# Does conditional entropy squeezing indicate normalized entropic uncertainty
relation steering?

A-S. F. Obada  Mathematics Department, Faculty of Science, Al-Azher
University, Nassr City 11884, Cairo, Egypt  M. Y. Abd-Rabbou
<EMAIL_ADDRESS>Mathematics Department, Faculty of Science, Al-Azher
University, Nassr City 11884, Cairo, Egypt  Saeed Haddadi<EMAIL_ADDRESS>Faculty of Physics, Semnan University, P.O.Box 35195-363, Semnan, Iran
Saeed’s Quantum Information Group, P.O.Box 19395-0560, Tehran, Iran

###### Abstract

A novel approach is introduced to assess one-way Normalized Entropic
Uncertainty Relation (NEUR)-steering in a two-qubit system by utilizing an
average of conditional entropy squeezing. The mathematical expressions of
conditional entropy squeezing and NEUR-steering are derived and presented. To
gain a better understanding of the relationship between the two measures, a
comparative analysis is conducted on a set of two-qubit states. Our results
reveal that the two measures exhibit complete similarity when applied to a
maximally entangled state, while they display comparable behavior with minor
deviations for partially entangled states. Additionally, it is observed that
the two measures are proportionally affected by some quantum processes such as
acceleration, noisy channels, and swapping. As a result, the average of
conditional entropy squeezing proves to be an effective indicator of NEUR-
steering.

NEUR-steering; Entropy squeezing; Steerability

##  I INTRODUCTION

In 1935, Schrödinger tried to interpret the Einstein-Podolsky-Rosen (EPR)
paradox by establishing correlations between two quantum systems that were too
strong to be explained classically, this phenomenon is commonly referred to as
EPR-steering PhysRev.47.777 ; schrodinger1935discussion . The concept of
steering involves one remote user using a pair of entangled states to
influence or steer their partner’s state through local measurements. As per
the hierarchy of quantum correlations, steerable states are a strict superset
of the states that can demonstrate Bell nonlocality and a strict subset of the
entangled states PhysRevLett.98.140402 ; PhysRevA.92.032107 ; andp2023 .
Quantum steering has recently received significant attention in the field of
quantum information research and has been the subject of both experimental and
theoretical investigations wittmann2012loophole ; RevModPhys.92.015001 . For
example, the experimental quantum steering has been studied through the
implementation of generalized entropic criteria and dimension-bounded steering
inequalities, where two or three measurement setups are used on each side
PhysRevLett.125.020404 . Steering game based on the all-versus-nothing
criterion has been experimentally demonstrated PhysRevLett.113.140402 . The
asymmetric property of EPR steering is relevant for experimental and potential
applications in quantum information as a one-sided device-independent quantum
key distribution PhysRevA.85.010301 , quantum teleportation
PhysRevLett.115.180502 , and optimal prepare-and-measure scenarios
passaro2015optimal . Moreover, for different quantum systems, the possibility
of quantum steering is experimentally interpreted, including photon
polarizations in a linear-optical setup bartkiewicz2016experimental , Bohmian
trajectories xiao2017experimental , a family of the natural two-qubit state
PhysRevResearch.4.013151 , and non-Gaussian state PhysRevLett.128.200401 .

In the theoretical framework, researchers have developed asymmetric criteria
of steering correlation for a pair of arbitrary continuous variables
PhysRevLett.98.140402 . Additionally, Walborn et al. PhysRevA.87.062103  have
utilized the entropic uncertainty relations to express the steering inequality
for arbitrary discrete observables. The violation of the Clauser-Horne-
Shimony-Holt inequality has also been employed to discuss the degree of
steerability cavalcanti2015analog ; ruzbehani2021simulation . Furthermore,
some investigations have been conducted on the violation of steering
inequality and its degree for various quantum systems, including a three-mode
optomechanical system PhysRevA.89.022332 , Heisenberg chain models
chen2018quantum ; li2018quantum , two-level or three-level detectors
liu2018influence ; wang2018generation , and qubit-qubit as well as qubit-
qutrit states sun2017exploration ; abd2022improving ; rahman2023bidirectional
.

On the other hand, the essential conceptions of squeezed spin systems were
introduced by Kitagawa and Ueda in 1993 PhysRevA.47.5138 . The entropy
squeezing for a bipartite system has been obtained for three discrete
observables in $N$-dimensional Hilbert space and employing the discrete
Shannon entropy fang2000entropy . The violation of two quadratures of entropy
squeezing inequality represents a magnificent indicator of entanglement
el2008single . Meanwhile, the entropy squeezing of multi-qubit inside a cavity
system has been considered a hot research topic, such as: two-qubit
interacting with two-mode cavity field khalil2020entanglement , qutrit state
in a cavity filed liu2019entropy , and the effect of classical field and non-
linear term on the qubit-field interaction obada2018influence .

Our motivation is to introduce how entropy squeezing can be employed as an
indicator of the degree of steerability. Overall, as the discrete conditional
Shannon entropy is used as a measure of steerability, so do the two
quadratures of conditional entropy squeezing express the steering? This paper
is organized as follows. In Section II, we present the steerability based on
conditional entropy squeezing. In Section III, the main results of our paper
are discussed in detail. Finally, the conclusion is given in Section IV.

##  II Steerability based on conditional entropy squeezing

In order to gain a better understanding of the relationship between entropy
squeezing and normalized entropic uncertainty relation (NEUR)-steering for
bipartite subsystems $A$ and $B$, we can take advantage of the definition
provided by Walborn et al. PhysRevA.87.062103 . The mathematical framework of
NEUR-steering inequality concerning an even $N$-dimensional Hilbert space
along with the local hidden state for a pair of arbitrary discrete observables
is expressed as PhysRevA.87.062103

$\sum_{i=1}^{N+1}H(R^{B}_{i}|R^{A}_{i})\geq\frac{N}{2}\ln(\frac{N}{2})+(1+\frac{N}{2})\ln(1+\frac{N}{2}),$
(1)

where $\\{R^{A}_{i}\\}$ and $\\{R^{B}_{i}\\}$ are the eigenvectors of the
discrete observables $\hat{R}^{A}$ and $\hat{R}^{B}$, respectively, and $N$ is
the total number of different eigenvectors.
$H(R^{B}|R^{A})\geq\sum_{\lambda}P(\lambda)H_{Q}(R^{B}|\lambda)$ denotes the
corresponding local hidden state constraint for discrete observables, which is
defined by the conditional information entropy $H_{Q}(R^{B}|Q)$ of the
probability distribution $P_{Q}(R^{B}|\lambda)$ with the hidden variable
$\lambda$. In two-dimensional Hilbert space $N=2$, by employing the Pauli spin
operators $\\{\sigma_{x},\sigma_{y},\sigma_{z}\\}$ as measurements, the NEUR-
steering from $A$ to $B$ is realized only if the following condition is
violated PhysRevA.87.062103 ; sun2017relativistic

$H(\sigma_{x}^{B}|\sigma_{x}^{A})+H(\sigma_{y}^{B}|\sigma_{y}^{A})+H(\sigma_{z}^{B}|\sigma_{z}^{A})\geq
2\ln 2,$ (2)

where

$\begin{split}H(\sigma_{i}^{B}|\sigma_{i}^{A})&=H(\hat{\rho}_{AB})_{i}-H(\hat{\rho}_{A})_{i}\\\
&=-\sum_{n,m=1}^{2}P_{i}^{n,m}\ln P_{i}^{n,m}+\sum_{l}^{2}P_{i}^{l}\ln
P_{i}^{l}.\end{split}$ (3)

Here,
$P_{i}^{n,m}=\langle\phi^{i}_{n},\phi^{i}_{m}|\rho_{AB}|\phi^{i}_{n},\phi^{i}_{m}\rangle$
and $P_{i}^{n}=\langle\phi^{i}_{n}|\rho_{A}|\phi^{i}_{n}\rangle$ are the
probability distribution of an arbitrary two-qubit state $\rho_{AB}$ and
reduced single qubit state $\rho_{A}$, respectively, where
$|\phi^{i}_{j}\rangle$ represent the two possible eigenvectors ($j=1,2$) of
$\sigma_{i}$, and $\rho_{A}=Tr_{B}[\rho_{AB}]$.

In this paper, we assume that the density state $\hat{\rho}_{AB}$ with real
components in the standard basis
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$ can be written as

$\hat{\rho}_{AB}=\begin{pmatrix}\rho_{11}&0&0&\rho_{14}\\\
0&\rho_{22}&\rho_{23}&0\\\ 0&\rho_{23}&\rho_{33}&0\\\
\rho_{14}&0&0&\rho_{44}\end{pmatrix}.$ (4)

Note that the operator $\hat{\rho}_{AB}$ satisfies the common conditions
$\hat{\rho}_{AB}\geq 0$ and $Tr[\hat{\rho}_{AB}]=1$.

By applying state (4) in Eq. (2) and violating NEUR-steering inequality, one
can obtain

$\begin{split}\mathcal{I}_{AB}=&\sum_{i=1}^{3}\sum_{j=1}^{4}\frac{1+x_{ij}}{2}\ln(1+x_{ij})\\\
&-\sum_{k=1}^{2}(1+a_{k})\ln(1+a_{k})\leq 2\ln 2,\end{split}$ (5)

where the summations in the first term are related to the three Pauli spin
operators and probability distribution of the two-qubit $\rho_{AB}$,
respectively, and $x_{ij}$ are obtained by

$\begin{split}&x_{11}=x_{12}=-x_{13}=-x_{14}=2(\rho_{14}+\rho_{23}),\\\
&x_{21}=x_{22}=-x_{23}=-x_{24}=2(\rho_{23}-\rho_{14}),\\\
&x_{31}=3\rho_{11}-(\rho_{22}+\rho_{33}+\rho_{44}),\\\
&x_{32}=3\rho_{22}-(\rho_{11}+\rho_{33}+\rho_{44}),\\\
&x_{33}=3\rho_{33}-(\rho_{11}+\rho_{22}+\rho_{44}),\\\
&x_{34}=3\rho_{44}-(\rho_{11}+\rho_{22}+\rho_{33}).\end{split}$ (6)

Likewise, the summation in the second term is related to the probability
distribution of the reduced state $\rho_{A}$, and $a_{k}$ is given by

$\begin{split}a_{k}=(-1)^{k}(\rho_{11}+\rho_{22}-\rho_{33}-\rho_{44}).\end{split}$
(7)

However, the one-way NEUR-steering is quantified based on observable $A$
measurements as follows PhysRevA.93.020103

$S^{A\longrightarrow B}=\max\bigg{\\{}0,\frac{\mathcal{I}_{AB}-2\ln
2}{\mathcal{I}_{max}-2\ln 2}\bigg{\\}},$ (8)

where $\mathcal{I}_{max}=6\ln 2$ when the system is prepared in Bell states.

On the other hand, if we define the function
$\Xi(\sigma_{i}^{B}|\sigma_{i}^{A})=e^{H(\sigma_{i}^{B}|\sigma_{i}^{A})}$,
then the inequality (2) can be reformulated as

$\Xi(\sigma_{x}^{B}|\sigma_{x}^{A})\Xi(\sigma_{y}^{B}|\sigma_{y}^{A})\geq\frac{4}{\Xi(\sigma_{z}^{B}|\sigma_{z}^{A})},$
(9)

where

$\Xi(\sigma_{i}^{B}|\sigma_{i}^{A})=\sum_{n,m=1}^{2}\bigg{(}P_{i}^{n,m}\bigg{)}^{P_{i}^{n,m}}\times\sum_{l=1}^{2}\bigg{(}P_{i}^{l}\bigg{)}^{P_{i}^{l}}.$
(10)

According to Ref. fang2000entropy , the fluctuations in component
$\Xi(\sigma_{i}^{B}|\sigma_{i}^{A})$ ($i=x,y$) are said to be “squeezed in
entropy” if the squeezing factor $E(\sigma_{i}^{B}|\sigma_{i}^{A})$ satisfies
the condition

$E(\sigma^{B}_{i}|\sigma^{A}_{i})=\max\big{\\{}0,\frac{2}{\sqrt{\Xi(\sigma_{z}^{B}|\sigma_{z}^{A})}}-e^{\Xi(\sigma_{i}^{B}|\sigma_{i}^{A})}\big{\\}},$
(11)

with $i=x,y$. From the previous condition, we can depict the upper bounds or
the lower bounds of the NEUR-steering degree. If the state is a maximum
entangled state, then the upper and lower bounds in condition (11) are
identical. Hence, bidirectional steerability and the average of conditional
entropy squeezing quadrature have similar behavior. In partially entangled
states, the NEUR steerability is restricted between the upper and lower
bounds. Therefore, the average of the two components of conditional entropy
squeezing $E(\sigma_{i}^{B}|\sigma_{i}^{A})$ represents an indicator for
quantum steerability. In any case, we can define the quantum steerability
based on the average of entropy squeezing as

$\mathcal{Z}^{A\longrightarrow
B}=\max\big{\\{}0,\frac{E(\sigma_{x}^{B}|\sigma_{x}^{A})+E(\sigma_{y}^{B}|\sigma_{y}^{A})}{2}\big{\\}},$
(12)

where $E(\sigma_{x}^{B}|\sigma_{x}^{A})$ and
$E(\sigma_{y}^{B}|\sigma_{y}^{A})$ are defined in Eq. (11). Hereinafter, we
provide a comparative study between the average of conditional entropy
squeezing and one-way quantum steering for some different quantum systems.

##  III Some Results and Discussion

Here, we study the relationship between one-way steering and the average
conditional entropy squeezing for a class of two-qubit state, which reads

$\hat{\rho}_{AB}=\nu|\phi\rangle\langle\phi|+(1-\nu)|\psi\rangle\langle\psi|,$
(13)

where $|\phi\rangle=\frac{|01\rangle+|10\rangle}{\sqrt{2}}$,
$|\psi\rangle=\frac{|00\rangle+|11\rangle}{\sqrt{2}}$, and $\nu$ is the
setting state parameter. The state (13) is maximally entangled for $\nu=1$ and
$\nu=0$, while partially entangled for $\nu\in(0,0.5)\cup(0.5,1)$.

($a$)$\nu$
($b$)$\nu$

Figure 1: Comparative study between (a) $E(\sigma_{x}^{B}|\sigma_{x}^{A})$
(dotted curve), $E(\sigma_{y}^{B}|\sigma_{y}^{A})$ (dashed curve), and one-way
NEUR-steering $S^{A\longrightarrow B}$ (solid curve) for the state (13), and
(b) the average of conditional entropy squeezing
$\mathcal{Z}^{A\longrightarrow B}$ (dashed curve) and one-way NEUR-steering
$S^{A\longrightarrow B}$ (solid curve).

In Figure 1, we have performed a comparative analysis of the NEUR-steering and
entropy squeezing for a composite system consisting of a two-qubit state
represented by Eq. (13). Through our analysis, we have observed interesting
relationships between these two measures. Figure 1(a) clearly demonstrates
that the extent of NEUR-steering is bounded by the two quadratures of entropy
squeezing. Specifically, when the NEUR-steering is maximized, we observe that
the two quadratures of entropy squeezing become identical. Conversely, when
the NEUR-steering is minimized, the two quadratures of entropy squeezing are
separated. This finding indicates that the values of
$E(\sigma_{x}^{B}|\sigma_{x}^{A})$ and $E(\sigma_{y}^{B}|\sigma_{y}^{A})$ can
be considered as upper and lower bounds of NEUR-steering, respectively.
Furthermore, we have investigated the average of the two quadratures of
entropy squeezing and its relation with the NEUR-steering. Figure 1(b)
illustrates that at maximally entanglement $\nu=0$ and $\nu=1$, the average of
entropy squeezing ($\mathcal{Z}^{A\longrightarrow B}$) aligns closely with the
NEUR-steering $S^{A\longrightarrow B}$. However, at a lower degree of
steering, corresponding to a partially entangled state, we observe deviations
between the behaviors of NEUR-steering and the average of entropy squeezing.
Nevertheless, even in these cases, $\mathcal{Z}^{A\longrightarrow B}$ remains
a reliable indicator for expressing the presence of steerability in the
system.

###  III.1 Some Quantum Processes

In this subsection, we will compare in detail the effect of some quantum
processes on the functions $\mathcal{Z}^{A\longrightarrow B}$ and
$S^{A\longrightarrow B}$, namely acceleration process, decoherence via a
stochastic dephasing channel, and swapping process.

####  III.1.1 Acceleration Process

Let two qubits be simultaneously or separately accelerated in Rindler space.
The computational basis $\\{0,1\\}$ in this space for regions $I$ and $II$ can
be defined as PhysRevA.74.032326

$\begin{split}&|0_{k}\rangle=\cos
r_{k}|0_{k}\rangle_{I}|0_{k}\rangle_{II}+\sin
r_{k}|1_{k}\rangle_{I}|1_{k}\rangle_{II},\\\
&|1_{k}\rangle=|1_{k}\rangle_{I}|0_{k}\rangle_{II},\end{split}$ (14)

where $r_{k}\in[0,\pi/4]$ is the acceleration parameter of the qubit $k=A,B$.
By substituting in the state (13) and tracing over the degrees of the region
$II$, one can get the accelerated state as

$\begin{split}\hat{\rho}_{AB}^{acc}&=\mathcal{A}_{11}|00\rangle\langle
00|+\mathcal{A}_{22}|01\rangle\langle 01|+\mathcal{A}_{33}|10\rangle\langle
10|\\\ &+\mathcal{A}_{44}|11\rangle\langle
11|+(\mathcal{A}_{14}|00\rangle\langle 11|+\mathcal{A}_{23}|10\rangle\langle
01|+h.c.),\end{split}$ (15)

where

$\begin{split}&\mathcal{A}_{11}=\frac{\nu}{2}\cos^{2}r_{a}\cos^{2}r_{b},\
\mathcal{A}_{22}=\cos^{2}r_{a}(\frac{\nu}{2}\sin^{2}r_{b}+\frac{1-\nu}{2}),\\\
&\mathcal{A}_{33}=\cos^{2}r_{b}(\frac{\nu}{2}\sin^{2}r_{a}+\frac{1-\nu}{2}),\\\
&\mathcal{A}_{44}=\sin^{2}r_{a}(\frac{\nu}{2}\sin^{2}r_{b}+\frac{1-\nu}{2})+\frac{1-\nu}{2}\sin^{2}r_{b}+\frac{\nu}{2},\\\
&\mathcal{A}_{14}=\frac{\nu}{2}\cos r_{a}\cos r_{b},\
\mathcal{A}_{23}=\frac{1-\nu}{2}\cos r_{a}\cos r_{b}.\end{split}$ (16)

($a$)$r$
($b$)$r$

Figure 2: Average of conditional entropy squeezing
$\mathcal{Z}^{A\longrightarrow B}$ (dashed curve) and one-way NEUR-steering
$S^{A\longrightarrow B}$ (solid curve) for accelerated state (15) with
$\nu=1$. (a) $r_{a}=r$, $r_{b}=0$ and (b) $r_{a}=r_{b}=r$.

In Figure 2, we present an investigation into the impact of the accelerated
process on a two-qubit state. Specifically, we aim to explore the relationship
between the NEUR-steering and average entropy-squeezing measures under this
process. We assumed that the two-qubit state is maximally entangled with
$\nu=1$ as a fixed parameter. In Figure 2(a), we observe that when only one
qubit is accelerated with $r_{a}=r$ and $r_{b}=0$, the NEUR-steering is
maximized at lower values of the acceleration parameter. As the acceleration
parameter increases, we see a decrease in the degree of steering.
Interestingly, we note that the NEUR-steering and the entropy squeezing are
identical across different values of the acceleration parameter $r$. This
indicates a consistent relationship between these measures regardless of the
acceleration applied to the system. On the other hand, when both qubits are
accelerated simultaneously with $r_{a}=r_{b}=r$, Figure 2(b) reveals an
intriguing trend. As the acceleration parameter increases, the rate of
decrease in steering becomes more pronounced. This finding suggests that
accelerating two qubits simultaneously increases the suppression of steering.
However, it is important to note that despite this trend, the two measures,
namely NEUR-steering and entropy squeezing, exhibit little variations with
respect to the acceleration parameter.

####  III.1.2 Noisy Channel Process

To examine the two functions (8) and (12) under noisy channel models, we can
express the temporal density operator in terms of Kraus operators as

$\hat{\rho}_{AB}(t)=\sum_{i,j}K_{i}^{A}(t)K_{j}^{B}(t)\hat{\rho}_{AB}(0)(K_{i}^{A}(t)K_{j}^{B}(t))^{\dagger},$
(17)

here $\hat{\rho}_{AB}(0)$ is defined in Eq. (13), while $K_{i}^{k}(t)$ and
$K_{j}^{k}(t)$ with $k=A,B$ are the time-dependent Kraus operators for
different noise channels. For example, we use the Kraus operators of
amplitude-damping noise, which are defined by thapliyal2017quantum

$K_{1}(t)=|0\rangle\langle 0|+\sqrt{1-P(t)}|1\rangle\langle 1|,\quad
K_{2}(t)=\sqrt{P(t)}|0\rangle\langle 1|,$ (18)

where $P(t)=e^{-gt}\left[\cos(\frac{\lambda
t}{2})+\frac{g}{\lambda}\sin(\frac{\lambda t}{2})\right]^{2}$ with
$\lambda=\sqrt{g(2\gamma-g)}$. Herein, $g$ is a decay rate which depends on
the reservoir correlation time, and $\gamma$ is the coupling strength related
to qubit relaxation time.

Likewise, the Kraus operators for purely dephasing noise channels can be
defined as yu2010entanglement

$K_{1}(t)=|0\rangle\langle 0|+P(t)|1\rangle\langle 1|,\quad
K_{2}(t)=\sqrt{1-P^{2}(t)}|1\rangle\langle 1|,$ (19)

where

$P(t)=\exp\left\\{-\dfrac{\gamma}{2}\left(t+g^{-1}[\exp(-gt)-1]\right)\right\\}.$

($a$)$\gamma t$
($b$)$\gamma t$
($c$)$\gamma t$

Figure 3: Average of conditional entropy squeezing
$\mathcal{Z}^{A\longrightarrow B}$ (dashed curve) and one-way NEUR-steering
$S^{A\longrightarrow B}$ (solid curve) for amplitude-damping noise, where (a)
$\nu=1$, $g=0.01$, (b) $\nu=1$, $g=0.1$ and (c) $\nu=0.1$, $g=0.1$.

Figure 3 presents a comprehensive examination of the impact of amplitude-
damping noise on the degree of steerability, with the NEUR-steering and
entropy squeezing serving as the quantifying measurements. In Figure 3(a), we
observe that, by selecting a small value for the damping rate ($g=0.01$), the
NEUR-steering oscillates between its maximum and lower bounds. This
oscillatory behaviour is consistent with the properties of steerability under
the influence of amplitude-damping noise. It is noteworthy that the two
measures, NEUR-steering and entropy squeezing, coincide perfectly with the
scaled time parameter. This convergence of the two measurements further
reinforces their equivalence in capturing the system’s dynamics. Moving to
Figure 3(b), we examine the effect of increasing decay rates within the range
of $g$ (specifically with $g=0.1$ and maximally entangled state with $\nu=1$).
As the scaled time escalates $\gamma t$, we gradually observe the NEUR-
steering oscillating with an increase in the upper bounds. On the other hand,
in the case of a partially entangled state with a parameter value of
$\nu=0.1$, it can be observed from Figure 3(c) that a clear separation between
the two measures over time. Furthermore, as the scaled time increases, the
upper bounds of the steering degree exhibit a decrease in value. Remarkably,
the maximum bounds of steering continue to expand as time progresses,
indicating a growing influence of the amplitude-damping noise on the
steerability of the system. Just like in the previous case, the NEUR-steering
and average entropy-squeezing measurements in this scenario remain parallel,
underscoring their identical nature. This consistent agreement can be
attributed to the initial state being maximally entangled.

($a$)$\gamma t$
($b$)$\gamma t$
($c$)$\gamma t$

Figure 4: Average of conditional entropy squeezing
$\mathcal{Z}^{A\longrightarrow B}$ (dashed curve) and one-way NEUR-steering
$S^{A\longrightarrow B}$ (solid curve) under purely dephasing channel, where
(a) $\nu=1$, $g=0.01$, (b) $\nu=1$, $g=0.1$ and (c) $\nu=0.1$, $g=0.1$.

Figure 4 provides a detailed analysis of the effect of dephasing noise on the
degree of steering, utilizing the NEUR-steering and entropy squeezing as the
measurement criteria. Moreover, the comparison between these two measures
under the influence of the dephasing noise channel is examined. In the first,
we consider an initial state that is maximally entangled with $\nu=1$.
Initially, we note that while the entropy squeezing and NEUR-steering measures
display similarities in their general behavior, they are not entirely
identical as the scaled time progresses. At the onset, both measures exhibit
their maximum bounds. However, as time increases, we observe a notable
distinction between the maximum bounds of entropy squeezing and NEUR-steering.
Interestingly, the maximum bounds of entropy squeezing surpass those of NEUR-
steering. This disparity suggests that the influence of dephasing noise
imposes a more pronounced impact on the entropy squeezing measure compared to
NEUR-steering. In the context of partial entanglement $\nu=0.1$ with $g=0.1$,
our observations indicate a rapid decay of steering and a decrease in the
upper bounds of steering during the initial stages of the interaction. This
suggests that partial entanglement has a significant impact on the dynamics of
steering. Furthermore, it is important to note that the time evolution of the
steered system is influenced by the degree of entanglement. As the
entanglement decreases, the decay of steering becomes more pronounced,
implying a decreasing ability to remotely control and influence the entangled
particles.

Notably, despite the difference in the maximum bounds between the two
measures, they still portray parallel trends. As the scaled time continues to
grow, we witness a decrease in the degrees of steering for both measures. This
observation implies that the detrimental effects of dephasing noise manifest
as a reduction in the correlation between the entangled subsystems. Besides,
it is evident that increasing the damping rate of the dephasing channel
exacerbates this decreasing effect on the steering degrees.

####  III.1.3 Swapping Process

Let us consider two different sources, $S_{12}$ and $S_{34}$, which generate
pairs of two-qubit state $\rho_{12}$ and $\rho_{34}$, respectively. Qubits 1
and 4 are far apart, while qubits 3 and 2 remain close. The swapping process
is aimed to measure the amount of quantum NEUR-steering between qubits 1 and 4
by performing a joint Bell measurement on qubits 2 and 3. The post-measurement
state $\rho_{14}$ is calculated by rosario2022quantum

$\rho_{14}=Tr_{23}\bigg{[}\frac{M_{i}.\rho_{1234}.M_{i}^{\dagger}}{Tr[M_{i}.\rho_{1234}.M_{i}^{\dagger}]}\bigg{]},$
(20)

where $\rho_{1234}=\rho_{AB}\otimes\rho_{AB}$, such that the two sources
generate the state $\rho_{AB}$ defined in Eq. (13). Moreover,
$M_{i}=I_{2}\otimes|\Phi_{i}\rangle\langle\Phi_{i}|\otimes I_{2}$ and
$|\Phi_{i}\rangle$ stand for the usual four Bell states.

$\nu$

Figure 5: Average of conditional entropy squeezing
$\mathcal{Z}^{A\longrightarrow B}$ (dashed curve) and one-way NEUR-steering
$S^{A\longrightarrow B}$ (solid curve) under swapping process with
$|\Phi_{i}\rangle=|\psi\rangle$.

Finally, Figure 5 focuses on analyzing the effects of the swapping process on
the behavior of NEUR-steering and conditional entropy squeezing concerning the
state parameter $\nu$. The post-measurement state $\rho_{14}$ at
$|\Phi_{i}\rangle=|\psi\rangle$ is studied to understand how the swapping
process influences the behavior of the quantum system. The findings indicate
that the steerability degree significantly decreases when the two-qubit state
is initially in a partially entangled state. Furthermore, when compared to
Figure 1, it is noticeable that the unsteerable region expands during the
swapping process. However, when the two-qubit state is maximally entangled,
the two measures are equal. Hence, conditional entropy squeezing serves as an
excellent indicator of the level of NEUR-steering present under this process.

##  IV Conclusion

We have proposed a new method for quantifying one-way quantum NEUR-steering in
an arbitrary two-qubit system using the average of conditional entropy
squeezing. We derived the explicit analytical expressions of NEUR-steering and
conditional entropy squeezing. A comparative analysis of the two measures was
conducted on a free maximally mixed two-qubit state, either restricted two-
qubit state by using acceleration, noisy channels, or swapping processes.

For the free maximally mixed two-qubit state, our results highlight the
interconnectedness between NEUR-steering and entropy squeezing. We
demonstrated that the quadratures of entropy squeezing serve as bounds for
NEUR-steering, which represented the upper and lower limits. Additionally, we
established the average of entropy squeezing as a valuable indicator of
steerability, particularly at a maximally entangled state.

The effects of the accelerated process on a two-qubit state have been
demonstrated. We observed that the behavior of NEUR-steering and entropy
squeezing depends on whether one or both qubits are accelerated. In the former
case, the degree of NEUR-steering decreases with increasing acceleration,
while in the latter case, the rate of decrease in steering is amplified.
Nonetheless, despite these variations, the measures of NEUR-steering and
entropy squeezing remain stable and invariant with respect to the acceleration
parameter.

Under the amplitude-damping noise, our results showed that under a specific
small damping rate, the NEUR-steering experiences oscillatory behavior. By
allowing the decay rate to increase, the NEUR-steering experiences oscillatory
behavior with expanding boundaries. Notably, the NEUR-steering and entropy-
squeezing measurements remained indistinguishable throughout these processes,
further validating their correlation. On the other hand, the effect of
dephasing noise on the degree of steering has been evaluated. While the two
measures differ in terms of their maximum bounds, they exhibit similar overall
trends. Specifically, as the scaled time increases, both measures demonstrate
a decrease in the degree of steering. This decreasing effect is amplified by
enhancing the damping rate of the dephasing channel.

Finally, the swapping process significantly diminishes the steerability degree
when the initial state of the two-qubit is partially entangled. For maximally
entangled states, the two measures coincide.

In conclusion, it is evident that entropy squeezing serves as a measure of
steering primarily in the context of a maximally entangled system. However,
when considering partially entangled states or situations where entanglement
is constrained by external factors, entropy squeezing remains a highly
reliable indicator of steering. By assessing the degree of entropy squeezing
in such scenarios, valuable insights can be gained regarding the presence and
extent of quantum steering. Entropy squeezing is a tool for indicating and
understanding quantum steering, even in cases where maximal entanglement may
not be achieved.

Data availability
All data generated during this study are included in this paper.

Competing interests
The authors declare no competing interests.

ORCID iDs
A-S. F. Obada
https://orcid.org/0000-0001-5862-7365
M. Y. Abd-Rabbou
https://orcid.org/0000-0003-3197-4724
Saeed Haddadi
https://orcid.org/0000-0002-1596-0763

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