# Characterization of the measurement uncertainty dynamics in an open system

Xi-Hao Fang  School of Physics and Astronomy, Sun Yat-Sen University,
Guangzhou 510275,People’s Republic of China  Fei Ming  School of Physics &
Material Science, Anhui University, Hefei 230601, People’s Republic of China
Dong Wang<EMAIL_ADDRESS>School of Physics & Material Science, Anhui
University, Hefei 230601, People’s Republic of China  CAS Key Laboratory of
Quantum Information,
University of Science and Technology of China, Hefei 230026, People’s Republic
of China

###### Abstract

Uncertainty principle plays a crucial role in quantum mechanics, because it
captures the essence of the inevitable randomness associated with the outcomes
of two incompatible quantum measurements. Information entropy can perfectly
describe this type of randomness in information theory, because entropy can
measure the degree of chaos in a given quantum system. However, the quantum-
assisted uncertainty of entropy eventually inflate inevitably as the quantum
correlations of the system are progressively corrupted by noise from the
surrounding environment. In this paper, we investigate the dynamical features
of the von Neumann entropic uncertainty in the presence of quantum memory,
exploring the time evolution of entropic uncertainty suffer noise from the
surrounding. Noteworthily, how the environmental noises affect the uncertainty
of entropy is revealed, and specifically we verify how two types of noise
environments, AD channel and BPF channel, influence the uncertainty.
Meanwhile, we put forward some effective operation strategies to reduce the
magnitude of the measurement uncertainty under the open systems. Furthermore,
we explore the applications of the uncertainty relation investigated on
entanglement witness and quantum channel capacity. Therefore , in open quantum
correlation systems, our investigations could provide an insight into quantum
measurement estimation

##  I  Introduction

The Heisenberg uncertainty relation is one of the fundamental concepts in
quantum theory, first proposed by German physicist Heisenberg in 1927 1 . In
quantum mechanics, wave function $\psi$ is used to describe the state of
particles, and the square of the wave function $|\psi(x)|^{2}$ describes the
probability that the particlebe found somewhere. After the probability of wave
of function in quantum mechanics is proposed, quantum mechanics partially
preserves the concept of waves and particles, and the uncertainty principle
sets limit on predicting the measurement outcomes of two incompatible
observables. Kennard 2  formalized Heisenberg original ideas in an uncertainty
relation by a pair of observables:

$\triangle p_{x}\cdot\triangle x\geq\frac{\hbar}{2}$ (1)

$p_{x}$ and $x$ are momentum and position of one particle, where uncertainties
in the measurements are quantified in terms of the standard deviations.

$\triangle x=\sqrt{\left\langle x^{2}\right\rangle-\left\langle
x\right\rangle^{2}}\qquad\triangle p_{x}=\sqrt{\left\langle
p_{x}^{2}\right\rangle-\left\langle p_{x}\right\rangle^{2}},$ (2)

Soon after, Robertson 3  and Schrodinger 4  generalized it to arbitrary pairs
of noncommuting observables $P$ and $Q$ and it is well known in the following
form:

$\triangle Q\cdot\triangle P\geq\frac{1}{2}|\left\langle Q,P\right\rangle|,$
(3)

uncertainties still quantified in terms of the standard deviations

$\triangle x=\sqrt{\left\langle Q^{2}\right\rangle-\left\langle
Q\right\rangle^{2}}\qquad\triangle P=\sqrt{\left\langle
P^{2}\right\rangle-\left\langle P\right\rangle^{2}}.$ (4)

Afterward , for characterizing the ”spread” in the outcomes of two
incompatible measurements was advocate dy Deutsch 5  . In recent years,
uncertainty relationship develops rapidly on entropy of information. Here,
suppose we use denote $p_{i}(x)$ the probability of the outcome, the Shannon
entropy can be expressed as $H(x)=-\sum_{i}p_{i}\log_{2}p_{i}(x)$. It
indicates the uncertainty of a system, large entropy represents large
uncertainty. When there are two random variables, mutual information, joint
entropy, conditional entropy can be used to describe the relationship between
the information they contain. Let the joint probability distribution of the
observables $x$ and $y$ is

$H(X,Y)=-\sum_{x,y}p(x,y)\log_{2}p(x,y).$ (5)

Conditional Shannon entropy is a generalization of conditional probability,
the uncertainty in system $(x,y)$ Conditional Shannon entropy is

$H(x|y)=H(x,y)-H(y).\\\ $ (6)

In the bipartite state, von Neumann entropy replac Shannon entropy as

$S(\rho)=-Tr(\rho\log_{2}\rho)=-\sum_{x}\lambda_{x}\log_{2}\lambda_{x},$ (7)

it also represents the uncertainty of the quantum system, $\lambda_{x}$ is the
eigenvalue of observables. At first, Bialynicki-Birula 6  proved the entropic
uncertainty relation(EUR) of observable position and momentum entropy in 1975.
Another conjecture put forth by Kraus 7  was proved by Maassen and Uffink 8 ,
leading to the following EUR fomular:

$H(P)+H(Q)\geq\log_{2}\frac{1}{c}:=q_{MU},$ (8)

here $c=\rm max_{i,j}c_{i,j}$ and
$c_{i,j}=|\left\langle{x_{i}|z_{i}}\right\rangle|^{2}$.

It is remarkably noting that the uncertainty principl can be described based
on an interesting game model between two participants Alice and Bob. In the
game, Bob prepares a particle $A$ being entangled with his quantum memory $B$
and sends it to Alice, and then Alice carries out two measurements $Q$ and $P$
on the particle $A$ and announces her choice to Bob. Bob’s task is to minimize
his uncertainty about measurement outcome of Alice. Mathematically, the
relation can be expressed as 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17

$S(P|B)+S(Q|B)\geq\log_{2}\frac{1}{c}+S(A|B)=U_{B},$ (9)

where $S(K|B)=S(\rho_{KB})-S(\rho_{B})$ is the post-measurement state
$\rho_{AB}$ after quantum system $A$ is measured in K-basis, the memory 18

$\rho_{XB}=\sum_{x}(|\varphi_{x}\rangle\langle\varphi_{x}|\otimes
I)\rho_{AB}(|\varphi_{x}\rangle\langle\varphi_{x}|\otimes I),$ (10)

respectively resulting after the POVMs $P$ and $Q$ are performed on the system
$A$.

In this scheme, we use two Pauli observables and as the measurement. Hence, it
is easy to obtain $c=\frac{1}{2}$ in this case.

Years later, several authors improved and tightened this bound , Pati 19
proved that the uncertainties are lower bounded by an additional term compared
to Eq. (9) as

$\displaystyle S(X|B)+S(Z|B)\geq$ $\displaystyle\log_{2}\frac{1}{c}+S(A|B)$
$\displaystyle+\max\\{0,D_{A}(\rho^{AB})-J_{A}(\rho^{AB})\\}$ (11)

$J_{A}(\rho^{AB})$ is the classical correlation, it can defined as

$J_{A}(\rho^{AB})\equiv\max_{\Pi_{k}}J(B|{\Pi_{k}}),$ (12)

where the optimization over all positive operator-valued measures acting on
measured particle $A$. The difference between the total and the classical
correlations can be expressed by quantum discord (QD). 20 ; 21 ; 22 ; 23

$D_{A}(\rho^{AB})=I(\rho^{AB})-J_{(}\rho^{AB}).$ (13)

where $I(a:b)$ called mutual information, representing the information
correlation between $b$ and $a$

$I(A:B)=H(a)+H(b)-H(a,b),$ (14)

total correlation is

$I(\rho_{AB})=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB}),$ (15)

The lower bound in Eq. (11) tightens the bound in Eq. (9) if the discord
$D_{A}(\rho^{{AB}})$ is larger than the classical correlation
$J_{A}(\rho^{{AB}})$.

Very recently, F. Adabi and S.Salimi 24  further improved the uncertainty
relationship based on the mutual information of the two-particle state .
Consider a bipartite state $\rho_{AB}$ shared between Alice and Bob. Alice
performs the $X$ or $Z$ measurement and announce her choice to Bob. Bob’s
uncertainty about both $X$ and $Z$ measurement outcomes is

$S(X|B)+S(Z|B)\geq q_{UM}+S(A|B)+\max\\{0,\delta\\},$ (16)

where

$\delta=I(a:b)-[I(X:b)+I(Z:b)].$ (17)

When Alice measures observable $P$ on her particle, she will obtain the $i$ th
outcome with probabilityand Bob’s particle will be left in the corresponding
state $\rho_{B}^{k}=\frac{Tr_{AB}(\Pi^{A}_{i}\rho\Pi^{A}_{i})}{p_{i}}$.

Then the Holevo quantity 25  is

$I(P:b)=S(\rho^{B})-\sum_{i}p_{i}S(\rho^{B}_{i}),$ (18)

it is equal to the upper bound of Bob’s accessible information about Alice’s
measurement results. Thereby, It can be seen that if the sum of the
information sent by Alice to Bob through her measurements is less than the
mutual information between $A$ and $B$, the above EUR represents an
improvement to Berta’s uncertainty relation by raising the lower bound limit
by the amount of . It is worth noting that if observables and are
complementary and subsystem $A$ is a maximally mixed state ,the inequality Eq.
(9) becomes an equality. Thus, for the class of states maximally mixed
subsystem $A$ (including Werner states, Bell diagonal states, and isotropic
states) ,the lower bound is perfectly tight.

This article’s layout provided as follow. In Section II, under two different
channel, we investigate the dynamical traits of the entropic uncertainty and
compare several different lower bounds Eq. (9,11,16). The results show that
the dynamic of uncertainty will exhibit non-monotonic behaviors over time. In
addition, we find that the lower bound proposed by Adabi’s bound is tighter
than the bound proposed by Berta and Pati. In Section III, by deriving the
optimal combination between local amplitude damping and bit-phase flip
channel, we offer two feasible and effective strategies to steer the
measurement uncertainty as a quantum weak measurement and filtering operation.
More over, in Section IV, we give two functional applications of the test
uncertainty relationship to entanglement witness and quantum channel capacity.
At last, we summarize our findings in Section V.

##  II  Entropy Quantum-memory-assisted entropic uncertainty under environment
noise

In order to further explore the uncertainty in different environments noise,
let us return to the uncertain game. Bob sends $A$ particle which is affected
by environmental noise to Alice initially, and particles $B$ remains as it is
because it’s not in the sending process. Kraus 7  used operators $E_{i}$ to
describe the process that the particle $A$ to be measured is exposed to the
environment characterized by evolution, and usually $B$ as quantum memory so
it is very robust and is not affected by any decoherence. Therefore, we can
give the evolution state of the system as

$\rho_{AB}(t)=T(\rho_{AB})=\sum_{i}(E_{i}\otimes I^{B})\rho_{AB}(E_{i}\otimes
I^{B})^{\dagger}.$ (19)

In the light of quantum-memory-assisted game mod, Bob hold a pair of particles
$A$ and $B$, and also $A$ and $B$ are entangled initially in the form of Bell-
diagonal state

$\rho_{AB}=(I^{A}\otimes I^{B}+\sum_{j=1}^{3}C_{\sigma
j}\sigma^{A}_{j}\otimes\sigma^{B}_{j}),$ (20)

where $\sigma$ are the standard Pauli matrices, $C_{\sigma
j}=Tr_{AB}(\rho_{AB}\sigma^{A}_{j}\otimes\sigma^{B}_{j})$ and the coefficients
of initial systematic state with $0\leq|C_{\sigma j}|\leq 1$ , $A$ is
maximally entangled with $B$.

###  II.1  SPMC condition

The right-hand side of Quantum-memory-assisted entropic uncertainty is

$U_{b}=\log_{2}\frac{1}{c}+S(A|B).$ (21)

If we measureing the uncertainty of $\sigma_{j}$ and $\sigma_{k}$, then when
the initial quantum state satisfy SPMC condition 26

$C_{\sigma i}=-C_{\sigma j}C_{\sigma k}(i\neq j\neq k),$ (22)

That is $U\equiv U_{b},$ the entropic uncertainty relation can always reach
the bound.

###  II.2  Amplitude damping channel

Amplitude damping channel is a description of the dissipation of energy in a
quantum system , the Kraus 12  operators can be given in sum-operator
representation as

$E^{AD}_{1}=\left(\begin{array}[]{cc}1&0\\\ 0&\sqrt{1-d}\\\
\end{array}\right)\qquad E^{AD}_{2}=\left(\begin{array}[]{cc}0&\sqrt{d}\\\
0&0\end{array}\right).$ (23)

Where $d=1-e^{-\lambda t}$ represents the parameter of the energy dissipation
rate, and $\lambda$ denotes the energy relaxation rate. time value $t$ range
is $[0,1]$ , under this channel, inital state $|0\rangle$ amplitude will
decrease with time. Amplitude damping channel is a spontaneous emission model
with zero ambient temperature, describe the energy consumption.

After go through the above environmental noise mentioned above, the matrix
elements of the bipartite state $\rho_{AB}(t)$ are taken as
$\rho_{11}=\frac{1}{4}(1+C_{3}+d-C_{3}d),\rho_{22}=\frac{1}{4}(1+C_{3}(-1+d)+d),\rho_{33}=\frac{1}{4}(-1+C_{3})(-1+d),\rho_{44}=-\frac{1}{4}(1+C_{3})(-1+d),\rho_{14}=\rho_{41}=\frac{1}{4}(C_{1}-C_{2})\sqrt{1-d},\rho_{23}=\rho_{23}=\frac{1}{4}(C_{1}+C_{2})\sqrt{1-d}.$

Four eigenvalues of density matrix are

$\displaystyle\lambda^{AD}_{1}$ $\displaystyle=\frac{1}{4}(1-C_{3}+C_{3}d$
$\displaystyle-\sqrt{C_{1}^{2}+2C_{1}C_{2}+C_{2}^{2}-C_{1}^{2}d-2C_{1}C_{2}d-C_{2}^{2}d+d^{2}}),$
$\displaystyle\lambda^{AD}_{2}$ $\displaystyle=\frac{1}{4}(1-C_{3}+C_{3}d$
$\displaystyle+\sqrt{C_{1}^{2}+2C_{1}C_{2}+C_{2}^{2}-C_{1}^{2}d-2C_{1}C_{2}d-C_{2}^{2}d+d^{2}}),$
$\displaystyle\lambda^{AD}_{3}$ $\displaystyle=\frac{1}{4}(1+C_{3}-C_{3}d$
$\displaystyle-\sqrt{C_{1}^{2}-2C_{1}C_{2}+C_{2}^{2}-C_{1}^{2}d+2C_{1}C_{2}d-C_{2}^{2}d+d^{2}}),$
$\displaystyle\lambda^{AD}_{4}$ $\displaystyle=\frac{1}{4}(1+C_{3}-C_{3}d$
$\displaystyle+\sqrt{C_{1}^{2}-2C_{1}C_{2}+C_{2}^{2}-C_{1}^{2}d+2C_{1}C_{2}d-C_{2}^{2}d+d^{2}}),$
(24)

To illustrate the feature of uncertainty, we may resort to $\sigma_{x}$ and
$\sigma_{z}$ as the measurement’s incompatibility, which is always used for
describing the spin $-\frac{1}{2}$ observables, with the eigenstates of
$|\sigma_{x}^{\pm}\rangle=\frac{(1,\pm
1)^{T}}{\sqrt{2}},|\sigma_{z}^{\pm}\rangle={(1,0)^{T}}.$

Therefore, the conditional von Neumann entropies representing the uncertainty
of the measurement outcomes are respectively given by
$S(\sigma_{x}|B)=S(\rho_{\sigma_{x}})-S(\rho_{B})$ and
$S(\sigma_{z}|B)=S(\rho_{\sigma_{z}})-S(\rho_{B}).$ Thus, the left-hand side
of Eq. (9) can be given by

$\displaystyle U=$ $\displaystyle-\frac{1}{4}\\{4\mu_{1}\rm
arctanh(\mu_{1})+2\mu_{2}\rm arctanh(\mu_{2})$ $\displaystyle-2\mu_{3}\rm
arctanh(\mu_{3})+2\log_{2}(1+\mu_{1})+2\log_{2}(1+\mu_{2})$
$\displaystyle+\log_{2}(1-\mu_{2})+\log_{2}(1+\mu_{2})+\log_{2}(1+\mu_{3})$
$\displaystyle+\log_{2}(1-\mu_{3})\\}$ (25)

where
$\mu_{1}=\sqrt{-(C_{1}-C_{2})(C_{1}+C_{2})(-1+d)},\mu_{2}=C_{3}+d-C_{3}d,\mu_{3}=C_{3}-C_{3}d-d.$

On the other hand, if the reduced density matrix of the Bell diagonal state
maintains maximally mixed ,the complementarity $c$ of the observables $x$ and
$z$ is exactly equal to $\frac{1}{2}$. Thus, the right-hand side of Eq. (9)
takes the form

$U_{B}=\log_{2}\frac{1}{c}+S(A|B)=S(\rho_{AB})-S(\rho)+1,$ (26)

In order to show traightly the evolution of entropy uncertain relation, we
show the diagrams of the uncertainty and the existed different bounds as
functions of the time-parameter $d=1-e^{-\lambda t}$ with
$C_{1}=-0.5,C_{2}=0.4,C_{3}=0.8$, in Figure 1. From the figure, we can find
that the entropic uncertainty will resonantly reduce with the growing time
$t$.

Obviously, with the $t$ increase, the measurement uncertainty and the referred
bounds will reveal non-monotonic feature and gradually close.In addition, it
is interesting to find that the bound of Adabi et al is perfectly coincident
with the left-hand side of the uncertainty expressed in Eq. (9). And one can
see that Pati et al’s bound is tighter than that proposed by Berta et al, and
Adabi et al’s bound is tighter than Pati et al’s one, as shown in Figure 1b.
This states that Adabi et al’s bound is optimal compared with the others,
which is, $U_{B}\leq U_{P}\leq U_{A}$ is hold.

Figure 1: Dynamics of entropic uncertainty under AD channel, several kinds of
lower bound and quantum correlation of the system with respect to the energy
dissipation rate $d=1-e^{-\lambda t}$ when the initial state is prepared with
$C_{1}=-0.5,C_{2}=0.4,C_{3}=0.8.$ Graph (a): $U_{l}$ denotes the entropic
uncertainty expressed in Eq. (9) , the yellow line denotes the bound proposed
by Berta et al., the blue line denotes the Pati et al.’s bound, the black
dashed line represents the bound by Adabi et al. The graph (b) and(d): $U_{A}$
denotes the Adabi et al.’s lower bound; $U_{B}$ denotes the Berta et al.’s
bound, $U_{P}$ denotes the Pati et al.’s bound. Graph (c):QD denotes quantum
discord expressed in Eq. (9) , $S^{A|B}_{\rm min}$ denotes the $A$’s minimal
condition entropy in Eq. (29).

In addition, QD has been shown in Figure 1c , it can quantify the dynamic of
the system’s quantum correlation under AD channel. Mathematically, in the
current bipartite system with an arbitrary X-structure density matrix, the QD
of the system can be accurately derived as

$D(\rho_{AB})=S_{\rm
bin}(\rho_{22}+\rho_{44}+\sum_{i=1}^{4}\lambda_{i}\log_{2}\lambda_{i}+\min\\{P_{1},P_{2}\\}),$
(27)

with

$P_{1}=S_{\rm
bin}(\Gamma),P_{2}=-\sum_{n}^{i=1}\rho_{ii}\log_{2}\rho_{ii}-S_{\rm
bin}(\rho_{11}+\rho_{33}).$ (28)

where
$\Gamma=\frac{1}{2}\\{1+\sqrt{[1-2(\rho_{33}+\rho_{44})]^{2}+4(|\rho_{14}|+|\rho_{23}|)^{2}}\\}$,
$\rho_{ij}$ is the element of the corresponding density matrix and
$\lambda_{i}$ denotes the eigenstates of $\rho_{AB}$.

We obviously find that this is counter-intuitive in the current situation,
that the bigger quantum correlation not induce the smaller uncertainty, as
shown in Figure 1c. In the following, we will render a physical explanation
for this phenomenon. To interpret this abnormal behavior, we can combine Eq.
(9) and Eq. (13), and it can derive that the QD is equal to

$U_{B}=1+S^{A|B}_{\min}-D(\rho_{AB})$ (29)

where $S^{A|B}_{\min}=\min_{B_{i}}\sum_{i}q_{i}S(\rho^{A}_{i})$ is $A$’s
minimal condition entropy. This directly indicates that the uncertainty’s
bound of interest is not only determined by $D(\rho_{AB})$, and also by
$S^{A|B}_{\min}.$

We define the discrepancy between the entropic uncertainty and the lower bound
as tightness ,itcan further explore the different bounds presented, let us
turn to focus on the differences between the entropic uncertainty and the
existed bounds,tightness define as

$T(Q,R)=U_{l}-U_{r},$ (30)

where $U_{l}$ denotes the left-hand-side of the uncertainty relation in Eq.
(9), and $U_{r}$ denotes the right-handside of the relation in Eq. (9), (11)
and (16). Therefore, Figure 1d shows the tightness $T(Q,R)$ for these
uncertainty relations with the increasing time $t$. accroding the figure, we
obtain the magnitude of the Berta et al’s tightness is larger than Pati et
al’s tightness and is larger than Adabi et al’s tightness , compared with the
other bounds ,Adabi et al’s bound is tightest. Moreover, it directly shows
that the tightness of Adabi et al is always maintained at zero,in other words,
all the relations’tightness mentioned above, except which proposed by Adabi,
are used to never fully synchronize with the measurement uncertainty.

###  II.3  Bit-phase flip channel

Bit flip, phase flip and bit-phase flip (BPF) also called $\sigma_{x}$ error,
$\sigma_{y}$ error and $\sigma_{z}$ error, generally used to describe the
effects of quantum systems on flipped noise: loss of information may occur
during the process but there is no dissipation of energy. After the particles
$A$ pass through the BPF channel, the density matrix still maintains maximally
entangled state. the Kraus 7  operators is

$E^{BPF}_{1}=\sqrt{p}\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right)\qquad E^{BPF}_{2}=\sqrt{1-p}\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right).$ (31)

Single-particle bits not change with a probability of $\sqrt{1-p}$ , change
with a probability of $p,$ where $p$ is the time parameter, value range also
is $[0,1]$ .

The state of the composite system consisting of AB would take a matrix form of
$\rho_{11}=\frac{1}{4}(1+C_{3}(-1+2p)),\rho_{22}=\frac{1}{4}(1+C_{3}-2C_{3}p),\rho_{33}=\frac{1}{4}(1+C_{3}-2C_{3}p),\rho_{44}=-\frac{1}{4}(1+C_{3}(-1+2p)),\rho_{14}=\rho_{41}=\frac{1}{4}(-C_{2}+C_{1}(-1+2p)),\rho_{23}=\rho_{23}=\frac{1}{4}(C_{2}+C_{1}(-1+2p)).$

Figure 2:  Dynamics of entropic uncertainty under BPF channel, several kinds
of lower bound and quantum correlation of the system as a function of time
parameter $p$ when the initial state is prepared with
$C_{1}=-0.5,C_{2}=0.4,C_{3}=0.8.$ Graph (a): $U_{l}$ denotes the entropic
uncertainty expressed in Eq. (9) , the black dashed line denotes the bound
proposed by Berta et al., the pink line denotes the Pati et al.’s bound, the
yellow dashed line represents the bound by Adabi et al. The graph (b) and(d):
$U_{A}$ denotes the Adabi et al.’s lower bound; $U_{B}$ denotes the Berta et
al.’s bound, $U_{P}$ denotes the Pati et al.’s bound. Graph (c):QD denotes
quantum discord expressed in Eq. (9) , $S^{A|B}_{\rm min}$ denotes the $A$’s
minimal condition entropy in Eq. (29).

The four eigenvalues of matrix are

$\displaystyle\lambda^{\rm
BPF}_{1}=\frac{1}{4}(1+C_{1}-C_{2}+C_{3}-2C_{1}p-2C_{3}p),$
$\displaystyle\lambda^{\rm
BPF}_{2}=\frac{1}{4}(1-C_{1}+C_{2}+C_{3}+2C_{1}p-2C_{3}p),$
$\displaystyle\lambda^{\rm
BPF}_{3}=\frac{1}{4}(1+C_{1}+C_{2}-C_{3}-2C_{1}p+2C_{3}p),$
$\displaystyle\lambda^{\rm
BPF}_{4}=\frac{1}{4}(1-C_{1}-C_{2}-C_{3}+2C_{1}p+2C_{3}p).$ (32)

Thus, the left-hand side of Eq. (9) can be given by

$U^{BPF}=-\frac{1}{2}\\{-2\nu_{1}\rm arctanh(\nu_{1})-\log_{2}(1+\nu_{1})$
$-\log_{2}(1-\nu_{1})-(1+\nu_{2})\log_{2}(1+\nu_{2})+(\nu_{2}-1)\log_{2}(1-\nu_{2}),$

where $\nu_{1}=C_{1}-2C_{1}p,\nu_{2}=C_{3}-2C_{3}p.$

Similarly, the right-hand side (i.e., bound) of equation takes the form

$U^{\rm
BPF}_{b}=-\frac{1}{4}\sum_{i=1}^{4}\zeta_{i}\log_{2}(\frac{\zeta_{i}}{4})$
(33)

where
$\zeta_{1}=1+C_{1}+C_{2}-C_{3}-2C_{1}p+2C_{3}p,\zeta_{2}=(1+C_{1}-C_{2}+C_{3}-2(C_{1}+C_{3})p),\zeta_{3}=1-C_{1}-C_{2}-C_{3}+2(C_{1}+C_{3})p,\zeta_{4}=1+C_{2}+C_{3}-2C_{3}p+C_{1}(-1+2p)),$
we plot the diagrams of evolution of entropy uncertain relation and the
existed bounds in Figure 2 . Likewise,the dynamics of the entropic uncertainty
expose non-monotonic feature and gradually close , Berta et al.’s bound is
largest, relatively that Pati et al.’s bound is between the other two, Adabi
et al.’s bound is still perfectly coincident with the left-hand side of the
uncertainty.And it is worth noting that different from AD channel , but in
Figure 2b Pati’s bound is coincident with the right-hand side of uncertainty
except channel parameter $p$ at 0.25 and 0.75.

##  III  A strategy to steer the uncertainty of measurement via a type of
flipping operations and quantum weak measurement

In open systems, the effects of decoherence or dissipation are very
significant?in fact, any quantum system is actually open and inevitably
affected by its surrounding environment noises. We can deduce in this respect
that when the environment and the system are coupled, the uncertainty will
increase more or less. Thus, here naturally comes a question:how to manipulate
the magnitude of the uncertainty with regard to a given system in order to
reduce the impact of the external environment? Up to now, some authors
presented some efficient methods to protect quantum states and quantum
correlations from the noises,for example, filtering operator (FO) and quantum
weak measurement (QWM). We are curious about that whether such operations are
valid to steer the quantity of the uncertainty. After our investigation, we
find the answer is positive. In the followings, we illustrate how these two
methods achieve the reduction of the magnitude of the entropic uncertainty.

Firstly we exploit a filtering operator on system experiences any noises to
decrease the uncertainty , and the Kraus operator of FO can be written as 27 ;
28

$\hat{M}=\left(\begin{array}[]{cc}\sqrt{1-k}&0\\\ 0&\sqrt{k}\\\
\end{array}\right),$ (34)

where $\hat{M}$ represents the operator acting on particles $A$ , and $k$ is
the measurement strength with $1>k>0$, and it can recover and increase
entanglement to some extent 29 ; 30 . Practically,we will expound the de the
details of how FO reduce the uncertainty of the measurement with pertain to
two incompatibility in the follows .

After the FO operation on the qubit $A$ to be detected, the final state of the
bipartite-system consisting AB can express as

$\rho^{M}_{AD}(t)=\frac{(\hat{M}\otimes I)\rho_{AB}(\hat{M}\otimes
I)^{\dagger}}{[Tr(\hat{M}\otimes I)\rho_{AB}(\hat{M}\otimes I)^{\dagger}]}.$
(35)

After normalization, one can obtain the density matrix $\rho^{M}_{AD}(t)$ of
the system

$\displaystyle\rho^{M}_{AD}(t)_{11}=$
$\displaystyle\frac{(1+C_{3}+d-C_{3}d)(1-k)}{2(1+d-2dk)},$
$\displaystyle\rho^{M}_{AD}(t)_{22}=$
$\displaystyle-\frac{(1+C_{3}(-1+d)+d)(-1+k)}{2(1+d-2dk)},$
$\displaystyle\rho^{M}_{AD}(t)_{22}=$
$\displaystyle-\frac{(1+C_{3}(-1+d)+d)(-1+k)}{2(1+d-2dk)},$
$\displaystyle\rho^{M}_{AD}(t)_{33}=$
$\displaystyle\frac{(-1+C_{3})(-1+d)k}{2(1+d-2dk)},$
$\displaystyle\rho^{M}_{AD}(t)_{14}=$
$\displaystyle\rho^{M}_{AD}(t)_{14}=\frac{(C_{1}-C_{2})\sqrt{1-d}\sqrt{(1-k)k}}{2(1+d-2dk)},$
$\displaystyle\rho^{M}_{AD}(t)_{23}=$
$\displaystyle\rho^{M}_{AD}(t)_{32}=\frac{(C_{1}+C_{2})\sqrt{1-d}\sqrt{(1-k)k}}{2(1+d-2dk)}.$
(36)

By calculating their eigenvalues, we can attain the corresponding von Neumann
entropies as

$\displaystyle
H(\rho^{M}_{XB})=-\sum_{i=1}^{4}\lambda^{M}_{XBi}\log_{2}(\lambda^{M}_{XBi}),$
$\displaystyle
H(\rho^{M}_{ZB})=-\sum_{i=1}^{4}\lambda^{M}_{ZBi}\log_{2}(\lambda^{M}_{ZBi}),$
(37)

with

$\displaystyle\lambda^{M}_{XB1}=\lambda^{M}_{XB3}=\frac{1}{4}-\delta,$
$\displaystyle\lambda^{M}_{XB2}=\lambda^{M}_{XB4}=\frac{1}{4}+\delta,$ (38)

$\delta=\frac{\sqrt{(-1+d)(C_{3}^{2}(-1+d)(1-2k)^{2}+4C_{1}^{2}(-1+k)k)}}{-4+d(-4+8k)}$
and $\lambda^{M}_{ZBi}=\rho^{M}_{AD}(t)_{ii}.$

In addition, we have

$H(\rho^{M}_{B})=-\sum_{i=1}^{2}\lambda^{M}_{Bi}\log_{2}(\lambda^{M}_{Bi}),$
(39)

where

$\displaystyle\lambda^{M}_{B1}=$
$\displaystyle\frac{\sqrt{1+d-2dk+C_{3}(-1+d)(-1+2k)}}{2+d(2-4k)},$
$\displaystyle\lambda^{M}_{B2}=$
$\displaystyle\frac{\sqrt{-1+C_{3}(-1+d)(-1+2k)+d(-1+2k)}}{-2+d(-2+4k)}.$ (40)

As a result, we can get the uncertainty of von Neumann entropy uncertainty is

$\displaystyle U^{M}=$
$\displaystyle-\sum_{i=1}^{4}\lambda^{M}_{ZBi}\log_{2}(\lambda^{M}_{ZBi})-2\sum_{i=1}^{2}\lambda^{M}_{XBi}\log_{2}(\lambda^{M}_{XBi})$
$\displaystyle+2\sum_{i=1}^{2}\lambda^{M}_{Bi}\log_{2}(\lambda^{M}_{Bi})$ (41)

In our consideration, the qubitt $A$ be measured suffers from a environment
noise, while qubit $B$ as quantum memory is free from any noises. Moreover, We
display in Fig. 3 how the measured entropy-based uncertainty changes with the
state parameter $d$ .

Figure 3: The entropy uncertainty $U$ is plotted as a function of the
operation strength $k,s$ and the states parameter $d$. the initial state under
AD channel is prepared same with $C_{1}=-0.5,C_{2}=0.4,C_{3}=0.8.$ In the Gaph
(a): the entropic uncertainty with $d$ with different via filtering operator
strengths with an initial Bell state constructed with the
$k_{1}=0.2,k_{2}=0.3,k_{3}=0.4,k_{4}=0.5,$ Graph (b): the entropic uncertainty
with $d$ with an initial Bell-diagonal state constructed with the
$s_{1}=0,s_{2}=0.4,s_{3}=0.6,s_{4}=0.8$ under different weak measurement
strengths.

Furthermore, we can exploit the quantum weak measurement to reduce the amount
of the measuring uncertainty, The quantum weak measurement is a non-trace-
preser map (NTPM) operation, which Kraus operator can be expressed by 31 ; 32
; 33

$\hat{W}=\left(\begin{array}[]{cc}1&0\\\ 0&\sqrt{1-s}\\\ \end{array}\right),$
(42)

where $s$ indicates the strength of the weak measurement operation and
satisfies $0<s<1$. When this operation is acted on qubit A, the final state of
the Quantum system would be written by

$\rho^{W}_{AD}(t)=\frac{(\hat{W}\otimes I)\rho_{AB}(\hat{W}\otimes
I)^{\dagger}}{[Tr(\hat{W}\otimes I)\rho_{AB}(\hat{W}\otimes I)^{\dagger}]}$
(43)

As same , we consider this case where $A$ experiencing AD channels. After the
operation acts on $A$ to be observed, the post-operation density matrix of
becomes

$\displaystyle\rho^{W}_{AD}(t)_{11}=$
$\displaystyle-\frac{1+C_{3}+d-C_{3}d}{4(1+1/2(-1+d)s)},$
$\displaystyle\rho^{W}_{AD}(t)_{22}=$
$\displaystyle-\frac{1+C_{3}(-1+d)+d)(-1+s)}{4(1+1/2(-1+d)s)},$
$\displaystyle\rho^{W}_{AD}(t)_{33}=$
$\displaystyle-\frac{(-1+C_{3})(-1+d)(-1+s)}{4(1+1/2(-1+d)s)},$
$\displaystyle\rho^{W}_{AD}(t)_{44}=$
$\displaystyle\frac{(1+C_{3})(-1+d)(-1+s)}{4(1+1/2(-1+d)s)},$
$\displaystyle\rho^{W}_{AD}(t)_{14}=$
$\displaystyle\rho^{W}_{AD}(t)_{14}=\frac{(C_{1}-C_{2})\sqrt{1-d}\sqrt{1-s}}{4(1+1/2(-1+d)s)},$
$\displaystyle\rho^{W}_{AD}(t)_{23}=$
$\displaystyle\rho^{W}_{AD}(t)_{32}=\frac{(C_{1}+C_{2})\sqrt{1-d}\sqrt{1-s}}{4(1+1/2(-1+d)s)}.$

and we can obtain the von Neumann entropy of state can be calculated as

$\displaystyle H(\rho^{W}_{XB})=$
$\displaystyle-\frac{1}{2}[\frac{1}{2}\sum_{i=1}^{4}\log_{2}(\lambda^{T}_{XB_{i}})$
$\displaystyle+4(\lambda^{T}_{XB_{2}}-1)\rm
arccoth\frac{1}{4(\lambda^{T}_{XB_{2}})}-\eta\rm arctanh(\eta)],$
$\displaystyle H(\rho^{W}_{ZB})=$
$\displaystyle-\sum_{i=1}^{4}\lambda^{T}_{ZBi}\log_{2}(\lambda^{T}_{ZBi}),$
(45)

with

$\displaystyle\lambda^{W}_{XB1}=\lambda^{W}_{XB3}=\frac{1}{4}-\epsilon,$
$\displaystyle\lambda^{W}_{XB2}=\lambda^{W}_{XB4}=\frac{1}{4}+\epsilon,$ (46)

$\epsilon=\frac{\sqrt{(-1+d)(4C_{1}^{2}(-1+s)+C_{3}^{2}(-1+d)s^{2})}}{8+4(-1+d)s},\eta=\frac{2C_{3}(-1+d)s}{2+(-1+d)s}$
and $\lambda^{W}_{ZBi}=\rho^{W}_{AD}(t)_{ii}.$

From the Fig. 3, one can easily get that the measured uncertainty decrease
monotonically with the increase of the weak measurement strength $k$ and the
FO operation intensity $s$ , which reflects that such an optimal measurement
choice to reduce the measured uncertainty is very effective, which is pretty
necessary during practical quantum-information-processing. Put another way,
the entropy uncertainty can be manipulated by optimal strategy satisfactorily,
best combination of a FO and QWM .

Figure 4:  The entropy uncertainty $U$ is plotted as a function of the
operation strength $k,s$ and the states under BPF noise with parameter $p$.
Gaph (a): the entropic uncertainty with different via filtering operator
strengths with an initial Bell state constructed with the
$k_{1}=0.1,k_{2}=0.25,k_{3}=0.5,$ Graph (b): the entropic uncertainty with $d$
with an initial Bell-diagonal state constructed with the
$s_{1}=0.7,s_{2}=0.4,s_{3}=0$ under different weak measurement strengths.

In addition, from the Fig. 4 , obviously,this is easy to achieve that the
state parameter $p$ and $s$ will also monotonically reduce the uncertainty of
BPF channel. This fact is because because we adjust the state parameters to
cause higher purity of the system, so the higher the purity, the smaller the
uncertainty. This shows that in order to obtain lower measurement uncertainty,
we can carry out this scheme in a composite system, therefore, we can
manipulate this operation to still gain high-precision measurements when
encountering various realistic noises caused by the external environment.

##  IV  Application

###  IV.1 Entanglement witness

One of the significant form of quantum correlation is quantum entanglement,
And has produced very important applications in both communication and quantum
information , such as quantum dense coding 34 , quantum telecloning 35 ,
teleportation 36 , quantum computing 37 , quantum electrodynamics (QED) 38 ;
39 ; 40 ; 41 ; 42 ; 43  ,and so on. Therefore, in the field of quantum
information processing, how to determine whether quantum states are entangled
has become a very important topic. Considering this , we briefly introduce an
effective criterion based on uncertainty relations in the current scenario.
Technically, $S(A|B)\leq 0$ shows the bound can be reduced contrast to the
previous one in Eq. (8), Therefore it is often used as an indicator of
entanglement witness.

According to the Eq. (9), if the inequality satisfy
$U=S(\hat{P}|B)+S(\hat{R}|B)\leq\log_{2}\frac{1}{c}=1$ which means $A$
entangle with $B$. In Fig. 5, we shown the relationship between
$S(\hat{P}|B)+S(\hat{R}|B)$ and $\log_{2}\frac{1}{c}=1$ with the maximal
purity of initial , therefore,by a rigorous calculation one can obtain the
entanglement condition via $S(\hat{P}|B)+S(\hat{R}|B)<1$

Figure 5: Evolution of entropic uncertainty $U$ when the particle $A$ is under
AD and BPF channel. Graph (a): Evolution of $U$ with respect $d$ to when $s$
takes different values. Graph (b): Evolution of $U$ with respect $p$ to when
$s$ takes different values, the pink line is plotted with s = 0, the yellow
line is plotted with s = 0.4, the red line is plotted with s = 0.8 and the red
line with p = 0.5. Here we set $C_{1}=-1,C_{2}=1,C_{3}=1$.

Here the case when particle A under the AD and BPF channel . By setting
$C_{1}=-1,C_{2}=1,C_{3}=1$, the initial state has maximal purity, and consider
the entropic uncertainty under quantum weak measurement Eq. (42), we then can
get the relationship with $d,p$ and $s$ ,when noisy streng of the AD channel

$d<d_{m},$ (47)

for instance, we can calculate the $d_{m}$ and its value is $d_{m}=0.4058$
with $s=0,$ and BPF noisy strength

$[0,p_{m})\cup(1-p_{m},1],$ (48)

when $U<1$ , the $p_{m}$ is $p_{m}=0.1125$ with $s=0,$ in an environment where
the initial system reaches maximum purity.

###  IV.2 Quantum channel capacities

Quantum channels generally describe the situation in which any physical
process acts on a quantum system 44 , including achieving non-eavesdropping
secure communication and sending secret classical information . And quantum
channel capacity means the maximal amount of information transfer via quantum
channel, which can be written as follows 45

$C(\rho_{AB})=\rm max\\{I(A:B)\\},$ (49)

where $I(A:B)$ indicate the mutual information of system $\rho_{AB}$. Here, we
are able to build the connection between the uncertainty’s bound and the
channel capacity , which can be derived as

$U_{b}=\log_{2}\frac{1}{c}+S(A|B)=1+S(A|B),$ (50)

by substituting Eq. (53) and Eq. (14) into the above equation, we can obtain
the channel capacity $C(\rho_{AB})$ can be written as

$C(\rho_{AB})=\rm max[S(\rho_{A})-U_{b}+1],$ (51)

which shows the connection between the uncertainty relation and the quantum
channel capacity. So we can get the exact expression of capacity as that

$C(\rho_{AB})=\sum_{i=1}^{4}\lambda^{AB}_{i}\log_{2}\lambda^{AB}_{i}-\sum_{i=1}^{2}\lambda^{A}_{i}\log_{2}\lambda^{A}_{i}-\sum_{i=1}^{2}\lambda^{B}_{i}\log_{2}\lambda^{B}_{i},$
(52)

and the $\lambda_{i}^{AB}$ are the eigenstates of state $\rho^{AB}$, which
have been given above. Clearly, we get the eigenstates of bipartite state
under AD channel $\rho_{A}$and $\rho_{B}$ are
$\lambda^{A}_{AD_{1}}=\frac{1-d}{2},\lambda^{A}_{AD_{2}}=\frac{1+d}{2},\lambda^{B}_{AD_{1}}=\lambda^{B}_{AD_{2}}=\frac{1}{2},$
and under BPF channel
$\lambda^{A}_{BPF_{1}}=\lambda^{A}_{BPF_{2}}=\lambda^{B}_{BPF_{1}}=\lambda^{B}_{BPF_{1}}=\frac{1}{2},$
and we thus have

$\displaystyle
C_{AD}(\rho_{AB})=\sum_{i=1}^{4}\lambda^{AB}_{AD_{i}}\log_{2}\lambda^{AB}_{AD_{i}}-\sum_{i=1}^{2}\lambda^{A}_{AD_{2}}\log_{2}\lambda^{A}_{AD_{i}}+1,$
$\displaystyle
C_{BPF}(\rho_{AB})=\sum_{i=1}^{4}\lambda^{AB}_{BPF_{i}}\log_{2}\lambda^{AB}_{BPF_{i}}+2.$
(53)

Figure 6: Evolution of channel capacity with maximal purity condition
$(C_{1}=C_{2}=1,C_{3}=-1).$ Graph (a):The quantum channel capacity $C_{AD}$ a
function of time $t$ and parameter $\lambda$ when takes different values, the
black line is drawed with $\lambda=0.1$, the yellow line is drawed with
$\lambda=0.3$ and the pink line is drawed with $\lambda=0.7.$ Graph (b):The
quantum channel capacity $C_{BPF}$ a function of $p.$

In order to show the relationship between the capacity and the bound more
explicitly, the channel capacity we provide is a function of time $t,p$ and
parameter $\lambda$ in terms of setting maximal purity condition
$(C_{1}=C_{2}=1,C_{3}=-1),$ following the Fig. 6 (a), it can be clearly found
that the evolution trend of capacity is similarly tocorrelated with the bound-
reducing ,but in Fig. 6 (b) , the capacity reducing firstly, when the
parameter equal 0.5, the capacity anti-correlated grow to maximum bound.
Considering these, due to the intrinsic connection between channel capacity
and the uncertainty’s bound, the channel capacity can be obtained through
analysis in the architecture.

##  V  Conclusions

We investigate the dynamics of the quantum memory-assisted entropic
uncertainty of bipartite system to be measured are coupled with two different
external noises in open environment. We considered two different noise
situations that particle A and B entangle in Bell-diagonal state suffers the
AD or BPF channels.And we choose mutual-noncommuting bases observation of
Pauli operators $\sigma_{x}$ and $\sigma_{z}$ as the incompatibility to
investigate dynamical properties of three different tighten bounds .Through
numerical analysis, we can draw the following conclusions: (I) When particles
A suffer from the different noise of environments , uncertainty and bunds have
non-monotonic feature and it can be obtained that its uncertainty will drop to
the lower bound for a long time. (II) The evolution of different bounds are as
expected that the bound of Adabi et al. is tighter than that of Pati et al.
,while the Pati et al.’s bound is tighter than Berta et al.’s bound. (III) the
correlated between quantum discord of the systemand bund is not completely
negtive. This can be explained as that the evolutionary characteristics of
bound is not only dependent of QD, but also the A’s minimal conditional
entropy $S^{A|B}_{\min}$ , which has been derived forenamed. We infer that the
quantum memory effect can essentially reduce the increase of uncertainty and
its lower bound. What?s more, we have put forward two effective operations to
manipulate the magnitude of the measurement’s uncertainty under the open
system via deriving quantum weak measurement and filtering operation
separately, it proves that the above operations can effectively suppress the
inflation of the uncertainty, and also reduce the magnitude to a large extent
. At last,we consider the applications on entanglement witness and quantum
channel capacity. Therefore, we claim the investigations we did may be of
great significance for the realization of practical quantum measurement
experment, and maymight be beneficial to help us understand the dynamical
behavior and operation of entropy uncertainty in open systems.

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