# Quantum Correlation Dynamics of Three Qubits in Non-Markovian Environments

M. A. Shukri1,111E-mail<EMAIL_ADDRESS>Fares S. S. Alzahri1222E-mail:
<EMAIL_ADDRESS>and Ali Saif M. Hassan1,333E-mail:
<EMAIL_ADDRESS>
1 Department of Physics, University of Sana’a, Sana’a, Yemen.
2 Department of Physics, University of Amran, Amran, Yemen

###### Abstract

We investigate the quantum correlation dynamics of three independent qubits
each locally interacting with a zero temperature non-Markovian reservoir by
using the Geometric measure of quantum discord (GQD). The dependence of
quantum correlation dynamics on amount of non-Markovian, the degree of initial
quantum correlation and purity of the initial states are studied in detail. It
is found that the quantum correlation of such three qubits system revives
after instantaneous disappearance period when a proper amount of non-Markovian
is present. A comparison to the pairwise quantum discord and entanglement
dynamics in three qubits system is also made.

Keywords: Geometric measure, Quantum discord, non-Markovian, Quantum
Correlation Dynamics

##  1 Introduction

Understanding quantum correlations in a multipartite quantum state is a
fundamental open problem. In quantum information theory, the problem of
characterization of correlations present in a quantum state has been a
fundamental problem generating intense research effort in the last two decades
[1, 2]. Correlations in quantum states, with far-reaching implications for
quantum information processing, are usually studied in the entanglement-
versus-separability scenario [1] leading to important insights in quantum
computing [3], quantum communication protocols like teleportation [4, 5],
superdense coding [6], cryptography [7], and so on. However, some results
showed that quantum correlations cannot only be limited to entanglement,
because separable quantum states can also have correlations which are
responsible for the improvement of some quantum tasks that cannot be achieved
by classical means [8, 9]. An alternative classification for correlations
based on quantum measurements has arisen in recent years and also plays an
important role in quantum information theory [10, 11]. This is the quantum-
versus-classical paradigm for correlations. The first attempts in this
direction were made by Ollivier and Zurek [12] and by Henderson and Vedral
[13], who studied quantum correlations from a measurement perspective and
introduced quantum discord as a measure of quantum correlations which has
generated increasing interest [14, 15]. Luo and Fu have suggested that the
quantum discord $D(\rho)$ can be expressed alternatively as the minimal loss
of correlations caused by the non-selective von Neumann projective measurement
given by the set of orthogonal $1D$ projectors $\\{\Pi_{i}^{a}\\}$ acting on
one part of the system [16],

$D(\rho)=\min_{\Pi^{a}}[I(\rho)-I(\Pi^{a}((\rho))],$ (1)

where

$\Pi^{a}(\rho)=\sum_{i}\left(\Pi_{i}^{a}\otimes
I^{b}\right)\rho\left(\Pi_{i}^{a}\otimes I^{b}\right)$

Here the minimum is over the von Neumann measurements
$\Pi^{a}=\left\\{\Pi_{i}^{a}\right\\}$ on a part say $a$ of a bipartite system
$ab$ in a state $\rho$ with reduced density operators $\rho^{a}$ and
$\rho^{b}$ and $\Pi^{a}(\rho)$ is the resulting state after the measurement.
$I(\rho)=S\left(\rho^{a}\right)+S\left(\rho^{b}\right)-S(\rho)$ is the quantum
mutual information, $S(\rho)=-Tr(\rho\ln\rho)$ is the von Neumann entropy and
$I^{b}$ is the identity operator on part $b$.
For tripartite and larger systems, several generalizations of discord have
been proposed. In Ref. [17] a symmetric multipartite discord was deﬁned based
on relative entropy and local measurements. Another deﬁnition of multipartite
discord was provided in Ref. [18], as the sum of bipartite discords after
making successive measurements. An approach using relative entropy was deﬁned
in Ref. [19] to deﬁne genuine quantum and classical correlations in
multipartite systems. Ref. [20] introduced the notion of quantum dissension
deﬁned as the difference between tripartite mutual information after a single
measurement.
It is difficult to generalize the quantum discord in terms of quantum mutual
information for multipartite cases [20, 21]. To overcome this hurdle, Dakic
_et al_ [22], have provided a geometric measure of quantum discord (GQD) as
the distance for a given state to the closest classical quantum state

$D(\rho)=\min_{\chi\in\Omega_{0}}||\rho-\chi||^{2},$ (2)

with the minimum taken over the set $\Omega_{0}$ of zero discord states and
where $\Omega_{0}$ is the set of classical-quantum states (set of zero-discord
states $D(\chi)={0}$) and $\|\rho-\chi\|^{2}:=Tr(\rho-\chi)^{2}$ is the square
Hilbert-Schmidt norm. Dakic _et al_ [22] also obtained an explicit formula for
GQD for a two qubit system based on the Hilbert-Schmidt distance. Consider a
two-qubit state $\rho$ expressed in its Bloch representation as

$\rho=\frac{1}{4}\left(I^{a}\otimes
I^{b}+\sum_{\alpha=1}^{3}\left(x_{\alpha}\sigma_{\alpha}\otimes
I^{b}+I^{a}\otimes
y_{\alpha}\sigma_{\alpha}\right)+\sum_{\alpha,\beta=1}^{3}t_{\alpha\beta}\sigma_{\alpha}\otimes\sigma_{\beta}\right)$

with $\\{\sigma_{\alpha}\\}$ being the Pauli operators. Then its geometric
measure of quantum discord is given by [22]

$D(\rho)=\frac{1}{4}\left(\|\mathrm{x}\|^{2}+\|T\|^{2}-\lambda_{\max}\right)$
(3)

Here $\vec{x}:=\left(x_{1},x_{2},x_{3}\right)^{t}$ and
$\vec{y}:=\left(y_{1},y_{2},y_{3}\right)^{t}$ are coherent (column) vectors
for single qubit reduced density operators, $T=\left(t_{\alpha\beta}\right)$
is the correlation matrix, and $\lambda_{\max}$ is the largest eigenvalue of
the matrix $\vec{x}\vec{x}^{t}+TT^{t}.$ The norms of vectors and matrices are
the Euclidean norms, for example, $\|x\|^{2}:=\sum_{\alpha}x_{\alpha}^{2}.$
Here and throughout this paper, the superscript $t$ denotes the transpose of
vectors and matrices and by the norm of any tensor, we mean its Euclidean
norm, that is, the square of the norm of a tensor is the sum of squares of its
elements.
Hassan and Joag [28] provided an analytical formula for calculation of
geometric measure of quantum discord for a $N$-qubit state corresponding to
the von Neumann measurement on the $k$th part.

Recently, a natural generalization of quantum discord as originally defined in
Ref. [12, 13] is made for multipartite systems using the concept of
conditional measurements, which satisﬁes all of the postulates of a
multipartite discord [4], but there is no analytical formula which can be
used.

Generally, an actual quantum system is not closed and therefore will
unavoidably be affected by surrounding environments [24]. Due to the
interaction with its environment, the quantum system is very fragile and easy
to lose its quantum correlation, which is the main problem for the
implementation of quantum information processing. In open composite quantum
systems, the dynamic behavior of the correlations strongly depends on the
noise generated by the surrounding environment. For a given quantitative
system, the characterization of the environment as Markovian (no memory) or
non-Markovian (with memory) is determined by the ratio between its typical
correlation time and the system relaxation time. It is believed that the
quantumness captured by discord is different from entanglement [12, 14]. It
was also investigated under the Markovian environment in [25] for dissipative
dynamics that the discord with an asymptotical decrease is more robust than
the entanglement with sudden death under the same conditions. Wang at el [26]
studied the dynamics of the quantum discord by exactly solving a model
consisting of two independent qubits subject to two zero-temperature non-
Markovian reservoirs, respectively. This implies that the quantum discord is
more useful than the entanglement to describe the quantum correlation involved
in quantum systems. It was also investigated the dynamics of pairwise quantum
discord (QD) for a mixed three-qubit $W$-type state in three independent non-
Markovian reservoirs at zero temperature [27].

The paper is organized as follows. In sec. 2, we present the geometric measure
of quantum correlation for multipartite systems. In sec. 3, we review the
dynamics of a single qubit in non-Markovian environment, the procedure to
solve the dynamics of three independent qubits is given and get the evolution
of three-qubit density matrix. In sec. 4, we study the three-qubit quantum
correlation dynamics in non-Markovian environment at zero temperature. A
conclusion is given in sec. 5.

##  2 Quantum Correlation

Geometric measure has been used to measure the multipartite discord of quantum
state with high dimensions. Hassan and Joag [28] gave an exact computable
formula for calculation of geometric measure of quantum discord for a
$N$-qubit states. Following the same notation in Ref. [28],

$\begin{array}[]{c}\sigma_{\alpha_{1}}^{(1)}=\left(\sigma_{\alpha_{1}}\otimes
I_{2}\otimes\otimes I_{2}\right)\\\
\sigma_{\alpha_{2}}^{(2)}=\left(I_{2}\otimes\sigma_{\alpha_{2}}\otimes
I_{2}\right)\\\
\sigma_{\alpha_{1}}^{(1)}\sigma_{\alpha_{2}}^{(2)}=\left(\sigma_{\alpha_{1}}\otimes\sigma_{\alpha_{2}}\otimes
I_{2}\right)\end{array}$ (4)

We can write a three qubits state $\rho_{123}$ in the Bloch representation as

$\rho_{123}=\frac{1}{8}\\{I_{8}+\sum_{k=1}^{3}\sum_{\alpha_{k}}s_{\alpha_{k}}\sigma_{\alpha_{k}}^{(k)}+\sum_{k\neq
k^{\prime}=1}^{3}\sum_{\alpha_{k},\alpha_{k^{\prime}}}t_{\alpha_{k}\alpha_{k^{\prime}}}\sigma_{\alpha_{k}}^{(k)}\sigma_{\alpha_{k^{\prime}}}^{(k^{\prime})}+\sum_{\alpha_{1}\alpha_{2}\alpha_{3}}t_{\alpha_{1}\alpha_{2}\alpha_{3}}\sigma_{\alpha_{1}}^{(1)}\sigma_{\alpha_{2}}^{\langle
2)}\sigma_{\alpha_{3}}^{(3)}\\}$ (5)

where $\mathbf{s}^{(k)}(k=1,2,3)$ is a Bloch (coherent) vector corresponding
to the $k$th qubit,
$\mathbf{s}^{(k)}=\left[s_{\alpha_{k}}\right]_{\alpha_{k}=1}^{3}$, which is a
tensor of order $1$ defined by

$s_{\alpha_{k}}=Tr\left[\rho\sigma_{\alpha_{k}}^{(k)}\right]=Tr\left[\rho_{k}\sigma_{\alpha_{k}}\right],$
(6)

We denote the tensors of order $2$ by
$T^{\\{k,k^{\prime}\\}}=\left[t_{\alpha_{k}\alpha_{k^{\prime}}}\right]$ which
are defined by

$t_{\alpha_{k}\alpha_{k^{\prime}}}=Tr\left[\rho_{123}\sigma_{\alpha_{k}}^{(k)}\sigma_{\alpha_{k^{\prime}}}^{\left(k^{\prime}\right)}\right]$
(7)

and the tensor of order $3$ by
$\mathcal{T}=\left[t_{\alpha_{1}\alpha_{2}\alpha_{3}}\right],$ which are
defined by

$t_{\alpha_{1}\alpha_{2}\alpha_{3}}=Tr\left[\rho_{123}\sigma_{\alpha_{1}}^{(1)}\sigma_{\alpha_{2}}^{(2)}\sigma_{\alpha_{3}}^{(3)}\right]$
(8)

Then the geometric measure of quantum discord for a three qubits state
corresponding to the von Neumann measurements $\\{\widetilde{\Pi}^{(k)}\\}$ on
the $k$th qubit is given by,

$D_{k}\left(\rho_{123}\right)=\frac{1}{8}\left\\{\left\|\textbf{s}^{(k)}\right\|^{2}+\sum_{k^{\prime}\neq
k=1}^{3}\left\|T^{\left\\{k,k^{\prime}\right\\}}\right\|^{2}+\|\mathcal{T}\|^{2}-\eta_{\max}^{(k)}\right\\};k=1,2,3.$
(9)

Here $\eta_{\max}^{(k)}$ is the largest eigenvalue of the matrix $G^{(k)}$
which is a $3\times 3$ real symmetric matrix, defined as

$G^{(k)}=\textbf{s}^{(k)}\left(\textbf{s}^{(k)}\right)^{t}+\sum_{k^{\prime}\neq
k=1}^{3}\left(T^{\left\\{k,k^{\prime}\right\\}}\right)^{t}T^{\left\\{k,k^{\prime}\right\\}}+\mathbf{T}^{(k)}$
(10)

where $\mathbf{T}^{(k)}=\left[\tau_{\alpha_{k}\beta_{k}}\right]$ is a $3\times
3$ real matrix defined element wise as

$\tau_{\alpha_{k}\beta_{k}}=\sum_{\alpha_{k_{1}}\alpha_{k_{2}}}t_{\alpha_{k_{1}}\alpha_{k_{2}}\alpha_{k}}t_{\alpha_{k_{1}}\alpha_{k_{2}}\beta_{k}}$
$\alpha_{k_{1}},\alpha_{k_{1}},\alpha_{k},\beta_{k}=1,2,3;\;k_{1},k_{2}=1,2,3\neq
k.$

Therefore, the geometric measure of quantum discord GQD (as represented bellow
by $D_{1},\;D_{2},\;D_{3}$) of the successive measurement states and total
quantum correlations TQC (as represented by $Q$) present in a three qubits
state $\rho_{123}$ is given by [28],

$D_{1}(\rho_{123})$

$D_{2}(\widetilde{\Pi}^{(1)}(\rho_{123}))$

$D_{3}(\widetilde{\Pi}^{(2)}(\widetilde{\Pi}^{(1)}(\rho_{123})))$
$Q(\rho_{123})=D_{1}(\rho_{123})+D_{2}({\widetilde{\Pi}}^{(1)}(\rho_{123}))+D_{3}({\widetilde{\Pi}}^{(2)}({\widetilde{\Pi}}^{(1)}(\rho_{123}))).$
(11)

The calculation of each term is shown in the appendix.

##  3 MODEL

We consider a system of three non-interacting qubits, that is, $A,\;B,$ and
$C$, locally interacting with the reservoirs $R_{A},\;R_{B},$ and $R_{C}$,
respectively, at zero-temperature. The single qubit-reservoir pair is
described by the following Hamiltonian

$\hat{H}=\omega_{0}\sigma_{+}\sigma_{-}+\sum_{k}\left(\omega_{k}b_{k}^{{\dagger}}b_{k}+g_{k}b_{k}\sigma_{+}+g_{k}^{*}b_{k}^{{\dagger}}\sigma_{-}\right)$
(12)

with $\omega_{0}$ being the transition frequency of the qubit and
$\sigma_{\pm}$ are the corresponding qubit’s raining and lowering operators.
The index $k$ labels different field modes of reservoir with frequencies
$w_{k},$ $b_{k}^{{\dagger}}$ and $b_{k}$ are the creation and annihilation
operators with $g_{k}$ being the coupling constant to the qubit [24, 29, 30].
This Hamiltonian represents one among the few open quantum systems amenable
for an explicit solution [31]. The dynamics of single qubit $s$ can be
described by the reduced density matrix $\rho^{s}(t)$ which can be written, in
the basis $\\{|0\rangle,|1\rangle\\}$ as [24, 29, 32]

$\hat{\rho}^{s}(t)=\left(\begin{array}[]{cc}P_{\mathrm{t}}\rho_{11}^{s}(0)&\sqrt{P_{\mathrm{t}}}\rho_{\mathrm{10}}^{s}(0)\\\
\sqrt{P_{\mathrm{t}}}\rho_{01}^{s}(0)&\rho_{00}^{s}(0)+\left(1-P_{\mathrm{t}}\right)\rho_{11}^{s}(0)\end{array}\right)$
(13)

where the function $P_{t}$ obeys the differential equation

$\dot{P}_{t}=-\int_{0}^{t}dt_{1}f\left(t-t_{1}\right)P_{t_{1}},$ (14)

and the correlation function is related to the spectral density $J(\omega)$ of
the reservoir by $f\left(t-t_{1}\right)=\int d\omega
J(\omega)e^{i(\omega_{0}-\omega)(t-t_{1})}$. To get the exact form of $P_{t}$
thus depends on the choice of spectral density of the reservoir [24]. In our
model, we use the Lorentzian spectral distribution

$J(\omega)=\frac{1}{2\pi}\frac{\gamma_{0}\lambda^{2}}{\left(\omega_{0}-\omega\right)^{2}+\lambda^{2}}$
(15)

where $\lambda$ denotes spectral width of the coupling, it is related to the
reservoir correlation time $\tau_{B}$ via $\tau_{B}\approx\lambda^{-1}$, while
$\gamma_{0}$ is related to the decay of the atomic excited state in the
Markovian limit of spectrum and is related to the qubit relaxation time by
$\tau_{R}\approx\gamma_{0}^{-1}$. The relation between parameters $\gamma_{0}$
and $\lambda$ distinguishes between Markovian and non-Markovian regimes. In
the Markovian regime there is $\gamma_{0}<\lambda/2$ or $\tau_{R}>2\tau_{B}$,
and the non-Markovian regime corresponds to $\gamma_{0}>\lambda/2$ or
$\tau_{R}<2\tau_{B},$ and the previously lost quantum information may be
feedback into the system again. According to spectral density function
$J(\omega)$ given in Eq.(15), the solution of Eq.(14) give

$P_{t}=e^{-\lambda
t}[\cos(\frac{dt}{2})+\frac{\lambda}{d}\sin(\frac{dt}{2})]^{2},$ (16)

where $d=\sqrt{2\gamma_{0}\lambda-\lambda^{2}}$ [24, 29], which is an
oscillating function that has discrete zeros $t=2[n\pi-\arctan(d/\lambda)]/d$
with $n$ being an arbitrary integer.

By using the evolution of the reduced density matrix elements for the single
qubit
$\rho_{i\acute{i}}^{s}(t)=\sum_{l\acute{l}}A_{i\acute{i}}^{l\acute{l}}(t)\rho_{l\acute{l}}^{s}(0),$
to construct the reduced density matrix $\rho(t)$ for the three-qubit system
as [29, 30],

$\rho_{ii^{\prime},jj^{\prime},kk^{\prime}}(t)=\sum_{ll^{\prime},mm^{\prime},nn^{\prime}}A_{ii^{\prime}}^{ll^{\prime}}(t)B_{jj^{\prime}}^{mm^{\prime}}(t)C_{kk^{\prime}}^{nn^{\prime}}(t)\rho_{ll^{\prime},mm^{\prime},nn^{\prime}}(0).$
(17)

By means of Eqs. (13), (16) and (17) and under the standard product basis
$|1\rangle$=$|111\rangle$, $|2\rangle$=$|110\rangle$,
$|3\rangle$=$|101\rangle$, $|4\rangle$=$|100\rangle$,
$|5\rangle$=$|011\rangle$, $|6\rangle$=$|010\rangle$,
$|7\rangle$=$|001\rangle$, $|8\rangle$=$|000\rangle\\}$, we obtain the
diagonal elements of the reduced density matrix for three-qubit system as

$\displaystyle\rho_{11}(t)$ $\displaystyle=$ $\displaystyle
P_{t}^{3}\rho_{11}(0),$ $\displaystyle\rho_{22}(t)$ $\displaystyle=$
$\displaystyle
P_{t}^{2}\rho_{22}(0)+P_{t}^{2}\left(1-P_{t}\right)\rho_{11}(0),$
$\displaystyle\rho_{33}(t)$ $\displaystyle=$ $\displaystyle
P_{t}^{2}\rho_{33}(0)+P_{t}^{2}\left(1-P_{t}\right)\rho_{11}(0),$
$\displaystyle\rho_{44}(t)$ $\displaystyle=$ $\displaystyle
P_{t}\rho_{44}(0)+P_{t}\left(1-P_{t}\right)\rho_{33}(0)+P_{t}\left(1-P_{t}\right)\rho_{22}(0)+P_{t}\left(1-P_{t}\right)^{2}\rho_{11}(0),$
$\displaystyle\rho_{55}(t)$ $\displaystyle=$ $\displaystyle
P_{t}^{2}\rho_{55}(0)+P_{t}^{2}\left(1-P_{t}\right)\rho_{11}(0),$
$\displaystyle\rho_{66}(t)$ $\displaystyle=$ $\displaystyle
P_{t}\rho_{66}(0)+P_{t}\left(1-P_{t}\right)\rho_{55}(0)+P_{t}\left(1-P_{t}\right)\rho_{22}(0)+P_{t}\left(1-P_{t}\right)^{2}\rho_{11}(0),$
$\displaystyle\rho_{77}(t)$ $\displaystyle=$ $\displaystyle
P_{t}\rho_{77}(0)+P_{t}\left(1-P_{t}\right)\rho_{55}(0)+P_{t}\left(1-P_{t}\right)\rho_{33}(0)+P_{t}\left(1-P_{t}\right)^{2}\rho_{11}(0),$
$\displaystyle\rho_{88}(t)$ $\displaystyle=$
$\displaystyle\rho_{88}(0)+\left(1-P_{t}\right)\left[\rho_{77}(0)+\rho_{66}(0)+\rho_{44}(0)\right]+\left(1-P_{t}\right)^{2}\left[\rho_{55}(0)+\rho_{33}(0)+\rho_{22}(0)\right]$
(18) $\displaystyle+\left(1-P_{t}\right)^{3}\rho_{11}(0)$

and the off-diagonal elements as:

$\displaystyle\rho_{12}(t)$ $\displaystyle=$ $\displaystyle
P_{t}^{2}\sqrt{P_{t}}\rho_{12}(0)$ $\displaystyle\rho_{13}(t)$
$\displaystyle=$ $\displaystyle P_{t}^{2}\sqrt{P_{t}}\rho_{13}(0)$
$\displaystyle\rho_{14}(t)$ $\displaystyle=$ $\displaystyle
P_{t}^{2}\rho_{14}(0)$ $\displaystyle\rho_{15}(t)$ $\displaystyle=$
$\displaystyle P_{t}^{2}\sqrt{P_{t}}\rho_{15}(0)$ $\displaystyle\rho_{16}(t)$
$\displaystyle=$ $\displaystyle P_{t}^{2}\rho_{16}(0)$
$\displaystyle\rho_{17}(t)$ $\displaystyle=$ $\displaystyle
P_{t}^{2}\rho_{17}(0)$ $\displaystyle\rho_{18}(t)$ $\displaystyle=$
$\displaystyle P_{t}\sqrt{P_{t}}\rho_{18}(0)$ $\displaystyle\rho_{23}(t)$
$\displaystyle=$ $\displaystyle P_{t}^{2}\rho_{23}(0)$
$\displaystyle\rho_{24}(t)$ $\displaystyle=$ $\displaystyle
P_{t}\sqrt{P_{t}}\rho_{24}(0)+P_{t}\sqrt{P_{t}}\left(1-P_{t}\right)\rho_{13}(0)$
$\displaystyle\rho_{25}(t)$ $\displaystyle=$ $\displaystyle
P_{t}^{2}\rho_{25}(0)$ $\displaystyle\rho_{26}(t)$ $\displaystyle=$
$\displaystyle
P_{t}\sqrt{P_{t}}\rho_{26}(0)+P_{t}\sqrt{P_{t}}\left(1-P_{t}\right)\rho_{15}(0)$
$\displaystyle\rho_{27}(t)$ $\displaystyle=$ $\displaystyle
P_{t}\sqrt{P_{t}}\rho_{27}(0)$ $\displaystyle\rho_{28}(t)$ $\displaystyle=$
$\displaystyle P_{t}\rho_{28}(0)+P_{t}(1-P_{t})\rho_{17}(0)$

$\displaystyle\rho_{34}(t)$ $\displaystyle=$ $\displaystyle
P_{t}\sqrt{P_{t}}\rho_{34}(0)+P_{t}\sqrt{P_{t}}\left(1-P_{t}\right)\rho_{12}(0)$
$\displaystyle\rho_{35}(t)$ $\displaystyle=$ $\displaystyle
P_{t}^{2}\rho_{35}(0)$ $\displaystyle\rho_{36}(t)$ $\displaystyle=$
$\displaystyle P_{t}\sqrt{P_{t}}\rho_{36}(0)$ $\displaystyle\rho_{37}(t)$
$\displaystyle=$ $\displaystyle
P_{t}\sqrt{P_{t}}\rho_{37}(0)+P_{t}\sqrt{P_{t}}\left(1-P_{t}\right)\rho_{15}(0)$
$\displaystyle\rho_{38}(t)$ $\displaystyle=$ $\displaystyle
P_{t}\rho_{38}(0)+P_{t}\left(1-P_{t}\right)\rho_{16}(0)$
$\displaystyle\rho_{45}(t)$ $\displaystyle=$ $\displaystyle
P_{t}\sqrt{P_{t}}\rho_{45}(0)$ $\displaystyle\rho_{46}(t)$ $\displaystyle=$
$\displaystyle P_{t}\rho_{46}(0)+P_{t}\left(1-P_{t}\right)\rho_{35}(0)$
$\displaystyle\rho_{47}(t)$ $\displaystyle=$ $\displaystyle
P_{t}\rho_{47}(0)+P_{t}\left(1-P_{t}\right)\rho_{25}(0)$
$\displaystyle\rho_{48}(t)$ $\displaystyle=$
$\displaystyle\sqrt{P_{t}}\rho_{48}(0)+\sqrt{P}_{t}\left(1-P_{t}\right)\rho_{37}(0)+\sqrt{P}_{t}\left(1-P_{t}\right)\rho_{26}(0)+\sqrt{P_{t}}(1-P_{t})^{2}\rho_{15}(0)$
$\displaystyle\rho_{56}(t)$ $\displaystyle=$ $\displaystyle
P_{t}\sqrt{P_{t}}(1-P_{t})\rho_{12}(0)+P_{t}\sqrt{P_{t}}\rho_{56}(0)$
$\displaystyle\rho_{57}(t)$ $\displaystyle=$ $\displaystyle
P_{t}\sqrt{P_{t}}(1-P_{t})\rho_{13}(0)+P_{t}\sqrt{P_{t}}\rho_{57}(0)$
$\displaystyle\rho_{58}(t)$ $\displaystyle=$ $\displaystyle
P_{t}\rho_{58}(0)+P_{t}(1-P_{t})\rho_{14}(0)$ $\displaystyle\rho_{67}(t)$
$\displaystyle=$ $\displaystyle P_{t}\rho_{67}(0)+P_{t}(1-P_{t})\rho_{23}(0)$
$\displaystyle\rho_{68}(t)$ $\displaystyle=$
$\displaystyle\sqrt{P_{t}}(1-P_{t})^{2}\rho_{13}(0)+\sqrt{P_{t}}(1-P_{t})\rho_{24}(0)+\sqrt{P_{t}}(1-P_{t})\rho_{57}(0)+\sqrt{P_{t}}\rho_{68}(0)$
$\displaystyle\rho_{78}(t)$ $\displaystyle=$
$\displaystyle\sqrt{P_{t}}(1-P_{t})^{2}\rho_{12}(0)+\sqrt{P_{t}}(1-P_{t})\rho_{34}(0)$
(19)
$\displaystyle+\sqrt{P_{t}}(1-P_{t})\rho_{56}(0)+\sqrt{P_{t}}\rho_{78}(0).$

##  4 Non-Markovian Quantum Correlation Dynamics

For three-qubit states, there are two type of nonequivalent entangled states,
which called $|GHZ\rangle$-state and $|W\rangle$-state [1], and any fully
entangled three qubits state is stochastic local operations and classical
communication $SLOCC$-equivalent to either $|GHZ\rangle$ and $|W\rangle$ [33].
So, this states are important in quantum information and computation
processing. We consider two initial states constructed with $|GHZ\rangle-$ and
$|W\rangle-$like states,

$\rho^{\psi}=r|\psi\rangle\langle\psi|+\frac{1-r}{8}I_{8},\;\rho^{\phi}=r|\phi\rangle\langle\phi|+\frac{1-r}{8}I_{8}.$
(20)

Here $r\;(0\leq r\leq 1)$ denotes the purity of the initial state and
$|\psi\rangle=\alpha_{\psi}|000\rangle+\beta_{\psi}|111\rangle,\;|\phi\rangle=\alpha_{\phi}|001\rangle+\beta_{\phi}|010\rangle+\eta_{\phi}|100\rangle.$
It is obvious that the states $|\psi\rangle,\;|\phi\rangle$ are the
$|GHZ\rangle$ and $|W\rangle$ respectively with $\alpha_{\psi(\phi)}$ real,
$\beta_{\psi(\phi)}=|\beta_{\psi(\phi)}|e^{i\delta}$ and
$\eta_{\phi}=|\eta|e^{i\epsilon}$ where
$\alpha_{\psi}^{2}+|\beta_{\psi}|^{2}=1,\;\alpha_{\phi}^{2}+|\beta_{\phi}|^{2}+|\eta_{\phi}|^{2}=1.$
As is known the $|GHZ\rangle$ and $|W\rangle$ states are different classes of
entanglement, we expect that the two initial states might have some difference
in quantum correlation dynamics measured by geometric measures. Since, the
exact expression of GQD and TQC are rather complicated and not very
enlightening, we study only numerically of the quantum discord and total
quantum correlation dynamics as follow.

_Case 1_ In this case we mainly study how the GQD and TQC dynamics is
influenced by the degree of non-Markovian. To this purpose, we plot, Figs. 1,
2, the GQD and TQC as functions of $\gamma_{0}t$ for four values of
$\lambda/\gamma_{0}$ ($\lambda/\gamma_{0}=2.5,0.1,0.05,0.01$), starting from
pure $\rho^{\psi},\;\rho^{\phi}$ states, that is, for $r=1$ and taking
$\alpha_{\psi}^{2}=\beta_{\psi}^{2}=1/2,\;\alpha_{\phi}^{2}=\beta_{\phi}^{2}=\eta_{\phi}^{2}=1/3.$
This choice of the parameters permits us to evidence quite different time
behaviors of GQD and TQC in terms of $\lambda/\gamma_{0}$ for the two initial
states $\rho^{\psi}(0)=|\psi\rangle\langle\psi|$ and
$\rho^{\phi}(0)=|\phi\rangle\langle\phi|.$ Geometric measure of quantum
discord and total quantum correlation (as stated in Eq. (11)) of state
$|\psi\rangle$, under this choice of parameters, as a function of
$\gamma_{0}t$ for the four values of $\lambda/\gamma_{0}$ is plotted in Figs.
1. The counter part of state $|\phi\rangle$ are correspondingly plotted in
Figs. 2. According to the conditions for Markovian and non-Markovian dynamics
regimes, Markovian dynamics take place when $\lambda/\gamma_{0}=2.5$, while
$\lambda/\gamma_{0}=0.1,\;0.05,\;0.01$ the non-Markovian dynamics is relevant.

Figures 1 and 2 show three similar characters. Firstly, the geometric measures
of quantum discord and total quantum correlation, measured by
$D_{1}(\rho_{123})$ and $Q(\rho_{123})$ respectively, evolution in non-
Markovian regime differs in essence from that in Markovian regime. Geometric
measures in Markovian regime vanishes only in an asympotical way without
revival. whereas, geometric measures in non-Markovian regime
($\lambda/\gamma_{0}=0.1,\;0.05,\;0.01$) decays gradually to zero but
reappears after a period of time with a damping amplitude, which is in good
agreement with the corresponding result described in Ref. [27] for three
qubits and in Ref. [26] for two qubits. In contrast to the Figs. 1, 2 in Ref.
[35] the entanglement in Markovian regime decays exponentially to zero and
finally vanishes. However, entanglement in non-Markovian regime also decays to
zero but reappears after a period of time with a damping amplitude. As there
is no interaction between three qubits initially, this revival phenomenon is
due to single qubit non-Markovian dynamics resulting from the memory effect of
the environment. Secondly, the non-Markovian decay rate and revival amplitude
depend on the amount of non-Markovian $\lambda/\gamma_{0}$ with
$\lambda/\gamma_{0}$ decreasing, which means degree of non-Markovian
increasing, non-Markovian decay rate decreases and the revival amplitute
increases and the numbers of revival amplitude increases with degree of non-
Markovian increasing. Thirdly, there is no decay rate of $D_{3}$ of both Figs.
1c (for GHZ-state) and 2c (for W-state) and it has anomalous behavior
different of that for $D_{1}$ and $D_{2}$ evolution.

Figure 1: The geometric measures of QD
$D_{1}\;\textbf{(a)},\;D_{2}\;\textbf{(b)},\;D_{3}\;\textbf{(c)},$ and
$TQC\;\textbf{(d)}$ vs.
$\gamma_{0}t\;\textrm{for}\;\lambda/\gamma_{0}=2.5\;\textrm{(dash-dotted
line)},\;\lambda/\gamma_{0}=0.1\;\textrm{(solid
line)},\;\lambda/\gamma_{0}=0.05\;\textrm{(dashed
line)},\;\lambda/\gamma_{0}=0.01\;\textrm{(dotted
line)},\;\textrm{for}\;\rho^{\psi}\;\textrm{where}\;\alpha_{\psi}^{2}=\beta_{\psi}^{2}=1/2\;\textrm{and}\;r=1.$

Figure 2: The geometric measures of QD $D_{1}\;\textbf{(a)},$
$D_{2}\;\textbf{(b)},$ $D_{3}\;\textbf{(c)},$ and $TQC\;\textbf{(d)}$ vs,
$\gamma_{0}t$ for $\lambda/\gamma_{0}=2.5$ (dash-dotted line),
$\lambda/\gamma_{0}=0.1$ (solid line), $\lambda/\gamma_{0}=0.05$ (dashed
line), $\lambda/\gamma_{0}=0.01$ (dotted line), for $\rho^{\phi}$ where
$\alpha_{\phi}^{2}=\beta_{\phi}^{2}=\eta_{\phi}^{2}=1/3,\;r=1.$

Moreover, Figures 1 and 2 show three different characters. Firstly, the number
of revival amplitude of Fig. 2 (for W-state) is more than that of Fig. 1 (for
GHZ-state) in non-Markovian dynamics regime, but the maximum values of the
revival amplitudes of Fig. 2 (for W-state) is smaller than that of Fig. 1 (for
GHZ-state). Secondly, the decay rate of Fig. 1b, $D_{2}$, (for GHZ-state), is
faster than that of Fig. 2b (for W-state) and there is no revival amplitude of
Fig. 1b, $D_{2}$, in non-Markovian dynamics regime for GHZ-state, while there
is of Fig. 2b (for W-state). Thirdly, the anomalous behavior of Fig. 1c (for
GHZ-state) differ from that of Fig. 2c (for W-state), the revival amplitude
reappears once in Fig. 1c, while in Fig. 2c it reappears more than one and
differ in their form.

_Case 2_ Another aspect of interest is how the three qubits geometric measure
of TQC dynamics is affected by the degree of initial quantum correlation
represent by $\alpha_{\psi(\phi)},\;\beta_{\psi(\phi)}$ and $\eta_{\phi}$. We
assume the initial states with:
$r=1,\;\beta_{\psi}=\sqrt{1-\alpha_{\psi}^{2}},\;\beta_{\phi}=\eta_{\phi}=\sqrt{(1-\alpha_{\phi}^{2})/2}$
and plot TQC as a function of $\gamma_{0}t$ and $\alpha^{2}$ in Fig. 3. Figure
3 presents the case for the non-Markovian regime $(\lambda/\gamma_{0}=0.01).$
We plot in Fig. 3-left panel and Fig. 3-right panel for states $\rho^{\psi}$
(GHZ-state) and $\rho^{\phi}$ (W-state) respectively. It can be seen that The
TQC of both $\rho^{\psi}=|\psi\rangle\langle\psi|$ and
$\rho^{\phi}=|\phi\rangle\langle\phi|$ periodically vanishes in accordance
with the zero points of the function $P_{t}$ following the asymptotically
damping. We can also see that the revival amplitudes of TQC increases with
$\alpha^{2}$ increasing from $0$ to $1/2$, reaching its maximum value at
$\alpha^{2}=1/2$, and decreases with $\alpha^{2}$ increasing from $1/2$ to $1$
in Fig. 3-left panel, while Fig. 3-right panel show that $TQC$ revival
amplitude occurs in all the region of $\alpha^{2}$ with periodically amplitude
damping. Comparing influence of initial quantum correlation on TQC dynamics in
three qubits system to that in two qubits system [26], we find that the
effects are in general similar. Comparing entanglement dynamics in Figs. 3, 4
of Ref. [27] to Figs. 3 in our work, we find that entanglement, for GHZ-state
as measured by negativity, shows entanglement sudden death (ESD) occurs when
$0\leq\alpha^{2}\leq 1/2$ and the dark period decreases with $\alpha^{2}$
increasing, while in Fig. 3-left panel in our work the dark period does not
occur in the whole region of $\alpha^{2}$. Besides, Fig. 4 of Ref. [27] (for
W-state) ESD does not occur in the all region of $\alpha^{2}$, whereas Fig.
3-right panel of this paper show similar behavior but revival amplitude is
larger and zero period is smaller.

Figure 3: TQC, ($Q(\rho_{123})$) in terms of $\gamma_{0}t$ and
$\alpha_{\psi(\phi)}^{2}$ for the initial GHZ-state left panel and W-state
right panel, parameter $\lambda/\gamma_{0}=0.01,$ where
$\beta_{\psi}=\sqrt{1-\alpha_{\psi}^{2}},$
$\beta_{\phi}=\eta_{\phi}=\sqrt{\left(1-\alpha_{\phi}^{2}\right)/2}.$

_Case 3_ In this case we analyze how the three qubits TQC dynamics is affected
by the presence of mixedness in initial states regulated by the purity
parameter $r$. We set the parameters in two initial states as
$\alpha_{\psi}^{2}=\beta_{\psi}^{2}=1/2,\;\alpha_{\phi}^{2}=\beta_{\phi}^{2}=\eta_{\phi}^{2}=1/3$
and plot the geometric measure of TQC as a function of $\gamma_{0}t$ and $r$
in Fig. 4-left panel and Fig. 4-right panel for GHZ and W states respectively.

Figures 4-left panel, 4-right panel show that TQC increases with purity
parameter $r$ increasing at $t=0$ for both states. Also, it con be seen that
the value and period of revival amplitude increase with purity $r$ increasing
and when $r=1$ the revival amplitudes get the highest value with largest
period.

Contrary, the zero periods of TQC increase with purity $r$ decreasing and when
$r=0$ the zero periods are the largest. The difference between two figures is
the top of first revival amplitude in Fig.4-left panel is norm, while in
Fig.4-right panel is warped and the warped revival amplitude increase with
$\lambda/\gamma_{0}$ increasing. Compared with Fig. 3 of Ref. [27] of pairwise
quantum discord dynamics for W-state, Fig. 4-right panel in this paper show
similar behaviors. In contrast to Fig. 7 and Fig. 8 of Ref. [35] in non-
Markovian regime, the entanglement dynamics sudden death occurs in both states
in almost all the region of $r$, the TQC in Figs. 4-left panel and 4-right
panel of our paper show that no dark period in nearly all region of $r$ for
both same initial states. This phenomenon is further evidence that TQC is more
robust than the entanglement against decoherence.

Figure 4: TQC, $Q(\rho_{123})$ in terms of $\gamma_{0}t$ and $r$ for the
initial GHZ-state left panel and W-state right panel, parameter
$\lambda/\gamma_{0}=0.01.$

##  5 Conclusion

We have studied the dynamics of quantum correlation of three qubits using
exactly solvable model where each qubit independently and locally interacts
with zero-temperature reservoir. We have discussed the different effects from
the Markovian and non-Markovian reservoir. Specially, in the first case, we
have analyzed the effects of the amount of non-Markovian $\lambda/\gamma_{0}$
on the geometric measure of quantum discord in detail. We obtain that the two
types initial states have some similar and different characters. The non-
Markovian decay rate decreases and revival amplitude, in both its value and
numbers, increases with the amount of non-Markovian increasing. Moreover,
there is no decay rate of $D_{3}$ for both initial GHZ and W states and
$D_{3}$ has anomalous behavior different from that for $D_{1}$ and $D_{2}$
evolution. Also, this anomalous behavior in $D_{3}$ evolution differs for
initial GHZ and W states. In the second case, we analyzed the influence of the
initial quantum correlation on the geometric measure of quantum discord in
non-Markovian regime. We observe that the effects of initial quantum
correlation on TQC dynamics displays different behaviors for the two types of
three qubits quantum states as expected. In the third case, we have discussed
the effects of the purity $r$ on quantum correlation dynamics and compared
with that of entanglement dynamics of three qubits using same conditions. This
implies that quantum discord is more robust than entanglement against
decoherence. Moreover, geometric measure of quantum discord have similar
behavior to that for corresponding initial state in pairwise quantum discord
of three qubits and that in two-qubit system.

## Appendix

If we have the state of three-qubits $\rho$ expressed in its Bloch
representation as in Eq.(5).

  1. 1.

We express $\rho$ in the orthonormal bases
$\left\\{X_{i_{m}}^{(m)}\right\\},i_{m}=1,2,3,4$ as the generators of
$SU(2_{m});\quad m=1,2,3$ labeling for qubit. $\rho_{12\cdots
N}=\sum_{i_{1}i_{2}i_{3}}C_{i_{1}i_{2}i_{3}}X_{i_{1}}^{(1)}\otimes
X_{i_{2}}^{(2)}\otimes X_{i_{3}}^{(3)}$, where
$X_{1}^{(m)}=\frac{1}{\sqrt{2}}I_{2}$ and
$X_{i_{m}}^{(m)}=\frac{1}{\sqrt{2}}\sigma_{i_{m}-1};\quad
i_{m}=2,3,4,\;\sigma_{i}$ being the Pauli operators [28].

  2. 2.

We calculate the tensor $C$ of $\rho$ state as

$C_{i_{1}i_{2}i_{3}}=\\\ Tr\left(\rho X_{i_{1}}^{(1)}\otimes
X_{i_{2}}^{(2)}\otimes
X_{i_{3}}^{(3)}\right);i_{1},i_{2},i_{3}=1,2,3,4.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(A1)$

the three-way array (tensor of order 3) with size $4\times 4\times 4$. The
norm of tensor C is

$\|C\|^{2}=\sum_{i_{1}i_{2}i_{3}}^{4}C_{i_{1}i_{2}i_{3}}^{2}.$

  3. 3.

We calculate $\textbf{s}^{(m)}$ a Blech vector corresponding to $m=$th qubit
$m=1,2,3$,

$\displaystyle\textbf{s}_{i}^{(1)}$ $\displaystyle=$ $\displaystyle
Tr\left(\rho\sigma_{i}\otimes I\otimes I\right),$
$\displaystyle\textbf{s}_{j}^{(2)}$ $\displaystyle=$ $\displaystyle
Tr\left(\rho I\otimes\sigma_{j}\otimes I\right),$
$\displaystyle\textbf{s}_{k}^{(3)}$ $\displaystyle=$ $\displaystyle
Tr\left(\rho I\otimes
I\otimes\sigma_{k}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(A2)$

the correlation matrices for two-qubit as

$\displaystyle\mathrm{T}^{\\{1,2\\}}$ $\displaystyle=$
$\displaystyle[t_{ij}^{(12)}]=[Tr\left(\rho\sigma_{i}\otimes\sigma_{j}\otimes
I\right)]=C_{i+1,j+1,1},$ $\displaystyle\mathrm{T}^{\\{1,3\\}}$
$\displaystyle=$ $\displaystyle[t_{ik}^{(13)}]=[Tr\left(\rho\sigma_{i}\otimes
I\otimes\sigma_{k}\right)]=C_{i+1,1,k+1}$
$\displaystyle\mathrm{T}^{\\{2,3\\}}$ $\displaystyle=$
$\displaystyle[t_{jk}^{(23)}]=[Tr\left(\rho
I\otimes\sigma_{j}\otimes\sigma_{k}\right)]=C_{1,j+1,k+1},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(A3)$

the three-way correlation array for $\rho$ state as

$\mathcal{T}=[t_{ijk}]=[Tr\left(\rho\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k}\right)]=C_{i+1,j+1,k+1},\;i,j,k=1,2,3~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(A4)$

$\mathbf{T}^{(3)}$ is $3\times 3$ matrix, defined elementwise as

$\mathbf{T}^{(3)}=[\tau_{i\ell}^{(3)}]=[\sum_{jk}t_{ijk}t_{\ell
jk}].~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(A5)$

  4. 4.

We calculate $3\times 3$ real symmetric matrix $G^{(1)}$ using eqs. A2, A3 and
A5 as

$G^{(1)}=\textbf{s}^{(1)}(\textbf{s}^{(1)})^{t}+\
(\mathrm{T}^{\\{1,2\\}})^{t}\mathrm{T}^{\\{1,2\\}}+(\mathrm{T}^{\\{1,3\\}})^{t}\mathrm{T}^{\\{1,3\\}}+\mathbf{T}^{(3)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(A6)$

and we find the largest eigenvalue of $G^{(1)}$, $\eta_{max}^{(1)}$ and
corresponding eigenvector of it $V_{max}^{(1)}$.

  5. 5.

We calculate $D_{1}$, quantum discord (corresponding to the Von Neumann
measurement on the first qubit), for a three-qubits quantum state using eqs.
A2, A3 and A4 as

$D_{1}(\rho)=\frac{1}{2^{3}}\left[\|\textbf{s}^{(1)}\|^{2}+\|\mathrm{T}^{\\{1,2\\}}\|^{2}+\|\mathrm{T}^{\\{1,3\\}}\|^{2}+\|\mathcal{T}\|^{2}-\eta_{max}^{(1)}\right].$

  6. 6.

We put $\hat{e}=V_{max}^{(1)}$ to calculate
$\textbf{a}_{1}=\frac{1}{\sqrt{2}}(1,\hat{e})$ and
$\textbf{a}_{2}=\frac{1}{\sqrt{2}}(1,-\hat{e})$.

  7. 7.

We calculate the $b_{1}$ and $b_{2}$ matrices Eqs. (13,14) in Ref. [28] of the
state $\widetilde{\Pi}^{(1)}(\rho)$ using Eq. A1, $\textbf{a}_{1}$ and
$\textbf{a}_{2}$ as

$b_{1jk}=\sum_{i}c_{ijk}a_{ii}\quad,\quad b_{2jk}=\sum_{i}c_{ijk}a_{2i}$

  8. 8.

We calculate the tensor $C^{\prime}$ of
$\widetilde{\Pi}^{(1)}(\rho)=\widetilde{\rho}$ as

$C_{ijk}^{\prime}=a_{1i}b_{1jk}+a_{2i}b_{2jk}$

proof:

$\displaystyle C_{ijk}^{\prime}$ $\displaystyle=$ $\displaystyle
Tr\left(\widetilde{\Pi}^{(1)}(\rho)X_{i}\otimes X_{j}\otimes X_{k}\right)$
$\displaystyle=$ $\displaystyle
Tr\left(\sum_{\ell=1}^{2}p_{\ell}|\ell\rangle\langle\ell|\otimes\rho_{23/\ell}\left(X_{i}\otimes
X_{j}\otimes X_{k}\right)\right)$ $\displaystyle=$
$\displaystyle\sum_{\ell=1}^{2}Tr\left(|\ell\rangle\langle\ell|X_{i}\right)Tr\left(p_{\ell}\rho_{23/\ell}X_{j}\otimes
X_{k}\right)$ $\displaystyle=$
$\displaystyle\sum_{\ell=1}^{2}\left\langle\ell\left|X_{i}\right|\ell\right\rangle
Tr\left(p_{\ell}\rho_{23/\ell}X_{j}\otimes X_{k}\right)$ $\displaystyle
C_{ijk}^{\prime}$ $\displaystyle=$ $\displaystyle\sum_{\ell=1}^{2}a_{\ell
i}b_{\ell
jk}=a_{1i}b_{1jk}+a_{2i}b_{2jk}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(A7)$

the norm of $C_{ijk}^{\prime}$ is
$\|C_{ijk}^{\prime}\|^{2}=\sum_{ijk}C_{ijk}^{\prime 2}$.

  9. 9.

We repeat the step (iii) for the state
$\widetilde{\Pi}^{(1)}(\rho)=\widetilde{\rho}$, to calculate the coherent
vectors $\textbf{s}^{\prime^{(m)}}$, the correlation matrix for two-qubit,
three-way correlation array and $\mathbf{T}^{(3)}$ as

$s_{j}^{\prime(1)}=Tr\left(\widetilde{\Pi}^{(1)}(\rho)\sigma_{j}\otimes
I\otimes I\right)=2^{\frac{3}{2}}C_{j+1,1,1}^{\prime}$
$s_{j}^{\prime(2)}=Tr\left(\widetilde{\Pi}^{(1)}(\rho)I\otimes\sigma_{j}\otimes
I\right)=2^{\frac{3}{2}}C_{1,j+1,1}^{\prime}$
$s_{j}^{\prime(3)}=Tr\left(\widetilde{\Pi}^{(1)}(\rho)I\otimes
I\otimes\sigma_{j}\right)=2^{\frac{3}{2}}C_{1,1j+1}^{\prime}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(A8)$

and

$\displaystyle t_{ij}^{\prime(12)}$ $\displaystyle=$ $\displaystyle
Tr\left(\widetilde{\Pi}^{(1)}(\rho)\sigma_{i}\otimes\sigma_{j}\otimes
I\right)=2^{\frac{3}{2}}C_{i+1,j+1,1}^{\prime}$ $\displaystyle
t_{ik}^{\prime(13)}$ $\displaystyle=$ $\displaystyle
Tr\left(\widetilde{\Pi}^{(1)}(\rho)\sigma_{i}\otimes
I\otimes\sigma_{k}\right)=2^{\frac{3}{2}}C_{i+1,1,k+1}^{\prime}$
$\displaystyle t_{jk}^{\prime(23)}$ $\displaystyle=$ $\displaystyle
Tr\left(\widetilde{\Pi}^{(1)}(\rho)I\otimes\sigma_{j}\otimes\sigma_{k}\right)=2^{\frac{3}{2}}C_{1,j+1,k+1}^{\prime}~{}~{}~{}~{}~{}~{}~{}~{}~{}(A9)$

and

$t_{ijk}^{\prime}=\\\
Tr\left(\widetilde{\Pi}^{(1)}(\rho)\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k}\right)=2^{\frac{3}{2}}C_{i+1,j+1,k+1}^{\prime}~{}~{}~{}~{}~{}~{}~{}~{}(A10)$

where $\|\mathcal{T}^{\prime}\|^{2}=\sum_{ijk}t_{ijk}^{\prime 2},$

$\widetilde{\mathbf{T}}_{i\ell}^{(3)}=\sum_{jk}t_{jik}^{\prime}t_{j\ell
k}^{\prime}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(A11)$

  10. 10.

We repeat steps (iv) and (v) using Eqs. A8, A9, A10 to calculate
$G^{\prime(2)}$ real symmetric matrix for density matrix
$\widetilde{\Pi}^{(1)}(\rho)$ as

$G^{\prime(2)}=S^{\prime(2)}(S^{\prime(2)})^{t}+\
(\mathrm{T}^{\prime\\{1,2\\}})^{t}\mathrm{T}^{\prime\\{1,2\\}}+(\mathrm{T}^{\prime\\{2,3\\}})^{t}\mathrm{T}^{\prime\\{2,3\\}}+\mathbf{T}^{\prime(3)},~{}~{}~{}~{}~{}(A11)$

we find the largest eigenvalue $\eta_{\max}^{(2)}$ of $G^{\prime(2)}$ and
corresponding eigenvector $V_{max}^{(2)}.$ We calculate $D_{2}$ as

$D_{2}(\widetilde{\Pi}^{(1)}(\rho))=\frac{1}{2^{3}}\left[\|\textbf{s}^{\prime(2)}\|^{2}+\|\mathrm{T}^{\prime\\{1,2\\}}\|^{2}+\|\mathrm{T}^{\prime\\{2,3\\}}\|^{2}+\|\mathcal{T}^{\prime}\|^{2}-\eta_{max}^{(2)}\right].$

  11. 11.

We put $\hat{e}_{2}=V_{max}^{(2)}$ to calculate

$\tilde{\textbf{a}}_{1}=\frac{1}{\sqrt{2}}\left(1,\hat{e}_{2}\right)\quad,\quad\tilde{\textbf{a}}_{2}=\frac{1}{\sqrt{2}}\left(1,-\hat{e}_{2}\right)$

  12. 12.

We repeat the step (vii) to calculate $\widetilde{b}_{1}$ and
$\widetilde{b}_{2}$ matrix of the state
$\widetilde{\Pi}^{(2)}\left(\widetilde{\Pi}^{(1)}\left(\rho\right)\right)=\widetilde{\widetilde{\rho}}$.

$\widetilde{b}_{1ik}=\sum_{j}c_{ijk}\tilde{a}_{1j}\quad,\quad
b_{2ik}=\sum_{j}c_{ijk}\tilde{a}_{2j}$

  13. 13.

We repeat the step (viii) to calculate

$C_{ijk}^{\prime\prime}=\widetilde{a}_{1j}\widetilde{b}_{1ik}+\widetilde{a}_{2j}\widetilde{b}_{2ik}$

.

  14. 14.

We repeat the step (ix) to calculate the coherent vectors
$\textbf{s}^{\prime\prime^{(m)}}$, the correlation matrix for two-qubit
$t^{\prime\prime(12)},\;t^{\prime\prime(23)},\;t^{\prime\prime(13)}$, three-
way correlation array $t^{\prime\prime}$ and $\mathbf{T}^{\prime\prime(3)}$
for state
$\widetilde{\Pi}^{(2)}\left(\widetilde{\Pi}^{(1)}\left(\rho\right)\right)$.

  15. 15.

We repeat step (x) to calculate $G^{\prime\prime(3)}$ for state
$\widetilde{\Pi}^{(2)}\left(\widetilde{\Pi}^{(1)}\left(\rho\right)\right)=\widetilde{\widetilde{\rho}}$
and find $\eta_{\max}^{(3)}$ and using it to calculate
$D_{3}\left(\widetilde{\Pi}^{(2)}\left(\widetilde{\Pi}^{(1)}\left(\rho\right)\right)\right).$

  16. 16.

We calculate TQC, $Q(\rho)$ as in Eq. (11).

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# Spin correlations in top physics at ATLAS and CMS in Run 2

Jacob Linacre  on behalf of the ATLAS and CMS Collaborations  STFC Rutherford
Appleton Laboratory, Harwell Campus, Didcot OX11 0QX, England

###### Abstract

Measurements of $\mathrm{t}\bar{\mathrm{t}}$ spin correlations are presented
in events with top quarks produced in $\mathrm{pp}$ collisions at the LHC. The
data correspond to an integrated luminosity of $36\>\mathrm{fb^{-1}}$ at
$\sqrt{s}=13\>\mathrm{TeV}$ collected at both the ATLAS and CMS detectors. The
spin correlations are measured using the angular distributions of the leptons
in dilepton channel $\mathrm{t}\bar{\mathrm{t}}$ events. The spin correlations
are probed both directly, using distributions measured in the top quark rest
frames that depend only on the top quark spin, and indirectly, using
distributions measured in the laboratory frame. The distributions are unfolded
to the parton level and extrapolated to the full phase space. Some of the
laboratory frame distributions are additionally unfolded to the particle level
in the fiducial phase space of the ATLAS detector. The spin correlation
measurements are used to search for new physics in the form of a light top
squark or an anomalous top quark chromo-magnetic dipole moment, and stringent
constraints are placed in both cases.

##  1 Introduction

The large mass of the top quark and its corresponding strong coupling to the
Higgs boson suggest a connection between the top quark and the mechanism of
electroweak symmetry breaking. New physics in this mechanism is likely to
modify the spin properties of top quark pair ($\mathrm{t}\bar{\mathrm{t}}$)
events from the standard model (SM) expectations, either via underlying direct
production modes or from interference effects from new physics at higher mass
scales [1]. Furthermore, the top quark is the only quark that decays before
hadronising, so the information about its spin is transferred to its decay
products undiluted by non-perturbative effects. The charged lepton in top
quark decay is a perfect spin analyser [2], meaning its angular distribution
retains the maximum amount of information about the top quark spin. Top quark
spin measurements in dilepton $\mathrm{t}\bar{\mathrm{t}}$ events therefore
provide an ideal laboratory to test perturbative QCD and probe for new
physics.

At the LHC $\mathrm{t}\bar{\mathrm{t}}$ production proceeds primarily via the
strong interaction (mostly $\mathrm{gg}\to\mathrm{t}\bar{\mathrm{t}}$), which
at the leading order (LO) produces unpolarised top quarks. A small top quark
polarisation, measured relative to the direction of the recoiling top quark,
arises when including electroweak corrections, while a small polarisation
transverse to the scattering plane arises from absorptive terms at one loop
(both ${<}\,1\%$ [3]). However, the spins of the top quarks and antiquarks are
strongly correlated. The SM predicts a rich structure of spin correlations,
where the configuration of quark-antiquark spins is dependent on both the
initial state and the top quark production kinematics [4]. The analysed LHC
$\mathrm{pp}$ collision data corresponds to an integrated luminosity of
$36\>\mathrm{fb^{-1}}$ at $\sqrt{s}=13\>\mathrm{TeV}$ collected at both the
ATLAS [5] and CMS [6] detectors.

##  2 Indirect measurements of spin correlations

The correlation between the top quark and antiquark spins induces a tendency
towards alignment of the decay angles of the daughter leptons. This
correlation is most strongly observed in the transverse plane, and is retained
in the separation in azimuthal angle of the two leptons in the laboratory
frame ($\left|\Delta\phi_{\ell\ell}\right|$). The approximately back-to-back
configuration of the parent top quark and antiquark results in preference for
large $\left|\Delta\phi_{\ell\ell}\right|$, while the spin correlations
generate a relative enhancement of ${\approx}\,15\%$ at low
$\left|\Delta\phi_{\ell\ell}\right|$ (see Fig. 1, right). The
$\left|\Delta\phi_{\ell\ell}\right|$ distribution can be very precisely
reconstructed, owing to the excellent experimental angular resolution of the
lepton measurements, and it is therefore an important probe of deviations from
the SM.

The results from ATLAS [7] and CMS [8], unfolded to the parton level and
extrapolated to the full phase space, are shown in Fig. 1, along with various
predictions from simulation [9, 10] and fixed-order calculations [3, 11].
There is a clear preference of the data for the predictions including spin
correlations, and the ATLAS experiment uses the predictions of the next-to-LO
(NLO) Powhegv2 generator [9] to fit the strength of the spin correlations as a
fraction of the SM prediction, $f_{\mathrm{SM}}$ (Fig. 2, left). The result is
$f_{\mathrm{SM}}=1.25\pm 0.02\,\mathrm{(stat)}\pm 0.06\,\mathrm{(syst)}\pm
0.04\,\mathrm{(theo)}$, where the combined uncertainty is $\pm 0.08$,
suggesting that the observed spin correlations are $3.2\sigma$ stronger than
those predicted by the SM. The dominant source of systematic uncertainty is
the choice of renormalisation and factorisation scales ($\mu_{\mathrm{R}}$ and
$\mu_{\mathrm{F}}$) and the amount of initial- and final-state radiation (ISR
and FSR) in the generated $\mathrm{t}\bar{\mathrm{t}}$ events. The discrepancy
remains when ATLAS repeats the measurement, limiting the extrapolation of the
measured distribution to the fiducial phase space of the detector, and a
similar discrepancy is observed in the $\left|\Delta\phi_{\ell\ell}\right|$
disribution measured by CMS in the full phase space (Fig. 1, right). A
possible resolution appeared recently in the first full next-to-NLO (NNLO) QCD
calculations of the $\left|\Delta\phi_{\ell\ell}\right|$ distribution [11].
The corrections are found to be small in the full phase space, but in a
fiducial phase space similar to that of the ATLAS and CMS detectors they are
large enough to account for much of the observed discrepancy (Fig. 2, centre).
Since the ratio of the full and fiducial phase space differential cross
sections is used in the extrapolation of the measured distribution to the full
phase space, this effect can account for the observed discrepancy in the full
phase space as well.

Figure 1: Parton-level $\left|\Delta\phi_{\ell\ell}\right|$ distributions
measured by ATLAS [7] (left) and CMS [8] (right), compared with various
predictions. In the centre, the ATLAS measurement is compared with fixed-order
calculations [3, 11].

Figure 2: Left: fit of ATLAS parton-level $\left|\Delta\phi_{\ell\ell}\right|$
distribution [7] to a combination of correlated and uncorrelated templates, to
measure $f_{\mathrm{SM}}$. Centre: NNLO QCD calculations of the
$\left|\Delta\phi_{\ell\ell}\right|$ distribution in the fiducial and
inclusive phase space [11]. Right: parton-level $\left|\Delta\eta\right|$
distribution measured by ATLAS [7]. This observable has minimal sensitivity to
spin correlations, but is sensitive to new physics in
$\mathrm{t}\bar{\mathrm{t}}$ production (see Section 4.2).

##  3 Direct measurements of spin correlations

Using the lepton directions measured in their parent top quark rest frames as
proxies for the top quark spins, all of the spin-dependent parts of the
$\mathrm{t}\bar{\mathrm{t}}$ production density matrix can be probed [3]:

$\frac{1}{\sigma}\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega_{1}\mathrm{d}\Omega_{2}}=\frac{1}{(4\pi)^{2}}\left(1+\mathbf{B_{1}}\cdot\hat{\ell}_{1}+\mathbf{B_{2}}\cdot\hat{\ell}_{2}-\hat{\ell}_{1}\cdot
C\cdot\hat{\ell}_{2}\right),$ (1)

where $\mathbf{{B}_{1,2}}$ are three-dimensional vectors that characterise the
degree of top quark or antiquark polarisation in each direction, and ${C}$ is
a $3\times 3$ matrix that characterises the correlation between the top quark
and antiquark spins. The spin is measured using a basis, illustrated in Fig.
3, chosen such that the $B_{1,2}^{i}$ and $C_{ij}$ coefficients (the elements
of the $\mathbf{B_{1}}$ and $\mathbf{B_{2}}$ vectors and of the $C$ matrix)
have definite properties with respect to discrete symmetries such as C and P
[3].

Each coefficient is probed by the CMS experiment by measuring [8] a one-
dimensional angular distribution derived from Eq. 1, which for each
coefficient has the form

$\frac{1}{\sigma}\frac{\mathrm{d}\sigma}{\mathrm{d}x}=\frac{1}{2}\left(1+\mathrm{Coefficient}\times
x\right)f(x).$ (2)

These distributions are sensitive only to the top quark spin (independent of
the top quark kinematics), and the measurements are therefore less affected by
theoretial uncertainties. However, compared to the indirect measurements the
statistical precision of the measurements is diluted by the poor resolution of
the top quark momentum reconstruction, caused largely by the presence of two
neutrinos and ambiguities in the assignments of measured jets to the b quarks
from top quark decay. The measured normalised differential cross sections are
unfolded to the parton level and extrapolated to the full phase space. The
known functional forms of Eq. 2, which are unaffected by new physics in
$\mathrm{t}\bar{\mathrm{t}}$ production, are used to construct an unbiased
regularisation of the unfolding. In addition to full statistical and
systematic covariance matrices for each measured distribution, matrices are
calculated for the set of all measured bins, allowing constraints to be placed
using several measured distributions simultaneously (an example can be seen in
Fig. 6). A selection of results for distributions probing the $C$ matrix is
shown in Fig. 3.

Figure 3: Left: coordinate system used for the top quark spin measurements,
shown in the $\mathrm{pp}\to\mathrm{t}\bar{\mathrm{t}}$ scattering plane.
Centre: distribution representing the correlation between the lepton
directions both measured with respect to the $\hat{n}$ axis, probing $C_{nn}$
[8]. Right: distribution representing the correlation between the lepton
directions, one measured with respect to the $\hat{r}$ axis and the other
measured with respect to the $\hat{k}$ axis, probing $C_{rk}+C_{kr}$ [8].

From each measured normalised differential cross section CMS extracts the
corresponding spin correlation coefficient, and the results are shown in Fig.
4. The top quark polarisation is also measured with respect to each reference
axis, but the measurements are all consistent with zero and not sensitive to
the small level of top quark polarisation predicted in the SM. The systematic
and statistical uncertainties are of comparable size for most of the measured
coefficients, and the dominant sources of systematic uncertainty are typically
in the jet energy scale and b quark fragmentation (which affect the
reconstruction of the top quark rest frame), the background subtraction, and
the $\mathrm{t}\bar{\mathrm{t}}$ simulation modelling (ISR and FSR,
$\mu_{\mathrm{R}}$ and $\mu_{\mathrm{F}}$, and top quark $p_{\mathrm{T}}$).

Figure 4: Summary of measured [8] diagonal (left) and off-diagonal (right)
coefficients of the matrix $C$.

The measured coefficients are converted into values of $f_{\mathrm{SM}}$ using
theoretical predictions at NLO in QCD with EW corrections [3] (all
coefficients are zero in the absence of spin correlations), and the results
are shown in Fig. 5 (right). The most precise measurement of spin correlations
comes from the distribution of the opening angle between the lepton directions
$\hat{\ell}_{1}$ and $\hat{\ell}_{2}$ measured in their parent top quark rest
frames,
$\frac{1}{\sigma}\frac{d\sigma}{d\cos\varphi}=\frac{1}{2}(1-D\cos\varphi)$,
where $\cos\varphi=\hat{\ell}_{1}\cdot\hat{\ell}_{2}$ and
$D=-\mathrm{tr}(C)/3$. The measured $\cos\varphi$ distribution is shown in
Fig. 5 (left). From the $D$ coefficient, CMS measures $f_{\mathrm{SM}}=0.97\pm
0.05$. This is the most precise measurement of $f_{\mathrm{SM}}$ to date,
indicating that the effect of the worse experimental resolution of
$\cos\varphi$ compared to $\left|\Delta\phi_{\ell\ell}\right|$ is outweighed
by the reduced theoretical uncertainties and the stronger dependence on spin
correlations of the direct observable.

Figure 5: Left: measured distribution [8] of the cosine of the angle between
the lepton directions measured in their parent top quark rest frames,
sensitive to the spin correlation coefficient $D$ (related to the trace of the
$C$ matrix). The prediction without spin correlations (uncorrelated) is
strongly disfavoured. Right: summary of $f_{\mathrm{SM}}$ results [8].

##  4 Interpretation

###  4.1 Anomalous chromomagnetic dipole moment

The chromomagnetic dipole moment (CMDM) of a colour-charged particle in colour
fields can be defined by analogy to the magnetic dipole moment of an
electrically charged particle. In the SM, the intrinsic spin of the top quark
and its colour charge give it a small CMDM [3]. Several beyond-the-SM models
such as two-Higgs-doublet models (e.g., supersymmetry), technicolor, and top
quark compositeness models [1], predict an anomalous CMDM, leading to
modifications of the $\mathrm{t}\bar{\mathrm{t}}$ production rates and spin
structure.

The measurement of the $\mathrm{t}\bar{\mathrm{t}}$ production spin density
matrix is a powerful probe of the top quark CMDM. Using the measured
distributions and covariance matrices [8], and their predicted dependence [1]
on the Wilson coefficient of the CMDM operator divided by the square of the
new physics scale, $C_{\mathrm{tG}}/\Lambda^{2}$, CMS performs a $\chi^{2}$
fit. The resulting constraints at 95% CL, illustrated in Fig. 6, are
$-0.07<C_{\mathrm{tG}}/\Lambda^{2}\ <0.16\>\mathrm{TeV}^{-2}$, the strongest
constraints on $C_{\mathrm{tG}}/\Lambda^{2}$ to date.

Figure 6: Left: systematic correlation matrix for all 132 bins measured in the
CMS analysis [8]. Right: result [8] of the $\chi^{2}$ fit for
$C_{\mathrm{tG}}/\Lambda^{2}$. The “all” result includes the full
$C_{\mathrm{tG}}/\Lambda^{2}$-dependence of the $\mathrm{t}\bar{\mathrm{t}}$
cross section, and the result when using only the terms “linear” in
$C_{\mathrm{tG}}/\Lambda^{2}$, which describe the interference with the SM, is
almost the same.

###  4.2 “Top corridor” SUSY

Light top squarks ($\tilde{t}$) are favoured in natural supersymmetry (SUSY),
and in many scenarios could decay to a top quark and the lightest SUSY
particle (LSP), which would not interact with SM particles and would therefore
escape the detector without producing a signal. When the SUSY particle masses
are such that the daughter top quark is produced almost at rest ($\Delta
m=m_{\mathrm{\tilde{t}}}-m_{\mathrm{LSP}}=m_{\mathrm{t}}$), pair-produced
$\tilde{t}\tilde{t^{*}}$ events can be difficult to distinguish from
$\mathrm{t}\bar{\mathrm{t}}$ events. Unlike in direct SUSY searches that rely
on the presence of missing transverse momentum from the undetected LSP, in
this scenario the main distinguishing characteristics are the spin
correlations, which are absent in the scalar case, and the typically more
central production kinematics of scalars, which translate to the separation in
psuedorapidity of the leptons ($\left|\Delta\eta\right|$) [12]. Using measured
double-differential distributions in $\left|\Delta\phi_{\ell\ell}\right|$ and
$\left|\Delta\eta\right|$ along with the total rate [7], ATLAS sets exclusion
limits at 95% CL (Fig. 7, left). These results push the exclusion beyond those
of existing direct searches in the region close to $\Delta m=m_{\mathrm{t}}$.
The $\left|\Delta\eta\right|$ information makes a greater contribution to the
overall sensitivity than the $\left|\Delta\phi_{\ell\ell}\right|$ information
(see Fig. 7, right). The ATLAS analysis also unfolds the
$\left|\Delta\eta\right|$ distribution to the parton level (Fig. 2, right).

Figure 7: Left: observed and expected 95% CL exclusion in the plane of
$m_{\mathrm{\tilde{t}}}$ and $m_{\mathrm{LSP}}$ [7]. Right: limits on the
$\tilde{t}\tilde{t^{*}}$ cross section at 95% CL as a function of
$m_{\mathrm{\tilde{t}}}$, assuming $m_{\mathrm{LSP}}=0.5\>\mathrm{GeV}$. The
expected limits when using the $\left|\Delta\phi_{\ell\ell}\right|$ and
$\left|\Delta\eta\right|$ distributions alone are shown by the magenta and
blue dashed lines, respectively [7].

##  5 Summary

Both ATLAS and CMS have presented measurements of $\mathrm{t}\bar{\mathrm{t}}$
spin correlations using LHC $\mathrm{pp}$ collision data corresponding to an
integrated luminosity of $36\>\mathrm{fb^{-1}}$ at
$\sqrt{s}=13\>\mathrm{TeV}$. The significant tension observed between the
$\left|\Delta\phi_{\ell\ell}\right|$ distributions measured by the ATLAS and
CMS experiments and the SM predictions is likely explained by missing higher
order corrections to the top quark kinematics, which become more important in
the fiducial phase space accessible to the experiments. The direct
measurements of spin correlations are in good agreement with the SM
predictions, and all spin-dependent coefficients of the
$\mathrm{t}\bar{\mathrm{t}}$ production density matrix have been probed for
the first time at $\sqrt{s}=13\>\mathrm{TeV}$. The spin correlation
measurements are used to search for new physics in the form of a light top
squark or an anomalous top quark chromo-magnetic dipole moment, and stringent
constraints are placed in both cases.

## Acknowledgments

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 752730.

00footnotetext: Copyright 2019 CERN for the benefit of the ATLAS and CMS
Collaborations. CC-BY-4.0 license.

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