# Information leak and incompatibility of physical context: A modified
approach

Arindam Mitra1,2<EMAIL_ADDRESS>1Optics and Quantum Information Group, The
Institute of Mathematical Sciences, C. I. T. Campus, Taramani, Chennai 600113,
India.
2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400094, India.  Gautam Sharma1,2<EMAIL_ADDRESS>1Optics and
Quantum Information Group, The Institute of Mathematical Sciences, C. I. T.
Campus, Taramani, Chennai 600113, India.
2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400094, India.  Sibasish Ghosh1,2<EMAIL_ADDRESS>1Optics and
Quantum Information Group, The Institute of Mathematical Sciences, C. I. T.
Campus, Taramani, Chennai 600113, India.
2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400094, India.

###### Abstract

A beautiful idea about the incompatibility of Physical Context(IPC) was
introduced in [Phys. Rev. A 102, 050201(R) (2020)]. Here, a context is defined
as a set of a quantum state and two sharp rank-one measurements, and the
incompatibility of physical context is defined as the leakage of information
while implementing those two measurements successively in that quantum state.
In this work, we show the limitations in their approach. The three primary
limitations are that, (i) their approach is not generalized for POVM
measurements and (ii), they restrict information theoretic agents Alice, Eve
and Bob to specific quantum operations and do not consider most general
quantum operations i.e., quantum instruments and (iii), their measure of IPC
can take negative values in specific cases in a more general scenario which
implies the limitation of their information measure. Thereby, we have
introduced a generalization and modification to their approach in more general
and convenient way, such that this idea is well-defined for generic
measurements, without these limitations. We also present a comparison of the
measure of the IPC through their and our method. Lastly, we show, how the IPC
reduces in the presence of memory using our modification, which further
validates our approach.

##  I Introduction

Measurement incompatibility is a key feature of Quantum theory which
distinguishes it from the classical world [1]. A pair of observables are
incompatible if they are not measurable simultaneously,i.e., their outcomes
can not be obtained jointly via a single joint measurement. Today, the
connections among incompatibility, non-locality and steering are well known
[2, 3]. Non-classical features like Bell inequality violation as well as
steering can be demonstrated only using incompatible measurements [4, 5]. It
is also well known that incompatible measurements provide an advantage over
compatible measurements in several information-theoretic tasks in quantum
information theory [6, 7]. Measurement compatibility can be characterized as
the existence of a common (i.e, constructed using same ancilla state and
Hilbert space) commuting Naimark extensions [8, 9, 10]. It has been recently
shown there are several layers of classicality inside the compatibility of
measurement[11, 12].

Recently, a novel idea was presented in ref.[13] to get a better understanding
of non-classicality associated with incompatibility. The authors of [13]
introduced the concept of incompatibility of the physical context(IPC), which
is a function of a given context, where a context comprises of a quantum state
and two measurements. In a way their measure of IPC captures the notion of
non-classicality associated with the context, as it vanishes when the state is
a maximally mixed state or the measurements are commuting with each other. It
was defined as the difference between the information remaining in a quantum
state after the first sharp measurement and after the second sharp
measurement. Moreover, the IPC is also linked with the information leakage
when an eavesdropper performs a measurement on the state being transferred in
a QKD like game [14].

However, as we will show that the approach of [13] has several limitations.
Firstly, They restrict information theoretic agents Alice, Eve and Bob to
specific quantum operations and did not consider most general quantum
operations i.e., quantum instruments. Secondly, if we do not restrict Alice,
Eve and Bob to specific quantum operations which they did, then their measure
of IPC can take negative value, which implies that the state after second
measurement by eavesdropper Eve has more information than the state after
first measurement which physically does to not make sense. Thirdly, it is not
possible to extend this idea to a generic POVM measurements through their
approach and without introducing quantum instruments. Fourthly, in presence of
memory the IPC can increase which is against the intuition that
incompatibility is non-increasing as we add memory.

In this work, we have generalized the idea of IPC for POVMs and modified the
corresponding information measure. Our measure of the IPC can never be
negative and it is non-increasing on addition of memory. We also demonstrate
the usefulness of the modified IPC measure through a QKD like scenario as an
example. In this way our approach has a wider applicability.

The rest of this paper is organised as follows: In section (II), we discuss
the preliminary concepts necessary for this paper. Then, we discuss the
limitations of the approach given in [13] and discuss our main results in
section (III). Further, in section (IV), we include the presence of memory in
our analysis. Finally, in section (V) we summarise our work and discuss about
future direction.

##  II Preliminaries

###  II.1 Observables and Channels

An observable $A$ with outcome set $\Omega_{A}$ in quantum mechanics, is a
collection of positive hermitian matrices $\\{A(x)\mid x\in\Omega_{A}\\}$ such
that $\sum_{x}A(x)=\mathbb{I}$. A pair of observables $(A,B)$ acting on same
$d$ dimensional Hilbert space $\mathcal{H}$ and with outcome sets $\Omega_{A}$
and $\Omega_{B}$ respectively, is compatible if there exist a joint observable
$\mathcal{G}$ acting on same Hilbert space $\mathcal{H}$ and outcome set
$\Omega_{A}\times\Omega_{B}$ such that for all
$\rho\in\mathcal{S}(\mathcal{H})$, $x\in\Omega_{A}$ and $y\in\Omega_{B}$

$\displaystyle A(x)=\sum_{y}\mathcal{G}(x,y);B(y)=\sum_{x}\mathcal{G}(x,y)$
(1)

where $\mathcal{S}(\mathcal{H})$ is the state space. Only for PVMs,
compatibility implies commutativity. We denote the set of all observables as
$\mathcal{O}$.

On the other hand, a quantum channel is a CPTP map from one state space
$\mathcal{S}(\mathcal{H}_{1})$ to another state space
$\mathcal{S}(\mathcal{H}_{2})$, where $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$
are two Hilbert spaces. We denote the concatenation of two quantum channels
$\Lambda_{1}$ and $\Lambda_{2}$ as $\Lambda_{1}\circ\Lambda_{2}$. Therefore,
for all $\rho\in\mathcal{S}(\mathcal{H})$
$(\Lambda_{1}\circ\Lambda_{2})(\rho)=\Lambda_{1}(\Lambda_{2}(\rho))$. Consider
two quantum channels
$\Gamma:\mathcal{S}(\mathcal{H}_{1})\rightarrow\mathcal{S}(\mathcal{H}_{2})$
and
$\Lambda:\mathcal{S}(\mathcal{H}_{1})\rightarrow\mathcal{S}(\mathcal{H}^{\prime}_{1})$.
If there exist a quantum channel
$\Theta:\mathcal{S}(\mathcal{H}^{\prime}_{1})\rightarrow\mathcal{S}(\mathcal{H}_{2})$
such that $\Gamma=\Theta\circ\Lambda$ holds, we denote it as
$\Gamma\preceq\Lambda$. If both $\Gamma\preceq\Lambda$ and
$\Lambda\preceq\Gamma$ hold, we denote it as $\Gamma\simeq\Lambda$ and we call
it as $\Gamma$ and $\Lambda$ are concatenation equivalent. We denote the set
of all channels concatenation equivalent to channel $\Lambda$ as $[\Lambda]$.

There exists a special type of channel known as completely depolarising
channel, which we will use in following section. A channel $\Sigma$ is called
completely depolarising channel if for all $T\in\mathcal{L}^{+}(\mathcal{H})$,
$\Sigma(T)=\text{Tr}(T)\eta$ for some fixed $\eta\in\mathcal{S}(\mathcal{H})$,
where $\mathcal{L}^{+}(\mathcal{H})$ is set of positive linear operators on
Hilbert Space $\mathcal{H}$. We denote the set of all channels as
$\mathcal{C}$.

###  II.2 Quantum Instruments and Measurement models

In quantum measurements, there are two equivalent concepts, namely measurement
models and quantum instruments [15, 16]. Measurement models are descriptions
of measurement process, where as instruments are the concise version of it.
Consider a measured system $S$ associated with a Hilbert space
$\mathcal{H}_{S}$ and with density matrix $\rho$ and a ancilla system
associated with another Hilbert space $\mathcal{H}_{a}$ and with density
matrix $\sigma_{a}$. To perform a measurement on a measured system, at first a
joint unitary $U$ have to be applied on the composite system where, $U$ is
acting on Hilbert space $\mathcal{H}_{S}\otimes\mathcal{H}_{a}$. Then, a
pointer observable $A^{\prime}$ with outcome set $\Omega_{A}^{\prime}$ have to
be measured on the ancilla system. Now in this process if the observable $A$
with same outcome set as $A^{\prime}$ has, is the measured on the system $S$
then for all $x\in\Omega_{A}$ and $\rho\in\mathcal{H}_{S}$ we have

$\text{Tr}[\rho
A(x)]=\text{Tr}[U(\rho\otimes\sigma_{a})U^{\dagger}(\mathbb{I}\otimes
A^{\prime}(x))].$ (2)

The average post measurement state is given by

$\Lambda(\rho)=\text{Tr}_{\mathcal{H}_{a}}[U(\rho\otimes\sigma_{a})U^{\dagger}].$
(3)

Here, $\Lambda$ is a quantum channel. This measurement model is specified by
the quadruple $(\mathcal{H}_{a},\sigma_{a},U,A^{\prime})$.
A quantum instrument $\mathcal{I}$ through which the measurement of an
observable $A$ can be implemented, is a collection of CP trace non-increasing
maps $\\{\Phi_{x}\\}$ such that for all $x\in\Omega_{A}$ and
$\rho\in\mathcal{H}_{S}$ we have

$\text{Tr}[\rho A(x)]=\text{Tr}[\Phi_{x}(\rho)]$ (4)

and

$\sum_{x}\Phi_{x}(\rho)=\Lambda(\rho)$ (5)

where $\Lambda$ is a quantum channel. We call such an instrument as
$A$-compatible instrument. If $\mathcal{I}=\\{\Phi_{x}\\}$ is an
$A$-compatible instrument, then another instrument
$\Theta\circ\mathcal{I}=\\{\Theta\circ\Phi_{x}\\}$ is also an $A$-compatible
instrument [17], where
$(\Theta\circ\Phi_{x})(\rho)=\text{Tr}[\Phi_{x}(\rho)]\Theta\left(\frac{\Phi_{x}(\rho)}{\text{Tr}[\Phi_{x}(\rho)]}\right)$.
We denote the set of all $A$-compatible instruments as $\mathcal{J}_{A}$.
Therefore, given a measurement model
$(\mathcal{H}_{a},\sigma_{a},U,A^{\prime})$, one can associate a quantum
instrument $\mathcal{I}$ such that for all $x\in\Omega_{A}$ and
$\rho\in\mathcal{H}_{S}$ we have

$\text{Tr}[\Phi_{x}(\rho)]=\text{Tr}[U(\rho\otimes\sigma_{a})U^{\dagger}(\mathbb{I}\otimes
A^{\prime}(x))].$ (6)

Similarly, given a quantum instrument it is possible to find out a measurement
model such that equation (6) holds [18]. This implies that these two concepts
are equivalent.

###  II.3 Observable-Channel compatibility

A quantum channel $\Lambda$ is compatible with an observable $A$ if there
exists a quantum instrument $\mathcal{I}=\\{\Phi_{x}\\}$ such that equations
(4) and (5) together hold. Otherwise, they are incompatible. If a channel
$\Lambda$ and an observable $A$ are compatible, we denote it as
$\Lambda\hbox{\hskip 2.13394pt$\circ\hskip-2.84526pt\circ$\hskip 2.13394pt}A$
[19]. We call $\Lambda$ as $A$-compatible channel. It is well known that
completely depolarising channels are compatible with any observable [16]. For
a quantum channel $\Lambda\in\mathcal{C}$ and an observable $A\in\mathcal{O}$,
following sets are introduced in [19]:

$\displaystyle\tau_{c}(\Lambda)=\\{X\in\mathcal{O}\mid\Lambda\hbox{\hskip
2.13394pt$\circ\hskip-2.84526pt\circ$\hskip 2.13394pt}X\\};$ (7)
$\displaystyle\sigma_{c}(A)=\\{\Gamma\in\mathcal{C}\mid\Gamma\hbox{\hskip
2.13394pt$\circ\hskip-2.84526pt\circ$\hskip 2.13394pt}A\\}.$ (8)

Let us now write down the following theorem which was originally proved in
ref.[17]:

######  Theorem 1.

Suppose $A\in\mathcal{O}$ be an observable acting on the state space
$\mathcal{S(H)})$ and $(V,\mathcal{K},\hat{A})$ be its Naimark extension, i.e,
$\mathcal{K}$ is a Hilbert space, $V:\mathcal{H}\rightarrow\mathcal{K}$ is an
isometry and $\hat{A}=\\{\hat{A}(x)\\}$ is a PVM such that
$V^{\dagger}\hat{A}(x)V=A(x)$ for all $x\in\Omega_{A}$. Then,

$\sigma_{c}(A)=\\{\Lambda\in\mathcal{C}\mid\Lambda\preceq\Lambda_{A}\\}$ (9)

where for any state $\rho$, $\Lambda_{A}(\rho)=\sum_{x}\hat{A}(x)V\rho
V^{\dagger}\hat{A}(x)$.

We call $\Lambda_{A}$ as parent channel of $\sigma_{c}(A)$ and we also call
the corresponding $A$-compatible instrument $\mathcal{I}_{A}$ a parent
instrument in $\mathcal{J}_{A}$. Clearly, $\Lambda_{A}$ depends on the choice
of the Naimark extension. But any two parent channels are concatenation
equivalent. Therefore, we have freedom to choose it.

###  II.4 Holevo Bound

The Holevo bound captures the maximum classical information that can be
extracted from an ensemble of quantum states[20]. Suppose, we have an ensemble
$\mathcal{E}$={$p_{X}(x),\rho_{x}$}, and our task is to determine the
classical index $x$ by doing some measurements. The density matrix operator
corresponding to this ensemble has the form $\rho=\sum_{x}p_{X}(x)\rho_{x}$.
Now, we can do a measurement $\Lambda_{y}$, so that the information gain after
doing the measurement is given by the mutual information $I=I(X;Y)$ after the
measurement, where $Y$ is the random variable corresponding to the outcome of
measurement. It is known that the maximum value of this mutual information is
given by the Holevo bound[20, 21], given by

$\displaystyle\chi(\mathcal{E})=S(\rho)-\sum_{x}p_{X}(x)S(\rho_{x})$ (10)

where $S(\rho)$ is the Von-Neumann entropy of the state $\rho$. It is
interesting to note that the holevo bound $\chi$ is also the mutual
information of a classical-quantum state of the form
$\rho_{CQ}=\sum_{x}p_{X}(x)\ket{x}\bra{x}\otimes\rho_{x}$. Under the action of
a channel $\Lambda$ the ensemble transforms as
$\mathcal{E}\rightarrow\mathcal{E}^{\prime}=\\{p_{X}(x),\rho_{x}^{\prime}\\}$.
But we know that the mutual information is non-increasing under the action of
channels[21], which implies that the Holevo information is also non-increasing
under the action of quantum channels, i.e.,

$\displaystyle\chi(\mathcal{E})\geq\chi(\mathcal{E}^{\prime}).$ (11)

###  II.5 Incompatibility of physical context

In a recent work [13], the concept of IPC was introduced which was further
used to show quantum resource covariance[22]. To define this idea, we need the
notion of context. A context is defined as $\mathbb{C}=\\{\rho,X,Y\\}$, where
$\rho$ is an arbitrary quantum state. Also, $X=\\{X_{i}\\}$ and
$Y=\\{Y_{j}\\}$ are two observables, with $X_{i}$ and $Y_{j}$ as the
respective eigen projectors.

Other than the definition of context, we also need a game using which we
define the incompatibility of a context $\mathbb{C}$. The game goes like this.
Alice prepares the quantum state $\rho$, and of course it has some information
content which can be quantified by using any known measure. The authors in
ref.[13] quantify the information of $\rho$ using the following

$\displaystyle I(\rho)=\ln d-S(\rho),$ (12)

where $S(\rho)$=-$Tr(\rho\ln\rho)$ is the von Neumann entropy of $\rho$ and
$d$ is the dimension of the Hilbert space. This information is non-negative,
i.e., $I(\rho)\geq 0$, is ensured because $S(\rho)\leq\ln d$. After state
preparation, Alice performs a noisy measurement with $X$ on the prepared
state, so that $\rho$ transforms as

$\displaystyle\rho\rightarrow\mathcal{N}_{X}(\rho)=\sum_{i=1}^{d}X_{i}\rho
X_{i}.$ (13)

So, after this operation the information content in the state
$\mathcal{N}_{X}(\rho)$ is $I_{1}=I(\mathcal{N}_{X}(\rho))$. This state
$\mathcal{N}_{X}(\rho)$ is then delivered to Bob, who verifies the information
content of the state. In case Bob finds that there is no loss of information,
Alice and Bob will agree that the channel is free from information leakage.

But it might happen that there is an eavesdropper, Eve, who performs a noisy
measurement Y on the state $\mathcal{N}_{X}(\rho)$, before it is delivered to
Bob. The state is then transformed as

$\displaystyle\mathcal{N}_{X}(\rho)\rightarrow(\mathcal{N}_{Y}\circ\mathcal{N}_{X})(\rho)=\mathcal{N}_{YX}(\rho)=\sum_{j=1}^{d}Y_{j}\mathcal{N}_{X}(\rho)Y_{j}.$
(14)

Thus, the information content in the state $\mathcal{N}_{XY}(\rho)$ is
$I_{2}=I(\mathcal{N}_{YX}(\rho))$. And therefore, the leakage in the
information content is given by

$\displaystyle\mathscr{I}_{C}=I_{1}-I_{2}$
$\displaystyle=I(\mathcal{N}_{X}(\rho))-I(\mathcal{N}_{YX}(\rho)),$
$\displaystyle=S(\mathcal{N}_{YX}(\rho))-S(\mathcal{N}_{X}(\rho)).$ (15)

Hence, only if $\mathscr{I}_{C}>0$, Alice and Bob will know that there is
information leakage from the channel. Notice that $\mathscr{I}_{C}=0$ in two
kind of scenarios: 1) If $X$ and $Y$ commute with each other and 2) if $\rho$
is a maximally mixed state. In the first scenario
$\mathcal{N}_{X}(\rho)=\mathcal{N}_{YX}(\rho)$ because the two operators are
compatible with each other. And in the second type of scenarios $I_{1}=0$ and
there is no information to loose which results in $I_{1}=I_{2}$. Thus, we
require the incompatibility $I_{C}$ to be non-zero for Bob to detect any
leakage of information

Hence, the concept of IPC can be defined as

######  Definition 1.

Context incompatibility is the resource encoded in a context
$\mathbb{C}=\\{\rho,X,Y\\}$ that allows one to test the safety of a
communication channel against information leakage. It is quantified as
$\mathscr{I}_{C}=I_{1}-I_{2}=I(\mathcal{N}_{X}(\rho))-I(\mathcal{N}_{YX}(\rho))$.
It is operationally related to the amount of information lost from the system
under an external measurement.

##  III Main Results

###  III.1 Limitations of incompatibility of physical context

In this section, we discuss the limitations of approach given in [13].

  1. 1.

First of all, according to the approach given in [13], the post-measurement
states after measuring sharp observable $X\in\mathcal{O}$ on a quantum state
$\rho$ is $\mathcal{N}_{X}(\rho)$. Therefore, to measure an observable $X$,
Alice and Eve both are restricted to use a particular channel
$\mathcal{N}_{X}\in\mathcal{C}$, or equivalently they are restricted to use a
particular quantum instrument $\mathcal{I}_{X}=\\{\Phi_{X}(x)\\}$ such that
$\sum_{x}\Phi_{X}(x)=\mathcal{N}_{X}$. Since, we have no control atleast over
eavesdropper Eve, there is no reason to assume such a restriction.

  2. 2.

Second, to generalize it, suppose we remove such restriction, i.e., to measure
an observable, now Alice and Eve can use all possible instruments that are
compatible with that observable. Then to measure the observable $X$ if Alice
uses an arbitrary instrument
$\mathcal{I}^{\prime}_{X}=\\{\Phi^{\prime}_{X}(x)\\}$ such that
$\Lambda^{\prime}=\sum_{x}\Phi^{\prime}_{X}(x)$ and to measure $Y$ Eve uses a
special instrument $\mathcal{I}^{depo}_{Y}=\\{\Phi^{depo}_{Y}(y)\\}$ such that
for all $\rho\in\mathcal{S}(\mathcal{H})$ and a fix pure state $\eta$,
$\Lambda^{depo}_{\eta}(\rho)=\sum_{y}\Phi^{depo}_{Y}(y)(\rho)=\eta$ is a
completely depolarising channel. Now, as $S(\eta)=0$, from equation (II.5) we
have

$\displaystyle\mathscr{I}_{\mathcal{C}}$
$\displaystyle=I(\Lambda^{\prime}(\rho))-I((\Lambda^{depo}_{\eta}\circ\Lambda^{\prime})(\rho))$
$\displaystyle=-S(\Lambda^{\prime}(\rho))$ $\displaystyle\leq 0.$ (16)

The negativity of IPC implies that the post-measurement state of Eve has more
information than the post-measurement state of Alice, which does not make
sense. This is because the information which Alice sends to Bob, can not be
increased by the Eavesdropper. Such a problem is occurring because Von Neumann
entropy is not monotonically non-increasing under action of a quantum channel.
Therefore, in this general context, their information measure is not a proper
information measure.

  3. 3.

Thirdly, as we know that for any POVM, post measurement state depends on the
quantum instrument used to implement that POVM, their results can not be
generalised for POVMs without introducing quantum instruments or equivalently
without introducing measurement models!

Therefore, in our attempt to generalize the idea of IPC for POVMs, we need to
modify idea and present it in a different way which we describe in following
subsections.

###  III.2 Modified measure of Information leakage

In this section we present a generalization of the game presented in Sec.II.5.
Now, in the game, after the state preparation of $\rho$, instead of only doing
a sharp measurement we allow Alice to perform a more generic measurement. Now,
Alice performs her measurement with the POVM, $A$ on the quantum state
$\rho\in\mathcal{S}(\mathcal{H})$ using the $A$-compatible instrument
$\mathcal{I}^{\prime}_{A}=\\{\Phi_{A,x}\\}$ such that
$\Lambda^{\prime}_{A}=\sum_{x}\Phi_{A,x}$ and generates the ensemble
$\mathcal{E}_{A}=\\{p_{x},\rho_{x}\\}$, where
$p_{x}=\text{Tr}[\Phi_{A,x}(\rho)]$ and
$\rho_{x}=\frac{\Phi_{A,x}(\rho)}{\text{Tr}[\Phi_{A,x}(\rho)]}$. Here,
$\Lambda^{\prime}_{A}$ is the quantum channel such that
$\Lambda^{\prime}_{A}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{K})$.
Furthermore, we quantify the information content of the ensemble
$\mathcal{E}_{A}$ via the Holevo bound as:

$\displaystyle\chi(\rho,\mathcal{I}^{\prime}_{A})=S(\Lambda_{A}^{\prime}(\rho))-\sum_{x}p_{x}S(\rho_{x}).$

This measure of information has been previously used to quantify information
gain in ref.[23]. Similarly, the eavesdropper Eve performs the POVM
measurement $B$ on the quantum state
$\Lambda_{A}^{\prime}(\rho)\in\mathcal{S}(\mathcal{K})$ using the
$B$-compatible instrument $\mathcal{I}^{\prime}_{B}=\\{\Phi_{B,y}\\}$ such
that $\Lambda^{\prime}_{B}=\sum_{y}\Phi_{B,y}$ and generates the ensemble
$\mathcal{E}_{B}=\\{p_{x},\Lambda^{\prime}_{B}(\rho_{x})\\}$. Here,
$\Lambda^{\prime}_{B}$ is the quantum channel such that
$\Lambda^{\prime}_{B}:\mathcal{S}(\mathcal{K})\rightarrow\mathcal{S}(\mathcal{K}^{\prime})$.
It should be noted that, Alice and Bob do not have access to Eve's measurement
outcomes, her measurement can be represented using a channel. Now, the
information remaining in the state
$(\Lambda^{\prime}_{B}\circ\Lambda^{\prime}_{A})(\rho)$ is given by its Holevo
bound, i.e.,

$\displaystyle\chi(\rho,\mathcal{I}^{\prime}_{A},\mathcal{I}^{\prime}_{B})=S((\Lambda^{\prime}_{B}\circ\Lambda^{\prime}_{A})(\rho))-\sum_{x}p_{x}S(\Lambda^{\prime}_{B}(\rho_{x})).$

Therefore, Bob who was expecting to receive an ensemble with information
$\chi_{(}\rho,\mathcal{I}^{\prime}_{A})$, would receive a different ensemble
with information content
$\chi(\rho,\mathcal{I}^{\prime}_{A},\mathcal{I}^{\prime}_{B})$. Thus, the new
form of information leakage of the channel is

$\displaystyle
I^{H}_{c}(\rho,\mathcal{I}^{\prime}_{A},\mathcal{I}^{\prime}_{B})$
$\displaystyle=\chi(\rho,\mathcal{I}^{\prime}_{A})-\chi(\rho,\mathcal{I}^{\prime}_{A},\mathcal{I}^{\prime}_{B})$
$\displaystyle=S(\Lambda_{A}^{\prime}(\rho))-S((\Lambda^{\prime}_{B}\circ\Lambda^{\prime}_{A})(\rho))$
$\displaystyle+\sum_{x}p_{x}S(\Lambda^{\prime}_{B}(\rho_{x}))-\sum_{x}p_{x}S(\rho_{x}).$
(17)

As Holevo bound is monotonically non-increasing under the action of quantum
channels,
$I^{H}_{c}(\rho,\mathcal{I}^{\prime}_{A},\mathcal{I}^{\prime}_{B})\geq 0$.
When $I^{H}_{c}(\rho,\mathcal{I}^{\prime}_{A},\mathcal{I}^{\prime}_{B})>0$,
Alice and Bob will be able to detect the information leakage in the channel.
Now, if Eve is rational, her goal will be to minimize leakage along with
collecting information. Therefore, to measure $B$ she will choose an
instrument such that
$\mathcal{I}^{H}_{c}(\rho,\mathcal{I}^{\prime}_{A},\mathcal{I}^{\prime}_{B})$
takes the minimum value. Now, let $\Lambda_{B}$ be a parent channel in
$\sigma_{c}(B)$ and corresponding $B$-compatible instrument be
$\mathcal{I}_{B}$. Then, as for any other channel
$\Lambda^{\prime}_{B}\in\sigma_{B}$, $\Lambda^{\prime}_{B}\preceq\Lambda_{B}$
holds and Holevo bound is monotonically decreasing under action of a quantum
channel,

$\displaystyle\chi(\rho,\mathcal{I}^{\prime}_{A},\mathcal{I}^{\prime}_{B})\leq\chi(\rho,\mathcal{I}^{\prime}_{A},\mathcal{I}_{B})~{}\forall\mathcal{I}_{B}.$
(18)

Therefore, implementation of a parent instrument keeps maximum amount of
accessible information or equivalently maximum Holevo bound! Therefore, for a
given instrument of Alice the minimum leakage of information is

$\displaystyle I^{H}_{c}(\rho,\mathcal{I}^{\prime}_{A},B)$
$\displaystyle=\min_{\mathcal{I}^{\prime}_{B}}I^{H}_{c}(\rho,\mathcal{I}^{\prime}_{A},\mathcal{I}^{\prime}_{B})$
$\displaystyle=\chi(\rho,\mathcal{I}^{\prime}_{A})-\max_{\mathcal{I}^{\prime}_{B}}\chi(\rho,\mathcal{I}^{\prime}_{A},\mathcal{I}^{\prime}_{B})$
$\displaystyle=\chi(\rho,\mathcal{I}^{\prime}_{A})-\chi(\rho,\mathcal{I}^{\prime}_{A},\mathcal{I}_{B})$
$\displaystyle=I^{H}_{c}(\rho,\mathcal{I}^{\prime}_{A},\mathcal{I}_{B}).$ (19)

Note that the choice of $B$ depends on output state space
$\mathcal{S}(\mathcal{K})$ of the quantum channel $\Lambda^{\prime}_{A}$ and
in that sense, it is arbitrary.

Now, if Alice is also rational and she does not know the presence of Eve, she
will try to create an ensemble with most accessible information such that the
receiver i.e, Bob can get best amount of information, or equivalently, she
will use an $A$-compatible instrument for which
$\chi(\rho,\mathcal{I}^{\prime}_{A})$ is maximum. Let, $\Lambda_{A}$ be a
parent channel in $\sigma_{c}(A)$ and corresponding $A$-compatible parent
instrument be $\mathcal{I}_{A}$. Then, using arguments as above

$\displaystyle\chi(\rho,\mathcal{I}^{\prime}_{A})\leq\chi(\rho,\mathcal{I}_{A})~{}\forall\mathcal{I}^{\prime}_{A}.$
(20)

Therefore, if Alice uses the instrument $\mathcal{I}_{A}$, in this case the
information leakage will be minimum when Alice uses a parent channel from
$\sigma_{c}(A)$, and is given by:

$\displaystyle
I^{H}_{c}(\rho,A,B)=I^{H}_{c}(\rho,\mathcal{I}_{A},\mathcal{I}_{B}).$ (21)

Clearly, if Eve uses any other quantum instrument (e.g., dimension preserving
instrument) $\mathcal{I}^{\prime}_{B}$, then
$I^{H}_{c}(\rho,\mathcal{I}_{A},\mathcal{I}^{\prime}_{B})\geq
I^{H}_{c}(\rho,A,B)$. Therefore, assuming both Alice and Eve to be rational,
$I^{H}_{c}(\rho,A,B)$ is the appropriate amount of information leak when the
parent instruments are used.

###  III.3 Incompatibility of physical context: A modified version

First of all we modify the notion of context so that,
$\mathbb{C}=\\{\rho,\mathbb{X},\mathbb{Y}\\}$, where $\mathbb{X}$ and
$\mathbb{Y}$ are POVM measurements acting on $\mathcal{S}(\mathcal{H})$ and
$\mathcal{S}(\mathcal{H}^{\prime})$ respectively. Since, $\mathbb{X}$ and
$\mathbb{Y}$ are given, to define IPC, we restrict Alice's instrument
$\mathcal{I}^{\prime,\mathcal{H}^{\prime}}_{\mathbb{X}}=\\{\Phi^{\prime,\mathcal{H}^{\prime}}_{\mathbb{X},x}\\}$
such that
$\Lambda^{\prime}_{\mathbb{X}}=\sum_{x}\Phi^{\prime,\mathcal{H}^{\prime}}_{\mathbb{X},x}$
and
$\Lambda^{\prime}_{\mathbb{X}}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{H}^{\prime})$.
We denote the set of all such $\mathbb{X}$-compatible instruments as
$\mathcal{J}^{\mathcal{H}^{\prime}}_{\mathbb{X}}$. With this restriction also,
being rational, Alice's goal will be to maximize
$\chi(\rho,\mathcal{I}^{\prime,\mathcal{H}^{\prime}}_{\mathbb{X}})$. Let, for
some
$\mathcal{I}_{\mathbb{X},max}^{\mathcal{H}^{\prime}}\in\mathcal{J}^{\mathcal{H}^{\prime}}_{\mathbb{X}}$,

$\max_{\mathcal{I}_{\mathbb{X}}^{\mathcal{H}^{\prime}}}\chi(\rho,\mathcal{I}_{\mathbb{X}}^{\mathcal{H}^{\prime}})=\chi(\rho,\mathcal{I}_{\mathbb{X},max}^{\mathcal{H}^{\prime}}).$
(22)

Therefore, similar to the previous subsection, in this case the appropriate
amount of information leak is

$\mathfrak{I}(\mathbb{C})=I^{H}_{c}(\rho,\mathcal{I}_{\mathbb{X},max}^{\mathcal{H}^{\prime}},\mathcal{I}_{\mathbb{Y}}).$
(23)

For the special case of, $\mathcal{H}^{\prime}=\mathcal{H}$, we have

$\mathfrak{I}(\mathbb{C})=I^{H}_{c}(\rho,\mathcal{I}_{\mathbb{X},max}^{\mathcal{H}},\mathcal{I}_{\mathbb{Y}}).$
(24)

Therefore, we can define the generalized version of IPC as:

######  Definition 2.

Context incompatibility is the resource encoded in a context
$\mathbb{C}=\\{\rho,\mathbb{X},\mathbb{Y}\\}$ that allows one to test the
safety of the channel against information leakage. This resource is quantified
via
$\mathfrak{I}(\mathbb{C})=I^{H}_{c}(\rho,\mathcal{I}_{\mathbb{X},max}^{\mathcal{H}},\mathcal{I}_{\mathbb{Y}})$,
where $\mathcal{I}_{\mathbb{X},max}^{\mathcal{H}}$ is the
$\mathbb{X}$-compatible instrument that maximizes
$\chi(\rho,\mathcal{I}^{\mathcal{H}}_{\mathbb{X}})$. Operationally, it is the
proper information leakage in the channel caused by an external measurement on
the state.

Clearly, if Eve uses any other quantum instrument (e.g., dimension preserving
instrument) $\mathcal{I}^{\prime}_{\mathbb{Y}}$, then
$I^{H}_{c}(\rho,\mathcal{I}_{\mathbb{X},max}^{\mathcal{H}},\mathcal{I}^{\prime}_{\mathbb{Y}})\geq\mathfrak{I}(\mathbb{C})$.
Moreover, if Alice performs a sharp measurement $X=\\{X_{i}\\}$, from theorem
(1), choosing $V=\mathbb{I}$ or equivalently choosing
$\mathcal{H}=\mathcal{K}$ and $X_{i}=\hat{X_{i}}$ we get a parent channel
$\Lambda_{A}=\mathcal{N}(X):\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{H})$.
Let $\mathcal{I}_{X}=\\{\Phi_{X}\\}$ be corresponding $X$-compatible parent
instrument. As, implementation of a parent channel keeps maximum amount of
accessible information or, equivalently maximum Holevo bound, we have
$\mathcal{I}_{\mathbb{X},max}^{\mathcal{H}}=\mathcal{I}_{X}$. Then the proper
information leakage will have the following form

$\displaystyle\mathfrak{I}(\mathbb{C})$
$\displaystyle=\chi(\rho,\mathcal{I}_{X})-\chi(\rho,\mathcal{I}_{X},\mathcal{I}_{\mathbb{Y}}),$
$\displaystyle=S(\mathcal{N}_{X}(\rho))-S((\Lambda_{\mathbb{Y}}\circ\mathcal{N}_{X})(\rho))$
$\displaystyle+\sum_{x}p_{x}S(\Lambda_{\mathbb{Y}}(\rho_{x}))-\sum_{x}p_{x}S(\rho_{x}).$
(25)

where $\Lambda_{\mathbb{Y}}$ is the $\mathbb{Y}$-compatible parent channel
corresponding to the $\mathbb{Y}$-compatible parent instrument
$\mathcal{I}_{\mathbb{Y}}$.

###  III.4 Relation between two definitions

Our generalization of the measure of IPC, gives a simplified form when we
demand that both Alice and Eve perform rank-1 sharp measurements $X$ and $Y$
using parent instruments $\mathcal{I}_{X}\in\mathcal{J}_{X}$ and
$\mathcal{I}_{Y}\in\mathcal{J}_{Y}$, where $\mathcal{N}_{X}\in\sigma_{c}(X)$
and $\mathcal{N}_{Y}\in\sigma_{c}(Y)$ are corresponding channels respectively.
In this case $\mathfrak{I}(\mathbb{C})$ reads as

$\displaystyle\mathfrak{I}(\mathbb{C})=\sum_{x}p_{x}S(\mathcal{N}_{Y}(\rho_{x}))-\mathscr{I}_{C}.$
(26)

The above equation relates our generalized measure of IPC with the measure of
IPC $\mathscr{I}_{C}$ defined in [13]. To compare the two measures of the IPC,
we remind the reader that $\mathscr{I}_{C}$ is zero when 1) $X$ and $Y$
commute or 2) $\rho$ is a maximally mixed state (see secII.5). Coming to the
new measure of IPC we find that $\mathfrak{I}(\mathbb{C})=0$ whenever $X$ and
$Y$ commute, because then $\mathcal{N}_{Y}(\rho_{x})$ are pure states.
However, $\mathfrak{I}(\mathbb{C})$ is not necessarily equal to zero when
$\rho$ is a maximally mixed state(as in the Eq. (26), $\mathscr{I}_{C}$ is
zero but $S(\mathcal{N}_{Y}(\rho_{x}))$'s are not zero).

This implies that our measure captures the incompatibility of a context even
when the state (belonging to the context) is a maximally mixed state. This is
unlike the previous measure of IPC $\mathscr{I}_{C}$ given in ref [13], which
says that the context is compatible if the state is a maximally mixed state.
This difference arises from the fact that the Holevo quantity(unlike the
information measure in Eq.(II.5), which represents the extractable
information, can be non-zero for an ensemble created from measurement on a
maximally mixed state. We show the importance of the new IPC measure through
the following example.

######  Example 1.

Consider a scenario in which Alice is randomly implementing $\sigma_{x}$ and
$\sigma_{z}$ measurements on the maximally mixed state with equal
probabilities and generates ensemble
$\\{\\{\frac{1}{2},\ket{0}\bra{0}\\},\\{\frac{1}{2},\ket{1}\bra{1}\\}\\}$ and
$\\{\\{\frac{1}{2},\ket{+}\bra{+}\\},\\{\frac{1}{2},\ket{-}\bra{-}\\}\\}$
respectively, for Bob. This is a QKD like situation. Now, the possible bases
of measurements are known for eavesdropper Eve. But she does not know which
measurement is exactly implemented in a particular run. Therefore, she is
randomly measuring $\sigma_{x}$ and $\sigma_{z}$ on the ensemble created by
Alice. In this case, if we use the measure of IPC from Eq.(II.5), we get the
following as the average IPC

$\displaystyle\mathscr{I}_{avg}$
$\displaystyle=\frac{1}{4}\mathscr{I}(\frac{\mathbb{I}}{2},\sigma_{z},\sigma_{z})+\frac{1}{4}\mathscr{I}(\frac{\mathbb{I}}{2},\sigma_{z},\sigma_{x})$
$\displaystyle+\frac{1}{4}\mathscr{I}(\frac{\mathbb{I}}{2},\sigma_{x},\sigma_{z})+\frac{1}{4}\mathscr{I}(\frac{\mathbb{I}}{2},\sigma_{x},\sigma_{x})$
$\displaystyle=0$ (27)

Therefore, according to this analysis the information leakage is not
detectable. But it is well established fact that if Alice and Bob declare the
basis of their measurements Eve will be detected since her operation disturbs
the ensemble. Therefore, this measure of IPC is not very useful here. Instead,
if we use the modified measure of IPC, we get

$\displaystyle\mathfrak{I}_{avg}$
$\displaystyle=\frac{1}{4}\mathfrak{I}(\frac{\mathbb{I}}{2},\sigma_{z},\sigma_{z})+\frac{1}{4}\mathfrak{I}(\frac{\mathbb{I}}{2},\sigma_{z},\sigma_{x})$
$\displaystyle+\frac{1}{4}\mathfrak{I}(\frac{\mathbb{I}}{2},\sigma_{x},\sigma_{z})+\frac{1}{4}\mathfrak{I}(\frac{\mathbb{I}}{2},\sigma_{x},\sigma_{x})$
$\displaystyle=\frac{1}{4}\mathfrak{I}(\frac{\mathbb{I}}{2},\sigma_{z},\sigma_{x})+\frac{1}{4}\mathfrak{I}(\frac{\mathbb{I}}{2},\sigma_{x},\sigma_{z})$
$\displaystyle=\frac{1}{2}\ln 2$ $\displaystyle\neq 0,$ (28)

where we have used the equation (26) to arrived at the last line. This non-
zero value suggests that, information leakage can be detected, as expected.
Hence, the modified IPC measure is useful in this scenario.

In this Example we have considered that Eve is using a parent instrument for
her measurement. If she uses any instrument other than the parent instrument
the average information leakage will be higher than $\mathfrak{I}_{avg}$.

##  IV Incompatibility of physical context in the presence of memory

Motivated by the work in [24], where it was shown that in the presence of
memory the total uncertainty of two measurements gets reduced, we ask the
question, how the IPC will change in the presence of memory? To accommodate
the presence of memory, we modify our game slightly, for the scenario where we
perform only rank-one projective measurements ${X}$ and ${Y}$.

In the modified game, our initial state $\sigma_{in}$ is the subsystem of the
bipartite state $\sigma_{in,M}$, where $M$ acts as the memory. After the ${X}$
measurement on the subsystem $\sigma_{in}$, Alice produces the bipartite
ensemble ${\rho}_{AM}=\sum_{x}p_{x}\rho^{x}_{AM}$, where
$\rho^{x}_{M}=\text{Tr}_{A}[\rho^{x}_{AM}]$ acts as the memory and Bob
receives the subsystem $A$ prepared in the state
$\rho_{A}=\text{Tr}_{M}[\rho_{AM}]$. On this ensemble, if we use the approach
in ref.[13], the information content of $\rho_{A}$ conditioned on memory
$\rho_{M}$ is given by

$\displaystyle I^{mem}_{1}=\ln d-S(A|M)=\ln d-S(\rho_{AM})+S(\rho_{M}),$

where $S(A|M)=S(\rho_{AM})-S(\rho_{M})$ is the conditional entropy [21]. After
the ${Y}$ measurement by Eve on $\rho_{A}$, the ensemble transforms as
$\sum_{x}p_{x}\rho^{x}_{AM}\rightarrow\sum_{x}p_{x}(\mathcal{N}_{Y}\otimes\mathbb{I})(\rho^{x}_{AM})=\sum_{x}p_{x}\rho^{x}_{A^{\prime}M}$,
so that the remaining information content of the state $\rho_{A^{\prime}}$ is

$\displaystyle I^{mem}_{2}=\ln d-S(A^{\prime}|M)=\ln
d-S(\rho_{A^{\prime}M})+S(\rho_{M}),$

where $\mathbb{I}$ is the identity channel acting on the memory. Therefore, in
the presence of memory, the expression of IPC takes the following form.

$\displaystyle\mathscr{I}^{mem}_{C}$ $\displaystyle=I^{mem}_{1}-I^{mem}_{2}$
$\displaystyle=S(\rho_{A^{\prime}M})-S(\rho_{AM}).$ (29)

To compare the IPC with and without memory, we compare Eq.(II.5) with Eq.(IV),
which gives the following

$\displaystyle\mathscr{I}_{C}-\mathscr{I}^{mem}_{C}$
$\displaystyle=[S(\rho_{A^{\prime}})-S(\rho_{A^{\prime}M})]-[S(\rho_{A})-S(\rho_{AM})]$
$\displaystyle=I^{coh}(M\rangle A^{\prime})-I^{coh}(M\rangle A)\leq 0.$ (30)

Here, $I^{coh}(M\rangle A)=S(\rho_{A})-S(\rho_{AM})$ is the coherent
information that is non-increasing under the action of quantum channels [25,
26, 21]. This analysis tells us that the IPC is increasing in the presence of
memory, which seems contrary to the intuition that memory reduces the
incompatibility.

Next, we compute the IPC in the modified game with our approach. In our case,
after the ${X}$ measurement, the extractable classical information from
$\rho_{A}$ is the mutual information of the quantum-classical ensemble
$\rho_{CA}=\sum_{x}\ket{x}_{C}\bra{x}\otimes\rho^{x}_{A}$ (see sec.II.4).
However, now it is conditioned on the memory $\rho_{M}$. Therefore, in the
presence of memory the extractable information will be the mutual information
between $\rho_{C}=\sum_{x}\ket{x}_{C}\bra{x}$ and $\rho_{A}$, conditioned on
the memory $\rho_{M}$ via the tripartite classical-quantum state
$\rho_{CAM}=\sum_{x}p_{x}\ket{x}_{C}\bra{x}\otimes\rho^{x}_{AM}$, i.e.,

$\displaystyle\mathcal{X}^{mem}_{1}$ $\displaystyle=S(A:C|M)$
$\displaystyle=S(A|M)+S(C|M)-S(AC|M)$
$\displaystyle=S(\rho_{AM})-S(\rho_{M})+S(\rho_{CM})-S(\rho_{CAM}).$

Here, we have simply expanded the conditional entropies to get the final form.
Also, after Eve performs her measurement ${Y}$ on the subsystem $\rho_{A}$,
the remaining mutual information between $\rho_{A^{\prime}}$ and $\rho_{C}$
conditioned on the memory $\rho_{M}$, via the classical-quantum ensemble
$\sum_{x}p_{x}\ket{x}_{C}\bra{x}\otimes\mathcal{\rho}^{x}_{A^{\prime}M}$ is
given by

$\displaystyle\mathcal{X}^{mem}_{2}$ $\displaystyle=S(A^{\prime}:C|M)$
$\displaystyle=S(A^{\prime}|M)+S(C|M)-S(A^{\prime}C|M)$
$\displaystyle=S(\rho_{A^{\prime}M})-S(\rho_{M})+S(\rho_{CM})-S(\rho_{CA^{\prime}M}).$

Therefore the IPC, using our approach in presence of memory, takes the
following form

$\displaystyle\mathfrak{I}^{mem}(\mathbb{C})$
$\displaystyle=\mathcal{X}^{mem}_{1}-\mathcal{X}^{mem}_{2}$
$\displaystyle=S(\rho_{AM})-S(\rho_{A^{\prime}M})-S(\rho_{ACM})+S(\rho_{A^{\prime}CM})$
$\displaystyle=S(\rho_{AM})-S(\rho_{A^{\prime}M})-\sum_{x}S(\rho^{x}_{AM})+\sum_{x}S(\rho^{x}_{A^{\prime}M})$
$\displaystyle=S(\rho_{AM})-S(\rho_{A^{\prime}M})+\sum_{x}S(\rho^{x}_{A^{\prime}}).$
(31)

In the above calculations we have used the fact that $\rho^{x}_{A}$ are pure
states so that $\rho^{x}_{AM}$ and $\rho^{x}_{A^{\prime}M}$ are bipartite
product states. Now, if we compare the IPC without and with memory in from
Eq.(26) and Eq.(IV) respectively, we have

$\displaystyle\mathfrak{I}(\mathbb{C})-\mathfrak{I}^{mem}(\mathbb{C})$
$\displaystyle=[S(\rho_{A})-S(\rho_{AM})]-[S(\rho_{A^{\prime}})-S(\rho_{A^{\prime}M})]$
$\displaystyle=I^{coh}(M\rangle A)-I^{coh}(M\rangle A^{\prime})\geq 0.$ (32)

Thus, we find that using our approach, the IPC is non-increasing in the
presence of memory. It follows the intuition that the presence of memory
should reduce the incompatibility, as the memory can be utilized to recover
the lost information. On comparing Eq.(IV) with Eq.IV, we find that
$\mathscr{I}_{C}-\mathscr{I}^{mem}_{C}=-(\mathfrak{I}(\mathbb{C})-\mathfrak{I}^{mem}(\mathbb{C}))$.
This relation strongly indicates that the information leakage content
$\mathscr{I}_{C}$ envisaged in ref.[13], is not capable of fully capturing the
problems we face in a typical quantum information processing scenarios. This
analysis also validates our approach for quantifying the IPC.

######  Example 2 (Comparison of incompatibilities of a physical context with
two different memories).

Suppose,
$\sigma_{in}=\alpha\ket{\lambda_{1}}\bra{\lambda_{1}}+\beta\ket{\lambda_{2}}\bra{\lambda_{2}}$
be a qubit state and $S_{x}=\\{\ket{+}\bra{+},\ket{-}\bra{-}\\}$ and
$S_{z}=\\{\ket{0}\bra{0},\ket{1}\bra{1}\\}$ be the sharp spin measurements
along $x$ and $z$ directions respectively, where
$\\{\ket{\lambda_{1}},\ket{\lambda_{2}}\\}$ is the eigen basis of
$\sigma_{in}$. Here $0\leq\alpha,\beta\leq 1$ and $\alpha+\beta=1$. Now, we
take our physical context as $\mathbb{C}_{1}=(\sigma_{in},S_{z},S_{x})$. We
will consider the following case where Alice is using memories $M$ keeping
input state $\sigma_{in}$ fixed:
Suppose, Alice is using a qubit memory $M$ such that
$\sigma_{in,M}=p\ket{\psi_{in,M}}\bra{\psi_{in,M}}+\frac{1-p}{4}\mathbb{I}_{in,M}$,
where $0\leq p\leq 1$ ,
$\ket{\psi}_{in,M}=\sqrt{\alpha^{\prime}}\ket{\lambda_{1}}\ket{\lambda^{\prime}_{1}}+\sqrt{\beta^{\prime}}\ket{\lambda_{2}}\ket{\lambda^{\prime}_{2}}$,
$\mathbb{I}_{AM}=\mathbb{I}_{4\times 4}$,
$0\leq\alpha^{\prime},\beta^{\prime}\leq 1$,
$\alpha^{\prime}+\beta^{\prime}=1$,
$\\{\ket{\lambda_{1}}^{\prime},\ket{\lambda_{2}}^{\prime}\\}$ is the eigen
basis of $\sigma_{M}$ and $\sigma_{M}=Tr_{in}[\sigma_{in,M}]$. Alice chooses
$\alpha^{\prime}$,$\beta^{\prime}$ and $p$ such that

$\displaystyle\alpha=p\alpha^{\prime}+\frac{1-p}{2}$ (33)
$\displaystyle\beta=p\beta^{\prime}+\frac{1-p}{2}$ (34)

hold. Then, $\text{Tr}_{M}[\sigma_{in,M}]=\sigma_{in}$. For example, when
$\alpha=\frac{1}{4}$ and $\beta=\frac{3}{4}$, one possible choice is
$p=\frac{3}{4}$, $\alpha^{\prime}=\frac{1}{6}$ and
$\beta^{\prime}=\frac{5}{6}$. The state of the memory is
$\sigma_{M}=\text{Tr}_{in}(\sigma_{in,M})=\alpha\ket{\lambda^{\prime}_{1}}\bra{\lambda^{\prime}_{1}}+\beta\ket{\lambda^{\prime}_{2}}\bra{\lambda^{\prime}_{2}}$.
Let, $q_{xy}={\braket{x}{\lambda_{y}}}$ where $x\in\\{0,1,+,-\\}$ and
$y\in\\{1,2\\}$. The bipartite ensemble, created by the $S_{z}$ measurement of
Alice, is $\\{p^{\prime}_{i},\sigma^{i}_{AM}\\}$ where,
$p^{\prime}_{i}=\text{Tr}[(\ket{i}\bra{i}\otimes\mathbb{I})\sigma_{in,M}]=p[\alpha^{\prime}\mid
q_{i1}\mid^{2}+\beta^{\prime}\mid q_{i2}\mid^{2}]+\frac{1-p}{2}$ and
$\sigma^{i}_{AM}=\frac{(\ket{i}\bra{i}\otimes\mathbb{I})\sigma_{in,M}(\ket{i}\bra{i}\otimes\mathbb{I})}{\text{Tr}[(\ket{i}\bra{i}\otimes\mathbb{I})\sigma_{in,M}]}$
and $i\in\\{0,1\\}$. Now it can be easily checked that
$\sigma^{i}_{AM}=\ket{i}\bra{i}\otimes[p\ket{\phi^{\prime}_{i}}\bra{\phi^{\prime}_{i}}+\frac{1-p}{2p^{\prime}_{i}}\frac{\mathbb{I}}{2}]$
where
$\ket{\phi_{i}}=\frac{1}{\sqrt{p_{i}}}(\sqrt{\alpha^{\prime}}q_{i1}\ket{\lambda_{1}}+\sqrt{\beta^{\prime}}q_{i2}\ket{\lambda_{2}})$.
The post-measurement average bipartite state is
$\sigma_{AM}=\sum_{i}p^{\prime}_{i}\sigma^{i}_{AM}$. Clearly,
$\sigma_{A}=\text{Tr}_{M}\sigma_{AM}=p\sum_{i}p^{\prime}_{i}\ket{i}\bra{i}+(1-p)\frac{\mathbb{I}}{2}$.
After, Eve's $S_{x}$ measurement on $A$ part, the average bipartite state will
become $\sigma_{A^{\prime}M}=\frac{\mathbb{I}}{2}\otimes\sigma_{M}$ where
$\sigma_{M}=[p\sum_{i}p_{i}\ket{\phi^{\prime}_{i}}\bra{\phi^{\prime}_{i}}+(1-p)\frac{\mathbb{I}}{2}]=\sigma_{M}$
and the average state of $A$ part becomes
$\sigma_{A^{\prime}}=\frac{\mathbb{I}}{2}$. So, the reduction in information
leak is given as

$\displaystyle\mathfrak{I}(\mathbb{C}_{1})-\mathfrak{I}^{M}(\mathbb{C}_{1})$
$\displaystyle=[S(\mathcal{\sigma_{A}})-S(\sigma_{AM})]\hskip
5.69046pt-[S(\sigma^{\prime}_{A})-S(\sigma_{A^{\prime}M})]$
$\displaystyle=[S(\mathcal{\sigma_{A}})-S(\sigma_{AM})]\hskip
5.69046pt-[S(\sigma^{\prime}_{A})-S(\sigma_{A^{\prime}})-S(\sigma_{M})]$
$\displaystyle=S(\sigma_{M})+S(\sigma_{A})-S(\sigma_{AM})=I(A:M)_{\sigma_{AM}}.$
(35)

Now, consider a special case where $\ket{\lambda_{1}},\ket{\lambda_{2}}$ are
the eigen basis of $\sigma_{y}$, $\alpha=\frac{1}{4}$ and $\beta=\frac{3}{4}$.
In this case, $\mid q_{ij}\mid^{2}=\frac{1}{2}$ $\forall i\in\\{0,1\\}$ and
$\forall j\in\\{1,2\\}$. Also, from equation (33) we get
$\alpha^{\prime}=\frac{2p-1}{4p}$. Clearly, $\alpha^{\prime}\geq 0$ only for
$p\geq\frac{1}{2}$. We plot the leakage difference
$\mathfrak{I}(\mathbb{C}_{1})-\mathfrak{I}^{M}(\mathbb{C}_{1})$ with respect
to $p$ in Fig. (1).

Figure 1: (Colour online) Plot of Concurrence and mutual information of
$\sigma_{in,M}$ and the leakage difference vs the parameter $p$. It can be
seen that Concurrence and mutual information of $\sigma_{in,M}$ and the
leakage difference is monotonically increasing with respect to the parameter
$p$. All quantities are normalized i.e., all of them have been divided by
their maximum values.

To quantify the amount of memory we use the concurrence measure(ref. [27]) and
the mutual information of the initial bipartite state $\sigma_{in,M}$. From
Fig. (1) we get that with increment of $p$, concurrence and mutual information
of $\sigma_{in,M}$ and the leakage difference, are monotonically increasing
with $p$. We can also say that the information leakage difference is a
monotonically increasing function of both concurrence and mutual information
in the state $\sigma_{in,M}$. Equivalently, we can say that the leakage with
memory is monotonically decreasing with increasing value concurrence and
mutual information. It can be observed from Fig.(1), the leakage difference is
non-zero for the region $p\lessapprox 0.548$ where the concurrence is
vanishing. In this region the non-zero leakage difference can be attributed to
the non-vanishing mutual information.

The example (2) suggests us to write down the following conjecture:

######  Conjecture 1.1.

With increment of correlation between the memory and the input state,
information leakage monotonically decreases.

Therefore, we conclude based on the validity of the conjecture, that the
presence of more memory correlation helps in reducing the leakage.

##  V conclusion

In this work, we have derived the measure of an appropriate information
leakage in all QKD like games. Moreover, introducing quantum instruments, we
have generalised the notion of IPC for POVMs. We have shown the relation
between previous and our approaches for sharp measurements. Our approach
always leads to a non-negative measure of IPC. We have also shown that the
modified IPC measure is more useful compared to the earlier IPC measure in
Eq.(II.5), in a QKD like scenario as an example. Also, on including memory,
our measure of IPC can never increase. In fact in example(2), we have shown
that information leakage monotonically decreases with increment of correlation
between input state and memory. Thus, we have successfully modified the notion
of IPC for generic measurements.

Our work opens up several future directions. Firstly, it would be useful to
construct the resource theory of IPC using our measure. Further, our measure
can be a useful tool for generic information-theoretic tasks, that involve
transmission of classical information over quantum channels. We would like to
explore how our generalized version of IPC can be related to incompatibility
of POVMs.

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