![The red solid line, black solid line and blue solid line represent the quantum correlation entropy of ε(ρ2), ε(ρ3), and ε(ρ4), respectively. The noisy channel is bit-flip channel. We take c1 = 0.8, c2=c12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{2}=\frac {c_{1}}{2}$\end{document}, and c3 = c1 ⋅ c2. The red dotted line represents the value of − f(c1). In this paper, we take (mA11,mA12,mA13)=(−0.0905,−0.8613,0.5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(m_{A_{1}1}, m_{A_{1}2}, m_{A_{1}3})=(-0.0905, -0.8613, 0.5)$\end{document}, (mA21,mA22,mA23)=(0.3933,−0.4368,0.81)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(m_{A_{2}1}, m_{A_{2}2}, m_{A_{2}3})=(0.3933, -0.4368, 0.81)$\end{document}, (mA31,mA32,mA33)=−13,13,13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(m_{A_{3}1}, m_{A_{3}2}, m_{A_{3}3})=\left (-\frac {1}{\sqrt {3}}, \frac {1}{\sqrt {3}}, \frac {1}{\sqrt {3}}\right )$\end{document}, and (mA41,mA42,mA43)=13,13,13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(m_{A_{4}1}, m_{A_{4}2}, m_{A_{4}3})=\left (\frac {1}{\sqrt {3}}, \frac {1}{\sqrt {3}}, \frac {1}{\sqrt {3}}\right )$\end{document}](https://www.researchgate.net/profile/Xiang-Zhou-96/publication/367451837/figure/fig1/AS:11431281124311949@1677898959459/The-red-solid-line-black-solid-line-and-blue-solid-line-represent-the-quantum_Q320.jpg)
![The red solid line, black solid line and blue solid line represent the quantum correlation entropy of ε(ρ2), ε(ρ3), and ε(ρ4), respectively. The noisy channel is bit-flip channel. We take c2 = 0.4, and c1 = c3 = 0. The red dotted line represents the value of − f(c1), where f(c1) = 0. The green diamond shows the intersection of quantum correlation entropy of 2-qubit state, 3-qubit state and 4-qubit state. Note that the horizontal coordinates of the green diamond, red diamond, black diamond and blue diamond be G◇, R◇, B◇ and B̂◇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat {B}_{\diamond }$\end{document}, respectively](https://www.researchgate.net/profile/Xiang-Zhou-96/publication/367451837/figure/fig2/AS:11431281124294323@1677898959486/The-red-solid-line-black-solid-line-and-blue-solid-line-represent-the-quantum_Q320.jpg)
![The red solid line, black solid line and blue solid line represent the quantum correlation entropy of ε(ρ2), ε(ρ3), and ε(ρ4), respectively. The noisy channel is phase-flip channel. We take c1 = 0.8, and c2 = c3 = 0. The red dotted line represents the value of − f(c3) and f(c3) = 0. Let the horizontal coordinates of the red diamond, black diamond and blue diamond be R◇, B◇ and B̂◇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat {B}_{\diamond }$\end{document}, respectively](https://www.researchgate.net/profile/Xiang-Zhou-96/publication/367451837/figure/fig4/AS:11431281124248578@1677898959750/The-red-solid-line-black-solid-line-and-blue-solid-line-represent-the-quantum_Q320.jpg)
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International Journal of Theoretical Physics (2023) 62:23
https://doi.org/10.1007/s10773-022-05270-z
Dynamical Behavior of Quantum Correlation Entropy
Under the Noisy Quantum Channel for Multiqubit
Systems
Xiang Zhou1
Received: 8 November 2022 / Accepted: 19 December 2022
©The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023
Abstract
Quantum correlation entropy is used to measure total non-classical correlation of multiple
states. It is based on a local coarse-grained measurement. Quantum noisy processes have a
theoretically and experimentally important role in quantum information tasks. We study the
dynamical behavior of quantum correlation entropy of output state under the effected of the
bit-flip channel, phase-flip channel and bit -phase flip channel. We find that quantum
correlation entropy of output state exists the frozen phenomenon and the phenomenon of
sudden death. The phenomenon of sudden death shows that under the influence of a noisy
channel, the output state becomes a classically correlated state.
Keywords Observational entropy ·Quantum correlation entropy ·
Noisy quantum channel ·Frozen phenomenon ·Sudden death
1 Introduction
The research on observational entropy has attracted the wide interest of many scholars,
and highly significant results are obtained [1–6]. Quantum correlation entropy is defined as
the difference between the infimum local observational entropy and von Neumann entropy,
where the infimum is taken over all local coarse-grainings. Quantum correlation entropy,
which depends on a partition into subsystems and quantifies the additional uncertainty in
a multipartite system with local coarse-grained measurement, is a measure of total non-
classical correlation [7]. It can be regarded as a natural generalization of entanglement
entropy to mixed state and multipartite system [7]. There are numerous properties for quan-
tum correlation entropy [7], such as additive over independent system, invariant under local
unitary operation, and reduces to the entanglement entropy for bipartite pure state. In finite
dimensional system, SQC
A1A2···AN(ρ) =0 if and only if ρis a classically correlated state [7].
Xiang Zhou
202010106112@mail.scut.edu.cn
1Department of Mathematics, South China University of Technology,
Guangzhou, 510640, People’s Republic of China
23 Page 2 of 23 Int J Theor Phys (2023) 62:23
Quantum correlation entropy of ρwith a local coarse-graining Cisgivenby[7]
SQC
A1A2···AN(ρ) =inf
CSO(C)(ρ) −S(ρ).(1)
where ρis a N-partite quantum state associated with subsystems A1,A
2,···,A
N.The
infimum is taken over all local coarse-grainings.
A noisy quantum channel is a noisy quantum process. It is a preserve qubit state
noisy operations [8,9], such as bit-flip channel, phase-flip channel, bit -phase flip
channel [8]. A noisy channel is a linear, CPTP map that maps the initial state ρto the
output state ε(ρ). The dynamical behavior of quantum discord and entanglement under
the noisy channels [10–18] becomes the motivation for studying the dynamical behavior
of quantum correlation entropy of output state in this paper. These dynamical behaviors
include the frozen phenomenon, the phenomenon of sudden death and the phenomenon
of sudden revival. The reasons for these dynamical behaviors are analyzed. We give some
understanding of these dynamical behaviors.
In this paper, we evaluate the local observational entropy and quantum correlation
entropy of initial state and output state. The output state is generated by the bit-flip
channel, phase-flip channel and bit -phase flip channel acting on the initial state. The
dynamical behavior of quantum correlation entropy of output state is studied. It is shown
that quantum correlation entropy of output state exists the frozen phenomenon for fixed cj.
For another fixed cj, quantum correlation entropy of output state exists the phenomenon of
sudden death. This shows that the output state becomes a classically correlated state after
a certain time. Since quantum correlation entropy of output state does not exist the phe-
nomenon of sudden revival, the output state never restores to a non-classically correlated
state. On the other hand, the relations between the quantum correlation entropies of different
output states is studied.
The paper is organized as follows. Section 2contains the main results of this paper.
Section 3we summarize our results and point out interesting avenues of future research.
Appendix A, a number of mathematical processes are left for the Appendix.
2 Dynamical Behavior of Quantum Correlation Entropy Under Noisy
Channels for Multiqubit State
2.1 Preparations
In this paper, we denote Pauli matrices as
ˆ
I2=10
01
,σ
1=σx=01
10
,(2)
σ2=σy=0−i
i0,σ
3=σz=10
0−1.(3)
We can verify that
σj·σj·σj=σj,σ
j·σj·σj=−σj,(4)
where j, j=1,2,3andj= j.
Moreover,
σ1·σ2=−i·σ3,σ
2·σ1=−i·σ3,σ
3·σ1=i·σ2,(5)
σ1·σ3=−i·σ2,σ
2·σ1=i·σ1,σ
3·σ2=−i·σ1.(6)
Int J Theor Phys (2023) 62:23 Page 3 of 23 23
Consider the following N-qubit states associated with systems A1,A2,···,AN
ρN=1
2N⎛
⎝ˆ
I+
3
j=1
cjσ⊗N
j⎞
⎠.(7)
where cj=Tr[(σj⊗σj)ρN]and |cj|≤1, and ˆ
I=ˆ
I⊗N
2.
For N=2, ρNreduces to the well-known Bell diagonal states. For general N,these
states are highly symmetric and can be considered as X-state [19–32].
In this paper, we define a class function as
f(x)=−1
2(1+x)log2(1+x) −1
2(1−x)log2(1−x), (8)
where |x|<1.
Let {Πk=|kk|,k =0,1}be a standard orthogonal basis of 2-dimensional Hilbert
space. We define a local coarse-graining as
C=CA1⊗CA2⊗···⊗CAN=ˆ
PA1
l⊗ˆ
PA2
m⊗···⊗ ˆ
PAN
n,(9)
where l,m,...,n =0,1.
Denote
ˆ
PAj
k=VAjΠkV†
Aj,(10)
where j=1,2,...,N and k=0,1.
Denote
VAj=tAj0ˆ
I+tAj1σ1i+tAj2σ2i+tAj3σ3i(11)
be a unitary matrix and tAjk∈R,3
k=0t2
Ajk=1. Hence, the set {VAjΠkV†
Aj:k=
0,1}constitutes the complete set of one-rank projection operators of 2-dimensional Hilbert
Spaces.
Denote
mAj1=2(tAj1tAj3−tAj2tAj0), (12)
mAj2=2(tAj2tAj3+tAj1tAj0), (13)
mAj3=t2
Aj0+t2
Aj3−t2
Aj1−t2
Aj2,(14)
where j=1,2,...,N.
We can verify that
m2
Aj1+m2
Aj2+m2
Aj3=1. (15)
From the Formula (4), we can verify that
VAjΠ0V†
Aj·σj·VAjΠ0V†
Aj=mAjj·σj,(16)
VAjΠ1V†
Aj·σj·VAjΠ1V†
Aj=−mAjj·σj,(17)
where j=0,1,2,...,N and j=1,2,3.
2.2 Dynamical Behavior of Quantum Correlation Entropy Under Bit Flip Channel
The bit-flip channel flips the state of a qubit from |0to |1(and vice versa) with the
degree of decoherence p[33]. This channel acting on a single qubit can be described by the
following Kraus operators
ΓAj
0=1−pj
2ˆ
I, ΓAj
1=pj
2σ1,(18)
23 Page 4 of 23 Int J Theor Phys (2023) 62:23
where Ajlabels the subsystems and p∈[0,1]. Here, we consider the symmetric situation
in which the decoherence rate is equal, so p1=p2=···=p.
In Appendix A, we evaluate the quantum correlation entropy with local coarse-graining
C(9) and give the condition that quantum correlation entropy is monotonically decreasing
with respect to p.
In this paper, we show the dynamical behavior of quantum correlation entropy of output
state under the bit -flip channel for N-qubit state, where N=2,3,4.
For N=2, quantum correlation entropy of ε(ρ2)isgivenby(74)
SQC
A1A2(ε(ρ2)) =inf
βε
2
f(βε
2)+
4
i=1
λi
4log2λi. (19)
Equation (76) implies that the quantum correlation entropy of ε(ρ2)is monotonically
decreasing with respect to p, namely,
2α2·log2
1+βε
2
1−βε
2
<(c
2+c3)log2
λ2
λ1+(c2−c3)log2
λ4
λ3
. (20)
For N=3, quantum correlation entropy of ε(ρ3)isgivenby(79)
SQC
A1A2A3(ε(ρ3)) =inf
βε
3[f(βε
3)]+3−S(ε(ρ)). (21)
Equation (85) implies that the quantum correlation entropy of ε(ρ3)is monotonically
decreasing with respect to p, namely,
θα3·log2
1+βε
3
1−βε
3
<c2
2+c2
3(1−p)3log2
1+θ
1−θ. (22)
For N=4, quantum correlation entropy of ε(ρ4)isgivenby(90)
SQC
A1A2A3A4(ε(ρ4)) =inf
βε
4
f(βε
4)+
4
i=1
λi
4log2λi. (23)
Equation (92) implies that the quantum correlation entropy of ε(ρ4)is monotonically
decreasing with respect to p,thatis,
2α4·log2
1+βε
4
1−βε
4
<(c
2+c3)log2
λ1
λ2+(c2−c3)log2
λ3
λ4
. (24)
For instance, we take c1=0.8, c2=c1
2,andc3=c1·c2. Figure 1shows the dynamical
behavior of the quantum correlation entropy of ε(ρN)under the action of the bit -flip
channel, where N=2,3,4. From Fig. 1, it shows that quantum correlation entropy of
output state exists the frozen phenomenon under the action of bit-flipchannel. This means
that the quantum correlation entropy of output state is invariant and equal to −f(c
1)after
a certain time. The frozen phenomenon indicates that the output state never becomes a
classically correlated state under the action of bit-flipchannel.
In Fig. 1, the black line is always higher than the blue line, which means that under
the operation of bit-flip channel, quantum correlation entropy of ε(ρ3)is not less than
quantum correlation entropy of ε(ρ4). The red line always stays on top, which means quan-
tum correlation entropy of ε(ρ2)is not less than quantum correlation entropy of ε(ρ3)and
ε(ρ4). We surprise to find that SQC
A1A2(ε(ρ2)) is not less than SQC
A1A2A3(ε(ρ3)). Meanwhile,
SQC
A1A2A3(ε(ρ3)) is not less than SQC
A1A2A3A4(ε(ρ4)). This shows that under the influence of
Int J Theor Phys (2023) 62:23 Page 5 of 23 23
Fig. 1 The red solid line, black
solid line and blue solid line
represent the quantum correlation
entropy of ε(ρ2),ε(ρ3),and
ε(ρ4), respectively. The noisy
channel is bit-flip channel. We
take c1=0.8, c2=c1
2,and
c3=c1·c2. The red dotted line
represents the value of −f(c
1).
In this paper, we take
(mA11,m
A12,m
A13)=
(−0.0905,−0.8613,0.5),
(mA21,m
A22,m
A23)=
(0.3933,−0.4368,0.81),
(mA31,m
A32,m
A33)=
−1
√3,1
√3,1
√3,and
(mA41,m
A42,m
A43)=
1
√3,1
√3,1
√30 0.2 0.4 0.6 0.8 1
p
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Quantum correlation entropy (S)
bit-flipchannel, the increase in the number of subsystems does not improve the total non-
classical correlation of output states. It is worth noting that when we fix the values of cj,the
eigenvalues of the output states are negative for some ranges of p. Therefore, the quantity
depicted by a red line is not defined for all values of the parameter p.
Noting that c1is independent on time, we consider the case that c2=0.4, and
c1=c3=0. Figure 2shows the dynamical behavior of the quantum correlation entropy
of output state under the action of the bit-flip channel. In Fig. 2, We surprise to find that
SQC
A1A2(ε(ρ2)) is less than SQC
A1A2A3(ε(ρ3)) and SQC
A1A2A3A4(ε(ρ4)),whenp<G
. Mean-
while, SQC
A1A2A3A4(ε(ρ4)) is not less than SQC
A1A2A3(ε(ρ3)),whenp≤G.However,when
G≤p≤1, we have SQC
A1A2A3A4(ε(ρ4)) ≤SQC
A1A2A3(ε(ρ3)) ≤SQC
A1A2(ε(ρ2)). This shows
that under the influence of bit-flipchannel, the increase of the number of subsystems does
not improve the total non-classical correlation of output states under a certain time.
Figure 2also shows that quantum correlation entropy of ε(ρ2),ε(ρ3)and ε(ρ4)exist the
phenomenon of sudden death. That is to say, quantum correlation entropy of output states
Fig. 2 The red solid line, black
solid line and blue solid line
represent the quantum correlation
entropy of ε(ρ2),ε(ρ3),and
ε(ρ4), respectively. The noisy
channel is bit-flip channel. We
take c2=0.4, and c1=c3=0.
The red dotted line represents the
value of −f(c
1),where
f(c
1)=0. The green diamond
shows the intersection of
quantum correlation entropy of
2-qubit state, 3-qubit state and
4-qubit state. Note that the
horizontal coordinates of the
green diamond, red diamond,
black diamond and blue diamond
be G,R,Band ˆ
B,
respectively 0 0.2 0.4 0.6 0.8 1
p
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
Quantum correlation entropy (S)
23 Page 6 of 23 Int J Theor Phys (2023) 62:23
is equal to zero, when p≥R,p≥B,andp≥ˆ
B, respectively. The phenomenon of
sudden death shown in Fig. 2is a new feature for physical dissipation [13,16]. The zero of
quantum correlation entropy of output state means that the output state becomes a classically
correlated state after a certain time. Moreover, the classically correlated state never restores
to a non-classically correlated state, since the quantum correlation entropy of output state
does not exist the phenomenon of sudden revival.
2.3 Dynamical Behavior of Quantum Correlation Entropy Under Phase Flip Channel
The phase-flip channel acting on a single qubit can be described by the following Kraus
operators
ΓAj
0=1−pj
2ˆ
I, ΓAj
1=pj
2σ3,(25)
where Ajlabels the subsystems and p∈[0,1]. Here, we consider the symmetric situation
in which the decoherence rate is equal, so p1=p2=···=p.
For N=2, quantum correlation entropy of ε(ρ2)isgivenby(102)
SQC
A1A2(ε(ρ2)) =inf
βε
2
f(βε
2)+
4
i=1
λi
4log2λi,(26)
Equation (104) implies that the quantum correlation entropy of ε(ρ2)is monotonically
decreasing with respect to p, namely,
2α2·log2
1+βε
2
1−βε
2
<(c
1+c2)log2
λ4
λ1+(c1−c2)log2
λ3
λ2
. (27)
For N=3, quantum correlation entropy of ε(ρ3)isgivenby(107)
SQC
A1A2A3(ε(ρ3)) =inf
βε
3f(βε
3)+3−S(ε(ρ)). (28)
Equation (113) implies that the quantum correlation entropy of ε(ρ3)is monotonically
decreasing with respect to p, namely,
θα3·log2
1+βε
3
1−βε
3
<c2
1+c2
2(1−p)3log2
1+θ
1−θ. (29)
For N=4, quantum correlation entropy of ε(ρ4)isgivenby(118)
SQC
A1A2A3A4(ε(ρ4)) =inf
βε
4
f(βε
4)+
4
i=1
λi
4log2λi,(30)
Equation (120) implies that the quantum correlation entropy of ε(ρ4)is monotonically
decreasing with respect to p, namely,
2α4·log2
1+βε
4
1−βε
4
<(c
1+c2)log2
λ1
λ4+(c1−c2)log2
λ2
λ3
. (31)
For instance, we take c1=0.8, c2=c1
2,andc3=0.5. Figure 3shows the dynamical
behavior of the quantum correlation entropy of output state under the action of the phase-
flip channel. From Fig. 3, it shows that quantum correlation entropy of output state exists
the frozen phenomenon under the action of phase-flip channel for N=2,3,4. Quantum
correlation entropy of output state is equal to −f(c
3)for N=3,4 after a certain time. But
for N=2, quantum correlation entropy of output state is equal to a,where0<a <−f(c
3)
Int J Theor Phys (2023) 62:23 Page 7 of 23 23
00.20.40.60.81
p
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Quantum correlation entropy (S)
Fig. 3 The red solid line, black solid line and blue solid line represent the quantum correlation entropy of
ε(ρ2),ε(ρ3),andε(ρ4), respectively. The noisy channel is phase-flipchannel. We take c1=0.8, c2=0.5,
and c2=c1·c3. The red dotted line represents the value of −f(c
3). The red diamond is the intersection of
quantum correlation entropy of ε(ρ3)and ε(ρ4). The green diamond is the intersection of quantum correlation
entropy of ε(ρ2)and ε(ρ3). The yellow diamond is the intersection of quantum correlation entropy of ε(ρ2)
and ε(ρ4). Let the horizontal coordinates of the red diamond, green diamond and yellow diamond be R,G
and Y, respectively
after a certain time. The frozen phenomenon indicates that the output state never becomes a
classically correlated state under the action of phase-flip channel.
In Fig. 3,ifp≤R, the black line is always higher than the blue line, which means
that under the operation of phase-flip channel, quantum correlation entropy of ε(ρ3)is
not less than quantum correlation entropy of ε(ρ4). And vice versa, if p≥R.Ifp≤G,
the red line always stays on top, which means quantum correlation entropy of ε(ρ2)is not
less than quantum correlation entropy of ε(ρ3)and ε(ρ4).However,ifp≥G, quantum
correlation entropy of ε(ρ2)is not larger than quantum correlation entropy of ε(ρ3).If
p≥Y, quantum correlation entropy of ε(ρ2)is not larger than quantum correlation entropy
of ε(ρ4). This shows that under the influence of phase-flip channel, the increase of the
number of subsystems can improve the total non-classical correlation of output states under
a certain time. It is worth noting that when we fix the values of cj, the eigenvalues of the
output states are negative for some ranges of p. Therefore, the quantities depicted by a black
and red line are not defined for all values of the parameter p.
Noting that c3is independent on time, we consider the case that c1=0.8, and c2=
c3=0. Figure 4shows the dynamical behavior of the quantum correlation entropy of out-
put state under the action of the phase-flipchannel for N=2,3,4. In Fig. 4, we surprise
to find that quantum correlation entropy of output state exists the phenomenon of sudden
death. Quantum correlation entropy of output state is equal to zero, when p≥R,p≥B,
and p≥ˆ
B, respectively. The phenomenon of sudden death indicates that the output state
becomes a classically correlated state after a certain time. Moreover, the classically corre-
lated state never restores to a non-classically correlated state, since the quantum correlation
entropy of output state does not exists the phenomenon of sudden revival. We also observer
that under the influence of phase-flip channel, the increase of the number of subsystems
can not improve the total non-classical correlation of output states under a certain time.
23 Page 8 of 23 Int J Theor Phys (2023) 62:23
Fig. 4 The red solid line, black
solid line and blue solid line
represent the quantum correlation
entropy of ε(ρ2),ε(ρ3),and
ε(ρ4), respectively. The noisy
channel is phase-flip channel.
We ta ke c1=0.8, and
c2=c3=0. The red dotted line
represents the value of −f(c
3)
and f(c
3)=0. Let the horizontal
coordinates of the red diamond,
black diamond and blue diamond
be R,Band ˆ
B, respectively
0 0.2 0.4 0.6 0.8 1
p
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Quantum correlation entropy (S)
2.4 Dynamical Behavior of Quantum Correlation Entropy Under Bit-Phase Flip
Channel
The bit-phase flip channel acting on a single qubit can be described by the following Kraus
operators
ΓAj
0=1−pj
2ˆ
I, ΓAj
1=pj
2σ2,(32)
where Ajlabels the subsystems and p∈[0,1]. Here, we consider the symmetric situation
in which the decoherence rate is equal, so p1=p2=···=p.
For N=2, quantum correlation entropy of ε(ρ2)isgivenby(130)
SQC
A1A2(ε(ρ2)) =inf
βε
2
f(βε
2)+
4
i=1
λi
4log2λi. (33)
Equation (132) implies that the quantum correlation entropy of ε(ρ2)is monotonically
decreasing with respect to p, namely,
2α2·log2
1+βε
2
1−βε
2
<(c
1+c3)log2
λ3
λ1+(c1−c3)log2
λ4
λ2
. (34)
For N=3, quantum correlation entropy of ε(ρ3)isgivenby(135)
SQC
A1A2A3(ε(ρ3)) =inf
βε
3[f(βε
3)]+3−S(ε(ρ)). (35)
Equation (141) implies that the quantum correlation entropy of ε(ρ3)is monotonically
decreasing with respect to p, namely,
θα3·log2
1+βε
3
1−βε
3
<(c
2
1+c2
3)(1−p)3log2
1+θ
1−θ. (36)
For N=4, quantum correlation entropy of ε(ρ4)isgivenby(146)
SQC
A1A2A3A4(ε(ρ4)) =inf
βε
4
f(βε
4)+
4
i=1
λi
4log2λi,(37)
Int J Theor Phys (2023) 62:23 Page 9 of 23 23
Equation (148) implies that the quantum correlation entropy of ε(ρ4)is monotonically
decreasing with respect to p, namely,
2α4·log2
1+βε
4
1−βε
4
<(c
1+c3)log2
λ1
λ4+(c1−c3)log2
λ2
λ3
. (38)
For instance, we take c2=0.8, c1=0.5, and c3=c2·c1. Figure 5shows the dynam-
ical behavior of quantum correlation entropy under the action of bit-phase flip channel.
From Fig. 5, it shows that quantum correlation entropy of output state exists the frozen
phenomenon under the action of bit -phase flip channel for N=2,3,4. Quantum cor-
relation entropy of output state is equal to ANfor N=3,4 after a certain time, where
0<A
N<−f(c
2),andA3<A
4.ButforN=2, quantum correlation entropy of output
state is equal to A2in a short time, where 0 <A
2<A
3. The frozen phenomenon indi-
cates that the output state never becomes a classically correlated state under the action of
bit-phase flip channel.
In Fig. 5,ifp≤R, the black line is always higher than the blue line, which means that
under the operation of bit-phase flip channel, quantum correlation entropy of ε(ρ3)is not
less than quantum correlation entropy of ε(ρ4). And vice versa, if p≥R.Ifp≤G,the
red line always stays on top, which means that quantum correlation entropy of ε(ρ2)is not
less than quantum correlation entropy of ε(ρ3)and ε(ρ4).However,ifp≥G, quantum
correlation entropy of ε(ρ2)is not larger than quantum correlation entropy of ε(ρ4).If
p≥Y, quantum correlation entropy of ε(ρ2)is not larger than quantum correlation entropy
of ε(ρ3). It is worth noting that when we fix the values of cj, the eigenvalues of the output
states are negative for some ranges of p. Therefore, the quantities depicted by a black and
red line are not defined for all values of the parameter p.
0 0.2 0.4 0.6 0.8 1
p
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
Quantum correlation entropy (S)
Fig. 5 The red solid line, black solid line and blue solid line represent the quantum correlation entropy of
ε(ρ2),ε(ρ3),andε(ρ4), respectively. The noisy channel is bit-phase flip channel. We take c2=0.8, c1=
0.5, and c3=c2·c1. The red dotted line represents the value of −f(c
2). The red diamond is the intersection of
quantum correlation entropy of ε(ρ3)and ε(ρ4). The green diamond is the intersection of quantum correlation
entropy of ε(ρ2)and ε(ρ4). The yellow diamond is the intersection of quantum correlation entropy of ε(ρ2)
and ε(ρ3). Let the horizontal coordinates of the red diamond, green diamond and yellow diamond be R,G
and Y, respectively
23 Page 10 of 23 Int J Theor Phys (2023) 62:23
Fig. 6 The red solid line, black
solid line and blue solid line
represent the quantum correlation
entropy of ε(ρ2),ε(ρ3),and
ε(ρ4), respectively. The noisy
channel is bit-phase flip
channel. We take c1=0.5, and
c2=c3=0. The red dotted line
represents the value of −f(c
2)
and f(c
2)=0. Let the horizontal
coordinates of the red diamond,
black diamond and blue diamond
be R,Band ˆ
B, respectively
0 0.2 0.4 0.6 0.8 1
p
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Quantum correlation entropy (S)
Note that c2is independent on time, we consider the case that c1=0.5, and c2=c3=0.
Figure 6shows the dynamical behavior of the quantum correlation entropy of output state
under the action of the phase-flip channel for N=2,3,4. In Fig. 6, we surprise to find
that quantum correlation entropy of output state exists the phenomenon of sudden death.
Quantum correlation entropy of output state is equal to zero, when p≥R,p≥B,
and p≥ˆ
B, respectively. The phenomenon of sudden death indicates that the output state
becomes a classically correlated state after a certain time. Moreover, the classically corre-
lated state never restores to a non-classically correlated state, since the quantum correlation
entropy of output state does not exists the phenomenon of sudden revival.
3 Conclusion
We study the dynamical behavior of quantum correlation entropy under the noisy chan-
nels. These noisy channels are bit -flip channel, phase-flip channel and bit -phase flip
channel.
Under the action of bit -flip channel, the quantum correlation entropy of output state
exists the frozen phenomenon, when we set c1=0.8, c2=c1
2,andc3=c1·c2.This
indicates that the quantum correlation entropy of output state is invariant after a certain
time. Meanwhile, the output state remains a non-classically correlated state, which shows
the influence of bit-flip channel. However, when we set c2=0.4, and c1=c3=0. The
quantum correlation entropy of output state exists the phenomenon of sudden death. This
means that the quantum correlation entropy of output state is equal to zero, and the output
state becomes a classically correlated state. Since the quantum correlation entropy does not
exist the phenomenon of sudden revival, the output state never restores to a non-classically
correlated state.
Under the action of phase-flipchannel, the quantum correlation entropy of output state
exists the frozen phenomenon, when we set c1=0.8, c2=c1
2,andc3=0.5. This indi-
cates that the quantum correlation entropy of output state is invariant after a certain time.
Meanwhile, the output state remains a non-classically correlated state, which shows the
influence of phase-flip channel. However, when we set c1=0.8, and c2=c3=0. The
Int J Theor Phys (2023) 62:23 Page 11 of 23 23
quantum correlation entropy of output state exists the phenomenon of sudden death. This
means that the quantum correlation entropy of output state is equal to zero, and the output
state becomes a classically correlated state. Since the quantum correlation entropy does not
exist the phenomenon of sudden revival, the output state never restores to a non-classically
correlated state.
Under the action of bit-phase flip channel, the quantum correlation entropy of output
state exists the frozen phenomenon, when we set c2=0.8, c1=0.5, and c3=c2·c1.This
indicates that the quantum correlation entropy of output state is invariant after a certain time.
Meanwhile, the output state remains a non-classically correlated state, which shows the
influence of bit-phase flip channel. However, when we set c1=0.5, and c2=c3=0. The
quantum correlation entropy of output state exists the phenomenon of sudden death. This
means that the quantum correlation entropy of output state is equal to zero, and the output
state become a classically correlated state. Since the quantum correlation entropy does not
exist the phenomenon of sudden revival, the output state never restores to a non-classically
correlated state.
We give the results of the dynamical behavior of quantum correlation entropy for N-
qubit state, but we only analyze the case for Nis equal to 2, 3, and 4. A discussion of
higher dimensions might be an interesting question. Our results may highlight further inves-
tigations on quantum correlation entropy and their applications in quantum information
processing.
Appendix
A.1: Observational Entropy and Quantum Correlation Entropy for Multiqubit State
For the initial state ρN(7), and under the action of local coarse-graining C(9), we obtain
the final state as the ensemble {ρlm...n ,p
lm...n }with
ρlm...n =1
plm...n ˆ
PA1
l⊗ˆ
PA2
m⊗···⊗ ˆ
PAN
nρNˆ
PA1
l⊗ˆ
PA2
m⊗···⊗ ˆ
PAN
n,(39)
and
plm...n =Tr (ˆ
PA1
l⊗ˆ
PA2
m⊗···⊗ ˆ
PAN
n)ρN. (40)
For instance, we take N=3. We can verify that
ρ000 =1
p000 ˆ
PA1
0⊗ˆ
PA2
0⊗ˆ
PA3
0)ρ3(ˆ
PA1
0⊗ˆ
PA2
0⊗ˆ
PA3
0=ˆ
PA1
0⊗ˆ
PA2
0⊗ˆ
PA3
0,p
000 =1
8(1+β3),
(41)
ρ001 =1
p001 ˆ
PA1
0⊗ˆ
PA2
0⊗ˆ
PA3
1)ρ3(ˆ
PA1
0⊗ˆ
PA2
0⊗ˆ
PA3
1=ˆ
PA1
0⊗ˆ
PA2
0⊗ˆ
PA3
1,p
001 =1
8(1−β3),
(42)
ρ010 =1
p010 ˆ
PA1
0⊗ˆ
PA2
1⊗ˆ
PA3
0ρ3ˆ
PA1
0⊗ˆ
PA2
1⊗ˆ
PA3
0=ˆ
PA1
0⊗ˆ
PA2
1⊗ˆ
PA3
0,p
010 =1
8(1−β3),
(43)
ρ011 =1
p011 ˆ
PA1
0⊗ˆ
PA2
1⊗ˆ
PA3
1ρ3ˆ
PA1
0⊗ˆ
PA2
1⊗ˆ
PA3
1=ˆ
PA1
0⊗ˆ
PA2
1⊗ˆ
PA3
1,p
011 =1
8(1+β3),
(44)
23 Page 12 of 23 Int J Theor Phys (2023) 62:23
ρ100 =1
p100 ˆ
PA1
1⊗ˆ
PA2
0⊗ˆ
PA3
0ρ3ˆ
PA1
1⊗ˆ
PA2
0⊗ˆ
PA3
0=ˆ
PA1
1⊗ˆ
PA2
0⊗ˆ
PA3
0,p
100 =1
8(1−β3),
(45)
ρ101 =1
p101 ˆ
PA1
1⊗ˆ
PA2
0⊗ˆ
PA3
1ρ3ˆ
PA1
1⊗ˆ
PA2
0⊗ˆ
PA3
1=ˆ
PA1
1⊗ˆ
PA2
0⊗ˆ
PA3
1,p
101 =1
8(1+β3),
(46)
ρ110 =1
p110 ˆ
PA1
1⊗ˆ
PA2
1⊗ˆ
PA3
0ρ3ˆ
PA1
1⊗ˆ
PA2
1⊗ˆ
PA3
0=ˆ
PA1
1⊗ˆ
PA2
1⊗ˆ
PA3
0,p
110 =1
8(1+β3),
(47)
ρ111 =1
p111 ˆ
PA1
1⊗ˆ
PA2
1⊗ˆ
PA3
1ρ3ˆ
PA1
1⊗ˆ
PA2
1⊗ˆ
PA3
1=ˆ
PA1
1⊗ˆ
PA2
1⊗ˆ
PA3
1,p
111 =1
8(1−β3),
(48)
where
1
l=0
1
m=0
1
n=0
plmn =1, and β3=3
j=1cjmA1jmA2jmA3j,|β3|≤1.
The local observational entropy of ρ3is given by
SO(C)(ρ3)=−
lmn
plmn log2
plmn
Vlmn
=−4·1+β3
8log2
1+β3
8−4·1−β3
8log2
1−β3
8(49)
=−1+β3
2log2(1+β3)−1−β3
2log2(1−β3)+3
=f(β
3)+3,
where Vlmn =Trˆ
PA1
l⊗ˆ
PA2
m⊗ˆ
PA3
nand
1
l=0
1
m=0
1
n=0
Vlmn =8.
The quantum correlation entropy of ρ3is given by
SQC
A1A2A3(ρ3)=inf
β3[f(β
3)]+3−S(ρ3), (50)
where S(ρ3)=−tr[ρ3log2ρ3]is the von Neumann entropy of ρ3.
Under the operation of a quantum channel, we assume that ρ3be mapped to ε(ρ3)by
this quantum channel ε(·).
For instance, we evaluate the observational entropy and quantum correlation entropy
of ε(ρ3)(78), where ε(ρ3)is the output state of ρ3(77) under the operation of bit-
flip channel. Let local coarse-graining Cbe performed on ε(ρ3)(78). The ε(ρ3)will
be updated to final states as the ensemble ρε
lmn,p
ε
lmnwith ρε
lmn := 1
plmn ˆ
PA1
l⊗
ˆ
PA2
m⊗ˆ
PA3
nε(ρ3)ˆ
PA1
l⊗ˆ
PA2
m⊗ˆ
PA3
nand pε
lmn =Tr ˆ
PA1
l⊗ˆ
PA2
m⊗ˆ
PA3
nε(ρ)
ˆ
PA1
l⊗ˆ
PA2
m⊗ˆ
PA3
n. We can verify that
ρε
000 =1
pε
000 ˆ
PA1
0⊗ˆ
PA2
0⊗ˆ
PA3
0ε(ρ3)ˆ
PA1
0⊗ˆ
PA2
0⊗ˆ
PA3
0=ˆ
PA1
0⊗ˆ
PA2
0⊗ˆ
PA3
0,p
ε
000 =1
8(1+βε
3),
(51)
ρε
001 =1
pε
001 ˆ
PA1
0⊗ˆ
PA2
0⊗ˆ
PA3
1ε(ρ3)ˆ
PA1
0⊗ˆ
PA2
0⊗ˆ
PA3
1=ˆ
PA1
0⊗ˆ
PA2
0⊗ˆ
PA3
1,p
ε
001 =1
8(1−βε
3),
(52)
ρε
010 =1
pε
010 ˆ
PA1
0⊗ˆ
PA2
1⊗ˆ
PA3
0ε(ρ3)ˆ
PA1
0⊗ˆ
PA2
1⊗ˆ
PA3
0=ˆ
PA1
0⊗ˆ
PA2
1⊗ˆ
PA3
0,p
ε
010 =1
8(1−βε
3),
(53)
Int J Theor Phys (2023) 62:23 Page 13 of 23 23
ρε
011 =1
pε
011 ˆ
PA1
0⊗ˆ
PA2
1⊗ˆ
PA3
1ε(ρ3)ˆ
PA1
0⊗ˆ
PA2
1⊗ˆ
PA3
1=ˆ
PA1
0⊗ˆ
PA2
1⊗ˆ
PA3
1,p
ε
011 =1
8(1+βε
3),
(54)
ρε
100 =1
pε
100 ˆ
PA1
1⊗ˆ
PA2
0⊗ˆ
PA3
0ε(ρ3)ˆ
PA1
1⊗ˆ
PA2
0⊗ˆ
PA3
0=ˆ
PA1
1⊗ˆ
PA2
0⊗ˆ
PA3
0,p
ε
100 =1
8(1−βε
3),
(55)
ρε
101 =1
pε
101 ˆ
PA1
1⊗ˆ
PA2
0⊗ˆ
PA3
1ε(ρ3)ˆ
PA1
1⊗ˆ
PA2
0⊗ˆ
PA3
1=ˆ
PA1
1⊗ˆ
PA2
0⊗ˆ
PA3
1,p
ε
101 =1
8(1+βε
3),
(56)
ρε
110 =1
pε
110 ˆ
PA1
1⊗ˆ
PA2
1⊗ˆ
PA3
0ε(ρ3)ˆ
PA1
1⊗ˆ
PA2
1⊗ˆ
PA3
0=ˆ
PA1
1⊗ˆ
PA2
1⊗ˆ
PA3
0,p
ε
110 =1
8(1+βε
3),
(57)
ρε
111 =1
pε
111 ˆ
PA1
1⊗ˆ
PA2
1⊗ˆ
PA3
1ε(ρ3)ˆ
PA1
1⊗ˆ
PA2
1⊗ˆ
PA3
1=ˆ
PA1
1⊗ˆ
PA2
1⊗ˆ
PA3
1,p
ε
111 =1
8(1−βε
3),
(58)
where
1
l=0
1
m=0
1
n=0
pε
lmn =1andβε
3=c1mA11mA21mA31+c2mA12mA22mA32(1−p)3+
c3mA13mA23mA33(1−p)3,|βε
3|≤1.
Local observational entropy of ε(ρ3)can be given by
SO(C)(ε(ρ3)) =f(βε
3)+3. (59)
Quantum correlation entropy of ε(ρ3)is given by
SQC
A1A2A3(ε(ρ3)) =inf
CSO(C)(ε(ρ3)) −S(ε(ρ3))
=inf
βε
3[f(βε
3)]+3−S(ε(ρ3)). (60)
In general, for N-qubit state, we can verify that the local observational entropy of ρN(7)
is given by
SO(C)(ρN)=f(β
N)+N. (61)
where βN=3
j=1cjmA1jmA2j···mANj,|βN|≤1.
Hence, the quantum correlation entropy of ρN(7)isgivenby
SQC
A1A2···AN(ρN)=inf
βN[f(β
N)]+N−S(ρN), (62)
where S(ρN)=−tr[ρNlog2ρN]is the von Neumann entropy of ρN.
Meanwhile, We also take ε(ρN)as the output state of ρNunder the operation of a bit-
flip channel. We can verify that the local observational entropy of ε(ρN)is given by
SO(C)(ε(ρN)) =f(βε
N)+N. (63)
where βε
N=c1mA11mA21···mAN1+c2mA12mA22···mAN2(1−p)N+c3mA13mA23
···mAN3(1−p)N,|βε
N|≤1.
Hence, the quantum correlation entropy of ε(ρN)is given by
SQC
A1A2···AN(ρN)=inf
βε
N[f(βε
N)]+N−S(ε(ρN)), (64)
where S(ε(ρN)) =−tr[ε(ρN)log2ε(ρN)]is the von Neumann entropy of ε(ρN).
23 Page 14 of 23 Int J Theor Phys (2023) 62:23
A.2: Quantum Correlation Entropy Under Bit Flip Channel
The bit-flip channel flips the state of a qubit from |0to |1(and vice versa) with the
degree of decoherence p[33]. This channel acting on a single qubit can be described by the
following Kraus operators
ΓAj
0=1−pj
2ˆ
I, ΓAj
1=pj
2σ1,(65)
where Ajlabels the subsystems and p∈[0,1]. Here, we consider the symmetric situation
in which the decoherence rate is equal, so p1=p2=···=p.
We can verify that
l,...,n
ΓA1
l···ΓAN
nσ⊗N
1ΓAN†
n···ΓA1†
l=σ⊗N
1,(66)
l,...,n
ΓA1
l···ΓAN
nσ⊗N
2ΓAN†
n···ΓA1†
l=(1−p)Nσ⊗N
2,(67)
l,...,n
ΓA1
l···ΓAN
nσ⊗N
3ΓAN†
n···ΓA1†
l=(1−p)Nσ⊗N
3,(68)
where l=···=n=0,1.
Consider the following 2-qubit state associated with subsystems A1,A
2
ρ2=1
22(ˆ
I+c1σ⊗2
1+c2σ⊗2
2+c3σ⊗2
3). (69)
Under the operation of bit-flip channel, the output state ε(ρ2)is given by
ε(ρ2)=1
4ˆ
I+c1σ⊗2
1+(1−p)2c2σ⊗2
2+(1−p)2c3σ⊗2
3. (70)
The nonzero and positive eigenvalues of ε(ρ2)are written as λ1=1−c1−(1−p)2c2−
(1−p)2c3,λ2=1−c1+(1−p)2c2+(1−p)2c3,λ3=1+c1−(1−p)2c2+(1−p)2c3,
λ4=1+c1+(1−p)2c2−(1−p)2c3. The derivative of λiwith respect to pcan be cast as
λ
1=∂λ1
∂p =2(c2+c3)(1−p), λ
2=∂λ2
∂p =−2(c2+c3)(1−p), (71)
λ
3=∂λ3
∂p =2(c2−c3)(1−p), λ
4=∂λ4
∂p =−2(c2−c3)(1−p). (72)
From (63), the local observational entropy of ε(ρ2)is given by
SO(C)(ρ2)=f(βε
2)+2. (73)
where βε
2=c1mA11mA21+α2(1−p)2,|βε
2|≤1andα2=
3
j=2
cjmA1jmA2j.
Quantum correlation entropy of ε(ρ2)is given by
SQC
A1A2(ε(ρ2)) =inf
βε
2
f(βε
2)+
4
i=1
λi
4log2λi,(74)
Denote F(p) =f(βε
2)+4
i=1
λi
4log2λi. The derivative of F(p)is given by
dF(p)
dp =−βε
2
2log2
1+βε
2
1−βε
2−λ
1
4log2
λ2
λ1−λ
3
4log2
λ4
λ3
,(75)
Int J Theor Phys (2023) 62:23 Page 15 of 23 23
where βε
2=∂βε
2
∂p =−2(1−p) ·α2.
In order to show dF(p)
dp <0, the following inequality must hold
2α2·log2
1+βε
2
1−βε
2
<(c
2+c3)log2
λ2
λ1+(c2−c3)log2
λ4
λ3
. (76)
Therefore, (76) implies that SQC
A1A2(ε(ρ2)) is monotonically decreasing with respect to p.
Consider the following 3-qubit states associated with subsystems A1,A2and A3,
ρ3=1
23⎛
⎝ˆ
I+
3
j=1
cjσj⊗σj⊗σj⎞
⎠. (77)
Under the action of bit-flipchannel, the initial state ρ3be updated to ε(ρ3)as
ε(ρ3)=1
8ˆ
I+c1σ⊗3
1+(1−p)3c2σ⊗3
2+(1−p)3c3σ⊗3
3. (78)
From (64), quantum correlation entropy of ε(ρ3)is given by
SQC
A1A2A3(ε(ρ3)) =inf
βε
3[f(βε
3)]+3−S(ε(ρ3)). (79)
where βε
3=c1mA11mA21mA31+α3(1−p)3,|βε
3|≤1andα3=
3
j=2
cjmA1jmA2jmA3j.
Set θ=c2
1+(1−p)6c2
2+(1−p)6c2
3,wehave
S(ε(ρ3)) =f(θ)+3. (80)
Hence, we have
SQC
A1A2A3(ε(ρ3)) =inf
βε
3[f(βε
3)]−f(θ). (81)
Since θ>0, this implies that log21+θ
1−θ>0.
We have
θ=∂θ
∂p =−3(c2
2+c2
3)(1−p)5
θ. (82)
For 0 <(1−p)5<1, then θ<0.
We assume that SQC
A1A2A3(ε(ρ3)) is monotonically decreasing with respect to p, i.e.,
∂SQC
A1A2A3(ε(ρ3))
∂p <0.
Denote
g(p) =f(βε
3)−f(θ). (83)
We have
∂g(p)
∂p =−βε
3
2log2
1+βε
3
1−βε
3+θ
2log2
1+θ
1−θ. (84)
where βε
3=∂βε
3
∂p =−3α3(1−p)2.
In order to show ∂g(p)
∂p <0, the following inequality must hold
θα3·log2
1+βε
3
1−βε
3
<(c
2
2+c2
3)(1−p)3log2
1+θ
1−θ. (85)
The c1,c2,c3and psatisfying (85) follows that g(p) is monotonically decreasing with
respect to p, i.e., SQC
A1A2A3(ε(ρ3)) is monotonically decreasing with respect to p.
23 Page 16 of 23 Int J Theor Phys (2023) 62:23
For 4-qubit state, the state ρ4associated with subsystems A1,A
2,A
3,A
4is given by
ρ4=1
16 ˆ
I+c1σ⊗4
1+c2σ⊗4
2+c3σ⊗4
3. (86)
Under the operation of bit-flip channel, the output state ε(ρ4)is given by
ε(ρ4)=1
16 ˆ
I+c1σ⊗4
1+(1−p)4c2σ⊗4
2+(1−p)4c3σ⊗4
3. (87)
The nonzero and positive eigenvalues of ε(ρ) are written as λ1=1+c1+(1−p)4c2+
(1−p)4c3,λ2=1+c1−(1−p)4c2−(1−p)4c3,λ3=1−c1−(1−p)4c2+(1−p)4c3,
λ4=1−c1+(1−p)4c2−(1−p)4c3. The derivative of λiwith respect to pcan be cast as
λ
1=∂λ1
∂p =−4(c2+c3)(1−p)3,λ
2=∂λ2
∂p =4(c2+c3)(1−p)3,(88)
λ
3=∂λ3
∂p =−4(c2−c3)(1−p)3,λ
4=∂λ4
∂p =4(c2−c3)(1−p)3. (89)
From (64), quantum correlation entropy of ε(ρ4)is given by
SQC
A1A2A3A4(ε(ρ4)) =inf
βε
4
f(βε
4)+
4
i=1
λi
4log2λi,(90)
where βε
4=c1mA11mA21mA31mA41+α4(1−p)4,|βε
4|≤1andα4=
3
j=2
cjmA1jmA2j
mA3jmA4j.
Denote h(p) =f(βε
4)+4
i=1
λi
4log2λi. The derivative of h(p) is given by
∂h(p)
∂p =−βε
4
2log2
1+βε
4
1−βε
4+λ
1
4log2
λ1
λ2+λ
3
4log2
λ3
λ4
,(91)
where βε
4=∂βε
4
∂p =−4(1−p)3·α4.
In order to show ∂h(p)
∂p <0, the following inequality must hold
2α4·log2
1+βε
4
1−βε
4
<(c
2+c3)log2
λ1
λ2+(c2−c3)log2
λ3
λ4
. (92)
Therefore, (92) implies that SQC
A1A2A3A4(ε(ρ4)) is monotonically decreasing with respect
to p.
A.3: Quantum Correlation Entropy Under Phase Flip Channel
The phase fl ip channel acting on a single qubit can be described by the following Kraus
operators
ΓAj
0=1−pj
2ˆ
I, ΓAj
1=pj
2σ3,(93)
where Ajlabels the subsystems and p∈[0,1]. Here, we consider the symmetric situation
in which the decoherence rate is equal, so p1=p2=···=p.
We can verify that
l,...,n
ΓA1
l···ΓAN
nσ⊗N
1ΓAN†
n···ΓA1†
l=(1−p)Nσ⊗N
1,(94)
Int J Theor Phys (2023) 62:23 Page 17 of 23 23
l,...,n
ΓA1
l···ΓAN
nσ⊗N
2ΓAN†
n···ΓA1†
l=(1−p)Nσ⊗N
2,(95)
l,...,n
ΓA1
l···ΓAN
nσ⊗N
3ΓAN†
n···ΓA1†
l=σ⊗N
3,(96)
where l=···=n=0,1.
Consider the following 2-qubit state with subsystems A1,A
2
ρ2=1
22ˆ
I+c1σ⊗2
1+c2σ⊗2
2+c3σ⊗2
3. (97)
Under the operation of phase-flip channel, the output state ε(ρ2)is given by
ε(ρ2)=1
4ˆ
I+(1−p)2c1σ⊗2
1+(1−p)2c2σ⊗2
2+c3σ⊗2
3. (98)
The nonzero and positive eigenvalues of ε(ρ2)are written as λ1=1−(1−p)2c1−(1−
p)2c2−c3,λ2=1−(1−p)2c1+(1−p)2c2+c3,λ3=1+(1−p)2c1−(1−p)2c2+c3,
λ4=1+(1−p)2c1+(1−p)2c2−c3. The derivative of λiwith respect to pcan be cast as
λ
1=∂λ1
∂p =2(c1+c2)(1−p), λ
2=∂λ2
∂p =2(c1−c2)(1−p), (99)
λ
3=∂λ3
∂p =−2(c1−c2)(1−p), λ
4=∂λ4
∂p =−2(c1+c2)(1−p). (100)
From (63), the local observational entropy of ε(ρ2)is given by
SO(C)(ρ2)=f(βε
2)+2. (101)
where βε
2=α2(1−p)2+c3mA13mA23,|βε
2|≤1andα2=
2
j=1
cjmA1jmA2j.
Quantum correlation entropy of ε(ρ2)is given by
SQC
A1A2(ε(ρ2)) =inf
βε
2
f(βε
2)+
4
i=1
λi
4log2λi,(102)
Denote F(p) =f(βε
2)+4
i=1
λi
4log2λi. The derivative of F(p)is given by
∂F(p)
∂p =−βε
2
2log2
1+βε
2
1−βε
2+λ
1
4log2
λ1
λ4+λ
2
4log2
λ2
λ3
,(103)
where βε
2=∂βε
2
∂p =−2(1−p) ·α2.
In order to show ∂F(p)
∂p <0, the following inequality must hold
2α2·log2
1+βε
2
1−βε
2
<(c
1+c2)log2
λ4
λ1+(c1−c2)log2
λ3
λ2
. (104)
Therefore, (104) implies that SQC
A1A2(ε(ρ)) is monotonically decreasing with respect to p.
Consider the following 3-qubit states associated with subsystems A1,A2and A3,
ρ3=1
23⎛
⎝ˆ
I+
3
j=1
cjσj⊗σj⊗σj⎞
⎠,(105)
Under the action of phase-flip channel, the output state of ρ3is given by
ε(ρ3)=1
8ˆ
I+(1−p)3c1σ⊗3
1+(1−p)3c2σ⊗3
2+c3σ⊗3
3. (106)
23 Page 18 of 23 Int J Theor Phys (2023) 62:23
From (64), quantum correlation entropy of ε(ρ3)is given by
SQC
A1A2A3(ε(ρ3)) =inf
βε
3[f(βε
3)]+3−S(ε(ρ3)). (107)
where βε
3=α3(1−p)3+c3mA13mA23mA33,|βε
3|≤1andα3=
2
j=1
cjmA1jmA2jmA3j.
Set θ=(1−p)6c2
1+(1−p)6c2
2+c2
3,wehave
S(ε(ρ3)) =f(θ)+3. (108)
Hence, we have
SQC
A1A2A3ε(ρ3)=inf
βε
3[f(βε
3)]−f(θ) (109)
Since θ>0, this implies that log21+θ
1−θ>0.
We have
θ=∂θ
∂p =−3(c2
1+c2
2)(1−p)5
θ,(110)
For 0 <(1−p)5<1, then θ<0.
We assume that SQC
A1A2A3(ε(ρ3)) is monotonically decreasing with respect to p, i.e.,
∂SQC
A1A2A3(ε(ρ))
∂p <0.
Denote
g(p) =f(βε
3)−f(θ). (111)
We have
∂g(p)
∂p =−βε
3
2log2
1+βε
3
1−βε
3+θ
2log2
1+θ
1−θ. (112)
In order to show ∂g(p)
∂p <0, the following inequality must hold
θα3·log2
1+βε
3
1−βε
3
<(c
2
1+c2
2)(1−p)3log2
1+θ
1−θ. (113)
The c1,c2,c3and psatisfying (113) follows that g(p) is monotonically decreasing with
respect to p, i.e., SQC
A1A2A3(ε(ρ3)) is monotonically decreasing.
For 4-qubit state, the state ρassociated with subsystems A1,A
2,A
3,A
4is given by
ρ4=1
16 ˆ
I+c1σ⊗4
1+c2σ⊗4
2+c3σ⊗4
3. (114)
Under the operation of phase-flip channel, the output state ε(ρ4)is given by
ε(ρ4)=1
16 ˆ
I+(1−p)4c1σ⊗4
1+(1−p)4c2σ⊗4
2+c3σ⊗4
3. (115)
The nonzero and positive eigenvalues of ε(ρ4)are written as λ1=1+(1−p)4c1+(1−
p)4c2+c3,λ2=1+(1−p)4c1−(1−p)4c2−c3,λ3=1−(1−p)4c1+(1−p)4c2−c3,
λ4=1−(1−p)4c1−(1−p)4c2+c3. The derivative of λiwith respect to pcan be cast as
λ
1=∂λ1
∂p =−4(c1+c2)(1−p)3,λ
2=∂λ2
∂p =−4(c1−c2)(1−p)3,(116)
λ
3=∂λ3
∂p =4(c1−c2)(1−p)3,λ
4=∂λ4
∂p =4(c1+c2)(1−p)3. (117)
Int J Theor Phys (2023) 62:23 Page 19 of 23 23
From (64), quantum correlation entropy of ε(ρ4)is given by
SQC
A1A2A3A4(ε(ρ4)) =inf
βε
4
f(βε
4)+
4
i=1
λi
4log2λi,(118)
where βε
4=α4(1−p)4+c3mA13mA23mA33mA43,|βε
4|≤1andα4=
2
j=1
cjmA1jmA2j
mA3jmA4j.
Denote