# Operational characterization of general quantum correlation via complex weak
value measurement

Agung Budiyono<EMAIL_ADDRESS>Hermawan K. Dipojono  Department of
Engineering Physics, Bandung Institute of Technology, Bandung, 40132,
Indonesia  Research Center for Nanoscience and Nanotechnology, Bandung
Institute of Technology, Bandung, 40132, Indonesia

###### Abstract

The last two decades have witnessed significant progress on the understanding
the quantum correlation more general than entanglement, wherein even a
separable state may yield correlation that cannot be emulated by any classical
object. Such a general nonclassical correlation is not only intriguing from
the fundamental point of view, but it has also been recognized as a resource
in a variety of quantum information processing tasks and quantum technology.
Here, we propose a characterization of the general quantum correlation in
bipartite system in terms of direct laboratory operations using weak
measurement with postselection. We define a quantity based on the imaginary
part of weak values obtained via weak measurement of a local basis followed by
a postselection of another local basis, and an optimization procedure over all
possible choices of the two bases. We show that it satisfies certain desirable
requirements for a quantifier of general quantum correlations. It may be
statistically interpreted as the minimum genuine quantum share of uncertainty.
It is a faithful witness of entanglement for general pure states, giving an
observable lower bound to a scaled square root of the linear entropy of
entanglement. We then suggest an information theoretic interpretation of the
general quantum correlation in a multipartite state as the minimum mean
absolute error in an optimal estimation of any local measurement basis, based
on the outcomes of local projective measurement on the state, in the worst
case scenario.

general quantum correlation, quantum incompatibility, weak value measurement

###### pacs:

03.65.Ta, 03.65.Ca

##  I Introduction

Nonclassical correlation in multipartite quantum systems was originally
associated with entangled states, namely states that cannot be prepared by any
set of local operation and classical communication (LOCC) Horodecki
entanglement review . Studies in the last couple of decades however show that
most unentangled (i.e., separable) states are somewhat nonclassically
correlated, manifested in the various forms of discord-like quantum
correlations Ollivier quantum discord paper ; Henderson quantum discord paper
; Oppenheim work deficit ; Horodecki work deficit 1 ; Horodecki work deficit 2
; Devetak work deficit ; Rajagopal entropy method for quantum correlation:
quantum deficit ; Luo measurement induced disturbance approach to quantum
correlation: quantum deficit ; Dakic necessary and sufficient condition for
geometric quantum discord ; Modi geometric discord ; Luo geometric discord ;
Monras geometric discord ; Piani geometric discord ; Luo measurement induced
nonlocality ; Piani discord generates entanglement via measurement ; Streltsov
discord generates entanglement ; Gharibian discord generates entanglement ;
Piani discord generates entanglement 2 ; Adesso discord generates entanglement
via measurement ; Giampaolo global change due to local unitary ; Roga global
change due to local unitary ; Gharibian global change due to local unitary ;
Seshadreesan global change induced by entanglement breaking channel ; Girolami
quantum correlation based on WY skew information ; Girolami quantum
correlation based on Fisher information . See Refs. Modi discord and the
boundary of quantum and classical correlation review ; Adesso quantum
correlation beyond entanglement review ; Bera quantum discord review  for
recent reviews. Conceptually, these general quantum correlations in which
entanglement is a subset, arise from the noncommutativity between the
multipartite quantum state and some set of local quantum observables, rather
than from the nonseparability of the state. The nonclassical correlations
beyond entanglement have received a growing attention both from the
fundamental view point and also practically to better understand the physical
origin of the quantum advantages in certain information processing tasks and
schemes of quantum technology which consume very little or no entanglement
Datta discord and one qubit quantum computation ; Lanyon discord and quantum
computation ; Passante discord and quantum computation ; Piani discord and no-
local broadcasting ; Dakic discord and remote state preparation ; Cavalcanti
discord and quantum communication ; Gu and discord quantum communication ;
Modi discord and quantum metrology . Different approaches have been proposed
to characterize and quantify the general quantum correlation, using quantum
information theoretical concepts which take on an input of the state of the
multipartite system. They thus indirectly need a full quantum state tomography
for its evaluation. It remains intriguing whether it is possible to
characterize and quantify general quantum correlation directly in terms of
transparent laboratory operations.

On the other hand, there is a measurement scheme in which the noncommutativity
between the state and the observable appears directly at the pointer of the
measuring device. Consider the weak measurement of a Hermitian observable $O$
with the preselected state $\varrho$ followed with the postselection on the
state $\ket{\phi}$ via a normal, i.e., strong projective measurement. One then
obtains the following complex weak value Aharonov weak value ; Wiseman weak
value ; Aharonov-Daniel book :

$\displaystyle O^{\rm
w}(\phi|\varrho)=\frac{\braket{\phi}{O\varrho}{\phi}}{\braket{\phi}{\varrho}{\phi}},$
(1)

with its real and imaginary parts that can be directly read-off respectively
from the average shift of the position and momentum of the pointer of the
measuring device Lundeen complex weak value ; Jozsa complex weak value . The
complex nature of the weak value has been exploited recently to devise a
method for direct measurement of a quantum state or its quasiprobability
representation Lundeen direct measurement of wave function ; Lundeen
measurement of KD distribution ; Bamber measurement of KD distribution .
Notice in particular that the imaginary part of the weak value, i.e.,

$\displaystyle{\rm Im}\\{O^{\rm
w}(\phi|\varrho)\\}=\frac{1}{2i}\frac{\braket{\phi}{[O,\varrho]}{\phi}}{\braket{\phi}{\varrho}{\phi}},$
(2)

directly captures the noncommutativity between the state and the observable
which is the origin of various nonclassical properties of quantum systems
encoded in the quantum state. It is thus natural to ask if one can use the
operational scheme of weak measurement with postselection to directly
characterize and quantify the quantum correlation beyond entanglement arising
from the noncommutativity between the multipartite state and some set of local
observables.

In this paper, we argue for an affirmative answer to the above question.
First, given a preselected multipartite state, we operationally define a
quantity from the imaginary part of the weak values obtained by making weak
measurement over a complete set of local orthonormal basis of the subsystems,
followed by postselection over the product of the local bases, and optimized
over all possible choices of these two bases. We demonstrate that it satisfies
certain plausible properties expected for a quantifier of quantum correlation
beyond entanglement. It sets a lower bound to the negativity of quantumness
Piani discord generates entanglement via measurement ; Nakano negativity of
quantumness trace norm l1 norm . We argue that it may be interpreted as the
minimum genuine quantum share out of the total uncertainty of the local
measurement basis in the multipartite state. For general pure states, it gives
an observable lower bound, hence a witness, to the scaled square root of the
linear entropy of entanglement. Our approach thus offers a definition of
general quantum correlation in terms of direct laboratory operations based on
weak measurement with postselection, which might suggest a fresh insight into
its role as a resource in quantum information processing. Moreover, it leads
to the following information theoretical interpretation of the general quantum
correlation: a multipartite state is quantumly correlated if and only if the
optimal estimation of any local basis, based on the outcomes of the local
projective measurement over the state, must yield a nonvanishing mean absolute
error in the worst case scenario, with the minimum value that is given by the
quantum correlation. On the other hand, a multipartite state is classical
devoid of quantum correlation if and only if, local measurement over the state
can be used to make a sharp optimal estimation of a certain local basis.

##  II Nonclassical states, noncommutativity, and nonreal weak value

Consider a composite of two subsystems $AB$ with a quantum state that is
represented by a density matrix $\varrho_{AB}$ acting on the Hilbert space
$\mathcal{H}_{AB}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, where
$\mathcal{H}_{A(B)}$ is the Hilbert space of the subsystem $A(B)$. In general
multipartite setting, a separable state is intuitively defined as a state that
can be prepared by LOCC Horodecki entanglement review . For the above
bipartite system, a separable state can thus be in general expressed as a
classical statistical mixture of the product states, i.e.,
$\varrho_{AB}^{\mathcal{S}}=\sum_{k}p_{k}\varrho_{A}^{k}\otimes\varrho_{B}^{k}$,
where $\varrho_{A(B)}^{k}$ is the density matrix of the subsystem $A(B)$ on
$\mathcal{H}_{A(B)}$, and $\\{p_{k}\\}$ are mixing probabilities: $p_{k}\geq
0$, $\sum_{k}p_{k}=1$. Any other states, i.e., nonseparable states, are
entangled. For pure states, separability is equivalent to the absence of
nonclassical correlation. Remarkably, allowing impurity to the bipartite state
may lead to nonclassical correlations even when it is separable. That is,
various different schemes have been revealed in the last couple of decades
which show that mixed separable states may contain correlation that cannot be
accessed by any classical object Modi discord and the boundary of quantum and
classical correlation review ; Adesso quantum correlation beyond entanglement
review ; Bera quantum discord review ; Ollivier quantum discord paper ;
Henderson quantum discord paper ; Oppenheim work deficit ; Horodecki work
deficit 1 ; Horodecki work deficit 2 ; Devetak work deficit ; Rajagopal
entropy method for quantum correlation: quantum deficit ; Luo measurement
induced disturbance approach to quantum correlation: quantum deficit ; Dakic
necessary and sufficient condition for geometric quantum discord ; Modi
geometric discord ; Luo geometric discord ; Monras geometric discord ; Piani
geometric discord ; Luo measurement induced nonlocality ; Piani discord
generates entanglement via measurement ; Streltsov discord generates
entanglement ; Gharibian discord generates entanglement ; Piani discord
generates entanglement 2 ; Adesso discord generates entanglement via
measurement ; Giampaolo global change due to local unitary ; Roga global
change due to local unitary ; Gharibian global change due to local unitary ;
Seshadreesan global change induced by entanglement breaking channel ; Girolami
quantum correlation based on WY skew information ; Girolami quantum
correlation based on Fisher information . For instance, such a separable state
may yield a global modification as a response to any local measurement
Henderson quantum discord paper ; Ollivier quantum discord paper ; Luo
measurement induced nonlocality .

By contrast, classical intuition suggests that for a genuine classical
multipartite state, there is at least a local measurement which does not lead
to a global modification to the state. In the bipartite setting, there are two
classes of states which conform to the above intuition Luo measurement induced
disturbance approach to quantum correlation: quantum deficit ; Piani discord
and no-local broadcasting . One is a class of so-called classical-quantum (or,
$A-$classically correlated) states, which are those that can be expressed as

$\displaystyle\varrho_{AB}^{\mathcal{C}Q}\doteq\sum_{k}p_{k}\ket{k}\bra{k}_{A}\otimes\varrho_{B}^{k},$
(3)

where $\\{\ket{k}_{A}\\}$ is an orthonormal basis of the Hilbert space
$\mathcal{H}_{A}$ of subsystem $A$, $\\{\varrho_{B}^{k}\\}\in\mathcal{H}_{B}$
are states of subsystem $B$, and $\\{p_{k}\\}$ are probabilities, i.e.,
$p_{k}\geq 0$, $\sum_{k}p_{k}=1$. The quantum-classical (or, $B$-classically
correlated) state is defined analogously by exchanging the role of the parties
$A$ and $B$. The other class consists of bipartite states with density matrix
which take the following form:

$\displaystyle\varrho_{AB}^{\mathcal{C}C}\doteq\sum_{k,l}p_{kl}\ket{k}\bra{k}_{A}\otimes\ket{l}\bra{l}_{B},$
(4)

where $\\{\ket{k}_{A}\\}$ and $\\{\ket{l}_{B}\\}$ are respectively orthonormal
basis of subsystem $A$ and $B$, and $\\{p_{kl}\\}$ are joint probabilities so
that $p_{kl}\geq 0$, $\sum_{k,l}p_{kl}=1$. Such states are called classical-
classical (or, totally classically correlated) states. One can see that
applying local projective (von Neumann) measurement
$\\{\Pi_{A}^{k}\otimes\mathbb{I}_{B}\\}$ and
$\\{\Pi_{A}^{k}\otimes\Pi_{B}^{l}\\}$ (where $\Pi^{x}\doteq\ket{x}\bra{x}$ is
one-dimensional projector onto the subspace spanned by $\ket{x}$),
respectively, to the classical-quantum states and classical-classical states,
without learning or recording the outcomes, leave them unmodified, i.e.,
$\sum_{k}(\Pi_{A}^{k}\otimes\mathbb{I}_{B})\varrho_{AB}^{\mathcal{C}Q}(\Pi_{A}^{k}\otimes\mathbb{I}_{B})=\varrho_{AB}^{\mathcal{C}Q}$
and
$\sum_{k,l}(\Pi_{A}^{k}\otimes\Pi_{B}^{l})\varrho_{AB}^{\mathcal{C}C}(\Pi_{A}^{k}\otimes\Pi_{B}^{l})=\varrho_{AB}^{\mathcal{C}C}$.
This can be seen as due to the fact that the local measurement basis and the
bipartite state commute, i.e.,
$[(\Pi_{A}^{k}\otimes\mathbb{I}_{B}),\varrho_{AB}^{\mathcal{C}Q}]=0$ and
$[(\Pi_{A}^{k}\otimes\Pi_{B}^{l}),\varrho_{AB}^{\mathcal{C}C}]=0$, for all
$k$, $l$. Any bipartite state that cannot be represented either in the forms
of Eqs. (3) or (4) thus contain nonclassical correlation arising from the
noncommutativity between the state and the local measurement basis, even if it
is separable.

Various mathematical characterizations of the classical bipartite states of
the types of Eqs. (3) and (4) have led researchers to develop different
quantifiers and measures of quantum correlation beyond entanglement, by
applying quantum information theoretical concepts: quantum mutual information,
von-Neumann entropy, distance and infidelity between two density matrices,
Wigner-Yanase skew information or quantum Fisher information Adesso quantum
correlation beyond entanglement review . All these quantifiers essentially
quantify the difference between the bipartite state under scrutiny and the
classical bipartite states of Eqs. (3) and (4). Moreover, they indirectly or
directly reflects the noncommutativity between the bipartite states and some
set of local observables. To evaluate the nonclassical correlation in a
bipartite quantum state using these quantifiers, we typically need an input of
the associated density matrix, thus requires a tomography of the full quantum
state. In order to better understand the operational meaning of the quantum
correlation beyond entanglement, it is desirable to have a quantifier of
general quantum correlation whose definition directly and transparently
translates into laboratory operations.

On the other hand, as noted in the previous section, the imaginary part of the
weak value can be used to directly probe the noncommutativity between a
preselected bipartite quantum state and the local measurement basis as
follows. Suppose we wish to probe the noncommutativity between a bipartite
state $\varrho_{AB}$ and a local measurement basis $\\{\Pi_{A}^{k}\\}$ on
subsystem $A$. To do this, one first makes a weak measurement of
$\Pi_{A}^{k}\otimes\mathbb{I}_{B}$ with the preselected state $\varrho_{AB}$
followed by a postselection on $\ket{\phi_{AB}}$, reads the imaginary part of
the weak value from the average shift of the pointer momentum, and multiply it
with the probability of the successful postselection, to get

$\displaystyle
C_{A}[\Pi_{A}^{k}\otimes\mathbb{I}_{B};\varrho_{AB},\ket{\phi_{AB}}]$ (5)
$\displaystyle\doteq$ $\displaystyle{\rm
Im}\Big{\\{}\frac{\braket{\phi_{AB}}{(\Pi_{A}^{k}\otimes\mathbb{I}_{B})\varrho_{AB}}{\phi_{AB}}}{\braket{\phi_{AB}}{\varrho_{AB}}{\phi_{AB}}}\Big{\\}}\braket{\phi_{AB}}{\varrho_{AB}}{\phi_{AB}}$
$\displaystyle=$
$\displaystyle\frac{1}{2i}\braket{\phi_{AB}}{[(\Pi_{A}^{k}\otimes\mathbb{I}_{B}),\varrho_{AB}]}{\phi_{AB}}.$

One can then see that for the classical-quantum state of Eq. (3), there is a
local measurement basis $\\{\Pi_{A}^{k}\\}$ such that
$[(\Pi_{A}^{k}\otimes\mathbb{I}_{B}),\varrho_{AB}^{\mathcal{C}Q}]=0$, and we
have
$C_{A}[\Pi_{A}^{k}\otimes\mathbb{I}_{B};\varrho_{AB}^{\mathcal{C}Q},\ket{\phi_{AB}}]=0$,
for all $k$. Moreover, choosing a postselection state $\ket{\phi_{AB}}$ such
that its associated density matrix does not commute with the preselected state
$\varrho_{AB}$ and the local measurement $\Pi_{A}^{a}\otimes\mathbb{I}_{B}$,
then $C_{A}[\Pi_{A}^{a}\otimes\mathbb{I}_{B};\varrho_{AB},\ket{\phi_{AB}}]=0$
implies $[\Pi_{A}^{a}\otimes\mathbb{I}_{B},\varrho_{AB}]=0$. Hence, in this
setting, a nonvanishing
$C_{A}[\Pi_{A}^{a}\otimes\mathbb{I}_{B};\varrho_{AB},\ket{\phi_{AB}}]$
captures the noncommutativity between $\Pi_{A}^{a}\otimes\mathbb{I}_{B}$ and
$\varrho_{AB}$. A similar observation applies to the set of classical-
classical state. Namely, for each classical-classical state of Eq. (4), there
must be a pair of local bases $\\{\Pi_{A}^{k}\\}$ on $A$ and
$\\{\Pi_{B}^{l}\\}$ on $B$ so that
$[\Pi_{A}^{k}\otimes\Pi_{B}^{l},\varrho_{AB}^{\mathcal{C}C}]=0$, and we have
$C_{A}[\Pi_{A}^{k}\otimes\Pi_{B}^{l};\varrho_{AB},\ket{\phi_{AB}}]=0$.
Moreover, choosing $\ket{\phi_{AB}}$ whose density matrix does not commute
with $\varrho_{AB}$ and $\Pi_{A}^{a}\otimes\Pi_{B}^{b}$, then
$C_{A}[\Pi_{A}^{a}\otimes\Pi_{B}^{b};\varrho_{AB},\ket{\phi_{AB}}]=0$ implies
$[\Pi_{A}^{a}\otimes\Pi_{B}^{b},\varrho_{AB}]=0$. We note that the
noncommutativity between the state and a Hermitian observable can also be
directly checked via quantum variance capturing the uncertainty of the
observable in the state. However, quantum variance also takes the contribution
of the uncertainty due to the statistical, i.e., classical mixing of states,
so that it is not genuinely quantum.

##  III General nonclassical correlation from local weak value measurement

We want to devise a quantifier of general quantum correlation in a bipartite
state from
$C_{A}[\Pi_{A}^{k}\otimes\mathbb{I}_{B};\varrho_{AB},\ket{\phi_{AB}}]$ defined
in Eq. (5) so that it can be observed directly using weak measurement with
postselection. To this end, given an arbitrary bipartite quantum state
$\varrho_{AB}$, let us define the following quantity which maps the state to a
nonnegative real number:

$\displaystyle C^{\mathcal{Q}}_{A}[\varrho_{AB}]$ (6) $\displaystyle\doteq$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,\phi_{A},\phi_{B}}\big{|}C_{A}[\Pi_{A}^{a}\otimes\mathbb{I}_{B};\varrho_{AB},\ket{\phi_{A},\phi_{B}}]\big{|}$
$\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,\phi_{A},\phi_{B}}\big{|}{\rm
Im}\Big{\\{}\frac{\braket{\phi_{A},\phi_{B}}{(\Pi_{A}^{a}\otimes\mathbb{I}_{B})\varrho_{AB}}{\phi_{A},\phi_{B}}}{\braket{\phi_{A},\phi_{B}}{\varrho_{AB}}{\phi_{A},\phi_{B}}}\Big{\\}}\big{|}$
$\displaystyle\hskip
76.82243pt\times\braket{\phi_{A},\phi_{B}}{\varrho_{AB}}{\phi_{A},\phi_{B}}$
$\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,\phi_{A},\phi_{B}}\big{|}{\rm
Im}\\{\braket{\phi_{A},\phi_{B}}{(\Pi_{A}^{a}\otimes\mathbb{I}_{B})\varrho_{AB}}{\phi_{A},\phi_{B}}\\}\big{|}$
$\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,\phi_{A},\phi_{B}}\frac{1}{2}\big{|}\braket{\phi_{A},\phi_{B}}{[(\Pi_{A}^{a}\otimes\mathbb{I}_{B}),\varrho_{AB}]}{\phi_{A},\phi_{B}}\big{|},$

where $\\{\ket{\phi_{A},\phi_{B}}\\}$ is the set of postselection basis of the
Hilbert space $\mathcal{H}_{AB}$, assumed to be given by the tensor product
$\\{\ket{\phi_{A},\phi_{B}}\\}=\\{\ket{\phi_{A}}\otimes\ket{\phi_{B}}\\}$,
with $\\{\ket{\phi_{A(B)}}\\}$ is the basis of the Hilbert space
$\mathcal{H}_{A(B)}$. From the second line,
$C^{\mathcal{Q}}_{A}[\varrho_{AB}]$ can thus be directly translated
operationally as follows. We first make weak measurement of the local
measurement basis $\\{\Pi_{A}^{a}\\}$ on the subsystem $A$ with a preselected
bipartite state $\varrho_{AB}$ and a postselection given by the product basis
$\\{\ket{\phi_{A},\phi_{B}}\\}$, read the absolute imaginary part from the
average shift of the pointer momentum, average over the probability of
successful postselection, maximize over all product of postselection bases
$\\{\ket{\phi_{A},\phi_{B}}\\}\in\mathcal{B}_{\\{\ket{\phi_{A}}\otimes\ket{\phi_{B}}\\}}$,
and finally minimize over all possible local measurement bases
$\\{\Pi_{A}^{a}\\}\in\mathcal{B}_{\\{\Pi_{A}^{a}\\}}$. Here,
$\mathcal{B}_{\\{\ket{\phi_{A}}\otimes\ket{\phi_{B}}\\}}$ and
$\mathcal{B}_{\\{\Pi_{A}^{a}\\}}$ denote respectively the set of all local
postselection bases $\\{\ket{\phi_{A},\phi_{B}}\\}$ of $\mathcal{H}_{AB}$ and
local measurement bases $\\{\Pi_{A}^{a}\\}$ of $\mathcal{H}_{A}$.
$C_{A}^{\mathcal{Q}}[\varrho_{AB}]$ is intended to quantify the nonclassical
correlation in the bipartite state $\varrho_{AB}$ arising from the
noncommutativity between $\varrho_{AB}$ and any local measurement basis
$\\{\Pi_{A}^{a}\\}$ of the subsystem $A$. Similarly, to quantify the
nonclassical correlation in the bipartite state $\varrho_{AB}$ which arises
from the noncommutativity between $\varrho_{AB}$ and any local measurement
basis $\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\\}$, we define the following quantity
which maps the density matrix to a nonnegative real number:

$\displaystyle C^{\mathcal{Q}}_{AB}[\varrho_{AB}]$ (7) $\displaystyle\doteq$
$\displaystyle\min_{\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,b,\phi_{A},\phi_{B}}\big{|}{\rm
Im}\Big{\\{}\frac{\braket{\phi_{A},\phi_{B}}{(\Pi_{A}^{a}\otimes\Pi_{B}^{b})\varrho_{AB}}{\phi_{A},\phi_{B}}}{\braket{\phi_{A},\phi_{B}}{\varrho_{AB}}{\phi_{A},\phi_{B}}}\Big{\\}}\big{|}$
$\displaystyle\hskip
93.89409pt\times\braket{\phi_{A},\phi_{B}}{\varrho_{AB}}{\phi_{A},\phi_{B}}$
$\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,b,\phi_{A},\phi_{B}}\big{|}{\rm
Im}\\{\braket{\phi_{A},\phi_{B}}{(\Pi_{A}^{a}\otimes\Pi_{B}^{b})\varrho_{AB}}{\phi_{A},\phi_{B}}\\}\big{|}$
$\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,b,\phi_{A},\phi_{B}}\frac{1}{2}\big{|}\braket{\phi_{A},\phi_{B}}{[(\Pi_{A}^{a}\otimes\Pi_{B}^{b}),\varrho_{AB}]}{\phi_{A},\phi_{B}}\big{|},$

where, in this case, the minimization is carried out over all possible local
measurement bases
$\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\\}\in\mathcal{B}_{\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\\}}$
of $\mathcal{H}_{AB}$.

Mathematically, the maximization over the postselection basis
$\\{\ket{\phi_{A},\phi_{B}}\\}$ is necessary to avoid trivial result that for
any local measurement basis $\\{\Pi_{A}^{a}\\}$, one can choose the
postselection basis as $\\{\ket{\phi_{A}}\\}=\\{\ket{a}\\}$, so that
$C_{A}[\Pi_{A}^{a}\otimes\mathbb{I}_{B};\varrho_{AB},\ket{\phi_{A},\phi_{B}}]=0$.
For a similar reason, we emphasize that the search for the minimum over the
local measurement bases is performed after the maximization over the
postselection bases is done. Otherwise, for each postselection basis
$\\{\ket{\phi_{A},\phi_{B}}\\}$, we can always choose the local measurement
basis $\\{\Pi_{A}^{a}\\}=\\{\ket{\phi_{A}}\bra{\phi_{A}}\\}$ to get
$C_{A}[\Pi_{A}^{a}\otimes\mathbb{I}_{B};\varrho_{AB},\ket{\phi_{A},\phi_{B}}]=0$.
Conceptually, for a given local measurement basis $\\{\Pi_{A}^{a}\\}$ (or,
$\\{(\Pi_{A}^{a}\otimes\Pi_{B}^{b})\\}$), maximizing over all possible
postselection bases $\\{\ket{\phi_{A},\phi_{B}}\\}$ means that we are seeking
for the largest incompatibility between
$\\{(\Pi_{A}^{a}\otimes\mathbb{I}_{B})\\}$ (or,
$\\{(\Pi_{A}^{a}\otimes\Pi_{B}^{b})\\}$) and $\varrho_{AB}$. On the other
hand, the minimization over the local measurement bases $\\{\Pi_{A}^{a}\\}$
for $C_{A}^{\mathcal{Q}}[\varrho_{AB}]$ (respectively, over
$\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\\}$ for
$C_{AB}^{\mathcal{Q}}[\varrho_{AB}]$) means that we search for those
classical-quantum state of Eq. (3) (respectively, classical-classical state of
Eq. (4)) that is least incompatible with $\varrho_{AB}$.

We further emphasize that the projective postselection measurement is over
product of local bases, i.e.,
$\\{\ket{\phi_{A},\phi_{B}}\\}=\\{\ket{\phi_{A}}\otimes\ket{\phi_{B}}\\}$, so
as not to generate nonclassical correlation. Hence, both the weak measurement
and the postselection are local operations, so that any nonclassical
correlation captured by $C_{A}^{\mathcal{Q}}[\varrho_{AB}]$ or
$C_{AB}^{\mathcal{Q}}[\varrho_{AB}]$ must originate inherently from the
bipartite state under scrutiny. We shall here on refer to
$C_{A}^{\mathcal{Q}}[\varrho_{AB}]$ of Eq. (6) and
$C_{AB}^{\mathcal{Q}}[\varrho_{AB}]$ of Eq. (7), respectively, as one-sided
and two-sided w-nonclassical correlation. Finally, the above definition can be
extended to more than two parties straightforwardly, while retaining clear
operational meaning. For example, the one-sided w-nonclassical correlation in
the state $\varrho_{ABC}$ of tripartite system $ABC$ is defined as
$C_{A}^{\mathcal{Q}}[\varrho_{ABC}]\doteq\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B},\phi_{C}}\\}}\sum_{a,\phi_{A},\phi_{B},\phi_{C}}|{\rm
Im}\\{\braket{\phi_{A},\phi_{B},\phi_{C}}{(\Pi_{A}^{a}\otimes\mathbb{I}_{B}\otimes\mathbb{I}_{C})\varrho_{ABC}}{\phi_{A},\phi_{B},\phi_{C}}\\}|$,
where the postselection on the subsystem $C$ is given by the orthornormal
basis of subsystem $C$. Let us mention that completely different approaches to
characterize general quantum correlation based on weak measurement but without
postselection, and thus independent of the concept of weak value, were
reported in Refs. Singh quantum correlation from weak measurement ; Wang
quantum correlation from weak measurement .

Consider first the one-sided w-nonclassical correlation
$C^{\mathcal{Q}}_{A}[\varrho_{AB}]$ defined in Eq. (6). We argue that it
satisfies the following desirable properties for a quantifier of quantum
correlation Adesso quantum correlation beyond entanglement review :

  1. (i)

Faithful, i.e., $C^{\mathcal{Q}}_{A}[\varrho_{AB}]$ is vanishing if and only
if the bipartite state $\varrho_{AB}$ belongs to the class of classical-
quantum state of Eq. (3).

  2. (ii)

Invariant under local unitary transformation, i.e.,
$C^{\mathcal{Q}}_{A}[(U_{A}\otimes U_{B})\varrho_{AB}(U_{A}^{\dagger}\otimes
U_{B}^{\dagger})]=C^{\mathcal{Q}}_{A}[\varrho_{AB}]$, where $U_{A(B)}$ is any
unitary operator applying locally on subsystem $A(B)$.

  3. (iii)

Nonincreasing under any local completely positive trace preserving (CPTP)
operation $\Lambda_{B}$ on subsystem $B$, namely on the party whose
nonclassical correlation is not being quantified, i.e.,
$C^{\mathcal{Q}}_{A}[(\mathbb{I}_{A}\otimes\Lambda_{B})\varrho_{AB}]\leq
C^{\mathcal{Q}}_{A}[\varrho_{AB}]$.

Let us sketch the proofs of the above properties and discuss a couple of
others.

To establish property (i), first, let us suppose that the bipartite state
$\varrho_{AB}$ belongs to the class of classical-quantum state, namely there
exists an orthonormal basis $\\{\Pi_{A}^{k}=\ket{k}\bra{k}_{A}\\}$ on the
subsystem $A$ so that $\varrho_{AB}$ can be expressed as in Eq. (3). Then, we
can choose $\\{\Pi_{A}^{k}\\}$ as the local measurement basis in the
definition of $C^{\mathcal{Q}}_{A}[\varrho_{AB}]$ of Eq. (6). In this case,
since $[(\Pi_{A}^{k}\otimes\mathbb{I}_{B}),\varrho_{AB}]=0$ for all $k$, we
have $C^{\mathcal{Q}}_{A}[\varrho_{AB}]=0$. Conversely, suppose
$C^{\mathcal{Q}}_{A}[\varrho_{AB}]=0$. Then, from the definition of
$C^{\mathcal{Q}}_{A}[\varrho_{AB}]$ in Eq. (6), there must be an orthonormal
measurement basis $\\{\Pi_{A}^{k}\\}$ of the subsystem $A$ so that
$\braket{\phi_{A},\phi_{B}}{[(\Pi_{A}^{k}\otimes\mathbb{I}_{B}),\varrho_{AB}]}{\phi_{A},\phi_{B}}=0$
for all $k$ and for all possible choices of the postselection bases
$\\{\ket{\phi_{A},\phi_{B}}\\}$. This can only be true if
$[(\Pi_{A}^{k}\otimes\mathbb{I}_{B}),\varrho_{AB}]=0$ for all $k$. It first
implies that $\varrho_{AB}$ is separable. Moreover, it also implies that
$[\Pi_{A}^{k},\varrho_{A}]=0$ for all $k$, where $\varrho_{A}={\rm
Tr}_{B}\varrho_{AB}$, which means that $\\{\Pi_{A}^{k}\\}$ is the
eigenprojectors of $\varrho_{A}$. Hence, we have
$\varrho_{A}=\sum_{k}p_{k}\Pi_{A}^{k}$ for some $\\{p_{k}\\}$, $p_{k}\geq 0$,
$\sum_{k}p_{k}=1$. Finally, any separable state $\varrho_{AB}$ with the
reduced density matrix $\varrho_{A}=\sum_{k}p_{k}\Pi_{A}^{k}$ must take the
form of classical-quantum state of Eq. (3).

The property (ii) of invariant under local unitary transformation can be shown
as follows:

$\displaystyle C^{\mathcal{Q}}_{A}[(U_{A}\otimes
U_{B})\varrho_{AB}(U_{A}^{\dagger}\otimes U_{B}^{\dagger})]$ (8)
$\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,\phi_{A},\phi_{B}}\big{|}{\rm
Im}\\{\langle\phi_{A},\phi_{B}|(U_{A}\otimes U_{B})(U_{A}^{\dagger}\otimes
U_{B}^{\dagger})$ $\displaystyle\hskip 56.9055pt\cdot\hskip
2.84526pt(\Pi_{A}^{a}\otimes\mathbb{I}_{B})(U_{A}\otimes
U_{B})\varrho_{AB}(U_{A}^{\dagger}\otimes
U_{B}^{\dagger})|\phi_{A},\phi_{B}\rangle\\}\big{|}$ $\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,\phi_{A}^{\prime},\phi_{B}^{\prime}}\big{|}{\rm
Im}\\{\langle\phi_{A}^{\prime},\phi_{B}^{\prime}|(U_{A}\Pi_{A}^{a}U_{A}^{\dagger}\otimes\mathbb{I}_{B})\varrho_{AB}|\phi_{A}^{\prime},\phi_{B}^{\prime}\rangle\\}\big{|}$
$\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a^{\prime}}\\}}\max_{\\{\ket{\phi_{A}^{\prime},\phi_{B}^{\prime}}\\}}\sum_{a,\phi_{A}^{\prime},\phi_{B}^{\prime}}\big{|}{\rm
Im}\\{\braket{\phi_{A}^{\prime},\phi_{B}^{\prime}}{(\Pi_{A}^{a^{\prime}}\otimes\mathbb{I}_{B})\varrho_{AB}}{\phi_{A}^{\prime},\phi_{B}^{\prime}}\\}\big{|}$
$\displaystyle=$ $\displaystyle C^{\mathcal{Q}}_{A}[\varrho_{AB}].$

Here, we have inserted an identity $(U_{A}\otimes
U_{B})(U_{A}^{\dagger}\otimes U_{B}^{\dagger})=\mathbb{I}$ in the second line.
To get the third line, we have defined a new postselection basis:
$\\{\ket{\phi^{\prime}_{A}}\otimes\ket{\phi^{\prime}_{B}}=(U_{A}^{\dagger}\otimes
U_{B}^{\dagger})\ket{\phi_{A}}\otimes\ket{\phi_{B}}\\}$, where we have made
use of the assumption that the original postselection basis is a tensor
product of the local bases of subsystems $A$ and $B$. Moreover, the fourth
line is obtained by identifying a new set of local measurement basis as
$\\{\Pi_{A}^{a^{\prime}}\\}=\\{U_{A}\Pi_{A}^{a}U_{A}^{\dagger}\\}$. Noting
that the above transformation of bases do not change the sets
$\mathcal{B}_{\\{\ket{\phi_{A}}\otimes\ket{\phi_{B}}\\}}$ and
$\mathcal{B}_{\\{\Pi_{A}^{a}\\}}$ over which we respectively perform the
maximization and the minimization in Eq. (6), we thus have
$\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}(\cdot)=\max_{\\{\ket{\phi_{A}^{\prime},\phi_{B}^{\prime}}\\}}(\cdot)$
and
$\min_{\\{\Pi_{A}^{a}\\}}(\cdot)=\min_{\\{\Pi_{A}^{a^{\prime}}\\}}(\cdot)$.
This observation gives the last equality in Eq. (8).

To establish the property (iii), first, we recall that according to the
Stinespring’s theorem, any CPTP operation or quantum channel can be
implemented by a dilation on a larger Hilbert space, wherein the system is
made contact with an ancilla in a state $\varrho_{E}$ on the Hilbert space
$\mathcal{H}_{E}$, let them interact via some global unitary, and then
followed by partial tracing over the ancilla as:
$(\mathbb{I}_{A}\otimes\Lambda_{B})[\varrho_{AB}]={\rm
Tr}_{E}\\{(\mathbb{I}_{A}\otimes
U_{BE})(\varrho_{AB}\otimes\varrho_{E})(\mathbb{I}_{A}\otimes
U_{BE})^{\dagger}\\}$, where $U_{BE}$ is the unitary interaction applying on
the subsystem $B$ and the ancilla $E$. Using this, we then have

$\displaystyle
C^{\mathcal{Q}}_{A}[(\mathbb{I}_{A}\otimes\Lambda_{B})\varrho_{AB}]$ (9)
$\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,\phi_{A},\phi_{B}}\big{|}{\rm
Im}\\{\langle\phi_{A},\phi_{B}|(\Pi_{A}^{a}\otimes\mathbb{I}_{B})$
$\displaystyle\hskip 56.9055pt\cdot\hskip 2.84526pt{\rm
Tr}_{E}\\{(\mathbb{I}_{A}\otimes
U_{BE})(\varrho_{AB}\otimes\varrho_{E})(\mathbb{I}_{A}\otimes
U_{BE})^{\dagger}\\}|\phi_{A},\phi_{B}\rangle\\}\big{|}$ $\displaystyle\leq$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,\phi_{A},\phi_{B},\phi_{E}}\big{|}{\rm
Im}\\{\langle\phi_{A},\phi_{B},\phi_{E}|(\Pi_{A}^{a}\otimes\mathbb{I}_{B}\otimes\mathbb{I}_{E})$
$\displaystyle\hskip 56.9055pt\cdot\hskip 2.84526pt(\mathbb{I}_{A}\otimes
U_{BE})(\varrho_{AB}\otimes\varrho_{E})(\mathbb{I}_{A}\otimes
U_{BE})^{\dagger}|\phi_{A},\phi_{B},\phi_{E}\rangle\\}\big{|}$
$\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,\phi_{A},\phi_{B},\phi_{E}}\big{|}{\rm
Im}\\{\langle\phi_{A},\phi_{B}^{\prime},\phi_{E}|(\Pi_{A}^{a}\otimes\mathbb{I}_{B}\otimes\mathbb{I}_{E})$
$\displaystyle\hskip 56.9055pt\cdot\hskip
2.84526pt(\varrho_{AB}\otimes\varrho_{E})|\phi_{A},\phi_{B}^{\prime},\phi_{E}\rangle\\}\big{|},$

where the basis $\\{\ket{\phi_{E}}\\}$ of the Hilbert space $\mathcal{H}_{E}$
of the ancilla inserted in the second line is arbitrary, and we have chosen
$\\{\ket{\phi_{E}}\\}$ so that the unitarily transformed postselection basis
of the whole system-ancilla is again factorizable, i.e.,
$\\{(\mathbb{I}_{A}\otimes
U_{BE})^{\dagger}\ket{\phi_{A}}\otimes\ket{\phi_{B}}\otimes\ket{\phi_{E}}\\}=\ket{\phi_{A}}\otimes\ket{\phi^{\prime}_{B}}\otimes\ket{\phi_{E}}\\}$.
Such a basis can always be chosen given the interaction unitary $U_{BE}$. To
see this, we note that the interaction unitary must take the form
$U_{BE}=e^{-ig(O_{B},O_{E})}$, where $g(O_{B},O_{E})$ is the Hermitian
generator, and $O_{B}$ and $O_{E}$ are Hermitian operators acting respectively
on the subsystem $B$ and the ancilla $E$. We then choose
$\\{\ket{\phi_{E}}\\}$ as the eigenvectors of $O_{E}$ with the eigenvalues
$\\{\phi_{E}\\}$ so that
$U_{BE}^{\dagger}\ket{\phi_{B}}\otimes\ket{\phi_{E}}=e^{ig(O_{B},\phi_{E})}\ket{\phi_{B}}\otimes\ket{\phi_{E}}=\ket{\phi_{B}^{\prime}}\otimes\ket{\phi_{E}}$,
where $\ket{\phi_{B}^{\prime}}\doteq e^{ig(O_{B},\phi_{E})}\ket{\phi_{B}}$, as
claimed. Note that $\ket{\phi_{B}^{\prime}}$ now depends on the index
$\phi_{E}$. Finally, from Eq. (9), noting that
$\braket{\phi_{E}}{\varrho_{E}}{\phi_{E}}$ is nonnegative and real, and
$\sum_{\phi_{E}}\braket{\phi_{E}}{\varrho_{E}}{\phi_{E}}=1$, we obtain:

$\displaystyle
C^{\mathcal{Q}}_{A}[(\mathbb{I}_{A}\otimes\Lambda_{B})\varrho_{AB}]$ (10)
$\displaystyle\leq$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}^{\prime}}\\}}\sum_{a,\phi_{A},\phi^{\prime}_{B},\phi_{E}}\big{|}{\rm
Im}\\{\braket{\phi_{A},\phi_{B}^{\prime}}{(\Pi_{A}^{a}\otimes\mathbb{I}_{B})\varrho_{AB}}{\phi_{A},\phi_{B}^{\prime}}\\}\big{|}\braket{\phi_{E}}{\varrho_{E}}{\phi_{E}}$
$\displaystyle=$
$\displaystyle\sum_{\phi_{E}}C^{\mathcal{Q}}_{A}[\varrho_{AB}]\braket{\phi_{E}}{\varrho_{E}}{\phi_{E}}=C^{\mathcal{Q}}_{A}[\varrho_{AB}],$

where, in the last line, we have made use of the fact that, for each
$\phi_{E}$, the set $\mathcal{B}_{\\{\ket{\phi_{A},\phi_{B}^{\prime}}\\}}$ of
the postselection bases $\\{\ket{\phi_{A},\phi_{B}^{\prime}}\\}$ is the same
as the set $\mathcal{B}_{\\{\ket{\phi_{A},\phi_{B}}\\}}$ of the postselection
bases $\\{\ket{\phi_{A},\phi_{B}}\\}$ so that
$\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}(\cdot)=\max_{\\{\ket{\phi_{A},\phi_{B}^{\prime}}\\}}(\cdot)$
which is independent of $\phi_{E}$.

Besides satisfying the above three plausible constraints, the one-sided
w-nonclassical correlation $C^{\mathcal{Q}}_{A}[\varrho_{AB}]$ also has the
following properties. First, it is convex, i.e.,
$C^{\mathcal{Q}}_{A}[\sum_{k}p_{k}\varrho_{AB}^{k}]\leq\sum_{k}p_{k}C^{\mathcal{Q}}_{A}[\varrho_{AB}^{k}]$,
where $\\{p_{k}\\}$ are probabilities, due to triangle inequality and the fact
that $p_{k}\geq 0$. Moreover, the one-sided w-nonclassical correlation is
nonincreasing when one or more of the parties are discarded (ignored). To show
this, without losing generality, consider for example a tripartite state
$\varrho_{ABC}$. One then has

$\displaystyle C_{A}^{\mathcal{Q}}[\varrho_{ABC}]$ (11) $\displaystyle\doteq$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B},\phi_{C}}\\}}\sum_{a,\phi_{A},\phi_{B},\phi_{C}}|{\rm
Im}\\{\braket{\phi_{A},\phi_{B},\phi_{C}}{(\Pi_{A}^{a}\otimes\mathbb{I}_{B}\otimes\mathbb{I}_{C})\varrho_{ABC}}{\phi_{A},\phi_{B},\phi_{C}}\\}|$
$\displaystyle\geq$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,\phi_{A},\phi_{B}}|{\rm
Im}\\{\braket{\phi_{A},\phi_{B}}{(\Pi_{A}^{a}\otimes\mathbb{I}_{B})\varrho_{AB}}{\phi_{A},\phi_{B}}\\}|$
$\displaystyle=$ $\displaystyle C_{A}^{\mathcal{Q}}[\varrho_{AB}],$

where
$\varrho_{AB}=\sum_{\phi_{C}}\braket{\phi_{C}}{\varrho_{ABC}}{\phi_{C}}={\rm
Tr}_{C}\\{\varrho_{ABC}\\}$. The equality is reached when the subsystem $C$ is
totally uncorrelated with the rest, i.e.,
$C_{A}^{\mathcal{Q}}[\varrho_{AB}\otimes\varrho_{C}]=C_{A}^{\mathcal{Q}}[\varrho_{AB}]$,
by virtue of the fact that $\braket{\phi_{C}}{\varrho_{C}}{\phi_{C}}$ is real
and nonnegative and
$\sum_{\phi_{C}}\braket{\phi_{C}}{\varrho_{C}}{\phi_{C}}=1$. Hence, adding or
removing an (independent) ancilla does not change the w-nonclassical
correlation as intuitively expected. The above property straightforwardly
generalizes to any number of parties as $C_{1}^{\mathcal{Q}}[\varrho_{12\dots
N}]\geq C_{1}^{\mathcal{Q}}[\varrho_{12\dots N-1}]\geq\cdots\geq
C_{1}^{\mathcal{Q}}[\varrho_{123}]\geq C_{1}^{\mathcal{Q}}[\varrho_{12}]$,
where $\varrho_{1\dots k}={\rm Tr}_{k+1\dots N}\\{\varrho_{1\dots N}\\}$.

One can check that the two-sided w-nonclassical correlation
$C_{AB}^{\mathcal{Q}}[\varrho_{AB}]$ defined in Eq. (7) also satisfies
property (i) of faithfulness, i.e., it is vanishing if and only if the
bipartite state $\varrho_{AB}$ takes the form of classical-classical state of
Eq. (4). Moreover, it is also invariant under local unitary transformation
satisfying property (ii). Property (iii), however, cannot be defined for the
two-sided quantum correlation $C_{AB}^{\mathcal{Q}}[\varrho_{AB}]$ in the
bipartite setting. Instead, one can prove for a tripartite state that
$C_{AB}^{\mathcal{Q}}[(\mathbb{I}_{A}\otimes\mathbb{I}_{B}\otimes\Lambda_{C})\varrho_{ABC}]\leq
C_{AB}^{\mathcal{Q}}[\varrho_{ABC}]$, where $\Lambda_{C}$ is a completely
positive trace preserving operation on $C$. One can also easily see that
$C_{AB}^{\mathcal{Q}}[\varrho_{AB}]$ is convex. Moreover, discarding one or
more parties does not increase the w-nonclassical correlation, e.g., we have
$C_{AB}^{\mathcal{Q}}[\varrho_{AB}]\leq C_{AB}^{\mathcal{Q}}[\varrho_{ABC}]$,
$\varrho_{AB}={\rm Tr}_{C}\\{\varrho_{ABC}\\}$, with equality when
$\varrho_{ABC}=\varrho_{AB}\otimes\varrho_{C}$. Beside these, the two-sided
w-nonclassical correlation satisfies the following inequality:

$\displaystyle C_{ABC}^{\mathcal{Q}}[\varrho_{ABC}]$ (12)
$\displaystyle\doteq$
$\displaystyle\min_{\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\otimes\Pi_{C}^{c}\\}}\max_{\\{\ket{\phi_{A},\phi_{B},\phi_{C}}\\}}\sum_{a,b,c,\phi_{A},\phi_{B},\phi_{C}}|{\rm
Im}\\{\braket{\phi_{A},\phi_{B},\phi_{C}}{(\Pi_{A}^{a}\otimes\Pi_{B}^{b}\otimes\Pi_{C}^{c})\varrho_{ABC}}{\phi_{A},\phi_{B},\phi_{C}}\\}|$
$\displaystyle\geq$
$\displaystyle\min_{\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,b,\phi_{A},\phi_{B}}|{\rm
Im}\\{\braket{\phi_{A},\phi_{B}}{(\Pi_{A}^{a}\otimes\Pi_{B}^{b})\varrho_{AB}}{\phi_{A},\phi_{B}}\\}|$
$\displaystyle=$ $\displaystyle C_{AB}^{\mathcal{Q}}[\varrho_{AB}],$

by noting that
$\sum_{c,\phi_{C}}\braket{\phi_{C}}{(\mathbb{I}_{A}\otimes\mathbb{I}_{B}\otimes\Pi_{C}^{c})\varrho_{ABC}}{\phi_{C}}={\rm
Tr}_{C}\varrho_{ABC}=\varrho_{AB}$. It is the state of $AB$ after the strong
projective measurement of the local basis $\\{\Pi_{C}^{c}\\}$ on
$\varrho_{ABC}$ without reading the outcomes.

Finally, another important requirement for a quantifier to be a bonafide
measure of quantum correlation is that for pure states, it should reduce to a
corresponding measure of entanglement Adesso quantum correlation beyond
entanglement review . This captures the intuition that for pure states,
separablity is equivalent to no nonclassical correlation. We have not yet been
able to clarify this important issue. However, in the next section, we prove
that for general pure states of arbitrary finite dimension, the one-sided
w-nonclassical correlation defined in Eq. (6) gives an observable lower bound
to a scaled linear entropy of entanglement, which for two qubits pure state
reduces to the entanglement concurrence Wootters entanglement concurrence .
Hence, for pure states, the w-coherence can be seen as a faithful witness of
entanglement.

Let us further show that, for any generic mixed state, the two-sided
w-nonclassical correlation sets an observable lower bound to the negativity of
quantumness, a measure of general quantum correlation which quantifies the
amount of entanglement that can be activated via a local measurement of the
subsystem Nakano negativity of quantumness trace norm l1 norm ; Piani discord
generates entanglement via measurement . First, starting from the definition
of the two-sided w-nonclassical correlation in Eq. (7), we have

$\displaystyle C^{\mathcal{Q}}_{AB}[\varrho_{AB}]$ (13) $\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,b,\phi_{A},\phi_{B}}\big{|}{\rm
Im}\\{\braket{\phi_{A},\phi_{B}}{(\Pi_{A}^{a}\otimes\Pi_{B}^{b})\varrho_{AB}}{\phi_{A},\phi_{B}}\\}\big{|}$
$\displaystyle\leq$
$\displaystyle\min_{\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,b,\phi_{A},\phi_{B}}\sum_{a^{\prime}\neq
a,b^{\prime}\neq
b}\big{|}\braket{a,b}{\varrho_{AB}}{a^{\prime},b^{\prime}}\big{|}$
$\displaystyle\times\big{|}\braket{\phi_{A}}{a}\braket{\phi_{B}}{b}\braket{a^{\prime}}{\phi_{A}}\braket{b^{\prime}}{\phi_{B}}\big{|}$
$\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\\}}\sum_{a^{\prime}\neq
a,b^{\prime}\neq
b}\big{|}\braket{a,b}{\varrho_{AB}}{a^{\prime},b^{\prime}}\big{|}$
$\displaystyle\times\max_{\\{\ket{\phi_{A}}\\}}\sum_{\phi_{A}}\big{|}\braket{\phi_{A}}{a}\braket{a^{\prime}}{\phi_{A}}\big{|}\max_{\\{\ket{\phi_{B}}\\}}\sum_{\phi_{B}}\big{|}\braket{\phi_{B}}{b}\braket{b^{\prime}}{\phi_{B}}\big{|},$

where we have inserted the completeness relations
$\sum_{a^{\prime}}\ket{a^{\prime}}\bra{a^{\prime}}_{A}=\mathbb{I}_{A}$ and
$\sum_{b^{\prime}}\ket{b^{\prime}}\bra{b^{\prime}}_{B}=\mathbb{I}_{B}$. One
the other hand, using Cauchy-Schwartz inequality, we have

$\displaystyle\sum_{\phi_{A}}|\braket{\phi_{A}}{a}||\braket{a^{\prime}}{\phi_{A}}|\leq\big{(}\sum_{\phi_{A}}|\braket{\phi_{A}}{a}|^{2}\sum_{\phi_{A}^{\prime}}|\braket{a^{\prime}}{\phi_{A}^{\prime}}|^{2}\big{)}^{1/2}=1,$
$\displaystyle\sum_{\phi_{B}}|\braket{\phi_{B}}{b}||\braket{b^{\prime}}{\phi_{B}}|\leq\big{(}\sum_{\phi_{B}}|\braket{\phi_{B}}{b}|^{2}\sum_{\phi_{B}^{\prime}}|\braket{b^{\prime}}{\phi_{B}^{\prime}}|^{2}\big{)}^{1/2}=1,$
(14)

where we have again used the completeness relation
$\sum_{\phi_{A}}\ket{\phi_{A}}\bra{\phi_{A}}=\mathbb{I}_{A}$ and
$\sum_{\phi_{B}}\ket{\phi_{B}}\bra{\phi_{B}}=\mathbb{I}_{B}$. Given the
measurement basis $\\{\ket{a(b)}\\}$, the equalities in Eq. (14) are attained
when the postselection basis $\\{\ket{\phi_{A(B)}}\\}$ is mutually unbiased
with the measurement basis so that
$|\braket{\phi_{A(B)}}{a(b)}|=1/\sqrt{d_{A(B)}}$ for all $a(b)$ and
$\phi_{A(B)}$, where $d_{A(B)}$ is the dimension of the Hilbert space of
subsystem $A(B)$. Upon inserting Eq. (14) into Eq. (13), we finally obtain

$\displaystyle C^{\mathcal{Q}}_{AB}[\varrho_{AB}]$ $\displaystyle\leq$
$\displaystyle\min_{\\{\ket{a}\otimes\ket{b}\\}}\sum_{a^{\prime}\neq
a,b^{\prime}\neq
b}\big{|}\braket{a,b}{\varrho_{AB}}{a^{\prime},b^{\prime}}\big{|}$ (15)
$\displaystyle\doteq$ $\displaystyle
C^{\mathcal{Q}}_{AB;l_{1}}[\varrho_{AB}].$

The quantity on the right-hand side is just the measure of general quantum
correlation based on $l_{1}$-norm measure of (local) coherence Adesso quantum
correlation beyond entanglement review  which is equal to twice of the total
(two-sided) negativity of quantumness Nakano negativity of quantumness trace
norm l1 norm ; Piani discord generates entanglement via measurement . We
emphasize that unlike the measure of general quantum correlation based on
$l_{1}$-norm (local) coherence and negativity of quantumness, the
w-nonclassical correlation is in principle directly observable via weak
measurement and postselection.

We proceed to give a concrete computation of the w-nonclassical correlation in
a simple bipartite state. We first note that, as is also the case for the
quantifiers of general quantum correlation proposed in the literature, the
calculation of the w-nonclassical correlation defined in Eqs. (6) and (7) is
in general analytically intractable, involving optimization over all possible
relevant bases, which in general takes the form of an optimization of
multivariable nonlinear function. For example, consider the following
maximally entangled two-qubit state

$\displaystyle\ket{\Psi_{AB}^{+}}=\frac{1}{\sqrt{2}}\ket{00}+\frac{1}{\sqrt{2}}\ket{11},$
(16)

where $\\{\ket{0},\ket{1}\\}$ is the computational basis, i.e., the
eigenstates of Pauli $\sigma_{z}$ matrix. For the purpose of computation, we
express the postselection bases for the Hilbert spaces of subsystem $A$ and
$B$, i.e., $\\{\ket{\phi_{A(B)}}\\}=(\ket{\phi_{A(B)}+},\ket{\phi_{A(B)}-})$,
using the following parameterization:

$\displaystyle\ket{\phi_{A(B)}+}$ $\displaystyle=$
$\displaystyle\cos\frac{\alpha_{A(B)}}{2}\ket{0}_{A(B)}+\sin\frac{\alpha_{A(B)}}{2}e^{i\beta_{A(B)}}\ket{1}_{A(B)},$
$\displaystyle\ket{\phi_{A(B)}-}$ $\displaystyle=$
$\displaystyle\sin\frac{\alpha_{A(B)}}{2}\ket{0}_{A(B)}-\cos\frac{\alpha_{A(B)}}{2}e^{i\beta_{A(B)}}\ket{1}_{A(B)},$
(17)

where $0\leq\alpha_{A(B)}\leq\pi$, and $0\leq\beta_{A(B)}\leq 2\pi$. One can
scan over all the tensor product of the postselection bases for the qubit $A$
and $B$, i.e., all the bases
$\\{\ket{\phi_{A},\phi_{B}}\\}\in\mathcal{B}_{\\{\ket{\phi_{A}}\otimes\ket{\phi_{B}}\\}}$,
by varying the angles $(\alpha_{A},\beta_{A})$ and $(\alpha_{B},\beta_{B})$
over their ranges of values. Furthermore, let us parameterize all the possible
local measurement bases on subsystem $A$ as
$\\{\Pi_{A}^{\mu}\\}=(\Pi_{A}^{\mu+},\Pi_{A}^{\mu-})$, where
$\Pi_{A}^{\mu\pm}\doteq\ket{\mu\pm}\bra{\mu\pm}_{A}$, with

$\displaystyle\ket{\mu+}_{A}$ $\displaystyle=$
$\displaystyle\cos\frac{\theta_{A}}{2}\ket{0}_{A}+\sin\frac{\theta_{A}}{2}e^{i\eta_{A}}\ket{1}_{A},$
$\displaystyle\ket{\mu-}_{A}$ $\displaystyle=$
$\displaystyle\sin\frac{\theta_{A}}{2}\ket{0}_{A}-\cos\frac{\theta_{A}}{2}e^{i\eta_{A}}\ket{1}_{A},$
(18)

$0\leq\theta_{A}\leq\pi$, $0\leq\eta_{A}\leq 2\pi$. Inserting all the above
ingredients into the one-sided nonclassical correlation
$C_{A}^{\mathcal{Q}}[\varrho_{AB}]$ of Eq. (6), we obtain

$\displaystyle C_{A}^{\mathcal{Q}}[\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}]$
(19) $\displaystyle=$
$\displaystyle\min_{\\{\ket{\mu_{A}}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{\mu_{A},\phi_{A},\phi_{B}}\big{|}{\rm
Im}\\{\braket{\phi_{A},\phi_{B}}{(\Pi_{A}^{\mu}\otimes\mathbb{I}_{B})}{\Psi_{AB}^{+}}\braket{\Psi_{AB}^{+}}{\phi_{A},\phi_{B}}\\}\big{|}$
$\displaystyle=$
$\displaystyle\min_{(\theta_{A},\eta_{A})}\max_{(\alpha_{A},\alpha_{B},\beta_{A},\beta_{B})}|\sin\alpha_{A}\sin\alpha_{B}\sin(\beta_{A}+\beta_{B})\cos\theta_{A}-\cos\alpha_{A}\sin\alpha_{B}\sin(\beta_{B}+\eta_{A})\sin\theta_{A}$
$\displaystyle-$
$\displaystyle\sin\alpha_{A}\cos\alpha_{B}\sin(\beta_{A}-\eta_{A})\sin\theta_{A}|=0.99999\cdots.$

See Appendix A for the detailed derivation of the above trigonometric function
that has to be optimized, i.e., maximized over the parameters
$(\alpha_{A},\alpha_{B},\beta_{A},\beta_{B})$ and then followed by choosing
the minimum over $(\theta_{A},\eta_{A})$.

Next, as an example of a separable state with a nonvanishing one-sided
w-nonclassical correlation, we consider the Werner state for $2\times 2$
dimension:

$\displaystyle\varrho_{AB}^{\rm
W}=\frac{1-p}{4}(\mathbb{I}_{A}\otimes\mathbb{I}_{B})+p\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}.$
(20)

It is known that the above Werner state is separable for $p<1/3$. The one-
sided w-nonclassical correlation can be straightforwardly computed to get,
noting Eq. (19),

$\displaystyle C^{\mathcal{Q}}_{A}[\varrho^{\rm
W}_{AB}]=pC^{\mathcal{Q}}_{A}[\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}]\approx
p\geq 0,$ (21)

where the first equality is obtained directly from the definition in Eq. (6).
Hence, it is nonvanishing even in the regime when the Werner state is
separable. We note that the above value of the w-nonclassical correlation for
the Werner state is equal to the square root of the geometric discord based on
Hilbert-Schmidt distance Dakic discord and remote state preparation .

##  IV Statistical and information theoretic meaning

###  IV.1 W-nonclassical correlation as the witness for genuine local quantum
uncertainty and pure state entanglement

Recall that the one-sided and two-sided w-nonclassical correlations are
defined via the noncommutativity between the local measurement bases and the
multipartite state. Such noncommutativity is one of the sources of uncertainty
in quantum measurement. It is therefore instructive to discuss the relation
between the w-nonclassical correlations and the local quantum uncertainty. Let
us first consider the one-sided w-nonclassical correlation. Then, as shown in
the Appendix B, we obtain the following inequality

$\displaystyle C^{\mathcal{Q}}_{A}[\varrho_{AB}]$ (22) $\displaystyle\leq$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\sum_{a}\big{(}{\rm
Tr}\\{(\Pi_{A}^{a}\otimes\mathbb{I}_{B})^{2}\varrho_{AB}\\}-{\rm
Tr}\\{(\Pi_{A}^{a}\otimes\mathbb{I}_{B})\varrho_{AB}\\}^{2}\big{)}^{1/2}$
$\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\sum_{a}\Delta_{(\Pi_{A}^{a}\otimes\mathbb{I}_{B})}[\varrho_{AB}],$

where $\Delta^{2}_{\hat{O}}[\varrho]\doteq{\rm
Tr}\\{\hat{O}^{2}\varrho\\}-({\rm Tr}\\{\hat{O}\varrho\\})^{2}$ is the quantum
variance of $\hat{O}$ in the state $\varrho$ which takes into account both the
genuine quantum uncertainty arising from the noncommutativity between $O$ and
$\varrho$, and the classical uncertainty associated with classical mixing in
the state $\varrho$. Similarly, for the two-sided w-nonclassical correlation,
we obtain
$C^{\mathcal{Q}}_{AB}[\varrho_{AB}]\leq\min_{\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\\}}\sum_{a,b}\Delta_{(\Pi_{A}^{a}\otimes\Pi_{B}^{b})}[\varrho_{AB}]$.

Hence, the strength of the w-nonclassical correlation is limited by the
minimum of the total sum of the uncertainty of each element of the local
measurement basis quantified by the quantum standard deviation. Conversely,
the nonclassical correlation captured by $C^{\mathcal{Q}}_{A}[\varrho_{AB}]$
and $C^{\mathcal{Q}}_{AB}[\varrho_{AB}]$ give the lower bounds to the quantum
uncertainty of any local basis, i.e., from Eq. (22), for any local basis
$\\{\Pi_{A}^{k}\\}$, we have

$\displaystyle\sum_{k}\Delta_{(\Pi_{A}^{k}\otimes\mathbb{I}_{B})}[\varrho_{AB}]\geq\min_{\\{\Pi_{A}^{a}\\}}\sum_{a}\Delta_{(\Pi_{A}^{a}\otimes\mathbb{I}_{B})}[\varrho_{AB}]\geq
C^{\mathcal{Q}}_{A}[\varrho_{AB}],$ (23)

and similarly, for any local measurement bases
$\\{\Pi_{A}^{k}\otimes\Pi_{B}^{l}\\}$, we have
$\sum_{k,l}\Delta_{(\Pi_{A}^{k}\otimes\Pi_{B}^{l})}[\varrho_{AB}]\geq\min_{\\{\Pi_{A}^{a}\otimes\Pi_{B}^{b}\\}}\sum_{a,b}\Delta_{(\Pi_{A}^{a}\otimes\Pi_{B}^{b})}[\varrho_{AB}]\geq
C^{\mathcal{Q}}_{AB}[\varrho_{AB}]$. The w-nonclassical correlation can
therefore be seen as one of the sources of the quantum uncertainty in the
measurement of the local basis which arises from the noncommutativity between
the local basis and the multipartite state.

The above observation suggests to interpret
$C^{\mathcal{Q}}_{A}[\varrho_{AB}]$ and $C^{\mathcal{Q}}_{AB}[\varrho_{AB}]$
as the minimum genuine quantum share of the local uncertainty, out of the
total local quantum uncertainty quantified by the quantum standard deviation.
Recall that the latter also includes the uncertainty arising from the
classical mixing of states, and possibly from other sources. To this end, it
is intriguing to compare the w-nonclassical correlation with the Wigner-Yanase
skew information Wigner-Yanase skew information , which is also based on the
noncommutativity between the state and a Hermitian observable and has been
argued to quantify the genuine quantum share of uncertainty arising from
quantum noncommutativity Luo Wigner-Yanase skew information as genuine quantum
uncertainty . Let us mention that the minimum value of the Wigner-Yanase skew
information over certain set of local nondegenerate observables is proposed in
Ref. Girolami quantum correlation based on WY skew information  as a
quantifier of nonclassical correlation beyond entanglement. We note that
unlike the Wigner-Yanase skew information, w-nonclassical correlation
$C^{\mathcal{Q}}_{A}[\varrho_{AB}]$ and $C^{\mathcal{Q}}_{AB}[\varrho_{AB}]$
are operationally appealing, in that they have transparent interpretation in
terms of direct laboratory operations based on weak measurement and
postselection.

Next, noting that ${\rm
Tr}\\{(\Pi_{A}^{a}\otimes\mathbb{I}_{B})^{2}\varrho_{AB}\\}={\rm
Tr}\\{(\Pi_{A}^{a}\otimes\mathbb{I}_{B})\varrho_{AB}\\}=p(a|\varrho_{A};\Pi_{A}^{a})$
is just the probability to get the outcome $a$ in the local projective
measurement $\\{\Pi_{A}^{a}\\}$ over the state $\varrho_{A}={\rm
Tr}_{B}\\{\varrho_{AB}\\}$ of the subsystem $A$, and inserting into Eq. (22),
we obtain

$\displaystyle C^{\mathcal{Q}}_{A}[\varrho_{AB}]$ $\displaystyle\leq$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\sum_{a}\big{[}p(a|\varrho_{A};\Pi_{A}^{a})-p(a|\varrho_{A};\Pi_{A}^{a})^{2}]^{1/2}$
(24) $\displaystyle\leq$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}d_{A}^{1/2}\big{[}1-\sum_{a}p(a|\varrho_{A};\Pi_{A}^{a})^{2}]^{1/2}$
$\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}d_{A}^{1/2}S_{2}^{1/2}[\\{p(a|\varrho_{A};\Pi_{A}^{a})\\}],$

where we have again used the Cauchy-Schwartz inequality and the normalization
$\sum_{a}p(a|\varrho_{A};\Pi_{A}^{a})=1$ to get the second line, and

$\displaystyle S_{2}[\\{p(a|\varrho_{A};\Pi_{A}^{a})\\}]\doteq
1-\sum_{a}p(a|\varrho_{A};\Pi_{A}^{a})^{2}$ (25)

is the Tsallis entropy with the entropy index $2$.

Let us further show that the minimum of the Tsallis entropy on the right-hand
side of Eq. (24) is obtained when the local measurement basis
$\\{\Pi_{A}^{a}\\}$ is just given by the eigenprojectors of the reduced
density matrix $\varrho_{A}$. Suppose that the reduced density matrix has the
following spectral decomposition:
$\varrho_{A}=\sum_{j}\pi_{j}(\varrho_{A})\Pi^{\eta_{j}}_{A}$, where
$\\{\Pi^{\eta_{j}}_{A}=\ket{\eta_{j}}\bra{\eta_{j}}_{A}\\}$ are the
eigenprojectors, and $\\{\pi_{j}(\varrho_{A})\\}$, $\pi_{j}(\varrho_{A})\geq
0$, $\sum_{j}\pi_{j}(\varrho_{A})=1$, are the associated eigenvalues. First,
we have $p(a|\varrho_{A};\Pi_{A}^{a})={\rm
Tr}\\{\Pi_{A}^{a}\varrho_{A}\\}=\sum_{j}\pi_{j}(\varrho_{A}){\rm
Tr}\\{\Pi_{A}^{\eta_{j}}\Pi_{A}^{a}\\}$. Inserting this into the definition of
the Tsallis entropy in Eq. (25), and noting the convexity of the quadratic
function, we get, using the Jensen inequality,

$\displaystyle S_{2}[\\{p(a|\varrho_{A};\Pi_{A}^{a})\\}]$ $\displaystyle=$
$\displaystyle 1-\sum_{a}\big{(}\sum_{j}\pi_{j}(\varrho_{A}){\rm
Tr}\\{\Pi_{A}^{\eta_{j}}\Pi_{A}^{a}\\}\big{)}^{2}$ (26) $\displaystyle\geq$
$\displaystyle 1-\sum_{a}\sum_{j}\pi_{j}(\varrho_{A})^{2}{\rm
Tr}\\{\Pi_{A}^{\eta_{j}}\Pi_{A}^{a}\\}$ $\displaystyle=$ $\displaystyle
1-\sum_{j}\pi_{j}(\varrho_{A})^{2}=S_{2}[\\{\pi_{j}(\varrho_{A})\\}],$

where we have used the normalization $\sum_{a}{\rm
Tr}\\{\Pi_{A}^{\eta_{j}}\Pi_{A}^{a}\\}=\braket{\eta_{j}}{\eta_{j}}=1$. By
comparing Eq. (25) to Eq. (26), one finds that the equality in Eq. (26),
namely, the minimum of $S_{2}[\\{p(a|\varrho_{A};\Pi_{A}^{a})\\}]$, is reached
when $p(a|\varrho_{A};\Pi_{A}^{a})=\pi_{a}(\varrho_{A})$, i.e., when
$\\{\Pi_{A}^{a}\\}=\\{\Pi_{A}^{\eta_{a}}\\}$, as claimed. From Eqs. (24) and
(26), we thus have

$\displaystyle C^{\mathcal{Q}}_{A}[\varrho_{AB}]\leq
d_{A}^{1/2}S_{2}^{1/2}[\\{\pi_{j}(\varrho_{A})\\}].$ (27)

Finally, notice that the Tsallis entropy $S_{2}[\\{\pi_{j}(\varrho_{A})\\}]$
on the right-hand side of Eq. (27) is just the linear entropy associated with
the reduced density matrix
$\varrho_{A}=\sum_{j}\pi_{j}(\varrho_{A})\Pi_{A}^{\eta_{j}}$, i.e.,

$\displaystyle
S_{2}[\\{\pi_{a}(\varrho_{A})\\}]=1-\sum_{a}\pi_{a}(\varrho_{A})^{2}=1-{\rm
Tr}[\varrho_{A}^{2}].$ (28)

It quantifies the mixedness or impurity of the reduced density matrix
$\varrho_{A}={\rm Tr}_{B}\varrho_{AB}$. For a pure bipartite state
$\ket{\psi_{AB}}$, the linear entropy can be used as a measure of
entanglement, i.e., $E_{l}[\ket{\psi_{AB}}\bra{\psi_{AB}}]\doteq 1-{\rm
Tr}[\varrho_{A}^{2}]=1-{\rm
Tr}[\varrho_{B}^{2}]=S_{2}[\\{\pi_{a}(\varrho_{A})\\}]$, $\varrho_{A(B)}={\rm
Tr}_{B(A)}\\{\ket{\psi_{AB}}\bra{\psi_{AB}}\\}$. Eqs. (27) and (28) therefore
show that for general pure bipartite states $\ket{\psi_{AB}}$, the one-sided
w-nonclassical correlation
$C_{A}^{\mathcal{Q}}[\ket{\psi_{AB}}\bra{\psi_{AB}}]$ can be seen as a witness
for the quantum entanglement in $\ket{\psi_{AB}}$, namely, it gives a lower
bound to the scaled square root of the linear entropy of entanglement as

$\displaystyle C^{\mathcal{Q}}_{A}[\ket{\psi_{AB}}\bra{\psi_{AB}}]\leq
d_{A}^{1/2}(1-{\rm Tr}\\{\varrho_{A}^{2}\\})^{1/2}.$ (29)

Moreover, since $C^{\mathcal{Q}}_{A}[\ket{\psi_{AB}}\bra{\psi_{AB}}]$ is
vanishing if and only if $\ket{\psi_{AB}}$ is factorizable, it is a faithful
witness for the pure state entanglement. We note that for two-qubit pure
state, denoting the Schmidt coefficients as $\lambda_{+}$ and $\lambda_{-}$,
the right-hand side of Eq. (29) is just given by the entanglement concurrence
Wootters entanglement concurrence , i.e., $d_{A}^{1/2}(1-{\rm
Tr}\\{\varrho_{A}^{2}\\})^{1/2}=\sqrt{2}\sqrt{1-(\lambda_{+}^{4}+\lambda_{-}^{4})}=2\lambda_{+}\lambda_{-}$,
where we have used $\lambda_{+}^{2}+\lambda_{-}^{2}=1$. We emphasize that
unlike the linear entropy of entanglement on the right-hand side of Eq. (29),
the w-nonclassical correlation on the left-hand side has a transparent
operational interpretation and, at least in principle, can be observed
directly via weak measurement with postselection. Finally, note that the
maximum value of the right-hand side of Eqs. (27) or (29) over all bipartite
quantum states is reached when $\pi_{j}(\varrho_{A})=1/d_{A}$ for all $j$, to
give $\max_{\varrho_{A}}d_{A}^{1/2}(1-{\rm
Tr}\\{\varrho_{A}^{2}\\})^{1/2}=\sqrt{d_{A}-1}$. For pure bipartite state,
this is the case when it is maximally entangled, i.e.,
$\ket{\psi_{AB}}=\sum_{i}\frac{1}{\sqrt{d_{A}}}\ket{a_{i}}\ket{b_{i}}$. It is
however not known if for the maximally entangled state, the w-nonclassical
correlation $C_{A}^{\mathcal{Q}}[\ket{\psi_{AB}}\bra{\psi_{AB}}]$ on the left-
hand side of Eq. (27) is also maximized.

###  IV.2 W-nonclassical correlation as minimum mean absolute error in optimal
estimation of local basis based on local measurement on the multipartite state

The expression of the w-nonclassical correlation of Eq. (6) and (7) in terms
of the imaginary part of the weak value naturally suggests an information
theoretic interpretation based on the statistical interpretation of weak
values. First, we note that as argued in Refs. Johansen weak value best
estimation ; Hall prior information ; Hofmann imaginary part of weak value in
optimal estimation ; Agung general c-valued physical quantities and
uncertainty relation , the real part of the weak value $O^{\rm
w}(\phi|\varrho)$ of Eq. (1) can be seen as the optimal estimate of the
Hermitian observable $O$ given prior information about preparation represented
by the quantum state $\varrho$, based on the outcomes of a projective
measurement $\\{\Pi^{\phi}\\}$, $\sum_{\phi}\Pi^{\phi}=\mathbb{I}$, on
$\varrho$. Here, optimal means that the estimate minimizes the associated
mean-squared estimation error. Moreover and importantly, in this case, the
minimum mean-squared estimation error is just given by the variance of the
imaginary part of the weak value. Noting this fact, it is argued in Ref. Agung
general c-valued physical quantities and uncertainty relation  that the
imaginary part of the weak value can be seen as the strength of the error in a
single shot estimation via the introduction of an unbiased global random
variable. See also Refs. Agung ERPS representation ; Agung quantum mechanics
as optimal estimation  for the case of continuous quadrature variables.

With this observation in mind, $C^{\mathcal{Q}}_{A}[\varrho_{AB}]$ defined in
Eq. (6) can then be interpreted as follows. Suppose we wish to estimate the
outcome of a local projector on subsystem $A$, i.e., an element of the local
measurement basis $\Pi_{A}^{k}\in\\{\Pi_{A}^{a}\\}$, based on the outcomes
$(\phi_{A},\phi_{B})$ of a complete set of projective measurement of
$\\{\Pi^{\phi}_{A}\otimes\Pi^{\phi}_{B}\\}$,
$\Pi_{A(B)}^{\phi}=\ket{\phi_{A(B)}}\bra{\phi_{A(B)}}$, on the prepared
bipartite state represented by $\varrho_{AB}$. Then, according to the
information theoretical interpretation of weak value alluded to above, the
mean absolute estimation error is given by the absolute imaginary part of the
weak value of $\Pi_{A}^{k}\otimes\mathbb{I}_{B}$ with the preselected state
$\varrho_{AB}$ and the postselected state $\ket{\phi_{A},\phi_{B}}$, averaged
over the probability of the successful postselection, i.e.,

$\displaystyle\braket{}{\mathcal{E}_{\Pi_{A}^{k}}[\varrho_{AB};\ket{\phi_{A},\phi_{B}}]}{}$
(30) $\displaystyle\doteq$ $\displaystyle\sum_{\phi_{A},\phi_{B}}\big{|}{\rm
Im}\Big{\\{}\frac{\braket{\phi_{A},\phi_{B}}{(\Pi_{A}^{k}\otimes\mathbb{I}_{B})\varrho_{AB}}{\phi_{A},\phi_{B}}}{\braket{\phi_{A},\phi_{B}}{\varrho_{AB}}{\phi_{A},\phi_{B}}}\Big{\\}}\big{|}\braket{\phi_{A},\phi_{B}}{\varrho_{AB}}{\phi_{A},\phi_{B}}.$

We then consider the worst case scenario wherein the total sum of mean
absolute error of the estimation of $\\{\Pi_{A}^{a}\\}$ is maximized over all
possible measurement
$\\{\Pi^{\phi}_{A}\otimes\Pi^{\phi}_{B}\\}\in\mathcal{B}_{\\{\ket{\phi_{A}}\otimes\ket{\phi_{B}}\\}}$,
i.e.,
$\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a}\braket{}{\mathcal{E}_{\Pi_{A}^{a}}[\varrho_{AB};\ket{\phi_{A},\phi_{B}}]}{}$.
Finally, among these worst case scenarios, we choose the local basis
$\\{\Pi_{A}^{a}\\}$ so that the total mean absolute estimation error is the
least (i.e., the best of the worst scenario). This gives us the one-sided
w-nonclassical correlation of Eq. (6), i.e.,

$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a}\braket{}{\mathcal{E}_{\Pi_{A}^{a}}[\varrho_{AB};\ket{\phi_{A},\phi_{B}}]}{}$
(31) $\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi_{A},\phi_{B}}\\}}\sum_{a,\phi_{A},\phi_{B}}\big{|}{\rm
Im}\\{\braket{\phi_{A},\phi_{B}}{(\Pi_{A}^{a}\otimes\mathbb{I}_{B})\varrho_{AB}}{\phi_{A},\phi_{B}}\\}\big{|}=C_{A}^{\mathcal{Q}}[\varrho_{AB}].$

This is just a minimax estimation problem. An analogous interpretation can be
made for the two-sided w-nonclassical correlation
$C_{AB}^{\mathcal{Q}}[\varrho_{AB}]$ of Eq. (7).

The above information theoretic interpretation in terms of optimal estimation
of local basis based on local measurement on the multipartite state, suggests
a new characterization of the quantum versus classically correlated
multipartite states of Eqs. (3) or (4) as follows. A multipartite state is
quantumly correlated if and only if the optimal estimation of any local basis
based on the local projective measurement on the state must yield a
nonvanishing mean absolute error in the worst case scenario, with the minimum
value that is given by the w-nonclassical correlation. By contrast, a
multipartite state is classical devoid of quantum correlation, if and only if
the outcomes of local projective measurement on it can be used to make a sharp
(i.e., with vanishing estimation error) optimal estimation of a certain local
basis. Such a negative characterization of general quantum correlation is
similar to the characterization of quantum correlation expressed by no-go
theorem for quantum local broadcasting Piani discord and no-local broadcasting
.

##  V Summary and Remarks

We have proposed a characterization of nonclassical correlation beyond
entanglement in terms of transparent and direct laboratory operations. We
operationally defined a quantity based on the imaginary part of weak values
obtained via the weak measurement and postselection, together with an
optimization procedure, which captures the minimum incompatibility between the
multipartite state under scrutiny and any local measurement basis. We showed
that it satisfies certain desirable requirements for a quantifier of general
nonclassical correlation in the multipartite state. We argued that it may be
statistically interpreted as the minimum genuine quantum part of local
uncertainty arising from the incompatibility between the local measurement
basis and the multipartite state. It is a faithful witness for pure state
entanglement giving an observable lower bound to a scaled square root of the
linear entropy of entanglement. Moreover, we have offered an information
theoretic interpretation of the nonclassical correlation as the minimum mean
absolute error of the optimal estimation of a local basis, based on the
outcomes of local projective measurement on the multipartite state, in the
worst case scenario.

Our results suggest intriguing fundamental links between the quantum-classical
division and contrast captured by general nonclassical correlation, and the
concept of weak value in the multipartite setting. Note that the ability of
the scheme of weak measurement with postselection to directly characterize
general nonclassical correlation is related to the fact that a similar scheme
can be used to directly observe the quantum state or its equivalent
quasiprobability representation Lundeen complex weak value ; Jozsa complex
weak value , in which, the nonclassical features of quantum systems, including
general nonclassical correlation, are encoded. In both schemes for the direct
measurement of quantum states and direct characterization of general quantum
correlation, the complex nature of the weak value, especially the imaginary
part which captures the noncommutativity between the quantum state and
observable, are crucial. It is hoped that the weak measurement with
postselection may offer a fresh operational framework to characterize and
quantify other nonclassical features of quantum mechanics.

###### Acknowledgements.

##  Appendix A Computation of Eq. (19)

Let us first compute each term in the sum in Eq. (19) for different elements
of the local measurement and postselection bases. From Eq. (18), it is
convenient to express the local measurement basis as

$\displaystyle\ket{\mu+}\bra{\mu+}_{A}=\cos^{2}\frac{\theta_{A}}{2}\ket{0}\bra{0}_{A}+\frac{1}{2}\sin\theta_{A}e^{-i\eta_{A}}\ket{0}\bra{1}_{A}$
$\displaystyle+\frac{1}{2}\sin\theta_{A}e^{i\eta_{A}}\ket{1}\bra{0}_{A}+\sin^{2}\frac{\theta}{2}\ket{1}\bra{1}_{A}.$
$\displaystyle\ket{\mu-}\bra{\mu-}_{A}=\sin^{2}\frac{\theta_{A}}{2}\ket{0}\bra{0}_{A}-\frac{1}{2}\sin\theta_{A}e^{-i\eta_{A}}\ket{0}\bra{1}_{A}$
$\displaystyle-\frac{1}{2}\sin\theta_{A}e^{i\eta_{A}}\ket{1}\bra{0}_{A}+\cos^{2}\frac{\theta}{2}\ket{1}\bra{1}_{A}.$
(32)

One then obtains, for the weak measurement of $\ket{\mu+}\bra{\mu+}_{A}$ with
the postselection state $\ket{\phi_{A}+,\phi_{B}+}$, using the expression for
$\ket{\mu+}\bra{\mu+}_{A}$ of Eq. (32) and noting the expression for the
postselection states in Eq. (17),

$\displaystyle
C^{\mu_{A}+}_{\phi_{A}+,\phi_{B}+}[\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}]$
(33) $\displaystyle\doteq$ $\displaystyle|{\rm
Im}\\{\braket{\phi_{A}+,\phi_{B}+}{(\ket{\mu+}\bra{\mu+}_{A}\otimes\mathbb{I}_{B})}{\Psi_{AB}^{+}}\braket{\Psi_{AB}^{+}}{\phi_{A}+,\phi_{B}+}\\}|$
$\displaystyle=$
$\displaystyle|I_{++}^{00}\cos^{2}\frac{\theta_{A}}{2}+I^{01}_{++}\sin\theta_{A}+I^{10}_{++}\sin\theta_{A}+I^{11}_{++}\sin^{2}\frac{\theta_{A}}{2}|,$

where each term on the right-hand side takes the form:

$\displaystyle I^{00}_{++}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\sin\alpha_{A}\sin\alpha_{B}\sin(\beta_{A}+\beta_{B});$
$\displaystyle I^{01}_{++}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\cos^{2}\frac{\alpha_{A}}{2}\sin\alpha_{B}\sin(-\beta_{B}-\eta_{A})+\frac{1}{8}\sin\alpha_{A}\sin^{2}\frac{\alpha_{B}}{2}\sin(\beta_{A}-\eta_{A});$
$\displaystyle I^{10}_{++}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\sin\alpha_{A}\cos^{2}\frac{\alpha_{B}}{2}\sin(-\beta_{A}+\eta_{A})+\frac{1}{8}\sin^{2}\frac{\alpha_{A}}{2}\sin\alpha_{B}\sin(\beta_{B}+\eta_{A});$
$\displaystyle I^{11}_{++}$ $\displaystyle=$
$\displaystyle-\frac{1}{8}\sin\alpha_{A}\sin\alpha_{B}\sin(\beta_{A}+\beta_{B}).$
(34)

On the other hand, for the weak measurement of $\ket{\mu-}\bra{\mu-}_{A}$ with
the postselection state $\ket{\phi_{A}+,\phi_{B}+}$, using the expression for
$\ket{\mu-}\bra{\mu-}_{A}$ of Eq. (32), we get

$\displaystyle
C^{\mu_{A}-}_{\phi_{A}+,\phi_{B}+}[\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}]$
(35) $\displaystyle\doteq$ $\displaystyle|{\rm
Im}\\{\braket{\phi_{A}+,\phi_{B}+}{(\ket{\mu-}\bra{\mu-}_{A}\otimes\mathbb{I}_{B})}{\Psi_{AB}^{+}}\braket{\Psi_{AB}^{+}}{\phi_{A}+,\phi_{B}+}\\}|$
$\displaystyle=$
$\displaystyle|I_{++}^{00}\sin^{2}\frac{\theta_{A}}{2}-I^{01}_{++}\sin\theta_{A}-I^{10}_{++}\sin\theta_{A}+I^{11}_{++}\cos^{2}\frac{\theta_{A}}{2}|.$

Similarly, one can compute the cases for the weak measurement of
$\ket{\mu_{A}\pm}\bra{\mu_{A}\pm}$ with the remaining postselection states
$\ket{\phi_{A}+,\phi_{B}-}$, $\ket{\phi_{A}-,\phi_{B}+}$, and
$\ket{\phi_{A}-,\phi_{B}-}$. One can further show by inspection that
$|C^{\mu_{A}+}_{\phi_{A}+,\phi_{B}+}[\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}]|=|C^{\mu_{A}-}_{\phi_{A}+,\phi_{B}+}[\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}]|=|C^{\mu_{A}+}_{\phi_{A}+,\phi_{B}-}[\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}]|=|C^{\mu_{A}-}_{\phi_{A}+,\phi_{B}-}[\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}]|=|C^{\mu_{A}+}_{\phi_{A}-,\phi_{B}+}[\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}]|=|C^{\mu_{A}-}_{\phi_{A}-,\phi_{B}+}[\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}]|=|C^{\mu_{A}+}_{\phi_{A}-,\phi_{B}-}[\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}]|=|C^{\mu_{A}-}_{\phi_{A}-,\phi_{B}-}[\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}]|$,
where, upon inserting Eq. (34) into Eq. (33), we have

$\displaystyle|C^{\mu_{A}+}_{\phi_{A}+,\phi_{B}+}[\ket{\Psi_{AB}^{+}}\bra{\Psi_{AB}^{+}}]|$
$\displaystyle=$
$\displaystyle\frac{1}{8}|\sin\alpha_{A}\sin\alpha_{B}\sin(\beta_{A}+\beta_{B})\cos\theta_{A}$
(36) $\displaystyle-$
$\displaystyle\cos\alpha_{A}\sin\alpha_{B}\sin(\beta_{B}+\eta_{A})\sin\theta_{A}$
$\displaystyle-$
$\displaystyle\sin\alpha_{A}\cos\alpha_{B}\sin(\beta_{A}-\eta_{A})\sin\theta_{A}|.$

Inserting these terms into the definition of the one-sided w-correlation of
Eq. (6), we obtain Eq. (19). Hence, we need to perform the maximization of
nonlinear function of Eq. (19) over
$(\alpha_{A},\alpha_{B},\beta_{A},\beta_{B})$, and followed by choosing the
minimum over $(\theta_{A},\eta_{A})$. We do this by using the subroutine
fminsearch in a simple matlab code to get the maximum over
$(\alpha_{A},\alpha_{B},\beta_{A},\beta_{B})$, followed with the application
of subroutine min of matlab to search for the minimum over
$(\theta_{A},\eta_{A})$.

##  Appendix B Proof of Eq. (22)

First, we have, from Eq. (6),

$\displaystyle C^{\mathcal{Q}}_{A}[\varrho_{AB}]$ (37) $\displaystyle=$
$\displaystyle\min_{\\{\Pi_{A}^{a}\\}}\max_{\\{\ket{\phi}\\}}\sum_{a}\sum_{\phi}\Big{|}{\rm
Im}\Big{\\{}\frac{{\rm
Tr}\\{\Pi_{AB}^{\phi}(\Pi_{A}^{a}\otimes\mathbb{I}_{B})\varrho_{AB}\\}}{{\rm
Tr}\\{\Pi_{AB}^{\phi}\varrho_{AB}\\}}\Big{\\}}\Big{|}{\rm
Tr}\\{\Pi_{AB}^{\phi}\varrho_{AB}\\}$ $\displaystyle\leq$
$\displaystyle\sum_{a^{\star}}\Big{[}\sum_{\phi^{\star}}\Big{(}\Big{|}\frac{{\rm
Tr}\\{\Pi_{AB}^{\phi^{\star}}(\Pi_{A}^{a^{\star}}\otimes\mathbb{I}_{B})\varrho_{AB}\\}}{{\rm
Tr}\\{\Pi_{AB}^{\phi^{\star}}\varrho_{AB}\\}}\Big{|}^{2}-{\rm
Re}\Big{\\{}\frac{{\rm
Tr}\\{\Pi_{AB}^{\phi^{\star}}(\Pi_{A}^{a^{\star}}\otimes\mathbb{I}_{B})\varrho_{AB}\\}}{{\rm
Tr}\\{\Pi_{AB}^{\phi^{\star}}\varrho_{AB}\\}}\Big{\\}}^{2}\Big{)}{\rm
Tr}\\{\Pi_{AB}^{\phi^{\star}}\varrho_{AB}\\}\Big{]}^{1/2}$ $\displaystyle\leq$
$\displaystyle\sum_{a^{\star}}\Big{[}\sum_{\phi^{\star}}\frac{({\rm
Tr}\\{\Pi_{AB}^{\phi^{\star}}(\Pi_{A}^{a^{\star}}\otimes\mathbb{I}_{B})\varrho_{AB}\\})^{2}}{{\rm
Tr}\\{\Pi_{AB}^{\phi^{\star}}\varrho_{AB}\\}}-\big{(}\sum_{\phi^{\star}}{\rm
Re}\big{\\{}{\rm
Tr}\\{\Pi_{AB}^{\phi^{\star}}(\Pi_{A}^{a^{\star}}\otimes\mathbb{I}_{B})\varrho_{AB}\\}\big{\\}}\big{)}^{2}\Big{]}^{1/2}.$

Here $\\{\ket{\phi^{\star}}\\}=\\{\ket{\phi_{A}^{\star},\phi_{B}^{\star}}\\}$
is the postselection basis which achieves the maximum,
$\Pi_{AB}^{\phi^{\star}}=\Pi_{A}^{\phi^{\star}}\otimes\Pi_{B}^{\phi^{\star}}=\ket{\phi_{A}^{\star}}\bra{\phi_{A}^{\star}}\otimes\ket{\phi_{B}^{\star}}\bra{\phi_{B}^{\star}}$,
and $\\{\Pi_{A}^{a^{\star}}\\}$ is the local measurement basis which achieves
the minimum. Moreover, we have applied the Cauchy-Schwartz inequality and the
completeness relation for the postselection basis, i.e.,
$\sum_{\phi^{\star}}{\rm Tr}\\{\Pi_{AB}^{\phi^{\star}}\varrho_{AB}\\}=1$, to
get the third and fourth lines. Next, applying again the Cauchy-Schwartz
uncertainty relation for bounded operators, $|{\rm
Tr}\\{XY^{\dagger}\\}|^{2}\leq{\rm Tr}\\{XX^{\dagger}\\}{\rm
Tr}\\{YY^{\dagger}\\}$, to the numerator in the first term on the right-hand
side of Eq. (37), i.e.,

$\displaystyle({\rm
Tr}\\{\Pi_{AB}^{\phi^{\star}}(\Pi_{A}^{a^{\star}}\otimes\mathbb{I}_{B})\varrho_{AB}\\})^{2}$
$\displaystyle=$ $\displaystyle({\rm
Tr}\\{([\Pi_{AB}^{\phi^{\star}}]^{1/2}(\Pi_{A}^{a^{\star}}\otimes\mathbb{I}_{B})\varrho_{AB}^{1/2})(\varrho_{AB}^{1/2}[\Pi_{AB}^{\phi^{\star}}]^{1/2})\\})^{2}$
(38) $\displaystyle\leq$ $\displaystyle{\rm
Tr}\\{\Pi_{AB}^{\phi^{\star}}(\Pi_{A}^{a^{\star}}\otimes\mathbb{I}_{B})\varrho_{AB}(\Pi_{A}^{a^{\star}}\otimes\mathbb{I}_{B})\\}{\rm
Tr}\\{\varrho_{AB}\Pi_{AB}^{\phi^{\star}}\\},$

and using the completeness relation
$\sum_{\phi^{\star}}\Pi_{AB}^{\phi^{\star}}=\mathbb{I}$, we finally obtain Eq.
(22), i.e.,

$\displaystyle
C^{\mathcal{Q}}_{A}[\varrho_{AB}]\leq\sum_{a^{\star}}\big{(}{\rm
Tr}\\{(\Pi_{A}^{a^{\star}}\otimes\mathbb{I}_{B})^{2}\varrho_{AB}\\}-{\rm
Tr}\\{(\Pi_{A}^{a^{\star}}\otimes\mathbb{I}_{B})\varrho_{AB}\\}^{2})^{1/2}.$
(39)

By following similar steps as above, we also obtain, for the two-sided
w-nonclassical correlation,

$\displaystyle
C^{\mathcal{Q}}_{AB}[\varrho_{AB}\\}]\leq\sum_{a^{\star},b^{\star}}\big{(}{\rm
Tr}\\{(\Pi_{A}^{a^{\star}}\otimes\Pi_{B}^{b^{\star}})^{2}\varrho_{AB}\\}-{\rm
Tr}\\{(\Pi_{A}^{a^{\star}}\otimes\Pi_{B}^{b^{\star}})\varrho_{AB}\\}^{2})^{1/2},$
(40)

where $\\{\Pi_{A}^{a^{\star}}\otimes\Pi_{B}^{b^{\star}}\\}$ is the product of
the local measurement bases for subsystems $A$ and $B$ which achieves the
minimum.

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