\star}_{1})^{\dagger}\xrightarrow{N^{0}_{c}}\chi_{\bf
A}=N^{0}_{c}(\tau_{\bf A})\xrightarrow{U_{\bf A}}\chi^{p}_{\bf A}=U_{\bf
A}\chi_{\bf A}U^{\dagger}_{\bf A}.$ (42)

Let $\sigma_{\bf A}=(V_{1}\otimes W_{2})\rho_{\bf A}(V_{1}^{\dagger}\otimes
W_{2}^{\dagger})$, where $V_{1}$ and $W_{2}$ are some arbitrary local unitary
on the $1$ and $2$ sub-parts respectively. The expression (41) in this case
reads as,

$\displaystyle\mathbb{H}_{GN^{0}_{c}}(\sigma_{\bf A})=E(\eta_{\bf
A}^{p})-E(\sigma_{\bf A}^{p}),$ (43)

and the state evolution [as describe in Eq.(42)] in this case becomes,

$\displaystyle\sigma_{\bf A}\xrightarrow{Z^{\star}_{1}}\zeta_{\bf
A}=(Z^{\star}_{1})\sigma_{\bf
A}(Z^{\star}_{1})^{\dagger}\xrightarrow{N^{0}_{c}}\eta_{\bf
A}=N^{0}_{c}(\zeta_{\bf A})\xrightarrow{U^{\prime}_{\bf A}}\eta^{p}_{\bf
A}=(U^{\prime}_{\bf A})\eta^{p}_{\bf A}(U^{\prime}_{\bf A})^{\dagger}.$ (44)

Note that $\sigma^{p}_{\bf A}=\rho^{p}_{\bf A}$ as $\rho_{\bf A}$ and
$\sigma_{\bf A}$ are unitarily connected. Therefore, to establish the claim of
the proposition, we only need to examine the invariance of $E(\eta^{p}_{\bf
A})$ and $E(\chi^{p}_{\bf A})$. The proof proceeds by discussing these aspects
in two parts.

  * P-I:

Consider that $\sigma_{\bf A}=(I_{1}\otimes W_{2})\rho_{\bf A}(I_{1}\otimes
W_{2}^{\dagger})$. The optimal state $\eta_{\bf A}$ achieved in Eq.(44) when
starting with the initial state $\sigma_{\bf A}$ can also be achieved staring
with the initial state $\rho_{\bf A}$. For that the second party applies the
unitary $W_{2}$ on the state $\chi_{\bf A}$ in Eq.(42), i.e., $\eta_{\bf
A}=(I_{1}\otimes W_{2})\chi_{\bf A}(I_{1}\otimes W_{2}^{\dagger})$, implying
$E(\chi^{p}_{\bf A})=E(\eta^{p}_{\bf A})$.

  * P-I:

Consider that $\sigma_{\bf A}=(V_{1}\otimes I_{2})\rho_{\bf
A}(V_{1}^{\dagger}\otimes I_{2}^{\dagger})$. Assume that, $E(\chi^{p}_{\bf
A})>E(\eta^{p}_{\bf A})$. On the other hand, we have

$\displaystyle\rho_{\bf A}\xrightarrow{V_{1}}\sigma_{\bf
A}\xrightarrow{Eq.(\ref{pr23})}\eta^{p}_{\bf A}.$ (45)

This implies that starting from $\rho_{\bf A}$, they can reach to the passive
state $\eta^{p}_{\bf A}$. This along with the assumption $E(\chi^{p}_{\bf
A})>E(\eta^{p}_{\bf A})$ contradicts the claim that $U^{\star}_{1}$ in Eq.(42)
is the optimal unitary, and thus $E(\chi^{p}_{\bf A})\ngtr E(\eta^{p}_{\bf
A})$.

Now, assume that $E(\chi^{p}_{\bf A})<E(\eta^{p}_{\bf A})$. Again this will
contradict the optimality of $Z^{\star}_{1}$ in Eq.(44). Therefore we have,
$E(\chi^{p}_{\bf A})=E(\eta^{p}_{\bf A})$.

(P-I) and (P-II) together thus implies $\mathbb{H}_{GN^{0}_{c}}(\rho_{\bf
A})=\mathbb{H}_{GN^{0}_{c}}\big{(}(V_{1}\otimes W_{2})\rho_{\bf
A}(V_{1}^{\dagger}\otimes W_{2}^{\dagger})\big{)}$. This completes the proof.
∎

##  VI Appendix VI: Proof of Theorem 3

###### Proof.

A general 2-qubit state may be written as

$\displaystyle\rho=\frac{1}{4}\left[\mathbf{I}+\bm{\sigma}_{1}^{\intercal}\mathbf{r}_{1}+\mathbf{r}_{2}^{\intercal}\bm{\sigma}_{2}+\bm{\sigma}_{1}^{\intercal}\bm{T}\bm{\sigma}_{2}\right],$
(46)

where $\mathbf{r}_{1}$ is the Bloch vector of subsystem $A,$
$\bm{\sigma}_{1}=(\sigma_{x},\sigma_{y},\sigma_{z})_{A}$ is the vector of 3
Pauli matrices for subsystem $A$ (same idea for $B$) and $\bm{T}$ is the
correlation matrix. These can be obtained via:

$\displaystyle\mathbf{r}_{1}={\sf
Tr}\left[\rho\bm{\sigma}_{1}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathbf{r}_{2}={\sf Tr}\left[\rho\bm{\sigma}_{2}\right],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \bm{T}={\sf
Tr}\left[\rho\bm{\sigma}_{1}\bm{\sigma}_{2}^{\intercal}\right].$ (47)

Application of local unitary $U_{1}$ is equivalent to applying a 3x3 real
rotation matrix to $\bm{\sigma}_{1}$, or equivalently replacing
$\mathbf{r}_{1}\to\bm{R}\mathbf{r}_{1}$ and $\bm{T}\to\bm{R}\bm{T}$. Then we
have

$\displaystyle\tau=U_{1}\rho
U_{1}^{\dagger}=\frac{1}{4}\left[\mathbf{I}+\bm{\sigma}_{1}^{\intercal}\bm{R}\mathbf{r}_{1}+\mathbf{r}_{2}^{\intercal}\bm{\sigma}_{2}+\bm{\sigma}_{1}^{\intercal}\bm{R}\bm{T}\bm{\sigma}_{2}\right],$
(48)

The classical channel $\mathcal{N}^{0}_{C}$ between Alice and Bob is a measure
and prepare channel. Any generic pair of qubit projectors can be written as
$P_{\pm\bm{n}}=\ket{\bm{n}_{\pm}}\bra{\bm{n}_{\pm}}=\frac{1}{2}\left(\mathbf{I}\pm\bm{\sigma}_{1}^{\intercal}\bm{n}\right),$
where $\bm{n}$ is a unit vector in the Bloch sphere (it identifies the 2
orthogonal pure states forming the eigenbasis). This projective measurement
also follows
$\bra{\bm{n}_{\pm}}\bm{\sigma}_{1}^{\intercal}\ket{\bm{n}_{\pm}}=\pm\bm{n}^{\intercal}$.
So the projected state of Bob after passing through $\mathcal{N}^{0}_{C}$ is

$\displaystyle\tau_{\pm}=\bra{\bm{n}_{\pm}}\tau\ket{\bm{n}_{\pm}}_{A}=\frac{1}{4}\left[\mathbf{I}\pm\bm{n}^{\intercal}\bm{R}\mathbf{r}_{1}+\left(\mathbf{r}_{2}^{\intercal}\pm\bm{n}^{\intercal}\bm{R}\bm{T}\right)\bm{\sigma}_{2}\right],$
(49)

The state accessible to Bob after all the above operations is
$\chi=P_{\bm{n}}\tau_{\bm{n}}+P_{-\bm{n}}\tau_{-\bm{n}}$. We know that the
eigenvalues of a qubit Hermitian operator of the form
$B=b_{0}\mathbf{I}+\bm{b}^{\intercal}\bm{\sigma}_{2}$ are
$\lambda_{\pm}=b_{0}\pm|\bm{b}|$. So we get the eigenvalues of the partial
projections:

$\displaystyle{\sf Spectrum}[\tau_{+\bm{n}}]$
$\displaystyle=\left\\{\frac{1}{4}\left[1+\bm{n}^{\intercal}\bm{R}\mathbf{r}_{1}\pm\left|\mathbf{r}_{2}^{\intercal}+\bm{n}^{\intercal}\bm{R}\bm{T}\right|\right]\right\\},$
(50a) $\displaystyle{\sf Spectrum}[\tau_{-\bm{n}}]$
$\displaystyle=\left\\{\frac{1}{4}\left[1-\bm{n}^{\intercal}\bm{R}\mathbf{r}_{1}\pm\left|\mathbf{r}_{2}^{\intercal}-\bm{n}^{\intercal}\bm{R}\bm{T}\right|\right]\right\\}.$
(50b)

Since the projectors $P_{\pm\bm{n}}$ are mutually orthogonal, the eigenvalues
of $\chi$ follow directly from our previous results:

$\displaystyle{\sf Spectrum}[\chi]=\left\\{\\!\begin{aligned}
\frac{1}{4}\left[1+\bm{n}^{\intercal}\bm{R}\mathbf{r}_{1}+\left|\mathbf{r}_{2}^{\intercal}+\bm{n}^{\intercal}\bm{R}\bm{T}\right|\right],\leavevmode\nobreak\
\frac{1}{4}\left[1+\bm{n}^{\intercal}\bm{R}\mathbf{r}_{1}-\left|\mathbf{r}_{2}^{\intercal}+\bm{n}^{\intercal}\bm{R}\bm{T}\right|\right]\\\
\frac{1}{4}\left[1-\bm{n}^{\intercal}\bm{R}\mathbf{r}_{1}+\left|\mathbf{r}_{2}^{\intercal}-\bm{n}^{\intercal}\bm{R}\bm{T}\right|\right],\leavevmode\nobreak\
\frac{1}{4}\left[1-\bm{n}^{\intercal}\bm{R}\mathbf{r}_{1}-\left|\mathbf{r}_{2}^{\intercal}-\bm{n}^{\intercal}\bm{R}\bm{T}\right|\right]\end{aligned}\right\\}.$
(51)

We note that $\bm{R}$ always appears in the form $\bm{n}^{\intercal}\bm{R}$,
so we may simplify the problem by defining a new unit vector ${\bf
m}\equiv\bm{R}^{\intercal}\bm{n}$. Now, we only need to optimize over the
vector ${\bf m}$, under the constraint $|{\bf m}|=1$, so this gives us

$\displaystyle{\sf Spectrum}[\chi]=\\{s_{ij}({\bf
m}):=\frac{1}{4}\left(1+(-1)^{i}{\bf m}^{\intercal}{\bf r}_{1}+(-1)^{j}|{\bf
r}^{\intercal}_{2}+(-1)^{i}{\bf m}^{\intercal}{\bf T}|\right)\\}_{i,j=0}^{1}.$
(52)

After arranging the spectrum in decreasing order we get $\\{s_{ij}({\bf
m})\\}\mapsto\\{q_{k}({\bf m})$}. Let $\\{E_{k}\\}$ be the non-degenerate
increasing eigenvalues of $\mathbb{H}$, $\\{p_{k}\\}$ are the decreasing
spectral of $\rho_{\bf A}$. Then the extractable ergotropy is.

$\displaystyle H_{GN^{0}_{c}}(\rho_{\bf A})$
$\displaystyle=\min_{u_{1}}E({\chi}_{\bf A}^{p})-E(\rho_{\bf
A}^{p})=\min_{m}\sum_{k=1}^{4}(q_{k}({\bf m})-p_{k})E_{k}.$ (53)

This completes the proof. ∎

###  VI.1 Proof of corollary 1

###### Proof.

A most general two-qubit state with maximally mixed marginals can be written
as $\rho_{\bf
A}=\frac{1}{4}\left[\mathbf{I}+\bm{\sigma}_{1}^{\intercal}\bm{T}\bm{\sigma}_{2}\right]$.
This state can always be written as convex mixture of Bell state by applying
local unitary $V_{1}\otimes W_{2}$, i.e.,

$\displaystyle\sigma_{\bf A}$ $\displaystyle=(V_{1}\otimes W_{2})\rho_{\bf
A}(V_{1}^{\dagger}\otimes
W_{2}^{\dagger})=\frac{1}{4}\left[\mathbf{I}+\bm{\sigma}_{1}^{\intercal}\bm{\tilde{T}}\bm{\sigma}_{2}\right]$
$\displaystyle=p_{1}\ket{\phi^{+}}\bra{\phi^{+}}+p_{2}\ket{\phi^{-}}\bra{\phi^{-}}+p_{3}\ket{\psi^{+}}\bra{\psi^{+}}+p_{4}\ket{\psi^{-}}\bra{\psi^{-}},$

where $\bm{\tilde{T}}={\sf
diag}(2(p_{1}+p_{3})-1,2(p_{2}+p_{3})-1),2(p_{1}+p_{2})-1)$. Denoting,
$\omega=\max\\{(p_{1}+p_{3}),(p_{2}+p_{3}),(p_{1}+p_{2})\\}$, we have
$\sqrt{\lambda_{\max}\left(\bm{T}\bm{T}^{\intercal}\right)}=2\omega-1$. Let us
arrange the probabilities in decreasing order,
$\\{p_{i}\\}\mapsto\\{p^{\downarrow}_{i}\\}$ with $p^{\downarrow}_{1}\geq
p^{\downarrow}_{2}\geq p^{\downarrow}_{3}\geq p^{\downarrow}_{4}$. The energy
of the $\sigma_{\bf A}^{p}$ and $\chi_{\bf A}^{p}$ are given by

$\displaystyle E(\sigma_{\bf A}^{p})=$
$\displaystyle\sum_{i=1}^{4}p_{i}^{\downarrow}E_{i},$ $\displaystyle
E(\chi_{\bf A}^{p})=$
$\displaystyle\frac{1}{4}\left(1+\sqrt{\lambda_{\max}\left(\bm{T}\bm{T}^{\intercal}\right)}\right)(E_{1}+E_{2})+\frac{1}{4}\left(1-\sqrt{\lambda_{\max}\left(\bm{T}\bm{T}^{\intercal}\right)}\right)(E_{3}+E_{4})$
$\displaystyle=$
$\displaystyle\frac{\omega}{2}(E_{1}+E_{2})+\frac{(1-\omega)}{2}(E_{3}+E_{4}).$

Thus we have,

$\displaystyle\mathbb{H}_{GN^{0}_{c}}(\rho_{\bf
A})=\frac{\omega}{2}(E_{1}+E_{2})+\frac{1-\omega}{2}\left(E_{3}+E_{4}\right)-\sum_{i}p_{i}^{\downarrow}E_{i}.$
(54)

This completes the proof. ∎