# Formation of Fano resonance in double quantum dot system

Grzegorz Michałek<EMAIL_ADDRESS>Bogdan R. Bułka  Institute of
Molecular Physics, Polish Academy of Sciences, ul. Mariana Smoluchowskiego 17,
60-179 Poznań, Poland

###### Abstract

Transient electron dynamics and the Fano resonance formation over time in
transport through a two quantum dot system in a T-shape geometry (2QD-T) are
analysed for free electrons. Time evolution of the transport characteristics
are different for a weak and a strong inter-dot coupling with respect to the
dot-electrode coupling. When the dot-dot coupling is weak, we find that
transport dynamics is governed by two channel tunneling processes with two
different relaxation times and buildup of the Fano resonance is developed much
slower, with the longer relaxation time. In the opposite limit, the
transmission evolves with a single relaxation time, from the single peak
resonance structure into the structure with two asymmetric peaks. One observes
large dot-electrode charge oscillations, which can temporarily change
direction of transient currents. We find also that the time evolution of
entanglement is different in both coupling regimes; in particular rapid Rabi
oscillations caused by coherent electron transfer between the 2QD states can
be seen for the strong dot-dot coupling.

##  I Introduction

Nowadays, one can experimentally observe dynamics of charge flow in
nanostructures under the non-stationary conditions, after applying short-time
voltage pulses or other sudden changes in system parameters. The ultra-fast
time-re-solved spectroscopy techniques enable real-time observation of, among
others, how the system returns to equilibrium from excited states. Just
recently the Fano resonance buildups were monitored in optical experiments
using the spectrally resolved electron interferometry and the atto-second
transient absorption spectroscopy [1, 2]. The Fano resonance was observed in
electron transport measurements in nanostructures, namely in the system of a
strongly coupled quantum dot with electrodes [3, 4]. Two quantum dots in a
T-shape geometry (2QD-T) seems to be a suitable system for theoretical studies
of the Fano resonance, because conductance shows a characteristic asymmetric
dip due to destructive interference of a travelling wave through the central
QD with an localized state at the side attached QD [5, 6] (see also [7, 8, 9]
and references therein).

However the analysis of transient dynamics in the 2QD-T system has been
limited to coherent ringing (oscillations) or beats in transient currents
after a switching on or off a bias voltage [10]. In Ref. [11] a charging
effect and transient currents were studied in a T-shaped wire of $N$ QDs after
a sudden change of the bias voltage. Therefore, in the paper we focus on an
unexplored topic: Fano resonance buildup over time in the 2QD-T system when
the second QD is suddenly side attached. In particular, we will study
transient currents and the time evolution of transmission to identify
relaxation processes in different time regimes.

##  II Description of DQD system and method of calculations

We consider two single-level QD in a T-shape geometry coupled with two normal-
metal left (L) and right (R) electrodes. We assume that in a distant past only
the central QD1 is coupled with the electrodes under bias and it is in a
stationary state, while the second QD2 is suddenly attached to the QD1 at the
time $t=0$. The Hamiltonian of the system (for the spinless noninteracting
electrons) can be written in the form
$H=H_{QD}+\sum_{\alpha=L,R}H_{\alpha}+H_{T}$, where

$\displaystyle
H_{QD}=\sum_{i=1,2}\epsilon_{i}d_{i}^{\dagger}d_{i}+\Theta(t)t_{12}(d_{1}^{\dagger}d_{2}+d_{2}^{\dagger}d_{1})\;,$
$\displaystyle H_{\alpha}=\sum_{k}(\epsilon_{\alpha k}-\mu_{\alpha})c_{\alpha
k}^{\dagger}c_{\alpha k}\;,$ (1) $\displaystyle H_{T}=\sum_{\alpha
k}t_{\alpha}(d_{1}^{\dagger}c_{\alpha k}+c_{\alpha k}^{\dagger}d_{1})\;.$

Here, $d_{i}^{\dagger}(d_{i})$ stands for creation (annihilation) operator for
an electron on the QDi and at the dot level $\epsilon_{i}$. An abrupt coupling
of the QD2 at $t=0$ is described by time-dependent hopping between the dots
$\Theta(t)t_{12}$, where $\Theta(t)$ is a Heaviside step function and $t_{12}$
denotes the inter-dot hopping parameter. $c_{\alpha k}^{\dagger}(c_{\alpha
k})$ describes electrons with the energy $\epsilon_{\alpha k}$ in the $\alpha$
electrode which has a chemical potential $\mu_{\alpha}$. The coupling of the
central QD1 with the electrodes is given by the hopping parameter
$t_{\alpha}$.

The time-dependent electrical currents flowing into the QD1 from the $\alpha$
electrode can be calculated from the evolution of the total number operator
using the equation of motion technique (EOM) for the non-equilibrium Green
function [12]

$J_{\alpha}(t)=-2\frac{iet_{\alpha}}{\hbar}\Im\sum_{k}\langle
d_{1}^{\dagger}(t)c_{\alpha k}(t)\rangle\;,$ (2)

where the electrodes are treated in the wide-band approximation. For the
creation (annihilation) operators, in the QDs and the electrodes, we write
appropriate Heisenberg equation of motion. Next we solve this system of
equations by performing a Laplace transform [13]. The time dependencies of the
operators are found by inverting the Laplace transform in the wide band limit
[13]. In general, the current $J_{L}(t)$ can be written in the form:

$J_{L}(t)=\frac{e}{h}\int_{-\infty}^{\infty}[T(E,t)\Delta
f(E)+A(E,t)f_{W}(E)]dE\;,$ (3)

where $\Delta f(E)\equiv f_{L}(E)-f_{R}(E)$,
$f_{W}(E)\equiv[\Gamma_{L}f_{L}(E)+\Gamma_{R}f_{R}(E)]/(\Gamma_{L}+\Gamma_{R})$
denotes the Wigner distribution function for electrons at the 2QD system and
$f_{\alpha}(E)\equiv f(E-\mu_{\alpha})$ is the Fermi-Dirac distribution in the
electrode $\alpha$ with a chemical potential $\mu_{\alpha}$. The tunneling
rates are $\Gamma_{\alpha}\equiv 2\pi\rho_{\alpha}t_{\alpha}^{2}$, where
$\rho_{\alpha}$ is a constant density of state in the electrode $\alpha$.
$T(E,t)$ denotes a transmission evolution, while $A(E,t)$ is related with a
local charge and entanglement evolution [14, 15] to the stationary state. Both
coefficients include contributions to quantum interference from various
tunneling processes.

##  III Results

To simplify considerations we assume that both QD are identical with
$\epsilon_{1}=\epsilon_{2}=\epsilon$ and there is a symmetrical coupling to
the electrodes, $\Gamma_{L}=\Gamma_{R}=\Gamma$. Since time evolution of
transport characteristics and electron dynamics are different for a weak and a
strong inter-dot coupling $t_{12}$ with respect to $\Gamma$, we present them
separately.

###  III.1 Weak dot-dot coupling

Figure 1: a) $T(E,t)$ as a function of energy $E$ for selected times
$t=\\{0,2\tau_{+},2\tau,2\tau_{+}^{osc},2\tau_{-},2\tau_{-}^{osc},3\tau_{-}^{osc},4\tau_{-}^{osc}\\}$
(black, red, blue, magenta, orange, brown, green, gray curves) for small
$t_{12}/\Gamma=0.4$. b) and c) The time evolution of $T(E,t)$ and its
components for $E=0$ and $E=1$, respectively. The other parameters are
$\epsilon=0$, $\Gamma=1$, $\tau=1/\Gamma=1$.

Figure 2: a) $A(E,t)$ as a function of energy $E$ for selected times $t$ for
small $t_{12}/\Gamma=0.4$. b) and c) The time evolution of $A(E,t)$ and its
components for $E=0$ and $E=1$, respectively. The other parameters are the
same as in Fig. 1.

We start from the weak dot-dot coupling case, i.e. when $t_{12}\ll\Gamma$. In
this case one level is strongly hybridized with the electrodes, while the
other one (on the QD2) is localized and only weakly coupled to a conducting
channel. Interference between these two conducting channels leads to a Fano
anti-resonance with a characteristic dip in the transmission. The time
evolution of the transmission can be expressed as

$\displaystyle T(E,t)$
$\displaystyle=\sum_{\nu=\pm}[T_{1QD}^{osc,\nu}(E,t)-T_{2QD}^{osc,\nu}(E,t)]e^{-t/\tau_{\nu}^{osc}}$
$\displaystyle+T_{2QD}^{s}(E)\;.$ (4)

The components

$\displaystyle
T_{1QD}^{osc,\nu}(E,t)=\frac{\nu\Gamma\Gamma_{\nu}}{2\delta}\frac{\Gamma\cos(\mathcal{E}t)-\mathcal{E}\sin(\mathcal{E}t)}{\mathcal{E}^{2}+\Gamma^{2}}\;,$
(5) $\displaystyle
T_{2QD}^{osc,\nu}(E,t)=\frac{\nu\Gamma\Gamma_{\nu}}{4\delta}\frac{\Gamma_{\nu}\cos(\mathcal{E}t)-2\mathcal{E}\sin(\mathcal{E}t)}{\mathcal{E}^{2}+\Gamma_{\nu}^{2}/4}\;$
(6)

and they describe a time evolution of the transmission related with the 1QD
system and the creation of the Fano resonance in the 2QD system with the same
two characteristic times $\tau_{\pm}^{osc}\equiv 2/\Gamma_{\pm}$, where
$\Gamma_{\pm}=\Gamma\pm\delta$ and $\delta=(\Gamma^{2}-4t_{12}^{2})^{1/2}$. In
this case the relaxation processes depend not only on the coupling $\Gamma$
with the external electrodes but they are also strongly modified by the
hopping $t_{12}$ between QDs, which significantly changes transport properties
of the system. Here, we denoted by $\mathcal{E}=E-\epsilon$, and took $e\equiv
1$ and $h\equiv 1$. Above we performed the spectral decomposition to show a
role of both relaxation processes with their characteristic times
$\tau_{\pm}^{osc}=2/\Gamma_{\pm}$. The last term corresponds to the
transmission in the stationary state (at $t=\infty$) and after the spectral
decomposition it can be expressed as
$T_{2QD}^{s}(E)=\sum_{\nu=\pm}T_{2QD}^{s,\nu}(E)$, where

$T_{2QD}^{s,\nu}(E)=\frac{\Gamma}{4\delta}\frac{\nu\Gamma_{\nu}^{2}}{\mathcal{E}^{2}+\Gamma_{\nu}^{2}/4}\;.$
(7)

This result can be interpreted as resonant transport through two conducting
channels with two different broadenings $\Gamma_{\pm}/2$. Notice that
$T_{2QD}^{s,-}(E)$ is negative and describes the destructive interference.

Fig. 1a) presents a 3D plot of $T(E,t)$, the function which is always
positive, although its components can be positive as well as negative. Fig.
1b) shows the evolution of the Fano dip (at $E=0$). In this case
$T_{2QD}^{s}(0)=0$, the components
$T_{1QD}^{osc,\pm}(0)=\pm\Gamma_{\pm}/(2\delta)$,
$T_{2QD}^{osc,\pm}(0)=\pm\Gamma/\delta$ and thus,
$T(0,t)=(-\Gamma_{-}e^{-t/\tau_{+}^{osc}}+\Gamma_{+}e^{-t/\tau_{-}^{osc}})/(2\delta)$
vanishes monotonically. One can say that the Fano dip evolves with both
relaxation times $\tau_{\pm}$, but destructive interference dominates in a
long time limit, for $t>\tau_{-}^{osc}$. For $\mathcal{E}\neq 0$ one finds
oscillations in $T(E,t)$ with the period $\omega=2\pi|\mathcal{E}|$. These
oscillations are related to coherent charge transfers between the QD system
and the electrodes. In general, these oscillations are dominated by
$T_{1QD}^{-}(E,t)$ and $T_{2QD}^{-}(E,t)$ component, while $T_{1QD}^{+}(E,t)$
and $T_{2QD}^{+}(E,t)$ component become relevant only in the short time
regime.

The spectral function $A(E,t)$ describes processes associated with
establishing a local chemical potential (related with the Wigner function) and
with changes of entanglement in the system, also in presence of the bias
voltage. It can be written in the form

$\displaystyle A(E,t)=A^{g}(E)e^{-t/\tau}$
$\displaystyle+\sum_{\nu=\pm}[A^{\nu}(E)e^{-t/\tau_{\nu}}+A^{osc,\nu}(E,t)e^{-t/\tau_{\nu}^{osc}}]\;,$
(8)

where components $A^{g}(E)$, $A^{\nu}(E)$ and $A^{osc,\nu}(E,t)$ are complex
and are given in Appendix. $A(E,t)$ vanishes in the stationary states, i.e.
for $t=0$ or $t=\infty$ (see Fig. 2), because we consider symmetric coupling
with the electrodes when local charge accumulation is absent. One can see that
in $A(E,t)$ new terms appear which are related with additional relaxation
processes (with a relaxation time $\tau\equiv 1/\Gamma$ and $\tau_{\pm}\equiv
1/\Gamma_{\pm}$). They are relevant only for short times ($t<\tau_{-}^{osc}$),
while for larger ones $A^{osc,-}(E,t)$ component plays a dominant role. In
contrast to $T(E,t)$, energy-dependent oscillations are clearly visible in
$A(E,t)$. Their period increase with time. Let us take a look at $E=0$.
$A(0,t)$ is positive but a non-monotonic function of time due to an interplay
of the components $A^{g}(0)=\Gamma_{+}\Gamma_{-}/\delta^{2}$,
$A^{\pm}(0)=-\Gamma_{\mp}^{2}/(2\delta^{2})$ and
$A^{osc,\pm}(0)=\mp\Gamma_{\mp}/\delta$ which vanish with the relaxation times
$\tau$, $\tau_{\pm}$ and $\tau_{\pm}^{osc}$, respectively.

###  III.2 Strong dot-dot coupling

Figure 3: a) $T(E,t)$ as a function of energy $E$ for selected
$t/\tau^{osc}=\\{0,1,2,4,6,8,10\\}$ (black, red, blue, magenta, orange, brown,
green curves) for large $t_{12}/\Gamma=3$. b) and c) The vanishing of the 1QD
peak and formation of the anti-bonding state for $E=0$ and $E=E_{-}$,
respectively. The other parameters are $\epsilon=0$, $\Gamma=1$ and
$\tau=1/\Gamma$.

Figure 4: a) $A(E,t)$ as a function of energy $E$ for selected times $t$. b)
and c) The time evolution of $A(E,t)$ and its components for $E=0$ and
$E=E_{-}$, respectively. The other parameters are the same as in Fig. 3.

For the strong dot-dot coupling, $t_{12}\gg\Gamma$, one can treat 2QD as an
artificial molecule and observe the time evolution of the transmission from
the single peak structure into the double peak structure, see Fig. 3. The
transmission $T(E,t)$ is expressed as

$\displaystyle T(E,t)$
$\displaystyle=\sum_{\nu=\pm}[T_{1QD}^{osc,\nu}(E,t)+T_{2QD}^{osc,\nu}(E,t)]e^{-t/\tau^{osc}}$
$\displaystyle+T_{2QD}^{s}(E)\;,$ (9)

where its components are

$\displaystyle
T_{1QD}^{osc,\nu}(E,t)=-\nu\frac{\Gamma}{2\Delta}\left\\{\frac{\Gamma(\mathcal{E}-\nu\Delta)\cos[(\mathcal{E}+\nu\Delta/2)t]}{\mathcal{E}^{2}+\Gamma^{2}}\right.$
$\displaystyle+\left.\frac{(\Gamma^{2}+\nu\mathcal{E}\Delta)\sin[(\mathcal{E}+\nu\Delta/2)t]}{\mathcal{E}^{2}+\Gamma^{2}}\right\\}\;,$
(10) $\displaystyle
T_{2QD}^{osc,\nu}(E,t)=\nu\frac{\Gamma}{4\Delta}\left\\{\frac{2\Gamma\mathcal{E}\cos[(\mathcal{E}+\nu\Delta/2)t]}{(\mathcal{E}+\nu\Delta/2)^{2}+\Gamma^{2}/4}\right.$
$\displaystyle+\left.\frac{(\Gamma^{2}+2\nu\Delta\mathcal{E}+\Delta^{2})\sin[(\mathcal{E}+\nu\Delta/2)t]}{(\mathcal{E}+\nu\Delta/2)^{2}+\Gamma^{2}/4}\right\\}\;,$
(11) $\displaystyle
T_{2QD}^{s}(E)=-\frac{1}{2\Delta}\sum_{\nu=\pm}\frac{\nu\Gamma^{2}\mathcal{E}}{(\mathcal{E}+\nu\Delta/2)^{2}+\Gamma^{2}/4}\;.$
(12)

Here, $\Delta=(4t_{12}^{2}-\Gamma^{2})^{1/2}$ denotes a separation between the
bonding and anti-bonding energy levels, $E_{\pm}=\epsilon\mp\Delta/2$. Now, in
the stationary limit $T_{2DQ}^{s}(E)$ has two asymmetric resonance peaks at
$E_{\pm}$ with the broadening $\Gamma/2$. It is in contrast to the weak
coupling case, Eq. (7), where two conducting channels have different
broadenings $\Gamma_{\pm}$ but the resonance is at $E=\epsilon$. Relaxation
processes are different for the strong dot-dot coupling. The transmission
evolves with a single relaxation time $\tau^{osc}\equiv 2/\Gamma$ which
describes coherent electron transfers to the external electrodes. Now, the
dot-electrode charge oscillations are more pronounced, which lead to a change
in the sign of the transmission and consequently, in a change in the direction
of the current flow, see Fig. 3. The strongest changes of $T(E,t)$ one can
observe for $E=0$, i.e. in the valley between bonding and anti-bonding states
when destructive interference takes place (in contrast with the previous case
for the weak dot-dot coupling). It is clear that at $E=0$ the major changes
are seen in the 1QD components,
$T_{1QD}^{osc,\pm}(t)=[\cos(t\Delta/2)-(\Gamma/\Delta)\sin(t\Delta/2)]/2$,
while bonding and anti-bonding states also contribute equally,
$T_{2QD}^{osc,\pm}(t)=(\Gamma/\Delta)\sin(t\Delta/2)$, but play a minor role,
see Fig. 3b). Formation of the anti-bonding state at
$E=E_{-}=\epsilon+\Delta/2$ is presented in Fig. 3c). It is caused mainly by
an exponential decay of the negative component $T_{2QD}^{osc,-}(t)$ while the
other components give small oscillations.

In the spectral function $A(E,t)$ one finds two components:

$A(E,t)=A^{osc}(E,t)e^{-t/\tau^{osc}}+A^{Rabi}(E,t)e^{-t/\tau}\;.$ (13)

Apart from the aforementioned charge oscillations there are also rapid Rabi
oscillations caused by coherent electron transfer between the 2QD states, see
Fig. 4. These components are proportional to $\sin(t\Delta/2)$ and
$\cos(t\Delta/2)$. The frequency of the Rabi oscillations
$\omega^{Rabi}=2\pi\Delta/2$ can be tuned by changing dot-dot coupling
$t_{12}$. In general, the Rabi component $A^{Rabi}(E,t)$ gives a negative
contribution to the spectral function $A(E,t)$ and vanish with a relaxation
time $\tau\equiv 1/\Gamma$, i.e. twice as quick as charge oscillation
component $A^{osc}(E,t)$, see Fig. 4 and for details Appendix.

##  IV Conclusions and outlook

In the paper, we studied a role of quantum interference in time formation of
the Fano resonance in the 2QD-T system. We assumed that at the initial time
$t=0$ the 1QD system, with a single QD coupled with the electrodes, was in a
stationary state, and next, the second QD was suddenly side attached to the
first QD. Transient tunneling processes were analyzed, in particular the time
evolution of transmission, $T(E,t)$ and local charge relaxation (described by
the local density $A(E,t)$ which is related with entanglement). We found two
regimes with different dynamics depending on the inter-dot coupling with
respect to the dot-electrode coupling. For the weak dot-dot coupling formation
of the Fano anti-resonance in $T(E,t)$ is governed by two channel tunnelling
processes with two characteristic relaxation times. In contrast, for the
strong dot-dot coupling $T(E,t)$ evolves with a single relaxation time to two
peak structure which corresponds to resonance transport through bonding and
anti-bonding states. One founds large charge oscillations between the
bonding/anti-bonding states and the electrodes, which can even change
direction of the transient current. Additionally, in the time evolution of
$A(E,t)$ one observes interplay of rapid Rabi oscillations (between the
bonding and the anti-bonding state) and charge oscillations to the electrodes.

It will be interesting to extend our time dependent studies of charging
effects and entanglement, analyzing entropy flow and showing various
thermalization processes caused by quantum coherence which one expects will be
different in the weak and strong coupling regime. Another fascinating
perspective will be the analysis of the current-current correlations in the
transient regime, e.g.: to determine the dominant microscopic relaxation
processes at different time scales; to show how quantum interference affect
short-time fluctuations.

## Acknowledgements

The research was financed by the National Science Centre, Poland - project
number 2016/21/B/ST3/02160.

##  Appendix A Appendix

The Appendix gives the explicit form of the components of the density function
$A(E,t)$.

###  A.1 Weak dot-dot coupling

The components are separated $A^{g}(E)=A_{1QD}^{g}(E)+A_{2QD}^{g}(E)$,
$A^{\nu}(E)=A_{1QD}^{\nu}(E)+A_{2QD}^{\nu}(E)$ and
$A^{osc,\nu}(E,t)=A_{1QD}^{osc,\nu}(E,t)+A_{2QD}^{osc,\nu}(E,t)$ to show their
time evolution from the 1QD to the 2QD system. These components are expressed
as:

$\displaystyle
A^{g}_{1QD}(E)=-\frac{\Gamma_{+}\Gamma_{-}}{2\delta^{2}}\frac{4\Gamma^{2}+\Gamma_{+}^{2}+\Gamma_{-}^{2}}{9\Gamma^{2}-\delta^{2}}\frac{\Gamma^{2}}{\mathcal{E}^{2}+\Gamma^{2}}\;,$
(A.1) $\displaystyle
A^{g}_{2QD}(E)=\frac{\Gamma}{2\delta^{2}}\sum_{\nu=\pm}\frac{\Gamma_{-}\Gamma_{+}}{2\Gamma+\Gamma_{\nu}}\frac{\Gamma_{\nu}^{2}}{\mathcal{E}^{2}+\Gamma_{\nu}^{2}/4}\;,$
(A.2) $\displaystyle
A_{1QD}^{\nu}(E)=\frac{\Gamma^{2}}{2\delta^{2}}\frac{\Gamma_{-}\Gamma_{+}}{2\Gamma+\Gamma_{\nu}}\frac{\Gamma_{\nu}}{\mathcal{E}^{2}+\Gamma^{2}}\;,$
(A.3) $\displaystyle
A_{2QD}^{\nu}(E)=-\frac{\Gamma^{2}}{2\delta^{2}}\frac{\Gamma_{-}\Gamma_{+}}{2\Gamma+\Gamma_{\nu}}\frac{\Gamma_{\nu}}{\mathcal{E}^{2}+\Gamma_{\nu}^{2}/4}\;,$
(A.4) $\displaystyle
A_{1QD}^{osc,\nu}(E,t)=\nu\frac{\Gamma}{\delta}\frac{\Gamma_{-}\Gamma_{+}\Gamma_{\nu}}{9\Gamma^{2}-\delta^{2}}\frac{\Gamma\cos(\mathcal{E}t)-\mathcal{E}\sin(\mathcal{E}t)}{\mathcal{E}^{2}+\Gamma^{2}}\;,$
(A.5) $\displaystyle
A_{2QD}^{osc,\nu}(E,t)=\frac{\Gamma^{2}}{2\delta^{2}}\frac{\Gamma_{-}\Gamma_{+}}{2\Gamma+\Gamma_{\nu}}\frac{\Gamma_{\nu}\cos(\mathcal{E}t)+2\mathcal{E}\sin(\mathcal{E}t)}{\mathcal{E}^{2}+\Gamma_{\nu}^{2}/4}$
$\displaystyle-\frac{\Gamma}{2\delta^{2}}\frac{\Gamma_{\nu}\Gamma_{\bar{\nu}}^{2}}{2\Gamma+\Gamma_{\bar{\nu}}}\frac{\Gamma_{\bar{\nu}}\cos(\mathcal{E}t)+2\mathcal{E}\sin(\mathcal{E}t)}{\mathcal{E}^{2}+\Gamma_{\bar{\nu}}^{2}/4}\;,$
(A.6)

where $\bar{\nu}\equiv-\nu$.

###  A.2 Strong dot-dot coupling

In the strong coupling limit the components corresponding to the Rabi
oscillations and related with the oscillations to the electrodes, are
separated $A^{Rabi}(E,t)=A_{1QD}^{Rabi}(E,t)+A_{2QD}^{Rabi}(E,t)$ and
$A^{osc}(E,t)=A_{1QD}^{osc}(E,t)+A_{2QD}^{osc}(E,t)$. Their components are
given by:

$\displaystyle
A^{Rabi}_{1QD}(E,t)=\frac{\Gamma^{2}}{\Delta^{2}}\frac{\Gamma^{2}+\Delta^{2}}{9\Gamma^{2}+\Delta^{2}}\Big{[}\frac{3\Gamma^{2}-\Delta^{2}}{\mathcal{E}^{2}+\Gamma^{2}}$
$\displaystyle-\frac{(3\Gamma^{2}+\Delta^{2})\cos(\Delta
t)+2\Gamma\Delta\sin(\Delta t)}{\mathcal{E}^{2}+\Gamma^{2}}\Big{]}\;,$ (A.7)
$\displaystyle
A^{Rabi}_{2QD}(E,t)=\frac{\Gamma^{2}}{2\Delta^{2}}\frac{\Gamma^{2}+\Delta^{2}}{9\Gamma^{2}+\Delta^{2}}\times$
$\displaystyle\sum_{\nu=\pm}\Big{\\{}\frac{-3\Gamma^{2}-4\nu\Delta\mathcal{E}-3\Delta^{2}}{(\mathcal{E}+\nu\Delta/2)^{2}+\Gamma^{2}/4}-\nu\frac{2\Gamma(3\mathcal{E}+2\nu\Delta)\sin(\Delta
t)}{(\mathcal{E}+\nu\Delta/2)^{2}+\Gamma^{2}/4}$
$\displaystyle+\frac{(3\Gamma^{2}-2\nu\Delta\mathcal{E}-\Delta^{2})\cos(\Delta
t)}{(\mathcal{E}+\nu\Delta/2)^{2}+\Gamma^{2}/4}\Big{\\}}\;,$ (A.8)
$\displaystyle
A^{osc}_{1QD}(E,t)=-\frac{\Gamma}{\Delta}\frac{\Gamma^{2}+\Delta^{2}}{9\Gamma^{2}+\Delta^{2}}\times$
$\displaystyle\sum_{\nu=\pm}\nu\Big{\\{}\frac{\Gamma(\mathcal{E}-\nu\Delta)\cos[(\mathcal{E}+\nu\Delta/2)t]}{\mathcal{E}^{2}+\Gamma^{2}}$
$\displaystyle+\frac{(\Gamma^{2}+\nu\Delta\mathcal{E})\sin[(\mathcal{E}+\nu\Delta/2)t]}{\mathcal{E}^{2}+\Gamma^{2}}\Big{\\}}\;,$
(A.9) $\displaystyle
A^{osc}_{2QD}(E,t)=\frac{\Gamma}{2\Delta^{2}}\frac{\Gamma^{2}+\Delta^{2}}{9\Gamma^{2}+\Delta^{2}}\times$
$\displaystyle\sum_{\nu=\pm}\Big{\\{}\frac{\Gamma(3\Gamma^{2}+4\nu\Delta\mathcal{E}+3\Delta^{2})\cos[(\mathcal{E}+\nu\Delta/2)t]}{(\mathcal{E}+\nu\Delta/2)^{2}+\Gamma^{2}/4}$
$\displaystyle+\nu\frac{[\Gamma^{2}\Delta+2\nu(3\Gamma^{2}+\Delta^{2})\mathcal{E}+\Delta^{3}]\sin[(\mathcal{E}+\nu\Delta/2)t]}{(\mathcal{E}+\nu\Delta/2)^{2}+\Gamma^{2}/4}$
$\displaystyle-\frac{\Gamma(3\Gamma^{2}-2\nu\Delta\mathcal{E}-\Delta^{2})\cos[(\mathcal{E}-\nu\Delta/2)t]}{(\mathcal{E}+\nu\Delta/2)^{2}+\Gamma^{2}/4}$
$\displaystyle-\frac{2\Gamma^{2}(3\mathcal{E}+2\nu\Delta)\sin[(\mathcal{E}-\nu\Delta/2)t]}{(\mathcal{E}+\nu\Delta/2)^{2}+\Gamma^{2}/4}\Big{\\}}\;.$
(A.10)

Notice that $A_{1QD}^{osc}(E,t)$ and $A_{2QD}^{osc}(E,t)$ have two components
related with charge oscillations to the bonding and anti-bonding states.

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# Exploring Bell nonlocality of quantum networks with stabilizing and logical
operators

Li-Yi Hsu  Department of Physics, Chung Yuan Christian University, Chungli,
32081, Taiwan  Ching-Hsu Chen  Department of Electrophysics, National Chiayi
University, Chiayi, 600, Taiwan

([; date; date; date; date)

###### Abstract

In practical quantum networks, a variety of multiqubit stabilized states
emitted from independent sources are distributed among the agents, and the
correlations across the entire network can be derived from each agent’s local
measurements on the shared composite quantum systems. To reveal the Bell
nonlocality in such cases as a quantum feature, minimal knowledge of the
emitted stabilizer state is required. Here, we demonstrate that knowing the
stabilizing and logical operators indeed provides a new way of exploring Bell
nonlocality in quantum networks. For the qubit distribution in quantum
networks, the associated nonlinear Bell inequalities are derived. On the other
hand, to violate these inequalities, one can design local incompatible
observables using minimal knowledge of the emitted states. The tilted
nonlinear Bell inequalities tailored for specific nonmaximal entangled
stabilizer states and a way of achieving the maximal violation are also
explored.

year number number identifier Date text]date

##  I Introduction

In the seminal work of John Bell, it was shown that the quantum correlations
arising from spatially separated systems can break the limits of classical
causal relations Bell ; EPR . In classical physics, the realism and locality
of spacelike events constrain the strength of classical correlations bounded
by the Bell inequalities. Quantum theory inconsistent with local realism
predicts stronger correlations that violate the Bell inequalities. Thanks to
quantum information science, two-particle and multi-particle quantum
correlations have been extensively investigated. As a distinct feature from
classical physics, Bell nonlocality has led to applications in quantum
information processing including private random number generation random ,
quantum cryptography DIC , and reductions in communication complexity
complexity .

In typical Bell experiments on statistical correlations, a source emits a
state comprising two or more particles that are shared between two or more
distant observers, who each perform local measurements with a random chosen
setting and then obtain the measurement outcomes. To reveal the Bell
nonlocality of an entangled state, the local observables should be set
deliberately. For example, to violate the Bell inequalities tailored to
stabilizer states such as graph states, it is helpful to take the stabilizers
as a reference for finding suitable measurement settings gf1 ; gf2 ; gf3 . To
emphasize the role of stabilizing operators and logical bit-flip operators, we
review the Clauser-Horne-Shimony-Holt (CHSH) inequality and its modification
below. Let the two-qubit entangled state ($0<\phi<\frac{\pi}{2}$)

$\left|\phi\right\rangle=\cos\phi\left|\overline{0}\right\rangle+\sin\phi\left|\overline{1}\right\rangle$
(1)

be the codeword of the [[2, 1, 2]] stabilizer-based quantum error-detecting
code with the logical states
$\left|\overline{0}\right\rangle=\left|00\right\rangle$ and
$\left|\overline{1}\right\rangle=\left|11\right\rangle$. The stabilizer
generator, logical bit-flip, and phase-flip operators are $\sigma_{z}\otimes$
$\sigma_{z}$, $\sigma_{x}\otimes$ $\sigma_{x}$, and $I\otimes\sigma_{z}$,
respectively ($\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$ are Pauli matrices
and $I$ is the identity operator). In the bipartite Bell test, the CHSH
operator is $\mathbf{B}_{CHSH}=\sum\nolimits_{i,\text{
}j=0}^{1}(-1)^{ij}A_{i}\otimes B_{j}$, where $A_{i}$ and $B_{j}$ are local
observables, and the CHSH inequality states that
$\left\langle\mathbf{B}_{CHSH}\right\rangle\overset{c}{\leq}2$, where the
$\left\langle\cdot\right\rangle$ denotes the expectation value of $\cdot.$ To
violate the CHSH inequality, for the first-qubit, we assign

$\sigma_{z}\rightarrow\frac{1}{2\cos\mu}(A_{0}+A_{1})\text{,
}\sigma_{x}\rightarrow\frac{1}{2\sin\mu}(A_{0}-A_{1}),$ (2)

and for the second-qubit, we assign

$\sigma_{z}\rightarrow B_{0}\text{, }\sigma_{x}\rightarrow B_{1\text{ }}$ (3)

It is easy to verify that

$\displaystyle\left\langle\phi\left|\mathbf{B}_{CHSH}\right|\phi\right\rangle$
$\displaystyle=2\cos\mu\left\langle\phi\left|\sigma_{z}\otimes\sigma_{z}\right|\phi\right\rangle+2\sin\mu\left\langle\phi\left|\sigma_{x}\otimes\sigma_{x}\right|\phi\right\rangle$
$\displaystyle=2(\cos\mu+\sin\mu\sin 2\phi)$
$\displaystyle=2\sqrt{1+\sin^{2}2\phi}\cos(\mu-\mu_{0})$ $\displaystyle\leq
2\sqrt{1+\sin^{2}2\phi},$ (4)

where $\tan\mu_{0}=\sin 2\phi$. Some remarks are in order. First, the
operators $(A_{0}+A_{1})\otimes B_{0}=\cos\mu\sigma_{z}\otimes\sigma_{z}$ and
$(A_{0}-A_{1})\otimes B_{1}=\sin\mu\sigma_{x}\otimes\sigma_{x}$, and hence
both terms in the first equality in (4) exemplify the usefulness of the
stabilizing operator and the logical (bit-flip) operator for finding the local
observables that violate the CHSH inequality. The maximal CHSH value larger
than two can be achieved by setting $\mu=\mu_{0}$. In particular,
$\left|\phi=\frac{\pi}{4}\right\rangle$ is maximally entangled,
$\sigma_{x}\otimes$ $\sigma_{x}$ becomes another stabilizing operator rather
than simply a logical bit-flip operator, and the CHSH inequality can be
maximally violated. There are a variety of Bell inequalities violated by the
graph states as a specific family of stabilizer states, where the associated
Bell operators can be reformulated as the sum of their stabilizing operators,
and hence the perfect/antiperfect correlations therein can reach the maximal
violation gf2 ; gf3 ; gs2 ; gs3 ; gs4 ; gs5 . If the multi-qubits mixed states
involve two stabilizing operators, their nonlocality can be verified by the
violation of stabilizer-based Bell-type inequalities mix1 ; mix2 . Second,
observables $A_{0}$ and $A_{1}$ can be regarded as the results of “cutting and
mixing” the stabilizing operator and the logical bit-flip operator into the
local observables:

$\sigma_{z}\otimes\sigma_{z},\sigma_{x}\otimes\sigma_{x}\overset{\text{cutting}}{\rightarrow}\sigma_{z},\sigma_{x}\overset{\text{mixing}}{\rightarrow}A_{x_{i}}=\cos\mu\sigma_{z}+(-1)^{{}_{x_{i}}}\sin\mu\sigma_{x}.$
(5)

Cutting means cutting the first qubit observables from
$\sigma_{z}\otimes\sigma_{z}$ and $\sigma_{x}\otimes\sigma_{x}$ ; mixing means
linearly superposing the two cut observables. In what follows, the local
observables on the source side will be constructed in a similar way. In
addition, note that the local observables anticommute; i.e.,
$\left\\{A_{0}\text{, }A_{1}\right\\}=\left\\{B_{0}\text{, }B_{1}\right\\}=0$.
Third, the observable $B_{0}$ is the phase-flip operator, which can be
exploited in the tilted CHSH operators $\mathbf{B}_{\beta\text{-}CHSH}=\beta
B_{0}+$ $\mathbf{B}_{CHSH}$ with the Bell inequality
$\left\langle\mathbf{B}_{\beta\text{-}CHSH}\right\rangle\overset{c}{\leq}2+\beta$.
By setting $\beta=2/\sqrt{1+2\tan^{2}2\phi}$, the maximal violation can be
achieved by $\left|\phi\right\rangle$ non1 ; non2 . That is, the nonmaximally
entangled state can maximally violate the tilted Bell inequalities involving
the phase-flip operators.

One can generalize the above approach to the multi-qubit case. For example,
let
$\left|\psi\right\rangle=\cos\psi\left|\overline{0}\right\rangle+\sin\psi\left|\overline{1}\right\rangle$
be the codeword of the [[$5$, $1$, $3$]] stabilizer-based quantum error-
correcting code (SQECC), and let Alice hold the first qubit and Bob hold the
other qubits. In this case, the useful stabilizing operator and logical bit-
flip operator are $g=\sigma_{z}\otimes\sigma_{z}\otimes\sigma_{x}\otimes
I\otimes\sigma_{x}$ and $\sigma_{x}^{\otimes 5}$, respectively. The
observables of the first and second qubits of the stabilizing operators and
the bit-flip operator are the same as those in (2) and (3), respectively. The
last three qubits are termed idle qubits since the observable $\sigma_{x}$ is
always measured on them, while the outcome of the fourth qubit is discarded if
Bob measures $B_{1}^{\prime}$. In the Bell test, Alice randomly measures one
of the observables in (2), whereas Bob randomly measures
$B_{0}^{\prime}=B_{0}\otimes\sigma_{x}^{\otimes 3}$ or
$B_{1}^{\prime}=B_{1}\otimes\sigma_{x}\otimes I\otimes\sigma_{x}$. The CHSH-
like inequality can be written as $\left\langle\sum\nolimits_{i,\text{
}j=0}^{1}(-1)^{ij}A_{i}\otimes B_{j}^{\prime}\right\rangle\overset{c}{\leq}2.$
As another example, let the logical states be
$\left|\overline{0}\right\rangle=\left|0\right\rangle^{\otimes n}$ and
$\left|\overline{1}\right\rangle=\left|1\right\rangle^{\otimes n}$, and denote
the $n$-qubit Greenberger–Horne–Zeilinger (GHZ) state ($n\geq 3$) as
$\left|GHZ_{\phi}\right\rangle=\cos\phi\left|\overline{0}\right\rangle+\sin\phi\left|\overline{1}\right\rangle$.
Alice holds the first $m$ qubits and Bob holds the other $n-m$ qubits. Here,
the useful stabilizing, logical bit-flip and phase-flip operators can be set
as $\sigma_{z}\otimes I^{\otimes m-1}\otimes\sigma_{z}\otimes I^{\otimes
n-m-1}$, $\sigma_{x}^{\otimes n}$, and $I^{\otimes m}\otimes\sigma_{z}\otimes
I^{\otimes n-m-1}$, respectively. One can construct the local observables
$A_{i}^{\prime\prime}=\cos\mu\sigma_{z}\otimes I^{\otimes
m-1}+(-1)^{i}\sin\mu\sigma_{x}^{\otimes m}$ using the cutting-and-mixing
scheme, and, $B_{j}^{\prime\prime}=(1-j)\sigma_{z}\otimes I^{\otimes
n-m-1}+j\sigma_{x}^{\otimes n-m}$ such that
$\left\\{A_{0}^{\prime\prime}\text{,
}A_{1}^{\prime\prime}\right\\}=\left\\{B_{0}^{\prime\prime}\text{,
}B_{1}^{\prime\prime}\right\\}=0$. Hence, we reach the CHSH-like inequality
$\left\langle\sum\nolimits_{i,\text{
}j=0}^{1}(-1)^{ij}A_{i}^{\prime\prime}\otimes
B_{j}^{\prime\prime}\right\rangle\overset{c}{\leq}2$. Here, the last $n-m-1$
qubits are idle since only the observable $\sigma_{x}$ is always measured on
each of them. Conditional on the qubit assignment, one can select the useful
stabilizing operators and logical operators to derive similar CHSH-like
inequalities.

Recently, Bell nonlocality in quantum networks as the generalized Bell
experiments have attracted much research attention. Long-distance quantum
networks of large-scale multi-users are the substantial goals of quantum
communication, so it is fundamental to study their nonlocal capacities. A
quantum network involves multiple independent quantum sources, where each of
them initially emits a two- or multi-qubit entangled state shared among a set
of observers or agents. There are several obstacles to the study of
nonlocality in a quantum network. The classical correlations of a network
indicate more sophisticated causal relations and lead to stronger constraints
than those in the typical (one-source) Bell scenario. In addition, each
observer can perform a joint measurement on the qubits at hand, which could
result in strong correlations across the network. Unlike the typical linear
Bell-type inequalities, most Bell-type inequalities for various classes of
networks are nonlinear, which implies the nonconvexity of the multipartite
correlation space star ; noncyclic ; Poly ; luo1 . In the two-source case as
the simplest quantum network, bilocal and nonlocal correlations were
thoroughly investigated bilocal1 ; bilocal2 . Next the Bell-type inequalities
for star-shaped and noncyclic networks were studied star ; noncyclic .
Recently, broader classes of quantum networks based on locally causal
structures have also been investigated Poly ; Poly1 ; luo1 ; luo2 . In
particular, computationally efficient algorithms for constructing Bell
inequalities have been proposed luo1 . These Bell-type inequalities are
tailored for networks with quantum sources emitting either two-qubit Bell
states or generalized GHZ states. The stabilizing operators and logical
operators implicitly play substantial roles in setting up local joint
observables. On the other hand, regarding the potential applications of
encrypted communication in quantum networks, stabilizer quantum error-
correcting codes (SQECCs) can be more useful in quantum network cryptography,
such as in quantum secret sharing QSS  and secure quantum key distribution
protocols Preskill . Revealing Bell nonlocality in a network is required for
detecting potential eavesdropping attacks. Moreover, a structured quantum
state with stabilizing operators and logical operators is more useful in the
engineering of quantum networks percolation ; memory ; repeater1 ; QSS1 ; QSS2
.

In this work, we study the Bell nonlocality of quantum networks. Hereafter, we
consider a $K$-locality network $\mathcal{N}$ as shown in Fig. (1). There are
$K+M$ agents of which $K$, $\mathcal{S}^{(1)}$…$\mathcal{S}^{(K)}$, are on the
source side and $M$, $\mathcal{R}^{(1)}$…$\mathcal{R}^{(M)}$, on the receiver
side. There are $N$ independent sources. Let $0=e_{0}<e_{1}<\cdots<e_{K}=N$.
The agent $\mathcal{S}^{(s)}$ holds the sources $e_{s-1}+1$,… , and $e_{s}$;
thus the number of sources that $\mathcal{S}^{(s)}$ holds is
$(e_{s}-e_{s-1})$. The source $i$ ($e_{s-1}+1\leq i\leq e_{s}$) held by
$\mathcal{S}^{(s)}$ emits $n_{i}$ particles of which $n_{i}^{(0)}$ ($\neq 0$)
are in possession of the agent $\mathcal{S}^{(s)\text{ }}$and
$n_{i}^{(m)}$($\geq$ $0$) are sent to the agent $\mathcal{R}^{(m)}$.
Consequently, $\sum_{j=0}^{M}n_{i}^{(j)}=n_{i}$. In the classical networks,
source $i$ emits $n_{i}$ particles described by hidden state $\lambda_{i}$; in
the quantum networks, it emits $n_{i}$ qubits in quantum state
$\left|\psi_{(i)}\right\rangle$, which can be either a stabilizer state or a
codeword of a SQECC. In the Bell test, observer
$\mathcal{S}^{(s)}$($\mathcal{R}^{(m)}$) measures observable $A_{x_{s}}^{(s)}$
($B_{y_{m}}^{(m)}$) with the associated outcome $a_{s}$ ($b_{m}$), where
$x_{s}$, $y_{m}\in\\{0,1\\}$ and $a_{s}$, $b_{m}\in\\{-1,1\\}$. In the
following, the index pair $(i,$ $j)$ denotes the $j$-th particle emitted from
source $i$, and $(i$, $j)\rightarrow\mathcal{X}^{(k)}$ indicates that particle
$(i$, $j)$ is at agent $\mathcal{X}^{(k)}$’s hand
($\mathcal{X}\in\\{\mathcal{R}$, $\mathcal{S}\\}$). Finally, we denote the
particle sets
$\mathbb{S}^{(k)}=\\{(i,j)|\forall(i,j)\rightarrow\mathcal{S}^{(k)}\\}$ and
$\mathbb{R}^{(k)}=\\{(i,j)|\forall(i,j)\rightarrow\mathcal{R}^{(k)}\\}$.

The remainder of this paper is organized as follows: In Sec. II, we
investigate the classical networks, where general local causal models (GLCMs)
are introduced. Then, Bell inequalities associated with classical networks are
proposed. We study Bell nonlocality of $K$-locality quantum networks in Sec.
III. First we review the stabilizer states and SQECCs. Then we demonstrate how
to violate the proposed Bell inequalities using deliberated local observables.
It will be shown that, the local observables can be made up of “cut-graft-
mixing” stabilizing operators and logical operators. Notably, there are two
alternative nonlocal correlations. One is due to the entanglement of the
logical states themselves, where two suitable stabilizing operators are
exploited to construct the local observables. The other nonlocality results
from the entanglement due to the superposition of logical states, in which a
stabilizing operator and a logical operator can be suitably exploited in this
case. We illustrate these two kinds of nonlocality in terms of the 5-qubit
code, which is the smallest stabilizer code that protects against single-qubit
errors. In Sec. IV, we propose Bell inequalities tailored for nonmaximally
entangled states distributed in a quantum network. Finally, conclusions are
drawn in Sec. V.

##  II Classical networks

###  II.1 General local causal models

The general local causal models (GLCMs) in classical networks can be described
as follows: The $i$-th source is associated with a local random variable as
the hidden state $\lambda_{i}$ in the measure space $(\Omega_{i}$,
$\Sigma_{i}$, $\mu_{i})$ with the normalization condition
$\int_{\Omega_{i}}d\mu_{i}(\lambda_{i})\ =1$. All systems in the Bell test
scenario are considered in the hidden state $\Lambda=(\lambda_{i},\cdots,$
$\lambda_{N})$ in the measure space $(\Omega$, $\Sigma$, $\mu)$, where
$\Omega=$ $\Omega_{1}\otimes\cdots\otimes\Omega_{N}$ and the measure of
$\Lambda$ is given by $\mu(\Lambda)=\prod\mu_{i}(\lambda_{i})$ with the
normalization condition $\int_{\Omega}d\mu(\Lambda)=1$. In the measurement
phase, agent $\mathcal{S}^{(s)}$ performs the measurement $A_{x_{s}}^{(s)}$ on
state $\Lambda_{S}^{(s)}$ with the corresponding outcome denoted by
$a_{s}\in\\{1,-1\\}$. The GLCM suggests a joint conditional probability
distribution of the measurement outcomes

$P(\mathbf{a}|\mathbf{x})=\int_{\Omega}d\mu(\Lambda)\prod_{s=1}^{K}P(a_{s}|x_{s},\Lambda),$
(6)

where $\mathbf{a}=(a_{1},a_{2}$, $\cdots$, $a_{K})$ and
$\mathbf{x}=(x_{1},x_{2}$, $\cdots$, $x_{K})$; hence, we have the correlation

$\left\langle A_{x_{1}}^{(1)}\ldots
A_{x_{K}}^{(K)}\right\rangle=\sum\nolimits_{\mathbf{a}}a_{1}\cdots
a_{K}P(\mathbf{a}|\mathbf{x},\Lambda).$ (7)

On the other hand, in the $K$-locality condition luo1 , $\mathcal{S}^{(s)}$
can access the hidden state $\Lambda_{S}^{(s)}=(\lambda_{e_{s-1}+1}$,$\cdots$,
$\lambda_{e_{s}})$ in the measure space
$\Omega_{S}^{(s)}=\Omega_{e_{s-1}+1}\otimes\cdots\otimes\Omega_{e_{s}}$, where

$\Lambda_{S}^{(s)}\cap\Lambda_{S}^{(s^{\prime})}=\emptyset\text{
}\Leftrightarrow\text{ }s\neq s^{\prime}$ (8)

and

$\cup_{i=1}^{K}\Lambda_{S}^{(i)}=\Lambda.$ (9)

Eq. (6) can be rewritten as

$P(\mathbf{a}|\mathbf{x})=\prod_{s=1}^{K}\int_{\Omega_{S}^{(s)}}d\mu_{s}(\Lambda_{S}^{(s)})P(a_{s}|x_{s}\text{,
}\Lambda_{S}^{(s)}),$ (10)

Denote the local expectation as

$\left\langle
A_{x_{s}}^{(s)}\right\rangle=\sum_{a_{s}=-1,1}P(a_{s}|x_{s}\text{,
}\Lambda_{S}^{(s)}),$ (11)

By $K$-locality with the GLCM, we have

$\left\langle A_{x_{1}}^{(1)}\ldots
A_{x_{K}}^{(K)}\right\rangle=\prod_{s=1}^{K}\left\langle
A_{x_{s}}^{(s)}\right\rangle.$ (12)

Denote $\Delta^{\pm}A^{(i)}=\frac{1}{2}(\left\langle
A_{0}^{(i)}\right\rangle\pm\left\langle A_{1}^{(i)}\right\rangle)$. Since
$-1\leq\left\langle A_{x_{i}}^{(i)}\right\rangle\leq 1$, we have

$-1\leq\Delta^{-}A^{(i)},\Delta^{+}A^{(i)}\leq 1$ (13)

and

$\left|\Delta^{+}A^{(i)}\right|+\left|\Delta^{-}A^{(i)}\right|=\max\\{\left|\left\langle
A_{0}^{(i)}\right\rangle\right|,\left|\left\langle
A_{1}^{(i)}\right\rangle\right|\\}\leq 1.$ (14)

On the receiving side, let $n_{{}_{j}}^{(m)}>0$ if
$j\in\\{j_{1},\cdots,j_{{}_{k_{m}}}\\}$ and $n_{{}_{j}}^{(m)}=0$ otherwise. In
this case, $\mathcal{R}^{(j)}$ receives the hidden states
$\Lambda_{R}^{(m)}=(\lambda_{j_{1}}$,$\cdots$, $\lambda_{j_{k_{m}}})$ in the
measure space $\Omega_{R}^{(m)}=$
$\Omega_{j_{1}}\otimes\cdots\otimes\Omega_{j_{k_{m}}}$, where $1\leq
j_{1}<j_{2}...<j_{k_{m}}\leq N$. In the measurement phase, $\mathcal{R}^{(m)}$
performs the measurement $B_{y_{m}}^{(m)},$ $y_{m}\in\\{0,1\\}$, on the state
$\Lambda_{R}^{(m)}$ with the corresponding outcome denoted by
$b_{m}\in\\{1,-1\\}$. We have

$\displaystyle\left|\left\langle B_{y_{m}}^{(m)}\right\rangle\right|$
$\displaystyle=\left|\int_{\Omega_{R}^{(m)}}\prod_{k}d\mu_{k}(\lambda_{k})\sum_{b_{m}=-1,1}b_{m}P(b_{m}|y_{m}\text{,
}\Lambda_{R}^{(m)})\right|$ $\displaystyle\leq 1.$ (15)

###  II.2 Bell inequalities

The correlation strength in the proposed $K$-locality network is evaluated by
two quantities:

$\mathbf{I}_{K,M}=\frac{1}{2^{K}}\left\langle\prod_{i=1}^{K}\prod_{j=1}^{M}(A_{0}^{(i)}+A_{1}^{(i)})B_{0}^{(j)}\right\rangle$
(16)

and

$\mathbf{J}_{K,M}=\frac{1}{2^{K}}\left\langle\prod_{i=1}^{K}\prod_{j=1}^{M}(A_{0}^{(i)}-A_{1}^{(i)})B_{1}^{(j)}\right\rangle.$
(17)

In the classical scenario, we have

$\displaystyle\left|\mathbf{I}_{K,M}\right|_{GLCM}$
$\displaystyle=\frac{1}{2^{K}}\int_{\Omega}\prod_{i=1}^{K}\left|\left\langle
A_{0}^{(i)}+A_{1}^{(i)}\right\rangle\right|\prod_{j=1}^{M}\left|\left\langle
B_{0}^{(j)}\right\rangle\right|\prod_{k=1}^{N}d\mu_{k}(\lambda_{k})$
$\displaystyle\leq\int_{\Omega}\prod_{i=1}^{K}\left|\Delta^{+}A^{(i)}\right|\prod_{k=1}^{N}d\mu_{k}(\lambda_{k})$
(18)

and

$\displaystyle\left|\mathbf{J}_{K,M}\right|_{GLCM}$
$\displaystyle=\frac{1}{2^{K}}\int_{\Omega}\prod_{i=1}^{K}\left|\left\langle
A_{0}^{(i)}-A_{1}^{(i)}\right\rangle\right|\prod_{j=1}^{M}\left|\left\langle
B_{1}^{(j)}\right\rangle\right|\prod_{k=1}^{N}d\mu_{k}(\lambda_{k})$
$\displaystyle\leq\int_{\Omega}\prod_{i=1}^{K}\left|\Delta^{-}A^{(i)}\right|\prod_{k=1}^{N}d\mu_{k}(\lambda_{k}),$
(19)

where the inequalities are from (15). Before proceeding further, two useful
lemmas are introduced as follows:

Lemma 1 (Mahler inequality) Let $\alpha_{k}$ and $\beta_{k}$ be nonnegative
real numbers, and let $p$ $\in\mathbb{N}$; then,

$\prod\nolimits_{k=1}^{p}\alpha_{k}^{1/p}+\prod\nolimits_{k=1}^{p}\beta_{k}^{1/p}\leq\prod\nolimits_{k=1}^{p}(\alpha_{k}+\beta_{k})^{1/p}.$
(20)

The proof can be found in star .

We obtain the following nonlinear Bell inequality:

$\displaystyle\left|\mathbf{I}_{K,M}\right|_{GLCM}^{\frac{1}{K}}+\left|\mathbf{J}_{K,M}\right|_{GLCM}^{\frac{1}{K}}$
$\displaystyle\leq\\{\int_{\Omega}\prod_{i=1}^{K}(\left|\Delta^{+}A^{(i)}\right|+\left|\Delta^{-}A^{(i)}\right|)\prod_{k=1}^{N}d\mu_{k}(\lambda_{k})\\}^{\frac{1}{K}}$
$\displaystyle=\\{\int_{\Omega}\prod_{i=1}^{K}(\max\\{\left|\left\langle
A_{x_{i}=0}\right\rangle\right|,\text{ }\left|\left\langle
A_{x_{i}=1}\right\rangle\right|\\})\prod_{k=1}^{N}d\mu_{k}(\lambda_{k})\\}^{\frac{1}{K}}$
$\displaystyle\leq(\int_{\Omega}\prod_{j=k}^{N}d\mu_{k}(\lambda_{k}))^{\frac{1}{K}}=1,$
(21)

where the first inequality is from Lemma 1, and the fourth line is a
consequence of (14).

##  III Bell nonlocality of a quantum network

###  III.1 Review of stabilizer states and stabilizer-based quantum error-
correcting code

Let the state $\left|\psi_{(i)}\right\rangle$ emitted from the quantum source
$i$ be an $n_{i}$-qubit stabilizer state. By definition, an $n_{i}$-qubit
stabilizer state is one that is stabilized by a stabilizer which is a
nontrivial subgroup of the Pauli group. In particular, if
$\left|\psi_{(i)}\right\rangle$ as a codeword of [[$n_{i}$, $k_{i}$, $d_{i}$]]
SQECC, denote the last ($k_{i}$-th) logical qubit as
$\left|\overline{0}_{i}\right\rangle$ and$\
\left|\overline{1}_{i}\right\rangle$ and the corresponding logical bit- and
phase-flip operators as $\overline{X}_{(i)}^{\text{ }}$ and
$\overline{Z}_{(i)}^{\text{ }}$. Without loss of generality, we have

$\left|\psi_{(i)}\right\rangle=\sum_{z\in\\{0,1\\}^{\otimes
k_{i}}}a_{z}\left|\overline{z}\right\rangle=\cos\phi_{i}\left|\varphi_{i}^{0}\right\rangle\left|\overline{0}_{i}\right\rangle+\sin\phi_{i}\left|\varphi_{i}^{1}\right\rangle\left|\overline{1}_{i}\right\rangle,$
(22)

where $\left\langle\varphi_{i}^{0}|\varphi_{i}^{1}\right\rangle\in R$, and
$\left\langle\varphi_{i}^{0}|\varphi_{i}^{0}\right\rangle=\left\langle\varphi_{i}^{1}|\varphi_{i}^{1}\right\rangle=1$.
In addition,

$\overline{X}_{(i)}^{\text{
}}\left|\overline{0}_{i}\right\rangle=\left|\overline{1}_{i}\right\rangle,\overline{X}_{(i)}^{\text{
}}\left|\overline{1}_{i}\right\rangle=\left|\overline{0}_{i}\right\rangle,$
(23) $\overline{Z}_{(i)}^{\text{
}}\left|\overline{c}_{i}\right\rangle=(-1)^{c}\left|\overline{c}_{i}\right\rangle(c\in\\{0,1\\}).$
(24)

In what follows, we exploit $g_{(i)}$ and $g_{(i)}^{\prime}$ as useful
stabilizing operators, and $\overline{X}_{(i)}^{\text{ }}$ and
$\overline{Z}_{(i)}^{\text{ }}$ as useful logical operators. Denote the Pauli
set of qubit $(i,$ $j)$ as $\mathbb{P}_{(i,\text{ }j)}=\\{X_{(i,\text{ }j)}$,
$Y_{(i,\text{ }j)}$, $Z_{(i,\text{ }j)}$, $I_{(i,\text{ }j)}\\}$, where
$X_{(i,\text{ }j)}=\sigma_{x},$ $Y_{(i,\text{ }j)}=\sigma_{y},$ $Z_{(i,\text{
}j)}=\sigma_{z}$, and $I_{(i,\text{ }j)}=I$. Let $h_{(i)}$
$\in\\{\overline{X}_{(i)}^{\text{ }},$ $g_{(i)}^{\prime}\\}$ and
$h_{(i)}^{\prime}$ $\in\\{\overline{X}_{(i)}^{\text{ }},$
$\overline{Z}_{(i)}^{\text{ }}\\}$. Note that

$\left[g_{(i)}\text{, }h_{(i)}\right]=\left[g_{(i)}\text{,
}h_{(i)}^{\prime}\right]=0,$ (25)

and since $g_{(i)}$, $h_{(i)}$, and $h_{(i)}^{\prime}$ are $n_{i}$-fold tensor
products of the Pauli operators, we have

$g_{(i)}=\prod_{j=1}^{n_{i}}\widehat{s}_{(i,\text{
}j)},h_{(i)}=\prod_{j=1}^{n_{i}}\widehat{t}_{(i,\text{
}j)},h_{(i)}^{\prime}=\prod_{j=1}^{n_{i}}\widehat{t}_{(i,\text{
}j)}^{\prime},$ (26)

where $\widehat{s}_{(i,\text{ }j)}$, $\widehat{t}_{(i,\text{ }j)}$, and
$\widehat{t}_{(i,\text{ }j)}^{\prime}\in\mathbb{P}_{(i,\text{ }j)}$ $\forall
i$, $j.$

Before proceeding further, some notations are introduced as follows: Denote
the qubit index sets as $\mathbb{D}_{i}=\\{j|\\{\widehat{s}_{(i,\text{
}j)},\widehat{t}_{(i,\text{ }j)}\\}=0\\}$ and
$\mathbb{H}_{i}=\\{j|[\widehat{s}_{(i,\text{ }j)},\widehat{t}_{(i,\text{
}j)}]=0\\}$, where $\mathbb{D}_{i}\cap\mathbb{H}_{i}=\emptyset$, and
$\left|\mathbb{D}_{i}\right|+\left|\mathbb{H}_{i}\right|=n_{i}$. The qubits
belonging to the sets $\mathbb{D}_{1}$, $\cdots$, $\mathbb{D}_{N}$ play
substantial roles in the proposed Bell inequalities of the quantum networks.
Note that qubit $(i$, $j)$ is called idle if $(i$, $j)\in$ $\mathbb{H}_{i}$
and $(i$, $j)\rightarrow\mathcal{R}^{(k)}$ for some $k$, and the nonidentity
operator of $\widehat{s}_{(i,\text{ }j)}\ $or $\widehat{t}_{(i,\text{ }j)}$ is
denoted by $\widehat{o}_{(i,\text{ }j)}$, which will be repeatedly measured on
qubit $(i$, $j)$ in the Bell test. Let the indicator $\delta_{(i,\text{
}j)}^{D}=1$ if qubit $(i$, $j)\in$ $\mathbb{D}_{i}$, and $0$ if $(i$, $j)\in$
$\mathbb{H}_{i}$. According to (25), we have
$\sum_{j=1}^{n_{i}}\delta_{(i,\text{ }j)}^{D}\operatorname{mod}2=0,$ and
hence,

$\sum_{i=1}^{N}\sum_{j=1}^{n_{i}}\delta_{(i,\text{
}j)}^{D}\operatorname{mod}2=0.$ (27)

Here, we focus on quantum networks fulfilling the following conditions:
Regarding the qubits held by $\mathcal{S}^{(k)}$,

$\sum\nolimits_{i=e_{k-1}+1}^{e_{k}}\sum\nolimits_{j=1,(i,\text{
}j)\in\mathbb{S}^{(k)}}^{n_{i}}\delta_{(i,\text{
}j)}^{D}\operatorname{mod}2=1\text{ }\forall k=1,2,\cdots,K\text{,}$ (28)

and regarding the qubits held by $\mathcal{R}^{(k)}$,

$\sum\nolimits_{i,\text{ }j,(i,\text{ }j)\in\mathbb{R}^{(k)}}\delta_{(i,\text{
}j)}^{D}\operatorname{mod}2=1\text{ }\forall j=1,\cdots,M.$ (29)

Combining the constraints (27), (28), and (29), the value $K+M$ must be even.
According to the qubit distribution, it is very flexible to choose suitable
$g_{(i)}$, $h_{(i)}$, and $h_{(i)}^{\prime}$ to implement local observables.

###  III.2 Violation of the Bell inequalities in a quantum network

To implement the local observables on the source side, we assign

$\widehat{S}^{(k)}=\prod\nolimits_{i,\text{ }j,\text{ }(i,\text{
}j)\in\mathbb{S}^{(k)}}\widehat{s}_{(i,\text{
}j)}\rightarrow\frac{1}{2\cos\mu}(A_{0}^{(k)}+A_{1}^{(k)})$ (30)

and

$\widehat{T}^{(k)}=\prod\nolimits_{i^{\prime},\text{ }j^{\prime},\text{
}(i^{\prime},\text{
}j^{\prime})\in\mathbb{S}^{(k)}}\widehat{t}_{(i^{\prime},\text{
}j^{\prime})}\rightarrow\frac{1}{2\sin\mu}(A_{0}^{(k)}-A_{1}^{(k)}).$ (31)

Since $A_{x_{k}}^{(k)}=A_{x_{k}}^{(k){\dagger}}$and
$\left(A_{xi}^{(i)}\right)^{2}=A_{x_{i}}^{(i)}A_{xi}^{(i){\dagger}}=I$,
$A_{x_{k}}^{(k)}$ is a unitary Hermitian with an eigenvalue of either $1$ or
$-1$. On the receiving side, the local observables $B_{y_{l}}^{(l)}$ for
observer $\mathcal{R}^{(k)}$ are

$B_{y_{l}}^{(l)}=(1-y_{l})\prod\nolimits_{i,\text{ }j,\text{ }(i,\text{
}j)\in\mathbb{R}^{(l)}}\widehat{s}_{(i,\text{
}j)}+y_{l}\prod\nolimits_{i^{\prime},\text{ }j^{\prime},\text{
}(i^{\prime},\text{
}j^{\prime})\in\mathbb{R}^{(l)}}\widehat{t}_{(i^{\prime},\text{
}j^{\prime})},$ (32)

where $\left\\{B_{0}^{(k)}\text{, }B_{1}^{(k)}\right\\}=0$ according to (29).
In the measuring phase, $\mathcal{S}^{(k)}$ ($\mathcal{R}^{(k)}$ ) randomly
measures either the observable $A_{0}^{(k)}$ or $A_{1}^{(k)}$ ($B_{0}^{(k)}$
or $B_{1}^{(k)}$) with an outcome of either $1$ or $-1$. In practice, if the
qubit $(i^{\prime\prime},$ $j^{\prime\prime})\rightarrow\mathcal{R}^{(k)}$ is
idle, $\mathcal{R}^{(k)}$ can always measure the observable
$\widehat{o}_{(i^{\prime\prime},j^{\prime\prime})}$ in each round and discard
the outcome in postprocessing if $\widehat{s}_{(i^{\prime\prime},\text{
}j^{\prime\prime})}=I$ or $\widehat{t}_{(i^{\prime\prime},\text{
}j^{\prime\prime})}=I$.

To evaluate the correlation strength, let $\left|\overline{\Psi}\right\rangle$
$=\prod\nolimits_{i\text{ }}^{N}\otimes\left|\psi_{(i)}\right\rangle,$ and we
have

$\displaystyle\left|\mathbf{I}_{K,M}\right|_{Q}$
$\displaystyle=\frac{1}{2^{K}}\left|\sum_{x_{1},\cdots
x_{M}=0}^{1}\left\langle A_{x_{1}}^{(1)}\cdots
A_{x_{K}}^{(K)}\prod_{j=1}^{M}B_{0}^{(j)}\right\rangle\right|$
$\displaystyle=\frac{1}{2^{K}}\left|\left\langle\prod_{i=1}^{K}(A_{0}^{(i)}+A_{1}^{(i)})\prod_{j=1}^{M}B_{0}^{(j)}\right\rangle\right|$
$\displaystyle=\left|\left\langle\prod_{k=1}^{K}\cos\theta_{k}\prod\nolimits_{i,\text{
}j,\text{ }(i,\text{ }j)\in\mathbb{S}^{(k)}}\widehat{s}_{(i,\text{
}j)}\prod_{l=1}^{M}\prod\nolimits_{i^{\prime},\text{ }j^{\prime},\text{
}(i^{\prime},\text{
}j^{\prime})\in\mathbb{R}^{(l)}}\widehat{s}_{(i^{\prime},\text{
}j^{\prime})}\right\rangle\right|$
$\displaystyle=\left|\prod_{k=1}^{K}\cos\theta_{k}\left\langle\prod_{i=1}^{N}g_{(i)}\right\rangle\right|$
$\displaystyle=\left|\prod_{k=1}^{K}\cos\theta_{k}\right|,$ (33)

where
$\left\langle\cdot\right\rangle=\left\langle\overline{\Psi}\left|\cdot\right|\overline{\Psi}\right\rangle$
and hence $\left\langle\prod_{i=1}^{N}g_{(i)}\right\rangle=1$.

In addition,

$\displaystyle\left|\mathbf{J}_{K,M}\right|_{Q}$
$\displaystyle=\frac{1}{2^{K}}\left|\sum_{x_{1},\cdots x_{M}=0}^{1}(-1)^{\sum
x_{j}}\left\langle A_{x_{1}}^{(1)}\cdots
A_{x_{n}}^{(n)}\prod_{j=1}^{M}B_{1}^{(j)}\right\rangle\right|$
$\displaystyle=\frac{1}{2^{K}}\left|\left\langle\prod_{i=1}^{K}(A_{0}^{(i)}-A_{1}^{(i)})\prod_{j=1}^{M}B_{1}^{(j)}\right\rangle\right|$
$\displaystyle=\left|\left\langle\prod_{k=1}^{K}\sin\theta_{k}\prod\nolimits_{i,\text{
}j,\text{ }(i,\text{ }j)\in\mathbb{S}^{(k)}}\widehat{t}_{(i,\text{
}j)}\prod_{l=1}^{M}\prod\nolimits_{i^{\prime},\text{ }j^{\prime},\text{
}(i^{\prime},\text{
}j^{\prime})\in\mathbb{R}^{(l)}}\widehat{t}_{(i^{\prime},\text{
}j^{\prime})}\right\rangle\right|$
$\displaystyle=\left|\prod_{j=1}^{K}\sin\theta_{j}\prod_{i=1}^{N}\left\langle
h_{(i)}\right\rangle\right|.$ (34)

A useful lemma is introduced as follows:

Lemma 2 For any $\theta_{1},\theta_{2},\cdots,\theta_{K}\in[0,\frac{\pi}{2}]$,

$\left|\prod\nolimits_{j=1}^{K}\sin\theta_{j}\right|^{\frac{1}{K}}\leq\sin\overline{\theta}\text{
and
}\left|\prod\nolimits_{j=1}^{K}\cos\theta_{j}\right|^{\frac{1}{K}}\leq\cos\overline{\theta},$
(35)

where $\overline{\theta}=\frac{1}{K}\sum_{j=1}^{K}\theta_{j}$. The proof can
be found in luo1 .

Let $c_{i}=\left\langle h_{(i)}\right\rangle\leq 1$ and
$C=\left|\prod_{i=1}^{N}c_{i}\right|^{\frac{1}{K}}$. We have

$\displaystyle\left|\mathbf{I}_{K,M}\right|_{Q}^{\frac{1}{K}}+\left|\mathbf{J}_{K,M}\right|_{Q}^{\frac{1}{K}}$
$\displaystyle=\left|\prod\nolimits_{k=1}^{K}\cos\theta_{j}\right|^{\frac{1}{K}}+\left|\prod\nolimits_{k=1}^{K}\sin\theta_{j}\right|^{\frac{1}{K}}\left|C\right|$
$\displaystyle\leq\left|\cos\overline{\theta}\right|+\left|\sin\overline{\theta}\right|\left|C\right|$
$\displaystyle\leq\sqrt{1+C^{2}},$ (36)

where the first inequality in (36) follows from Lemma 2 and the second
inequality follows from the Cauchy-Schwarz inequality. To reach the maximal
violation, by setting $\theta_{1}=\ldots=\theta_{K}=\overline{\theta}$, and
$\tan\overline{\theta}=C$, we obtain

$\max(\left|\mathbf{I}_{K,M}\right|_{Q}^{\frac{1}{K}}+\left|\mathbf{J}_{K,M}\right|_{Q}^{\frac{1}{K}})=\sqrt{1+C^{2}}>1.$
(37)

Some discussion is in order. The state $\left|\psi_{(i)}\right\rangle$ can
contribute to the nonlocality of the quantum network through different facets
of its nonlocality. Loosely speaking, there are $C_{2}^{2^{n_{i}-k_{i}}}$ ways
of choosing suitable $\ g_{(i)}$ and $g_{(i)}^{\prime}$ and
$k_{i}(2^{2(n_{i}-k_{i})})$ ways of choosing suitable $\ g_{(i)}$ and
$\overline{X}_{(i)}^{\text{ }}$, which reflect the flexibility of testing Bell
nonlocality in the quantum networks. The maximum value of $C=1$
($\overline{\theta}=\frac{\pi}{4}$), and hence
$\max(\left|I_{K,M}\right|_{Q}^{\frac{1}{K}}+\left|J_{K,M}\right|_{Q}^{\frac{1}{K}})=\sqrt{2}$,
can be reached if $c_{i}=1$ $\forall$ $i$. That is, $h_{(i)}$ stabilizes
$\left|\psi_{(i)}\right\rangle$ $\forall$ $i$. As shown in (30) and (31), one
can benefit from the stabilizing operators in designing local observables to
achieve the maximal violations of Bell inequalities in $K$-locality quantum
networks. In detail, $c_{i}=1$ if $h_{(i)}=$ $\overline{X}_{(i)}^{\text{ }}$,
$\left|\left\langle\varphi_{i}^{0}|\varphi_{i}^{1}\right\rangle\right|=1$, and
$\phi_{i}=\frac{\pi}{4}$. The Bell nonlocality here is due to the
superposition of $\left|\overline{0}_{i}\right\rangle$ and
$\left|\overline{1}_{i}\right\rangle$; if $h_{(i)}=$ $g_{(i)}^{\prime}$,
$c_{i}$ is certain to be $1$. The Bell nonlocality in a network is due to that
of either the stabilizer state or the logical states of the stabilizer code
[[$n_{i}$, $k_{i}$, $d_{i}$]]. Note that such nonlocality involving the
stabilizing operators $g_{(i)}$ and $g_{(i)}^{\prime}$ can be also obtained
even by using specific states stabilized by the same stabilizing operators.
For example, given source $i$ emitting a four-qubit state with stabilizing
operators $g_{(i)}=$ $\sigma_{z}^{\otimes 4}$ and $g_{(i)}^{\prime}=$
$\sigma_{x}^{\otimes 4}$, which both can also stabilize the 4-qubit Smolin
state
$\rho_{\text{Smolin}}=\frac{1}{4}\sum\nolimits_{ij=0,1}\left|\Psi_{ij}\right\rangle\left\langle\Psi_{ij}\right|$,
where
$\left|\Psi_{ij}\right\rangle=\sigma_{z}^{i}\otimes\sigma_{x}^{j}[\frac{1}{\sqrt{2}}(\left|00\right\rangle+\left|11\right\rangle)]$.
Eventually, Bell nonlocality of at most $\left|\mathbb{D}_{i}\right|$-qubit
entanglement in $\left|\psi_{(i)}\right\rangle$ is involved.

As an illustration, let $K=N=2$ and $M=1$, and let sources 1 and 2 each emit
the codeword states of $[[5$, $1$, $3]]$ SQECC with four stabilizer generators
(the subscript qubit index $(i,$ $j)$ is shortened as $j$)

$\displaystyle\mathfrak{g}_{1}$
$\displaystyle=X_{1}Z_{2}Z_{3}X_{4}I_{5},\mathfrak{g}_{2}=I_{1}X_{2}Z_{3}Z_{4}X_{5},$
$\displaystyle\mathfrak{g}_{3}$
$\displaystyle=X_{1}I_{2}X_{3}Z_{4}Z_{5},\mathfrak{g}_{4}=Z_{1}X_{2}I_{3}X_{4}Z_{5}.$
(38)

The well-known logical bit-flip and phase-flip operators are
$\overline{X}^{\prime\text{ }}=\prod\nolimits_{i=1}^{5}X_{i}$ and
$\overline{Z}^{\prime\text{ }}=\prod\nolimits_{i=1}^{5}Z_{i}$. Observers
$\mathcal{S}^{(1)}$ and $\mathcal{S}^{(2)}$ hold qubits $(1,$ $1)$ and $(2,$
$1)$, respectively, while observer $\mathcal{R}^{(k)}$ holds the other 8
qubits, as shown in Fig. (2). Note that, in the following examples, the
bilocal inequality is

$\sqrt{\mathbf{I}_{2,1}}+\sqrt{\mathbf{J}_{2,1}}\leq 1,$ (39)

which is exactly the bilocal inequality for binary inputs and outputs in
bilocal1 .

Example (a): Let
$\left|\psi_{(1)}\right\rangle=\left|\psi_{(2)}\right\rangle=\cos\phi\left|\overline{0}\right\rangle+\sin\phi\left|\overline{1}\right\rangle$.
Here, we choose the useful operators
$g_{(i)}=\mathfrak{g}_{1}\mathfrak{g}_{2}\mathfrak{g}_{3}\mathfrak{g}_{4}=Z_{1}Z_{2}X_{3}I_{4}X_{5}$
and $h_{(i)}=\overline{X}^{\text{ }}=\overline{X}^{\prime}$. In this case,
$\left|\mathbb{D}_{i}\right|=2$, and $(i,$ $3)$, $(i,$ $4)$, $(i,$ $5)$ are
idle qubits.
$A_{x_{i}}^{(i)}=\frac{1}{\sqrt{2}}(Z_{(i,1)}+(-1)^{x_{i}}X_{(i,1)})$,
$B_{0}^{(1)}=\prod\nolimits_{i=1}^{2}Z_{(i,2)}X_{(i,3)}X_{(i,5)}$, and
$B_{1}^{(1)}=\prod\nolimits_{i=1}^{2}X_{(i,2)}X_{(i,3)}X_{(i,4)}X_{(i,5)}$.
Note that $\mathbf{I}_{2,1}$ and $\mathbf{J}_{2,1}$ here are formally
equivalent to $I^{22}$ and $J^{22}$ in [bilocal1 ]. Denote the 5-qubit state
$\left|\varphi\right\rangle=(\cos\phi\left|0\right\rangle_{1}\left|0\right\rangle_{2}+\sin\phi\left|1\right\rangle_{1}\left|1\right\rangle_{2})\left|+\right\rangle_{3}\left|+\right\rangle_{4}\left|+\right\rangle_{5}$.
It is easy to verify that
$g_{(i)}\left|\varphi\right\rangle=\left|\varphi\right\rangle$ and
$\left\langle\psi_{(i)}\left|h_{(i)}\right|\psi_{(i)}\right\rangle=\left\langle\varphi\left|h_{(i)}\right|\varphi\right\rangle$.
As a result, the same violation can be obtained using either
$\left|\psi_{(1)}\right\rangle\left|\psi_{(2)}\right\rangle$ or
$\left|\varphi\right\rangle^{\otimes 2}$.

Example (b): Let
$\left|\psi_{(1)}\right\rangle=\left|\overline{0}\right\rangle$,
$\left|\psi_{(2)}\right\rangle=\left|\overline{1}\right\rangle$. Here, we
choose the useful operators
$g_{(i)}=\mathfrak{g}_{1}\mathfrak{g}_{2}\mathfrak{g}_{3}\mathfrak{g}_{4}=Z_{1}Z_{2}X_{3}I_{4}X_{5}$
and $h_{(i)}=\mathfrak{g}_{1}$ for any $i\in\\{1,2\\}$. In this case, the
local observables $A_{x_{i}}^{(i)}$ and $B_{0}^{(1)}$ are the same as those in
example (a), while
$B_{1}^{(1)}=\prod\nolimits_{i=1}^{2}Z_{(i,2)}Z_{(i,3)}X_{(i,4)}$. Here,
$\left|\mathbb{D}_{i}\right|=2$ and $(i,$ $2)$, $(i,$ $4)$, $(i,$ $5)$ are
idle qubits. However, note that
$g_{(i)}\left|\varphi^{\prime}\right\rangle=h_{(i)}\left|\varphi^{\prime}\right\rangle=\left|\varphi^{\prime}\right\rangle$,
where
$\left|\varphi^{\prime}\right\rangle=\frac{1}{\sqrt{2}}(\left|0\right\rangle_{1}\left|+\right\rangle_{3}+\left|1\right\rangle_{1}\left|-\right\rangle_{3})\left|0\right\rangle_{2}\left|+\right\rangle_{4}\left|+\right\rangle_{5}$.
The maximal violation can be obtained using either
$\left|\psi_{(1)}\right\rangle\left|\psi_{(2)}\right\rangle$ or
$\left|\varphi^{\prime}\right\rangle^{\otimes 2}$.

Consequently, although the two 5-qubit two logical states
$\left|\overline{0}\right\rangle$and $\left|\overline{1}\right\rangle$ and the
codeword states are genuinely entangled Add ; bdd , one can replace the
genuinely entangled state with a state involving two-qubit entanglement,
either $\left|\varphi\right\rangle$ or $\left|\varphi^{\prime}\right\rangle$,
to reach the same correlation strength. It is not known whether the Bell
nonlocality of genuine entanglement can be revealed in a $K$-locality quantum
network. Specifically, it is not known whether the genuine entanglement of
$\left|\psi_{(i)}\right\rangle$ can be deduced from violations of variant Bell
inequalities of a quantum network involving different stabilizing operators.

##  IV Bell inequalities tailored for nonmaximal entangled states in a quantum
network

If $c_{i}<1$ for at least one $i$, $\left|\overline{\Psi}\right\rangle$ cannot
achieve the maximal violation of the nonlinear Bell inequality (21). To
explore the Bell inequalities maximally violated by
$\left|\overline{\Psi}\right\rangle$, recall that in the two-qubit case
($N=K=M=1$, $n_{1}=2$), the tilted CHSH operator
$\mathbf{B}_{\beta\text{-}CHSH}=\beta B_{0}+$ $\mathbf{B}_{CHSH}$ is exploited
using the logical phase-flip operator $B_{0}$ with appropriate parameter
$\beta$. Although it is unlikely that $\prod_{m=1}^{M}B_{0}^{(m)\text{ }}=$
$\prod_{i=1}^{N}\overline{Z}_{(i)}^{\text{ }}$ in quantum networks, it will be
shown that the logical phase-flip operators are still useful in finding the
tilted Bell inequalities. Denote two index sets $\mathfrak{C}=\\{i|c_{i}=1\\}$
and $\mathfrak{c}=\\{i^{\prime}|c_{i^{\prime}}<1\\}$, where $\mathfrak{C\cap
c=}\emptyset$ and $\left|\mathfrak{C}\right|+\left|\mathfrak{c}\right|=N$.
Without loss of generality, let $i\in\mathfrak{C}$ for $1\leq
i\leq\left|\mathfrak{C}\right|$ and $i\in\mathfrak{c}$ for
$\left|\mathfrak{C}\right|+1\leq
i\leq\left|\mathfrak{C}\right|+\left|\mathfrak{c}\right|=N$. Denote
$\left|\overline{\Psi}_{\mathfrak{C}}\right\rangle=\prod\nolimits_{i=1\text{
}}^{\left|\mathfrak{C}\right|}\left|\psi_{(i)}\right\rangle$ and
$\left|\overline{\Psi}_{\mathfrak{c}}\right\rangle=\prod\nolimits_{i=\left|\mathfrak{C}\right|+1\text{
}}^{N}\left|\psi_{(i)}\right\rangle$; hence, we have
$\left|\overline{\Psi}\right\rangle=\left|\overline{\Psi}_{\mathfrak{C}}\right\rangle\otimes$
$\left|\overline{\Psi}_{\mathfrak{c}}\right\rangle$. Given
$i^{\prime}\in\mathfrak{c}$, set the suitable operator
$h_{(i^{\prime})}^{\prime}$ $=$ $\overline{Z}_{(i^{\prime})}^{\text{
}}=\prod_{j^{\prime}=1}^{n_{i^{\prime}}}\widehat{t^{\prime}}_{(i^{\prime},\text{
}j^{\prime})}$ that fulfills the following conditions: (i) if $(i^{\prime},$
$j^{\prime})\rightarrow\mathcal{S}^{(k)},$ then
$\widehat{t^{\prime}}_{(i^{\prime},\text{ }j^{\prime})}=I;$ and (ii) if
$(i^{\prime},$ $j^{\prime})\rightarrow\mathcal{R}^{(k)}$, then either qubit
$(i^{\prime},$ $j^{\prime})$ is idle or
$\widehat{t^{\prime}}_{(i^{\prime},\text{
}j^{\prime})}=\widehat{s}_{(i^{\prime},\text{ }j^{\prime})}$ if qubit
$(i^{\prime},$ $j^{\prime})$ is not idle. The logical phase-flip operator can
be revised as

$\overline{Z}_{(i^{\prime})}^{\text{ }}=\underset{j^{\prime},\text{
}(i^{\prime},\text{ }j^{\prime})\text{: on the source side
}}{\prod}\widehat{s}_{(i^{\prime},\text{
}j^{\prime})}\prod_{j^{\prime\prime},\text{ }(i^{\prime},\text{
}j^{\prime\prime}):\text{idle}}\widehat{o^{\prime}}_{(i^{\prime},\text{
}j^{\prime\prime})},$ (40)

where $\widehat{o^{\prime}}_{(i^{\prime},\text{
}j^{\prime\prime})}\in\\{\widehat{o}_{(i^{\prime},\text{ }j^{\prime\prime})}$,
$I\\}$. Instead of measuring $B_{0}^{(k)\text{ }}$directly, the agent
$\mathcal{R}^{(k)}$ measures

$\overline{B}_{0}^{(k)\text{ }}=B_{0}^{(k)\text{ }}\underset{\text{
}}{\underset{\widehat{s}_{(i^{\prime},\text{
}j^{\prime})}=I}{\prod_{(i^{\prime},\text{
}j^{\prime})\rightarrow\mathcal{R}^{(k)}}}}\widehat{t^{\prime}}_{(i^{\prime},j^{\prime})}.$
(41)

The outcome of $B_{0}^{(k)\text{ }}$can be obtained from that of
$\overline{B}_{0}^{(k)\text{ }}$by dropping any outcome of the local
observables $\widehat{t^{\prime}}_{(i^{\prime},j^{\prime})}$ in (41); the
outcome of
$\prod_{i^{\prime},i^{\prime}\in\mathfrak{c}}\overline{Z}_{(i^{\prime})}^{\text{
}}$ can be obtained from that of $\prod_{k=1}^{K}\overline{B}_{0}^{(k)\text{
}}$by dropping any outcome of the qubit $(i^{\prime},$ $j^{\prime\prime})$ if
(i) $i^{\prime}\in\mathfrak{C}$ or (ii) $i^{\prime}\in\mathfrak{c}$ and
$\widehat{o^{\prime}}_{(i^{\prime},\text{ }j^{\prime\prime})}$ in (40) is the
unit matrix. In this case, we propose the tilted Bell inequalities tailored
for $\left|\overline{\Psi}\right\rangle$

$G_{N,K}^{\beta}=\beta\left|P_{N,K}\right|^{\frac{1}{K}}+\left|I_{N,K}\right|^{\frac{1}{K}}+\left|J_{N,K}\right|^{\frac{1}{K}}\overset{c}{\leq}\beta+1,$
(42)

where

$P_{N,K}=\prod\nolimits_{i=1\text{ }}^{\left|\mathfrak{C}\right|}I^{\otimes
n_{i}}\prod\nolimits_{i^{\prime}=\left|\mathfrak{C}\right|+1\text{
}}^{N}\overline{Z}_{(i^{\prime})}^{\text{ }}$. However, it is very difficult
to exploit sum-of-squares decomposition to find the tailored Bell operators in
quantum networks with an extremely large Hilbert space non2 ; SOS1 . Our
strategy is to simplify $G_{N,K}^{\beta}$ as tilted CHSH inequalities. In
detail, according to Lemma 2, we have

$G_{N,K}^{\beta}\leq\overline{G}_{N,K}(\beta,\overline{\phi},\overline{\theta})=\beta(\cos
2\overline{\phi})^{\frac{\left|\mathfrak{c}\right|}{K}}+\cos\overline{\theta}+\sin\overline{\theta}(\sin
2\overline{\phi})^{\frac{\left|\mathfrak{c}\right|}{K}},$ (43)

where equality holds when $\theta_{1}=\ldots=\theta_{N}=\overline{\theta}$.
Let the parameter $\beta$ satisfying
$\frac{\partial}{\partial\overline{\theta}}\overline{G}_{N,K}|_{(\overline{\theta}_{\max},\overline{\phi})}=\frac{\partial}{\partial\overline{\phi}}\overline{G}_{N,K}|_{(\overline{\theta}_{\max},\overline{\phi})}=0$
be $\beta_{\max}$. We have

$\tan\overline{\theta}_{\max}=(\sin
2\overline{\phi})^{\frac{\left|\mathfrak{c}\right|}{K}},$ (44)

and

$\beta_{\max}=\frac{(\tan
2\overline{\phi})^{{}^{\frac{2\left|\mathfrak{c}\right|}{K}-2}}}{\sqrt{(1+\tan^{{}^{2}}2\overline{\phi})^{\frac{\left|\mathfrak{c}\right|}{K}}+(\tan
2\overline{\phi})^{\frac{2\left|\mathfrak{c}\right|}{K}}}}.$ (45)

As a result, the state
$\left|\overline{\Psi}_{\mathfrak{C}}\right\rangle\otimes$
$\left|\overline{\Psi}_{\mathfrak{c}}\right\rangle$ can maximally violate the
tailored Bell inequality
$G_{N,K}^{\beta_{\max}}=\overset{c}{\overline{G}_{N,K}(\beta_{\max},\overline{\phi},\overline{\theta}_{\max})\leq}\beta_{\max}+1\
$by setting $\theta_{1}=\ldots=\theta_{N}=\overline{\theta}_{\max}$. Note that
$G_{N,K}^{\beta_{\max}}$ coincides with the tilted CHSH inequality in the case
that $\frac{\left|\mathfrak{c}\right|}{K}=1$ non2 .

As an example, we consider the following ”star-like” quantum network as shown
in Fig. 2. Here, set $K=N$ as an odd integer and $M=1$. For any $i$, let
$\left|\psi_{(i)}\right\rangle=\cos\phi_{i}\left|\overline{0}_{i}\right\rangle+\sin\phi_{i}\left|\overline{1}_{i}\right\rangle$
be the codeword of the $[[5$, $1$, $3]]$ QECC, where $i\in\mathfrak{C}$ if
$\phi_{i}=\frac{\pi}{4}$ and $i\in\mathfrak{c}$ if
$\phi_{i}=\overline{\phi}\neq\frac{\pi}{4}$. The useful operators are
$g_{(i)}=\mathfrak{g}_{4}$, $h_{(i)}=\overline{X}_{(i)}^{\prime\text{ }}$, and
$h_{(i)}^{\prime}=\overline{Z}_{(i)}^{\text{
}}=\mathfrak{g}_{1}\mathfrak{g}_{3}\overline{Z}_{(i)}^{\prime\text{
}}=-I_{(i,2)}X_{(i,3)}X_{(i,4)}I_{(i,5)}Z_{(i,1)}$. On the source side, agent
$\mathcal{S}^{(i)}$ possesses qubit $(i$, $2)$, and using the cut-and-mix
method, the local observables are set as
$A_{x_{i}}^{(i)}=\cos\theta_{i}Z_{(i,\text{
}2)}+(-1)^{x_{i}}\sin\theta_{i}X_{(i,\text{ }2)}$ for any $i=1,..,N$. On the
receiving side, the only agent $\mathcal{R}^{(1)}$ possesses qubits $(j$,
$1)$, $(j$, $3),(j$, $4)$, and $(j$, $5)$, $1\leq j\leq N$. Notably,
$\widehat{t^{\prime}}_{(j,\text{ }2)}=I$, $\widehat{t^{\prime}}_{(j,\text{
}1)}=$ $\widehat{s}_{(j,\text{ }1)}=\sigma_{z}$, and qubits $(j$, $3),(j$,
$4)$, and $(j$, $5)$ are idle. Here two local observables for
$\mathcal{R}^{(1)}$ are
$B_{0}^{(1)}=\prod\nolimits_{j=1}^{N}X_{(j,3)}I_{(j,4)}X_{(j,5)}Z_{(j,1)}$ and
$B_{1}^{(1)}=\prod\nolimits_{j=1}^{N}X_{(j,3)}X_{(j,4)}X_{(j,5)}X_{(j,1)}$. In
this case, one can set
$\overline{B}_{0}^{(1)}=-B_{0}^{(1)}\prod\nolimits_{j=1}^{N}X_{(j,4)}$. In
practice,$\ \mathcal{R}^{(1)}$ randomly measures $\sigma_{x}$ or $\sigma_{z}$
on the qubits $(j$, $1)$, …, $(N$, $1)$, and always measures $\sigma_{x}$ on
qubits $(j$, $3),(j$, $4)$, and $(j$, $5)$, $1\leq j\leq N$. In this scenario,
the numerical simulation shows that $\max G_{N,K}^{\beta}=\beta_{\max}$.

##  V Conclusions

In conclusion, we study the quantum networks with sources emitting different
stabilizer states. To characterize Bell nonlocality, knowledge of the emitted
entangled states is demonstrated to be quite useful. Regarding qubit
distributions in quantum networks, nonlinear Bell inequalities are proposed,
which can reveal variant facets of Bell nonlocality. On the other hand, by
fully exploiting the logical bit-flip and phase-flip operators, we derive
tilted nonlinear Bell inequalities tailored for the codewords of [[5, 1, 3]]
QECC with a specific qubit distribution. It is interesting to construct the
tilted nonlinear Bell inequalities maximally violated by
$\left|\overline{\Psi}\right\rangle$ comprising the generic codewords of QECCs
in quantum networks.

## References

  * (1) J. S. Bell, On the Einstein–Podolsky–Rosen paradox, Physics 1 19 (1964).
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##  VI Figure Captions

Figure 1. The particle distribution of the state emitted from the source $i$
held by $\mathcal{S}^{(s)}$ if $e_{s-1}+1\leq i\leq e_{s}$. The particles
assigned on the source (receiving) side are inside (outside) the circle. The
state can be (a) the classical hidden state $\lambda_{i}$ or (b) the quantum
state $\left|\psi_{(i)}\right\rangle$. Here the particle ($i$, 1) assigned on
the source side is determined as the center of the star graph. Qubit $(i,$
$j)$ in (b) is denoted by a black dot if $\\{\widehat{s}_{(i,\text{ }j)},$
$\widehat{t}_{(i,\text{ }j)}\\}=0$. A quantum network is depicted in (c). The
qubits in a rounded rectangle are locally accessible by an agent. According to
(28) and (29), there is an odd number of black dots in each rounded rectangle.

Figure 2. A star-like quantum network with all states
$\left|\psi_{(1)}\right\rangle,\ldots,\left|\psi_{(N)}\right\rangle$ being
[[5, 1, 3]] codeword states.