FORMATION OF THE THEBE EXTENSION IN THE RING SYSTEM OF JUPITER

N. Borisov1 and H. Krüger2<EMAIL_ADDRESS>
(1) IZMIRAN, 142190, Kaluzshskoe Hwy 4,Troitsk, Moscow, Russia
(2) MPI for Solar System Research, 37077 Göttingen, Germany

## Abstract

Jupiter’s tenuous dust ring system is embedded in the planet’s inner
magnetosphere, and – among other structures – contains a very tenuous
protrusion called the Thebe extension. In an attempt to explain the existence
of this swath of particles beyond Thebe’s orbit, Hamilton and Krüger (2008)
proposed that the dust particle motion is driven by a shadow resonance caused
by variable dust charging on the day and night side of Jupiter. However, the
model by Divine and Garrett (1983) together with recent observations by the
Juno spacecraft indicates a warm and rather dense inner magnetosphere of
Jupiter which implies that the mechanism of the shadow resonance does not
work. Instead we argue that dust grains ejected from Thebe due to
micrometeoroid bombardment become the source of dust in the Thebe extension.
We show that large (grain radii of a few micrometers up to multi-micrometers)
charged dust grains having significant initial velocities oscillate in the
Thebe extension. Smaller charged grains (with sub-micrometer radii) ejected
from Thebe do not spend much time in the Thebe extension and migrate into the
Thebe ring. At the same time, if such grains are ejected from larger dust
grains in the Thebe extension due to fragmentation, they continue to oscillate
within the Thebe extension for years. We argue that fragmentation of large
dust grains in the Thebe extensions could be the main source of sub-micrometer
grains detected in the Thebe extension.

Keywords: Thebe; Cosmic Dust; Space Plasma; Electric fields, Jupiter ring
system, Planetary Rings

##  1 Introduction

Jupiter has a highly structured dust ring system that extends along the
planet’s equatorial plane. It consists of the main ring, the halo, two
gossamer rings and the Thebe extension (Burns, 1984; Showalter et al., 1987;
Burns et al., 2004). This dust system was investigated by the space missions
Voyager 1,2, Galileo, Cassini and New Horizons (Smith et al., 1979; Owen et
al., 1979; Ockert-Bell et al., 1999; Porco et al., 2003; Burns et al., 2004;
Showalter et al., 2007; Throop et al., 2016) and also by telescopes in space
(Hubble) and from the Earth (Keck) (de Pater et al., 1999, 2008; Showalter et
al., 2008).

Usually it is assumed that the gossamer rings are formed by dust grains
ejected from the surfaces of the corresponding moons (Thebe and Amalthea) due
to micrometeorite bombardment (Krivov et al., 2002, 2003; Dikarev et al.,
2006). Recently it has been argued that a supplementary mechanism of dust
grain ejection from the polar regions of Thebe and Amalthea, i.e.
electrostatic lofting, contributes to the formation of the gossamer rings
(Borisov and Krüger, 2020). At the same time the mechanism of the formation of
the Thebe extension is not so clear. Indeed, it is known that Jupiter’s
corotational radial electric field acts on dust grains in the planet’s inner
magnetosphere (Horányi and Burns, 1991). The corresponding electric force for
negatively charged grains in the gossamer rings is directed inwards (towards
Jupiter) for radial distances $r>R_{\rm synch}$, where $R_{\rm
synch}=2.27\,\mathrm{R_{J}}$ is the radius of the synchronous orbit, i.e. the
orbit at which the angular frequency of a body orbiting Jupiter is equal to
the angular frequency of Jupiter’s rotation ($\mathrm{R_{J}=71,492\,km}$ is
the radius of Jupiter). On the contrary, positively charged grains are pulled
outwards by this electric field (into the Thebe extension).

The sign and the magnitude of the electric charge on a dust grain in Jupiter’s
shadow is determined only by the plasma parameters in the inner magnetosphere
of Jupiter, and additionally on the sun-lit side by the action of the solar UV
radiation. Unfortunately, real plasma parameters of Jupiter’s inner
magnetosphere are still not well-known. Earlier in the theoretical
investigations devoted to dust grains dynamics in the gossamer rings the
authors often estimated the electron concentration in the inner magnetosphere
of Jupiter as of the order of $N_{e}\sim 1\,\mathrm{cm^{-3}}$ or even less,
and the electron temperature of a few electron volts, see, e.g. Horányi and
Burns (1991); Hamilton and Krüger (2008). In such a plasma environment, the
electric potential on dust grains on the sun-lit side can be positive, thus
driving such charged dust grains outwards. This mechanism was suggested by
Hamilton and Krüger (2008) to explain the formation of the Thebe extension. At
the same time, according to the well-known theoretical model of thermal plasma
in the inner magnetosphere of Jupiter by Divine and Garrett (1983), the
electron concentration in the vicinity of Thebe is predicted to be
$N_{e}\approx 50\,\mathrm{cm^{-3}}$ and the electron temperature $T_{e}\approx
45\,$eV. This model is based on Pioneer and Voyager data and was confirmed
later by Garrett et al. (2005) (only slightly corrected). For such large
values of the electron concentration and temperature the electric charges on
dust grains are definitely negative and rather high, both on the sun-lit side
and in the planet’s shadow. Furthermore, for such large concentrations of
thermal electrons the variation of the electric potential on the sun-lit side
and in the shadow of Jupiter should be almost negligible (as an example how
the electric potential changes with the growth of the plasma concentration see
Horányi and Burns (1991), their Fig. 5). As a result, assuming that the Divine
model is correct, the above mentioned explanation of the Thebe extension does
not work, and a new physical mechanism for its formation has to be developed.

In relation with the model by Divine and Garrett (1983) two aspects should be
emphasized. First, new experimental data obtained by the Juno mission at
$R\approx 10\,\mathrm{R_{J}}$ clearly demonstrate that the concentration of
ions grows rather quickly towards smaller distances from Jupiter (Kim et al.,
2020). Second, it is expected that later on (at the end of its mission) Juno
will provide us with the real concentrations of charged particles at $R\approx
3\,\mathrm{R_{J}}$.

According to the experimental data, the Thebe extension is a very faint
structure (dust concentration is smaller than in the gossamer rings, see Burns
et al. (1999); Ockert-Bell et al. (1999); Krüger et al. (2009)). This means
that possibly not all dust grains ejected from Thebe penetrate into the Thebe
extension but only some fraction. The radial range of the Thebe extension as
seen in images (Ockert-Bell et al., 1999) comprises at least approximately
$0.64\,\mathrm{R_{J}}$ (from the orbit of Thebe at $3.11\,\mathrm{R_{J}}$ out
to $3.75\,\mathrm{R_{J}}$) which is in agreement with in situ dust
measurements by the Galileo spacecraft (Krüger et al., 2009).

In this paper we take into account that dust grains lofted from the moons’
surface can have rather high initial velocities. Velocities of the ejected
dust grains and their directions of propagation were investigated by various
authors, e.g, Gault et al. (1963); Eichhorn (1978); O’Keefe and Ahrens (1985);
Koschny and Grün (2001); Sachse et al. (2015), and a recent review by Szalay
et al. (2018). Laboratory measurements together with numerical simulations
were applied to obtain the distributions of masses, velocities and their
directions as functions of a meteorite (micrometeorite) parameters. Various
authors obtained quite different maximum ejection speeds, ranging from
approximately $\mathrm{700-800\,m\,s^{-1}}$ up to $\mathrm{20\,km\,s^{-1}}$.

In addition to these so far inconclusive results based on laboratory
measurements, another line of evidence comes from in situ dust measurements in
the Jovian system: Impact ejecta dust clouds were directly measured at the
Galilean moons (Krüger et al., 1999, 2003). These tenuous clouds were detected
in the vicinity of these moons up to an altitude of approximately eight
satellite radii. It implies that the particles must have been ejected from the
surfaces of these moons with speeds not too far below their escape speeds,
which are in the range of approximately $\mathrm{2.5\,km\,s^{-1}}$.
Furthermore, the Galileo detector measured a very tenuous dust ring around the
planet in the region between the Galilean moons. These particles were
interpreted (at least partially) as being ejecta from these moons (Krivov et
al., 2002). Therefore, they must have reached ejection speeds even exceeding
the escape speeds of the Galilean moons. Given that the surface properties of
Amalthea and Thebe are likely comparable (dirty ice), we assume similar
ejection speeds also for these small moons.

In our calculations we shall use some intermediate values of dust grains
ejected velocities (a few $\mathrm{km\,s^{-1}}$) which are in the range of the
escape speeds from the Galilean moons. Previously, while discussing dust
dynamics in the gossamer rings and Thebe extension, the role of a dust grain
initial velocity was neglected. Taking into account the initial velocity makes
it possible to explain the penetration of some fraction of dust grains ejected
from Thebe into the Thebe extension. There are some peculiarities of the
discussed process. The ejected grain should have a significant initial
velocity with the appropriate direction of propagation. The most favourable
situation appears when a given large dust grain is ejected from the surface of
Thebe in the direction of Thebe’s instant azimuthal velocity or, more
generally, the grain at the moment of ejection has a significant velocity
component in the direction of the instantaneous azimuthal speed of Thebe and
the radial velocity directed into the Thebe extension. The discussed
requirements are easily realized if Thebe and a micrometeorite hitting the
surface move towards each other with antiparallel speed vectors. Note that in
such case the relative velocity of two bodies can exceed
$\mathrm{50\,km\,s^{-1}}$.

According to our calculations dust grains with sizes of the order of several
micrometers ejected with velocities $V_{\theta}\geq\mathrm{1.5\,km\,s^{-1}}$
penetrate deep into the Thebe extension. At the same time smaller grains
(fraction of a micrometer) penetrate into the Thebe extension only at the
initial stage after ejection from Thebe. Later on their radial localization
shifts more and more into the Thebe ring. That is why tiny grains appear in
the Thebe extension mainly due to some other mechanism. According to Burns et
al. (2004) tiny dust grains are produced in Jupiter’s rings due to ”sputtering
of surrounding plasma and collisions with gravitationally focused
interplanetary micrometeorites”. Later on Dikarev et al. (2006) – based on
these ideas – developed the model of two-stage dust delivery from satellites
to planetary rings. According to this model large (multi-micrometer-sized)
dust grains ejected from the surfaces of the moons supply the rings with tiny
dust grains due to collisions between themselves and with other
micrometeorites. Note that the influence of electric charging on the dust
grains dynamics was neglected in this model.

In this paper we argue that some part of large (from a few micrometers up to
multi-micrometer sizes) electrically charged grains ejected from Thebe due to
collisions with micrometeorites penetrate deep into the Thebe extension.
Smaller dust grains appear subsequently due to fragmentation of large grains
in the Thebe extension (caused by sputtering, mutual collisions and collisions
with micrometeorites or by electric disruption). Note that small electrically
charged dust grains ejected from Thebe migrate rather quickly into the Thebe
ring. At the same time we show that similar grains that appear in the Thebe
extension due to fragmentation of large grains remain in the extension for
years.

##  2 Basic Equations

In order to discuss the motion of dust grains in the equatorial inner
magnetosphere of Jupiter we use a cylindrical coordinate system centered at
Jupiter. Suppose that the z-axis is directed upwards while the magnetic field
lines of Jupiter are directed downwards. Two other coordinates are the radial
coordinate in the equatorial plane $r$ and the angular coordinate $\theta$.
The equations of motion for charged dust grains in this coordinate system were
investigated earlier by Horányi and Burns (1991). We present them in the
following form:

$\displaystyle\frac{d^{2}r}{dt^{2}}=r\Omega^{2}+\frac{q}{r^{2}}(\Omega-\Omega_{J})-\frac{\mu}{r^{2}}$
$\displaystyle\frac{d\Omega}{dt}=-\frac{d\ln
r}{dt}\left(\frac{q}{r^{3}}+2\Omega\right).$ (1)

Here $\Omega=d\theta/dt$ is the angular frequency of the orbiting dust grain,
$\Omega_{J}\approx\mathrm{0.000176\,s^{-1}}$ is the angular frequency of
Jupiter’s rotation, $q=Z\Omega_{He}R_{\rm Th}^{3}m_{e}/M_{d}$, $\Omega_{He}$
is the Larmour frequency of electrons on the surface of Jupiter,
$\Omega_{He}\approx\mathrm{8.5\cdot 10^{7}\,s^{-1}}$, $m_{e},M_{d}$ are the
masses of an electron and a dust grain, respectively, $Z=Q_{d}/e$ is the
amount of elementary charges on a given dust grain, $e$ is the charge of the
electron, $Q_{d}$ is the charge on a dust grain, $\mu=GM_{J}$,
$G=\mathrm{6.68\cdot 10^{-8}\,cm^{3}\,g^{-1}\,s^{-2}}$ is the gravitational
constant, $M_{J}\approx\mathrm{1.9\cdot 10^{30}\,g}$ is the mass of Jupiter,
and $R_{\rm Th}=3.11\,\mathrm{R_{J}}$ is the radial distance of Thebe’s orbit
from Jupiter.

Equation 1 describes the motion in the equatorial plane of Jupiter of a grain
with the electric charge $Q_{d}$ and mass $M_{d}$. The magnetic field has the
dipole form. It is assumed that the inner magnetosphere rigidly corotates with
the planet. Two forces – the gravitational force (term $\mu/r^{2}$) and the
electric force (term $q(\Omega-\Omega_{J})/r^{2}$) – determine the motion of a
dust grain. Previously, the dynamics of a charged grain ejected with zero
initial velocity with respect to the moon was considered (see, e.g. Horányi
and Burns (1991)). Here we discuss the more general and more realistic case
that ejected grains have initial radial and azimuthal velocities (Sections 3
and 5). This is quite natural because dust grains are ejected from the moon
due to micro-explosions caused by micrometeorite bombardment. It is important
that some ejected grains have higher initial azimuthal velocities than the
azimuthal velocity of the moon. We show that such large dust grains (with
small ratio $Q_{d}/M_{d}$) have equilibrium orbits shifted outwards with
respect to the orbit of the moon (see Figs. 1 and 3).

Note that the electric charge on a dust grain $Q_{d}$ is not constant. Usually
dust grains lying on the surface of the moon (Thebe in our case) have electric
charges much smaller than the electric charges in equilibrium in space
(Borisov and Mall, 2006). That is why after ejection from the surface dust
grains continue to acquire electric charge approaching an equilibrium value
$Q_{\rm st}$.

Let us proceed with the second of Eq. (1). It is convenient to seek for
$\Omega(t)$ in the form

$\Omega(t)=\Omega_{1}(t)\frac{R_{\rm Th}^{2}}{r^{2}}.$ (2)

The equation for $\Omega_{1}$ takes the form

$\frac{d\Omega_{1}}{dt}=-\frac{q(t)}{R_{\rm Th}^{2}r^{2}}\frac{dr}{dt}.$ (3)

This equation can also be written in the equivalent form which is more
convenient for an approximate solution in case of a changing charge $q(t)$:

$\frac{d\Omega_{1}}{dt}=\frac{1}{R_{\rm
Th}^{2}}\frac{d}{dt}\left(\frac{q}{r}\right)-\frac{1}{rR_{\rm
Th}^{2}}\frac{dq}{dt}.$ (4)

If a dust grain starts its motion from the surface of Thebe with negligible
velocity, the initial condition for $\Omega_{1}$ is $\Omega_{1}(0)=\Omega_{\rm
Th}$, where $\Omega_{\rm Th}\approx\mathrm{0.000107\,s^{-1}}$ is the angular
orbital rotation frequency of Thebe. If the initial azimuthal velocity
$V_{\theta}$ in the system moving with Thebe is finite
$\Omega_{1}(0)=\Omega_{\rm Th}+\Delta\Omega$, where
$\Delta\Omega=V_{\theta}/R_{\rm Th}.$ As a result the frequency $\Omega$ takes
the form

$\Omega(t)=(\Omega_{\rm Th}+\Delta\Omega)\frac{R_{\rm
Th}^{2}}{r^{2}}+\frac{1}{r^{2}}\left[\frac{q(t)}{r}\right]_{0}^{t}-\frac{1}{r^{2}}\int_{0}^{t}\frac{1}{r}\frac{dq}{dt}\,dt.$
(5)

Here $q(0)=Z_{0}\Omega_{He}R_{\rm Th}^{3}m_{e}/M_{d}$, $Z_{0}$ is the amount
of electrons on a given dust grain lying on the surface of the moon
$Z_{0}=Q_{0}/e$, $r_{0}=R_{\rm Th}$ is the initial radial coordinate.

If the electric charge is constant, $q=q_{\rm st}$, the derivative is
$dq/dt=0$. In this case we find

$\Omega(t)=\Omega_{0}\frac{R_{\rm Th}^{2}}{r^{2}}+\frac{q_{\rm
st}}{r^{2}}\left(\frac{1}{r}-\frac{1}{R_{\rm Th}}\right),$ (6)

where $\Omega_{0}$ is the initial angular frequency. If the radial coordinate
of a given dust grain $r(t)$ changes more significantly than the electric
charge $q(t)$ ($|\frac{dr}{rdt}|>>|\frac{dq}{qdt}|$) then we receive an
approximate solution which is more general than the result given in Eq.(6):

$\Omega(t)=\Omega_{0}\frac{R_{\rm
Th}^{2}}{r^{2}}+\frac{q(t)}{r^{2}}\left(\frac{1}{r}-\frac{1}{R_{\rm
Th}}\right).$ (7)

Here $\Omega_{0}=\Omega_{\rm Th}+\Delta\Omega$. With increasing time $q(t)\to
q_{\rm st}$, where $q_{\rm st}$ is the equilibrium charge on a dust grain in
space. It follows from Eqs. (5) and (7) that for the moment $t=0$ the
frequency $\Omega=\Omega_{\rm Th}+\Delta\Omega$, as it should be. Note that
the sign of $q$ is positive for negatively charged grains because the magnetic
field of Jupiter is directed downwards.

It is easy to check that the expression for $\Omega(t)$ given by Eq. (5) is
the solution of the second of Eq. (1). As a result we may conclude the
following. First, the last term on the right-hand side of Eq. (7) is positive
for radial distances $r$ less than the orbit of Thebe $R_{\rm Th}$. Second,
the smaller the variations $|R_{\rm Th}-r|$, the less significant is the
influence of the electric charge on the angular frequency. For a rather large
time $t\geq t_{\rm st}$ when $Q(t)=Q_{\rm st}$ we may designate
$\Omega_{0}R_{\rm Th}^{2}-q_{\rm st}/R_{\rm Th}=J$ and express the angular
frequency as

$\Omega(t)=\frac{J}{r^{2}}+\frac{q_{\rm st}}{r^{3}}.$ (8)

It can be shown that our Eq. (8) coincides with Eq. (3b) of Horányi and Burns
(1991). But for a smaller period of time $t\leq t_{\rm st}$ we may use a more
general expression:

$\Omega(t)=\frac{J}{r^{2}}+\frac{q(t)}{r^{3}}-\frac{q(t)-q_{\rm
st}}{r^{2}R_{\rm Th}}.$ (9)

The frequency $\Omega(t)$ given by Eq. (9) can be substituted into the first
Eq. (1) to obtain an equation describing the radial motion of a dust grain.
Assuming as in the paper by Horányi and Burns (1991) that the terms with the
electric charge are small and retaining only the terms of the first order with
respect to $q$, we arrive at an approximate equation:

$\frac{d^{2}r}{dt^{2}}=\frac{J^{2}}{r^{3}}+\frac{3qJ}{r^{4}}-\frac{2(q-q_{\rm
st})J}{r^{3}R_{\rm Th}}-\frac{\mu+q\Omega_{J}}{r^{2}}.$ (10)

##  3 Neutral Dust Grain Dynamics

Let us introduce the dimensionless coordinate $\rho=r/R_{\rm Th}$ instead of
the radial distance $r$. If the electric charge is negligible ($q(t)\to 0$)
Eq. (11) describes the radial motion of a neutral dust grain:

$\frac{d^{2}\rho}{dt^{2}}=\frac{J^{2}}{R_{\rm
Th}^{4}\rho^{3}}-\frac{\mu}{R_{\rm Th}^{3}\rho^{2}}.$ (11)

It can be shown with the help of Eq. (11) that at the moment $t=0$ when
$\rho=1$ the radial force is positive if $\Delta\Omega>0$. With increasing
$\rho$ this force decreases and at $\rho=\rho_{*}$, where
$\rho_{*}=J^{2}(R_{\rm Th}\mu)^{-1}$ it becomes equal to zero. For larger
distances $\rho>\rho_{*}$ this force is negative and it pulls dust grains
radially inwards into the Thebe ring.

Let us discuss the radial motion of a neutral dust grain analytically. It
follows from Eq. (11):

$\left(\frac{d\rho}{dt}\right)^{2}=\frac{2\mu}{R_{\rm
Th}^{3}\rho}-\frac{\Omega_{0}^{2}}{\rho^{2}}+W_{0},$ (12)

where $W_{0}$ is added on the right-hand side of Eq. (12) to include the
correct value of the initial radial velocity $V_{r}$

$W_{0}=\Omega_{0}^{2}-\frac{2\mu}{R_{\rm Th}^{3}}+\frac{V_{r}^{2}}{R_{\rm
Th}^{2}}.$ (13)

From Eq. (13) we find the maximal radial excursion $\rho_{\rm max}$ for a
given dust grain:

$\rho_{\rm max}=-\frac{\mu}{R_{\rm Th}^{3}W_{0}}+\sqrt{\left(\frac{\mu}{R_{\rm
Th}^{3}W_{0}}\right)^{2}+\frac{\Omega_{0}^{2}}{W_{0}}}.$ (14)

Figure 1: Radial motion of a dust grain without electric charge. The
horizontal axis gives the time in seconds and the vertical axis the relative
radial coordinate $r/R_{\rm Th}$. The initial radial velocity directed into
the Thebe extension is $V_{r}(t=0)=\mathrm{1.5\,km\,s^{-1}}$. The initial
azimuthal velocity with respect to the azimuthal velocity of Thebe is
$V_{\theta}=\mathrm{2.2\,km\,s^{-1}}$. The grain starts its motion at $t=0$
from the orbit of Thebe.

In Figure 1 we show the radial distance of a dust grain for an initial
azimuthal $V_{\theta}=\mathrm{2.2\,km\,s^{-1}}$ and radial
$V_{r}=1.5\,\mathrm{km\,s^{-1}}$ velocities. The initial azimuthal velocity is
taken in the direction of Thebe’s orbital motion, and the radial velocity is
directed towards the Thebe extension. It can be seen that neutral dust grains
with such significant initial velocities propagate rather far outside of the
Thebe orbit regardless of their size. In our case, the grain propagates up to
$\rho_{\rm max}=1.52$, which is equivalent to approximately 115,000 km.
According to Fig. 1 it takes a significant amount of time $\Delta t\approx 22$
hours for a neutral dust grain to penetrate into the Thebe extension and to
return to its initial radius $\rho=1$. The maximum radial velocity reaches the
value $V_{\rm max}\approx\mathrm{6\,km\,s^{-1}}$. Note that such dynamics is
valid for all neutral dust grains (with given initial velocities) regardless
of their mass. The situation changes if we take into account that dust grains
are negatively charged and the ratio of the electric charge to mass
$Q_{d}/M_{d}$ varies significantly for grains with different sizes $a$.
Indeed, under equilibrium conditions the electric charge on a given dust grain
with radius $a$ is $Q_{d}=\phi_{\rm st}a$, while the mass of such grain is
$M_{d}=(4\pi/3)\rho_{\rm dust}a^{3}$, where $\phi_{\rm st}$ is the electric
potential in the medium, $\rho_{\rm dust}$ is the dust density. The ratio
$Q_{d}/M_{d}$ is inversely proportional to $a^{2}$. Thus, the electric force
influences more significantly the motion of small dust grains. That is why the
motion of neutral dust grains discussed in this section can be considered as a
good approximation for the dynamics of large multi-micrometer sized dust
grains. We will discuss the influence of the electric charge and mass on the
dust grains dynamics in more detail in the next sections.

##  4 Charging of a Dust Grain on the Surface of the Moon and in Space

In this section we discuss charging of dust grains in the inner magnetosphere
of Jupiter taking into account the Divine model (Divine and Garrett, 1983). At
the beginning we estimate the initial electric charge on a grain lying on the
surface of the moon (Thebe). Then we consider the equilibrium charge on the
same grain ejected from the surface. At the end we investigate how the
electric charge on a given grain grows in time after ejection from the moon.

The problem of dust grains charging was discussed in many papers, see,
e.g.,Whipple (1981); Goertz (1989); Whipple et al. (1985); Havnes et al.
(1987); Horányi (1996); Borisov and Mall (2006). We present some estimates for
the typical electric charge on a dust grain with radius $a$ lying on the
surface of the moon and also in space after ejection.

The process of dust grain charging is considered in the system of coordinates
rotated with the magnetic field. This means that there is no corotating
electric field acting on electrons and ions but instead there exists an
azimuthal velocity component of a dust grain with respect to the plasma
$V_{0}=R_{d}(\Omega_{d}-\Omega_{J})$, where $R_{d}$ is the radial distance of
a dust grain from Jupiter, $\Omega_{d}$ is its angular velocity, $\Omega_{J}$
is the angular velocity of the planet (and the magnetic field). In general
this velocity should be taken into account when discussing dust grain charging
by ions. Below it is shown that in our case the additional flux of ions
associated with this velocity is smaller than the well-known flux of ions in
the vicinity of a grain (see, e.g. Whipple (1981); Goertz (1989); Whipple et
al. (1985); Havnes et al. (1987); Horányi (1996)) that provides its charging.
Hence, to a first approximation the velocity $V_{0}$ can be neglected.

In order to calculate the equilibrium potential we assume that far away from
the moon electrons have a Boltzmann distribution with the temperature $T_{e}$
and concentration $N_{e}$:

$F_{e}=\frac{N_{e}}{\pi^{3/2}V_{T_{e}}^{3}}\exp\left(-\frac{v^{2}}{V_{T_{e}}^{2}}\right),$
(15)

where $V_{T_{e}}=\sqrt{2T_{e}/m_{e}}$ is the thermal velocity, and $m_{e}$ is
the mass of an electron. Ions are assumed to have a similar distribution:

$F_{i}=\frac{N_{i}}{\pi^{3/2}V_{T_{i}}^{3}}\exp\left(-\frac{v^{2}}{V_{T_{i}}^{2}}\right),$
(16)

where $N_{i}=N_{e}=N_{0}$ is the plasma concentration, $T_{i}$ is the
temperature and $V_{T_{i}}=\sqrt{2T_{i}/M_{i}}$ is the thermal velocity of
ions, $M_{i}$ is the mass of ions.

Our aim is to estimate the electric potential $\phi$ on the surface of the
moon in a plasma that contains electrons and ions with the distribution
functions (15), (16) far away from the surface. This potential should be
negative because the thermal speed of electrons is much higher than the
thermal speed of ions $V_{Te}>>V_{Ti}$. Let us introduce the Debye radius
$R_{D}=V_{Te}/\Omega_{Pe}$. Here $\Omega_{Pe}=(4\pi e^{2}N_{e}/m_{e})^{1/2}$
is the plasma frequency, $e$ is the charge of an electron. The Debye radius in
the vicinity of Thebe according to the Divine model is of the order of
$R_{d}\approx 1-2$ m.

Electrons are retarded by the negative potential. Their concentration in the
plasma with the potential $\phi$ is $N_{e}=N_{0}\exp(e\phi/T_{e})$ while the
velocity distribution is still determined by Eq. (15). The flux of electrons
on to the negatively charged surface according to Havnes et al. (1987) is:

$\Phi_{e}=\sqrt{\frac{2}{\pi}}V_{Te}N_{0}\exp\left(\frac{e\phi}{T_{e}}\right).$
(17)

At the same time the flux of ions on to the same surface is:

$\Phi_{i}=\sqrt{\frac{2}{\pi}}V_{Ti}N_{0}\left(1-\frac{e\phi}{T_{i}}\right).$
(18)

Equating the fluxes of electrons and ions we find for the electric potential
under equilibrium conditions and equal temperatures $T_{e}=T_{i}$:

$(1+F_{st}(a))\exp(F_{st}(a))=\frac{V_{Te}}{V_{Ti}},$ (19)

where $F_{st}=-e\phi_{st}/T_{e}$. For equal temperatures of electrons and ions
and heavy ions ($M_{i}/m_{e}\approx 3\cdot 10^{4}$) the electric potential on
the surface is $e\phi_{s}/T_{e}\approx-3.6$. The electric field above the
surface of the moon is described by the Poisson equation:

$\frac{d^{2}\phi}{dz^{2}}=4\pi e(N_{e}(\phi)-N_{i}(\phi)),$ (20)

where $z$ is the coordinate across the surface of the moon. A detailed
calculation of the surface electric field can be found in Borisov and Mall
(2006). According to this paper the characteristic length that determines the
electric field is $L_{z}\approx(3-4)R_{D}$. So we estimate the vertical
surface electric field as $E_{s}\approx(0.3\div 0.4)\,\mathrm{V\,cm^{-1}}$.
This field is connected with the density $\sigma$ of the electric charge on
the surface that provides such electric field $E_{s}$, see Borisov and Mall
(2006). Indeed, after integration of Eq. (20) across the surface we arrive at
the relation

$\frac{d\phi}{dz}|_{+0}-\frac{d\phi}{dz}|_{-0}=4\pi\sigma.$ (21)

This equation allows us to estimate the charge on a dust grain with the radius
a lying on the flat surface. Taking into account that the cross-section of a
dust grain is $\Sigma=\pi a^{2}$ we find the electric charge on a grain lying
on the surface as $Q_{d}=0.5a^{2}E_{s}$. For example, the grain with the
radius $a=5~{}\mu\mathrm{m}$ lying on the surface of the moon (Thebe) contains
5 to 6 electrons ($Z=5-6$).

After ejection from the moon a dust grain acquires additional negative
electric charges until its potential $\phi_{st}$ reaches an equilibrium value
(which is achieved when the flux of electrons hitting a given grain becomes
equal to the flux of ions). It is more convenient to use the dimensionless
parameter $F=-e\phi/T_{e}$ instead of the potential $\phi$.

It is interesting that the electric potential determined above by Eq. (19) in
the vicinity of a grain can accelerate ions up to the velocity
$V_{b}\approx(-2e\phi_{st}/M_{i})^{1/2}$ towards a grain. This velocity
comprises $V_{b}\approx 35\,\mathrm{km\,s^{-1}}$ and it is higher than the
thermal speed of ions ($V_{b}\approx 1.9\,V_{Ti}$). At the same time the
azimuthal velocity of a dust grain with respect to the magnetic field
(mentioned at the beginning of this section) in the vicinity of Thebe is
$V_{0}\leq 15\,\mathrm{km\,s^{-1}}$. As a result the flux of ions given by Eq.
(18) is significantly larger than the flux $\Phi_{0}$ associated with the
azimuthal velocity $V_{0}$ which provides additional charging mainly from one
side $\Phi_{0}\approx\pi a^{2}N_{0}V_{0}$.

Note that while calculating the electric potential we have not included the
flux of secondary electrons. This flux contains several parameters whose
values are rather uncertain. For the same values of parameters that were used
earlier in relation with the inner magnetosphere of Jupiter (Horányi and
Burns, 1991) we find that the deviation of the surface potential from the one
estimated above is only approximately $1.5\,\%$. Taking into account that the
electric potential $\phi_{st}$ is connected with the equilibrium electric
charge on a given dust grain by the relation $\phi_{st}=Q_{d}/a$, we are able
to estimate the equilibrium electric charge on a dust grain with radius a in
space. This calculation shows that in the vicinity of Thebe a dust grain with
radius a should have the charge $Q_{d}\approx 3.6\,aT_{e}/e$ under equilibrium
conditions. For a grain with radius $a=5~{}\mu\mathrm{m}$ the charge is
$Q_{d}\approx 5\cdot 10^{5}e$ ($Z=5\cdot 10^{5}$). It should be mentioned that
the electric charge on a grain lying on the surface is proportional to the
square of its radius while in space the equilibrium charge is proportional
only to its radius. Nevertheless for $a=5~{}\mu\mathrm{m}$ the charge in space
is approximately five orders of magnitude higher than the charge on the same
grain lying on the surface.

Now we need to investigate how the electric charge on a given dust grain
changes in time. For this purpose we take into account the difference of the
fluxes of electrons and ions that provide the growth of the electric charge

$\frac{dQ_{d}}{dt}=4\pi a^{2}e(\Phi_{i}(a)-\Phi_{e}(a)).$ (22)

This equation rewritten in terms of the dimensionless function $F$ takes the
form

$\frac{dF(a)}{dt}=\frac{\sqrt{2\pi}V_{Te}a}{R_{d}^{2}}\left(\exp(-F(a))-\frac{V_{Ti}}{V_{Te}}(1+F(a))\right).$
(23)

The dimensionless potential $F(a)$ grows in time approaching its equilibrium
value $F_{st}(a)\approx 3.6$. With the help of Eq. (24) it is possible to
estimate the characteristic period $\Delta t$ of charging:

$\Delta t\approx\frac{R_{d}^{2}}{\sqrt{2\pi}V_{Te}a}.$ (24)

It follows form Eq. (24) that the period $\Delta t$ is inversely proportional
to the radius of a grain. For a dust grain with radius $a=1~{}\mu\mathrm{m}$
this time is of the order of one second. Note that $\Delta t$ describes the
characteristic period for the initial stage of charging. More accurately, the
variation of the potential $F(a)$ can be obtained solving Eq. (24)
numerically. In Fig. 2 we show the growth in time of the potential $F(a)$ for
the grain size $a=1~{}\mu\mathrm{m}$. On the horizontal axis the dimensionless
time $t_{1}=t/\Delta t$ is given. It is seen that the process of charging
becomes slower and slower with time. It takes the time $15\Delta t$ to achieve
$0.7F_{st}(a)$.

##  5 Dust Grain Dynamics in the Thebe Extension

Figure 2: Growth of the electric potential towards its equilibrium value in
time on a given dust grain with the radius $a=1\,\mathrm{\mu m}$ after its
ejection from Thebe at $t=0$. On the horizontal axis time in dimensionless
units $t/\Delta t$ is presented.

The dust grain dynamics after ejection from the moon with some finite initial
velocity can be investigated numerically. The radial displacement in time of
the charged dust grains with different sizes $a=5~{}\mu\mathrm{m}$ and
$a=0.3~{}\mu\mathrm{m}$ are calculated taking into account the equations of
motion (see Section 2). The results are shown in Figs. 3 and 4. The initial
azimuthal and radial velocities are $V_{\theta}=2.2\,\mathrm{km\,s^{-1}}$ and
$V_{r}=1.5\,\mathrm{km\,s^{-1}}$ (the same for both grains). The calculations
are carried out based on the Divine model. It is seen that the heavy charged
dust grain penetrates rather deep into the Thebe extension (approximately
100,000 km). This result qualitatively corresponds to the motion of a neutral
dust grain (see Fig. 1). Only the radial excursion into the Thebe extension is
somewhat smaller due to the action of the electric force. At the same time the
submicrometer dust grain at the beginning slightly penetrates into the Thebe
extension and later on its radial motion shifts into the Thebe ring. It is
interesting to mention that the period of radial oscillations for a smaller
dust grain is much shorter than that for the larger one. Such an effect was
not discussed before. It happens because strong electric charging of a dust
grain (that exists due to high plasma concentration and temperature)
influences the period of oscillations.

Figure 3: Radial oscillations in time of a charged dust grain with the radius
$a=5\,\mathrm{\mu m}$ ejected from Thebe. The horizontal axis shows the time
in seconds and the vertical axis the radial coordinate in dimensionless units
$r/R_{\rm Th}$. The initial radial velocity directed outwards is
$V_{r}=\mathrm{1.5\,km\,s^{-1}}$ and the azimuthal velocity with respect to
the azimuthal velocity of Thebe is $V_{\theta}=\mathrm{2.2\,km\,s^{-1}}$.
Figure 4: The same as in Fig. 3, except that the radius of a dust grain is
$a=0.3\,\mathrm{\mu m}$. Figure 5: Radial oscillations of a small dust grain
($a=0.3\,\mathrm{\mu m}$) ejected at $r_{0}=1.22\,R_{\rm Th}$ with the initial
radial velocity equal to the corresponding velocity of a large dust grain
$V_{r}(t=0)=2.8\,\mathrm{km\,s^{-1}}$. The initial azimuthal frequency
$\Omega(t=0)=0.000078\,\mathrm{s}^{-1}$ coincides with the angular frequency
of a large dust grain located at $r_{0}=1.22\,R_{\rm Th}$.

All calculations above were carried out assuming that dust grains (small and
large) were ejected from the surface of Thebe. Now we suppose that a small
dust grain appears in the Thebe extension due to fragmentation of a larger
dust grain (e.g. in collision with a micrometeoroid). In Fig. 5 we present one
such example. A dust grain with the size $a=0.3\,\mathrm{\mu m}$ is released
at t=0 from a large grain in the vicinity of $r=1.22R_{\rm Th}$. It is assumed
that the small grain is ejected with the radial velocity
$V_{r}(t=0)=\mathrm{2.8\,km\,s^{-1}}$ and the angular frequency
$\Omega(t=0)=\mathrm{0.000078\,s^{-1}}$.

In contrast to the results presented in Fig. 4 this small dust grain does not
migrate into the Thebe ring but continues to oscillate in the Thebe extension.
In the equilibrium state the oscillations take place between $r_{1}\approx
0.96R_{\rm Th}$ and $r_{2}\leq 1.22R_{\rm Th}$. This is a new result because
earlier it was found that the stationary orbit of dust grains should be
shifted inwards from the moon due to the action of the electric field, see,
e.g. Horányi and Burns (1991). Such distinction appears because of the
different initial conditions.

To better understand the situation, we have calculated the potential
describing the radial motion of a small dust grain with a constant electric
charge:

$\frac{1}{2}\left(\frac{d\xi}{dt}\right)^{2}=U(\xi)+W_{0}.$ (25)

Figure 6: Distribution in space of the potential $U$ for a small dust grain
($a=0.3\,\mathrm{\mu m}$) with an equilibrium electric charge. The initial
velocities are the same as in Fig. 5. On the horizontal axis the dimensionless
coordinate $r/R_{\rm Th}$ is given. The region with $U>0$ corresponds to the
range of radii where dust the grain oscillates.

Here $U(\xi)$ is the potential

$\displaystyle
U(\xi)=-\frac{\Omega_{0}^{2}\xi_{0}^{4}}{2\xi^{2}}-\frac{\Omega_{0}q}{R_{\rm
Th}^{3}\xi^{2}}\left(\frac{1}{\xi}-\frac{1}{\xi_{0}}\right)-\frac{q^{2}}{2R_{\rm
Th}^{6}\xi^{2}}\left(\frac{1}{\xi}-\frac{1}{\xi_{0}}\right)^{2}+\frac{\mu+q\Omega_{J}}{R_{\rm
Th}^{3}\xi},$ (26)

where $\xi=r/R_{\rm Th}$, $\Omega_{0}$ is the initial angular frequency
$\Omega_{0}=\Omega(t=0)$, $\xi_{0}$ is the initial radial position of a given
dust grain, $W_{0}$ is a constant added in Eq. (25) to provide the correct
initial velocity.

In Fig. 6 we present the potential $U_{1}(\xi)=U(\xi)+W_{0}$ for a dust grain
starting from $\xi_{0}=1.22$ with the angular frequency
$\Omega_{0}=\mathrm{0.000078\,s^{-1}}$. The dust grain oscillates between
$r_{1}\approx 0.96R_{\rm Th}$ and $r_{2}\approx 1.22R_{\rm Th}$ which
corresponds to the results of the numerical computations presented in Fig. 5.

##  6 Discussion and Conclusion

We have suggested a mechanism for the filling of the Thebe extension by dust
grains. Our calculations are based on the Divine model which assumes that
thermal plasma in the inner magnetosphere of Jupiter is rather warm and dense,
i.e. at the orbit of Thebe $T_{e}=45~{}\mathrm{eV}$ and
$N_{e}=50\,\mathrm{cm^{-3}}$. As a result the equilibrium electric potential
on dust grains in the Thebe extension should be high, $\phi\approx-162~{}\rm
V$, and the total electric charge on a micrometer-size grain is of the order
of $Q_{d}\approx 10^{5}e$. Due to such high values of the potential and the
electric charge, dust grains orbiting Jupiter are almost insensitive to the
solar UV radiation at the sunlit side of the planet. Hence, the mechanism of
shadow resonances as suggested by Hamilton and Krüger (2008) does not work
under these conditions.

According to our analysis the dynamics of strongly charged small dust grains
(fraction of a micrometer) in the vicinity of Thebe’s orbit can be quite
different depending on the initial conditions. Small dust grains ejected from
Thebe even with rather high radial velocity only slightly penetrate into the
Thebe extension and later on gradually migrate into the Thebe ring. Examples
of radial oscillations for grains with different masses are presented in Figs.
3 and 4. At the same time, negatively charged small grains starting their
motion in the Thebe extension remain there for a long time oscillating between
two turning points $r_{1}$ and $r_{2}$. The corresponding example is presented
in Fig. 5. The positions of these turning points depend on the initial
conditions with which a given small grain starts its motion (radial position
and initial velocities). For example, if a small grain ($a=0.3\,\mathrm{\mu
m}$) starts from a large dust grain due to fragmentation at $r_{0}=1.22R_{\rm
Th}$ with the radial velocity and the azimuthal frequency which coincide with
the corresponding parameters of a large grain
($V_{r}=\mathrm{2.8\,km\,s^{-1}}$, $\Omega_{0}=\mathrm{0.000078\,s^{-1}}$), it
oscillates in the equilibrium case between $r_{1}\approx 0.96~{}R_{\rm Th}$
and $r_{2}\approx 1.22~{}R_{\rm Th}$. This means that such grain penetrates
rather deep into the Thebe extension (up to approximately 50,000 km).

Our results have a quite clear physical meaning. Assume that an additional
moon U orbits Jupiter at some distance $R_{U}>R_{Th}$. At the moment $t=0$ a
small dust grain is ejected from this moon and begins to move with some
negative electric charge. It was shown by Horányi and Burns (1991) that
negatively charged grains ejected outside the synchronous orbit of Jupiter
oscillate at radial distances slightly less than the orbit of the moon. In our
case a large dust grain plays the role of an additional moon. As shown in Fig.
5, if the angular velocity of a small dust grain coincides with the that of a
large grain (the moon U), after some time it oscillates at radial distances
slightly less than the instant radial distance of the large grain at t=0. Only
at the beginning due to its finite initial radial velocity, the small grain
makes somewhat larger radial excursions. As for large grains they are weakly
influenced by the corotating electric field due to the small ratio of their
electric charge to mass $Q_{d}/M_{d}$. At the same time it is important to
note that we have considered the case when large dust grains were ejected from
Thebe with a non-zero velocity component parallel to the angular velocity of
Thebe. This additional velocity stems from the ejection of the particles from
Thebe’s surface. Due to this, equilibrium orbits for such grains are shifted
towards larger radial distances which is supported by numerical modelling (see
Figs. 1 and 3).

As mentioned before, the equilibrium electric charge on a dust grain in the
Thebe extension should be high. Hence, the local electric field in the
vicinity of such a grain is also high. After ejection of a grain from the
surface the electric charge and the electric field grow in magnitude,
approaching their equilibrium values (see Fig. 2). If a large grain is porous
it might disintegrate into smaller fragments in case the electric force
becomes strong enough (i.e. the electrostatic stress exceeds the tensile
strength), see, e.g. Stark et al. (2015) and Böhnhardt and Fechtig (1987).
Rather crude estimates confirm that the electric disruption could be efficient
in filling the Thebe extension with small dust grains. Note that such a
process should operate also in the gossamer rings. The other sources of small
dust grains in the Thebe extension are plasma sputtering, mutual collisions of
large grains and their collisions with micrometeorites. All these processes
are slow in comparison with the electric disruption. Note that due to electric
disruption tiny grains (hundreds and tens of nanometer) could appear, see
Stark et al. (2015) and Böhnhardt and Fechtig (1987). This means that not only
Io is a source of tiny grains in the inner magnetosphere of Jupiter (as it was
discussed by Grün et al. (1998)) but also electrostatic disruption of rather
large negatively charged dust grains can occur in the vicinity of Thebe’s and
Amalthea’s orbits about Jupiter.

So far, the measurements by the Juno spacecraft probe the Jovian magnetosphere
only down to $R\approx 10\,\mathrm{R_{J}}$, showing that the ion
concentrations strongly increase towards Jupiter (Kim et al., 2020). In the
future Juno will hopefully provide us with measurements of charged particles
much closer to Jupiter that will allow us to better constrain the physical
processes driving dust particle motion in the Thebe extension.

## Acknowledgments

NB likes to thank the Max Planck Institute for Solar System Research (MPS) for
supporting his visits at MPS during which a significant part of the work for
this manuscript was performed.

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Entropy of Quantum Markov states on Cayley trees

Farrukh Mukhamedov∗
Department of Mathematical Sciences,
College of Science, United Arab Emirates University,
P.O. Box 15551,Al Ain, Abu Dhabi, UAE
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
∗ Corresponding author.

Abdessatar Souissi∗
Department of Accounting, College of Business Management
Qassim University, Buraydah, Saudi Arabia

E-mail<EMAIL_ADDRESS>
∗ Corresponding author.

###### Abstract

In this paper, we continue the investigation of quantum Markov states (QMS)
and define their mean entropies. Such entropies are explicitly computed under
certain conditions. The present work takes a huge leap forward at tackling one
of the most important open problems in quantum probability which concerns the
calculations of mean entropies of quantum Markov fields. Moreover, it opens
new perspective for the generalization of many interesting results related to
the one dimensional quantum Markov states and chains to multi-dimensional
cases.
Mathematics Subject Classification: 46L53, 60J99, 46L60, 60G50.
Key words: Quantum Markov state; Cayley tree; mean entropy;

##  1 Introduction

It is well-known that the classical and quantum dynamical entropies are
important tools in many areas of mathematics and play a huge role in
information sciences [37] and note that entropies of classical Markov chains
are just as significant in statistical mechanics [10]. Quantum dynamical
entropy has been studied by Connes and Sormer[17], Connes, Narnhofer, Thirring
[18] and many others. Note that recently, the quantum dynamical entropy and
the quantum dynamical mutual entropy were defined by Ohya in terms of
complexity [37]. The notion of quantum Markov chain (QMC) was formulated by
means of the transition expectation introduced by Accardi [1]. Therefore, in
[8, 35] dynamical entropy through a quantum Markov chain has been introduced
and calculated for some simple models. In this similar fashion and direction a
mutual entropy in quantum Markov chains scheme has been investigated in [47].
Note the significance of these contributions as these calculations are vital
in both physics and information theory.

On the other hand, the study of quantum many-body systems has lived an
explosion of results. This is specifically true in the field of Tensor
Networks. Recently, ”Matrix Product States”, and more generally ”Tensor
Network States” have played a crucial role in the description of the whole
quantum systems [16, 38]. We point out that an interesting mathematical
approach to quantum states on tensor networks has closely tied up with QMC on
tensor product of matrix algebras [1, 24]. Therefore, QMC have found massive
applications in several research domains of quantum statistical physics and
information [19, 20],[15],[44].

As we have mentioned, it is very important to develop a method which enables
to compute the entropy for quantum systems (([18, 40]). In [39, 41] the
dynamical entropy was computed in the sense of Connes, Narnhofer, and Thirring
[18] of shift automorphism of QMC. Moreover, the dynamical entropy was proved
to be equal to the mean entropy. It is important to emphasize that mean
entropies calculations of QMC have been obtained in several papers [21, 36,
42]. However, in all these investigates QMC were considered over a one
dimensional lattice.

On the other hand, it is crucial to know certain formulas for the mean
entropies of the Gibbs measures on each of a sequence of large finite graphs.
When the graph is non-amenable, then such a calculation becomes very tricky
[10]. Recently, in [11] the mentioned formula has been found for Gibbs
measured defined over regular trees (in particularly, Cayley trees).

We point out that, in [6, 27, 28, 29] we have started to investigate
particular classes of quantum Markov chains (QMC) associated to the Ising
types models on the Cayley trees. It turned out that the above considered QMCs
fall into a special class called quantum Markov states (QMS) (see [30, 31]).
Furthermore, in [22, 30, 31] a description of QMS has been carried out. It is
worth to stress that the considered QMS had the Markov property not only with
respect to levels of the considered tree, but also with regard to the
interaction domain at each site, which has a finer structure, and through a
family of suitable quasi-conditional expectations which are localized [30,
32]. Such a localization property is essential for the integral decomposition
of QMS, since it takes into account the finer structure of conditional
expectations and filtration [2, 9].

In this paper, we continue to investigate such kind of QMS and define for
these states their mean entropies and under some conditions we are able to
explicitly compute them. The present work is another step towards one of the
most important open problems in quantum probability, which concerns the
calculations of means entropies of quantum Markov fields. Moreover, it opens a
new perspective for the generalization of many interesting results related to
one dimensional quantum Markov states and chains to multi-dimensional cases.

##  2 Preliminaries

Let $\Gamma^{k}_{+}=(V,E)$ be a semi-infinite Cayley tree of order $k\geq 1$
with the root $x^{0}$ (i.e. each vertex of $\Gamma^{k}_{+}$ has exactly $k+1$
edges, except for the root $x^{0}$, which has $k$ edges). Here $L$ is the set
of vertices and $E$ is the set of edges. The vertices $x$ and $y$ are called
nearest neighbors and they are denoted by $l=<x,y>$ if there exists an edge
connecting them. A collection of the pairs $<x,x_{1}>,\dots,<x_{d-1},y>$ is
called a path from the point $x$ to the point $y$. The distance $d(x,y),x,y\in
V$, on the Cayley tree, is the length of the shortest path from $x$ to $y$.

Recall a coordinate structure in $\Gamma^{k}_{+}$: every vertex $x$ (except
for $x^{0}$) of $\Gamma^{k}_{+}$ has coordinates $(i_{1},\dots,i_{n})$, here
$i_{m}\in\\{1,\dots,k\\}$, $1\leq m\leq n$ and for the vertex $x^{0}$ we put
$(0)$. Namely, the symbol $(0)$ constitutes level 0, and the sites
$(i_{1},\dots,i_{n})$ form level $n$ (i.e. $d(x^{0},x)=n$) of the lattice.

The hierarchical structure on the considered tree is expressed as follows.

$W_{n}=\\{x\in
V\,:\,d(x,x_{0})=n\\}\qquad\Lambda_{n}=\bigcup_{k=0}^{n}W_{k},\qquad\Lambda_{[n,m]}=\bigcup_{k=n}^{m}W_{k},\
(n<m).$

For $x\in\Gamma^{k}_{+}$, $x=(i_{1},\dots,i_{n})$ (i.e. $x\in W_{n}$) define
the set of its direct successors vertices by

$S(x):=\\{y\in\Lambda_{n}^{c}:\quad<x,y>\\}\subset W_{n+1}$

and its $k^{th}$ successors on the rooted tree $\Gamma^{k}_{+}$ is defined by
the induction as follows

$\displaystyle S_{1}(x)$ $\displaystyle=$ $\displaystyle S(x)$ $\displaystyle
S_{k+1}(x)$ $\displaystyle=$ $\displaystyle S(S_{k}(x)),\,\,\forall k\geq 1.$

Put

$T(x)=\bigcup_{k\geq 0}S_{k}(x)$ (1)

where $S_{0}(x)=x.$ Denote

$\overrightarrow{S}(x)=\\{(x,1),(x,2),\cdots,(x,k)\\},$

here $(x,i)$ means that $(i_{1},\dots,i_{n},i)$.

Since the Cayley tree $\Gamma^{k}_{+}$ is regular and each vertex has exactly
$k$ direct successors then according to the above enumeration, we define $k$
shifts on the tree as follows: for each $j\in\\{1,2,\dots,k\\}$ the $j^{th}$
shift is given by

$\alpha_{j}(x)=(j,x)=(j,i_{1},i_{2},\cdots,i_{n})\in W_{n+1}.$ (2)

for every $x=(i_{1},i_{2},\dots,i_{n})\in W_{n}$. Let
$g=(j_{1},j_{2},\cdots,j_{N})\in V$ one defines

$\alpha_{g}(x):=\alpha_{j_{1}}\circ\alpha_{j_{2}}\circ\cdots\circ\alpha_{j_{N}}(x)=(j_{1},j_{2},\cdots,j_{N},i_{1},i_{2},\cdots,i_{n}).$

The shift $\alpha_{g}$ maps the Cayley tree $\Gamma^{k}_{+}$ onto its sub-tree
$T_{x}$ defined by (1). It follows that $\alpha_{x}(0)=x$ and
$\alpha_{x}(\Lambda_{n})=S_{n}(x)$.

Let $\mathcal{A}$ be a $C^{*}$-algebra with unit ${\bf 1}\\!\\!{\rm I}$. For
each $x\in V$ we denote $\mathcal{A}_{x}$ the $C^{*}$-algebra which is the
same as $\mathcal{A}$, i.e. $\mathcal{A}_{x}\equiv\mathcal{A}$. For each
$\Lambda\subset V$ we denote by $\mathcal{A}_{\Lambda}$ the algebra generated
by $\\{\mathcal{A}_{x}:\,x\in\Lambda\\}$. In these notations, one has

$\mathcal{A}_{\Lambda_{n}}=\bigotimes_{x\in\Lambda_{n}}\equiv\mathcal{A}_{\Lambda_{n}}\otimes{\bf
1}\\!\\!{\rm I}_{W_{n+1}}\subset\mathcal{A}_{\Lambda_{n+1}}.$

Consider the local $*-$algebra

$\mathcal{A}_{V;\,loc}:=\bigcup_{n\in\mathbb{N}}\mathcal{A}_{\Lambda_{n}}$ (3)

One has

$\mathcal{A}_{V}:=\overline{\mathcal{A}_{V;\,loc}}^{C^{*}}.$

By $\mathcal{S}(\mathcal{B})$ we denote the set of states on a C∗–algebra
$\mathcal{B}$.

##  3 Quantum Markov states and chains on Trees

Recall that a transition expectation is a completely positive identity
preserving map from a C∗–algebra into its C∗-algebra.

######  Definition 3.1.

Let $\mathcal{C}\subset\mathcal{B}\subset\mathcal{A}$ be a triplet of unitary
C∗–algebras. A map $E:\mathcal{A}\to\mathcal{B}$ is called quasi-conditional
expectation w.r.t. the given triplet if it is completely positive identity
preserving and satisfies the following property

$E(ca)=cE(a);\quad\forall a\in\mathcal{A},\forall c\in\mathcal{C}.$ (4)

The typical form of transitions expectations $E_{\Lambda_{n}}$ w.r.t. the
triplet
$\mathcal{A}_{\Lambda_{n-1}}\subset\mathcal{A}_{\Lambda_{n}}\subset\mathcal{A}_{\Lambda_{n+1}}$
that we are going to investigate in the have the following form

${E}_{\Lambda_{n}}=id_{\Lambda_{n-1}}\otimes\mathcal{E}_{W_{n}}$ (5)

where $\mathcal{E}_{W_{n}}$ is a transition expectation from
$\mathcal{A}_{[N,n+1]}\to\mathcal{A}_{W_{n}}$. Note that $\mathcal{E}_{W_{n}}$
is assumed to be localized in the sense of [30]. i.e. for each $x\in W_{n}$
there exists a transition expectation $\mathcal{E}_{x}$ from
$\mathcal{A}_{\\{x\\}\cup S(x)}$ into $\mathcal{A}_{x}$ such that

$\mathcal{E}_{W_{n}}=\bigotimes_{x\in W_{n}}\mathcal{E}_{x}.$ (6)

The shifts $\alpha_{j}$ given by (2) acts on the algebra the algebra
$\mathcal{A}_{V}$ as follows:

$\alpha_{j}\left(\bigotimes_{x\in\Lambda_{n}}a_{x}\right):={\bf 1}\\!\\!{\rm
I}^{(o)}\otimes\bigotimes_{i=0}^{n}a_{x}^{(j,x)}.$ (7)

######  Definition 3.2.

A state $\varphi$ on $\mathcal{A}_{L}$ is said to be translation-invariant if

$\varphi\circ\alpha_{j}=\varphi$ (8)

for every $j\in\\{1,2,\cdots,k\\}$.

######  Remark 3.3.

The above definition generalizes the notion of translation-invariant states on
the one-dimensional lattice which corresponds to the case $k=1$. Note that the
entropy of quantum Markov chains on the quantum spin lattice has been studied
in many papers (see [39, 8, 23])

######  Definition 3.4.

[9] Let $\phi_{o}\in\mathcal{S}(\mathcal{A}_{o})$ be a (initial) state,
$\\{E_{\Lambda_{n}}\\}$ a sequence of quasi-conditional expectations w.r.t.
the triple
$\mathcal{A}_{{\Lambda}_{n-1}}\subseteq\mathcal{A}_{{\Lambda}_{n}}\subseteq\mathcal{A}_{{\Lambda}_{n+1}}$
and a sequence $h_{n}\in\mathcal{A}_{V_{n},+}$ of boundary conditions such
that for each $a\in\mathcal{A}_{V}$ the limit

$\varphi(a):=\lim_{n\to\infty}\phi_{0}\circ E_{\Lambda_{0}}\circ
E_{\Lambda_{1}}\circ\cdots\circ E_{\Lambda_{n}}(h_{n+1}^{1/2}ah_{n+1}^{1/2})$
(9)

exists in the weak-*-topology and defines a state. Then the triplet
$(\phi_{o},(E_{\Lambda_{n}})_{n},(h_{n})_{n})$ is called a quantum Markov
chain on $\mathcal{A}_{V}$. In this case the limiting state $\varphi$ given by
(9) is also called quantum Markov chain (QMC).

######  Definition 3.5.

A quantum Markov chain $\varphi$ on $\mathcal{A}_{V}$ is called quantum Markov
state(QMS) if one has

$\varphi\lceil_{\mathcal{A}_{\Lambda_{n}}}\circ
E_{\Lambda_{n}}=\varphi\lceil_{\mathcal{A}_{\Lambda_{n+1}}}.$ (10)

Due to the contributions made in [3, 24], the conditional expectations can be
used in place of quasi conditional expectations. However, the boundary
condition plays a more physically significant role in the existence of phase
transitions (see for instance [5, 27, 28, 46]). In [30, 31, 32], the authors
studied in detail the structure of quantum Markov states associated with
localized transitions expectations on the Cayley tree.

Let us now deal with the entropy of states. Assume that $\varphi$ is a state
on $\mathcal{A}_{V}$ and $\Lambda\subset V$ is a finite region. Let
$S(\varphi\lceil_{\mathcal{A}_{\Lambda}}$) be the von Neumann entropy of the
of the state $\varphi$ on the algebra $\mathcal{A}_{\Lambda}$ (see [37]),
where $\varphi\lceil_{\mathcal{A}_{\Lambda}}$ is the state on
$\mathcal{A}_{\Lambda}$ defined by the restriction of $\varphi$ on
$\mathcal{A}_{\Lambda}$. If $\mathcal{D}^{\varphi}_{\Lambda}$ is the density
matrix of the state $\varphi\lceil\mathcal{A}_{\Lambda}$, then

$S(\varphi\lceil_{\mathcal{A}_{\Lambda}})=-\mathrm{Tr}(D_{\Lambda}^{\varphi}\log
D_{\Lambda}^{\varphi})$ (11)

In the one dimensional case, where $V=\mathbb{N}$, using the subadditivity of
$S(\varphi\lceil_{\mathcal{A}_{\Lambda}})$, it was shown that the limit

$s(\varphi):=\lim_{n\to\infty}\frac{1}{n+1}S(\varphi\lceil_{\mathcal{A}_{[0,n]}})$
(12)

exists. The quantity $s(\varphi)$ is called the mean entropy of the state
$\varphi$ [21, 42]. In the next section, we are going to discuss such an
entropy for quantum Markov states on the Cayley trees.

##  4  Mean Entropy for quantum Markov states on Trees

In this section, we introduce the mean entropy for quantum Markov states
defined on trees. Let $\varphi$ be a locally faithful quantum Markov state on
$\mathcal{A}_{V}$, i.e. $\varphi\lceil_{\mathcal{A}_{\Lambda_{n}}}$ is
faithful for all $n\geq 1$. Indeed, by means of [36] one can establish that if
$\varphi$ is a locally faithful QMS, then $\varphi$ is faithful. From the tree
structure, it is quite natural to define a generalization of (12) as follows

$s(\varphi):=\lim_{n\to\infty}\frac{1}{|\Lambda_{n}|}S(\varphi\lceil_{\mathcal{A}_{\Lambda_{n}}})$
(13)

Let $\\{E_{\Lambda_{n}}\\}_{n\geq 0}$ be the sequence conditional expectations
w.r.t. the triplet
$\mathcal{A}_{\Lambda_{n-1}}\subset\mathcal{A}_{\Lambda_{n}}\subset\mathcal{A}_{\Lambda_{n+1}}$
associated with $\varphi$. We consider the Radon-Nikodym derivatives
$D^{\varphi}_{\Lambda_{n}}$ of the state
$\varphi_{\Lambda_{n}}:=\varphi\lceil_{\mathcal{A}_{\Lambda_{n}}}$ with
respect to the trace $\mathrm{Tr}_{\mathcal{A}_{\Lambda_{n}}}$ on
$\mathcal{A}_{\Lambda_{n}}$, i.e.

$\varphi_{\Lambda_{n}}(\cdot)=\mathrm{Tr}_{\mathcal{A}_{\Lambda_{n}}}(D^{\varphi}_{\Lambda_{n}}\cdot)$

The transition expectation $\mathcal{E}_{W_{n}}$ associated with
$E_{\Lambda_{n}}$ through (5) is localized in the sense of [30, 31]. Then it
satisfies (6) for certain localized transition expectations $\mathcal{E}_{x}$
from $\mathcal{A}_{\\{x\\}\cup S(x)}$ into $\mathcal{A}_{x}$ with $x\in
W_{n}$.

For $X\subseteq Y$ two finite sub-sets of $V$, we denote $T^{Y}_{X}$ the map
defined by linear extension of

$T^{Y}_{X}(a_{X}\otimes a_{Y\setminus X})=\mathrm{Tr}(a_{Y\setminus X})a_{X}$

One has

$T^{Y}_{X}(D^{\varphi}_{Y})=D^{\varphi}_{X}$

According to [26] the map $T_{X}^{Y}$ is the unique Umegaki conditional
expectation from $\mathcal{A}_{Y}$ to $\mathcal{A}_{X}$ with respect to the
tracial state.

Let $A,B$ and $C$ are mutually disjoint finite subsets of the vertex $V$. One
can easily check that

$\displaystyle T^{A,B,C}_{A,B}\lceil_{\mathcal{A}_{B\cup C}}=T^{B,C}_{B}$ (14)
$\displaystyle\mathcal{A}_{B}=\mathcal{A}_{A\cup B}\cap\mathcal{A}_{B\cap C}$
(15) $\displaystyle
T^{A,B,C}_{B,C}T^{A,B,C}_{A,B}=T^{A,B,C}_{A,B}T^{A,B,C}_{B,C}=T^{A,B,C}_{C}$
(16)

For each $x\in W_{n}$, assume that $K_{\\{x\\}\cup
S(x)}\in\mathcal{A}_{\\{x\\}\cup S(x)}$ is the conditional density amplitude
of the transition expectation $\mathcal{E}_{x}$, i.e.

$\mathcal{E}_{x}(\,\cdot\,)=T_{\\{x\\}}^{\\{x\\},S(x)}\left(K_{\\{x\\}\cup
S(x)}^{*}\,\cdot\,K_{\\{x\\}\cup S(x)}\right)$

Using (6), the conditional density amplitude of the transition expectation
$\mathcal{E}_{W_{n}}$ has the following form

$K_{[n,n+1]}:=\bigotimes_{x\in W_{n}}K_{\\{x\\}\cup
S(x)}\in\mathcal{A}_{\Lambda_{[n,n+1]}}$ (17)

######  Lemma 4.1.

Let $\varphi$ be a QMS on $\mathcal{A}_{V}$ together with its sequence of
quasi-conditional expectations $E_{\Lambda_{n}}$. In the above notations, one
has

$D^{\varphi}_{\Lambda_{n+1}}=K_{[n,n+1]}D^{\varphi}_{\Lambda_{n}}K_{[n,n+1]}^{\ast}$
(18)

###### Proof.

Let $a\in\mathcal{A}_{\Lambda_{n+1}}$, one has

$\displaystyle\varphi\lceil_{\mathcal{A}_{\Lambda_{n}}}(E_{\Lambda_{n}}(a))$
$\displaystyle=$
$\displaystyle\mathrm{Tr}_{\Lambda_{n}}\left(D^{\varphi}_{\Lambda_{n}}\mathrm{T}^{W_{n},W_{n+1}}_{W_{n}}\left(K_{[N,n+1]}^{\ast}aK_{[N,n+1]}\right)\right)$
$\displaystyle=$
$\displaystyle\mathrm{Tr}_{\Lambda_{n+1}}\left(D^{\varphi}_{\Lambda_{n}}K_{[N,n+1]}^{\ast}aK_{[N,n+1]}\right)$
$\displaystyle=$
$\displaystyle\mathrm{Tr}_{\Lambda_{n+1}}\left(K_{[N,n+1]}D^{\varphi}_{\Lambda_{n}}K_{[N,n+1]}^{\ast}a\right)$

Using (10), we arrive at the required identity. ∎

######  Theorem 4.2.

Let $\varphi$ be a QMS on $\mathcal{A}_{V}$. Then

$S(\varphi\lceil_{\mathcal{A}_{\Lambda_{n+1}}})+S(\varphi\lceil_{\mathcal{A}_{W_{n}}})=S(\varphi\lceil\mathcal{A}_{\Lambda_{n}})+S(\varphi\lceil_{\mathcal{A}_{\Lambda_{[n,n+1]}}}).$
(19)

If in addition $K_{[n,n+1]}$ and $D^{\varphi}_{\Lambda_{n]}}$ commute, then

$S(\varphi\lceil_{\mathcal{A}_{\Lambda_{n+1}}})-S(\varphi\lceil_{\mathcal{A}_{\Lambda_{n}}})=-2\sum_{x\in
W_{n}}\varphi_{\mathcal{A}_{\Lambda_{[n,n+1]}}}\left(\log|K_{\\{x\\}\cup\overrightarrow{S}(x)}|\right).$
(20)

###### Proof.

Since $\varphi$ is a Markov state on $\mathcal{A}_{V}$ with respect to the
filtration $(\mathcal{A}_{\Lambda_{n}})_{n}$, for each $n$ there exists a
quasi-conditional expectation $E_{n}$ with respect to the triplet
$\mathcal{A}_{\Lambda_{n-1}}\subset\mathcal{A}_{\Lambda_{n}}\subset\mathcal{A}_{\Lambda_{n+1}}$
satisfying

$\varphi\lceil_{\mathcal{A}_{\Lambda_{n}}}\circ
T_{\Lambda_{n}}^{\Lambda_{n+1}}\circ
E_{n}=\varphi\lceil_{\mathcal{A}_{\Lambda_{n}}}\circ
E_{n}=\varphi\lceil_{\mathcal{A}_{\Lambda_{n+1}}}.$

The quasi-conditional expectation $E_{n}$ can be written as follows

$E_{n}=id_{\Lambda_{n-1}}\otimes E_{[n,n+1]}$

where $E_{[n,n+1]}$ is a transition expectation from
$\mathcal{A}_{\Lambda_{[n,n+1]}}$ into $\mathcal{A}_{W_{n}}$. It follows that

$\varphi\lceil_{\mathcal{A}_{W_{n}}}\circ T_{W_{n}}^{W_{n+1},W_{n+1}}\circ
E_{[n,n+1]}=\varphi\lceil_{\mathcal{A}_{\Lambda_{[n,n+1]}}}\circ
E_{n}=\varphi\lceil_{\mathcal{A}_{\Lambda_{[n,n+1]}}}\circ
E_{n}=\varphi\lceil_{\mathcal{A}_{\Lambda_{[n,n+1]}}}.$

Then in the notations of (14), (15) and (16) for
$A=W_{n+1},B=W_{n},C=\Lambda_{n-1}$, the conditional expectation
$T^{\Lambda_{n+1}}_{\Lambda_{n}}$ is enough (in the sense of [43, 26]) for the
states $\varphi\lceil_{\mathcal{A}_{\Lambda_{n+1}}}$ and
$\varphi\lceil_{\mathcal{A}_{\Lambda_{[n,n+1]}}}$. Hence, by a result of [26],
we obtain

$S(\varphi\lceil_{\mathcal{A}_{\Lambda_{n+1}}})-S(\varphi_{\mathcal{A}_{W_{n},W_{n+1}}})-S(\varphi\lceil_{\mathcal{A}_{\Lambda_{n}}})+S(\varphi\lceil_{\mathcal{A}_{W_{n}}})=0$

This proves (19).

Now assume that $K_{[N,n+1]}$ and $D^{\varphi}_{\Lambda_{n}}$ commute. Since
$T^{\Lambda_{n+1}}_{W_{n},W_{n+1}}(D^{\varphi}_{\Lambda_{n}})=D^{\varphi}_{W_{n}}\otimes{\bf
1}\\!\\!{\rm I}_{W_{n+1}}$, by Lemma 4.1 the density matrices of
$\varphi_{\Lambda_{n+1}}$ and $\varphi_{W_{n},W_{n+1}}$ are
$D^{\varphi}_{\Lambda_{n}}|K_{[n,n+1]}|^{2}$ and
$D^{\varphi}_{\Lambda_{[}n,n+1]}=D^{\varphi}_{\Lambda_{n}}|K_{[n,n+1]}|^{2}$,
respectively. This leads to

$\displaystyle S(\varphi\lceil_{\mathcal{A}_{\Lambda_{n+1}}})$
$\displaystyle=$
$\displaystyle-\mathrm{Tr}\left(D^{\varphi}_{\Lambda_{n}}|K_{[N,n+1]}|^{2}\log(D^{\varphi}_{\Lambda_{n}}|K_{[N,n+1]}|^{2})\right)$
$\displaystyle=$ $\displaystyle-\mathrm{Tr}\left(D^{\varphi}_{\Lambda_{n}}\log
D^{\varphi}_{\Lambda_{n}}\mathcal{E}_{[n,n+1]}({\bf 1}\\!\\!{\rm
I})\right)-2\mathrm{Tr}\left(D^{\varphi}_{\Lambda_{n}}|K_{[n,n+1]}|^{2}\log|K_{[n,n+1]}|\right)$
$\displaystyle=$ $\displaystyle
S(\varphi\lceil_{\mathcal{A}_{\Lambda_{n}}})-2\varphi\lceil_{\mathcal{A}_{{[n,n+1]}}}\left(\log|K_{{[n,n+1]}}|\right).$

On the other hand, we find

$\displaystyle S(\varphi\lceil_{\mathcal{A}_{\Lambda_{[n,n+1]}}})$
$\displaystyle=$
$\displaystyle-\mathrm{Tr}(D^{\varphi}_{W_{n}}|K_{[n,n+1]}|^{2}\log(D^{\varphi}_{W_{n}}|K_{[n,n+1]}|^{2}))$
$\displaystyle=$
$\displaystyle-\mathrm{Tr}\left(D^{\varphi}_{W_{n}}\log\left(D^{\varphi}_{W_{n}}\right)\mathcal{E}_{[N,n+1]}({\bf
1}\\!\\!{\rm
I})\right)-2\mathrm{Tr}\left(D^{\varphi}_{W_{n}}|K_{[n,n+1]}|^{2}\log|K_{[n,n+1]}|\right)$
$\displaystyle=$ $\displaystyle
S(\varphi\lceil\mathcal{A}_{W_{n}})-2\varphi\lceil_{\mathcal{A}_{[N,n+1]}}\left(\log|K_{[N,n+1]}|\right)$
$\displaystyle\overset{(\ref{Knn+1=prodKxs(x)})}{=}$ $\displaystyle
S(\varphi\lceil\mathcal{A}_{W_{n}})-2\sum_{x\in
W_{n}}\varphi_{\mathcal{A}_{\Lambda_{[n,n+1]}}}\left(\log|K_{\\{x\\}\cup\overrightarrow{S}(x)}|\right).$

which proves (20). ∎

The following result consists a generalization to trees of a formula of the
mean entropy for translation-invariant quantum Markov states on the quantum
spin lattice [37].

######  Corollary 4.3.

Assume that conditions of Theorem 4.2 are satisfied. If a QMS $\varphi$ is
translation-invariant then

$\frac{S(\varphi_{\Lambda_{n+1}})-S(\varphi_{\Lambda_{n}})}{|W_{n}|}=S(\varphi_{\Lambda_{1]}})-S(\varphi_{\Lambda_{0]}}).$
(21)

###### Proof.

One has

$\varphi_{\Lambda_{n+1}}(\log|K_{\\{x\\}\cup\overrightarrow{S}(x)}|)=\varphi_{\Lambda_{1]}}(\log|K_{\\{o\\}\cup\overrightarrow{S}(o)}|),\quad\forall
x\in L.$

Then (20) leads to (21). ∎

It is known that, the mean entropy (12) of a QMS on the one-dimensional spin
lattice is obtained by the following limit

$s(\varphi)=\lim_{n\to\infty}S(\varphi\lceil_{\mathcal{A}_{[0,n+1]}})-S(\varphi\lceil_{\mathcal{A}_{[0,n]}})$

The following theorem proposes an extension of this result to the case of
trees.

######  Theorem 4.4.

The mean entropy of the Quantum Markov state $\varphi$ satisfies

$s(\varphi)=\lim_{n\to\infty}\frac{S(\varphi\lceil_{\mathcal{A}_{\Lambda_{n+1}}})-S(\varphi\lceil_{\mathcal{A}_{\Lambda_{n}}})}{|\Lambda_{n+1}|-|\Lambda_{n}|}$
(22)

If in addition, $\varphi$ is translation-invariant then

$s(\varphi)=\frac{S(\varphi\lceil_{\mathcal{A}_{\Lambda_{1}}})-S(\varphi\lceil_{\mathcal{A}_{\Lambda_{0}}})}{k}.$
(23)

###### Proof.

One has

$\displaystyle\frac{S(\varphi\lceil\mathcal{A}_{\Lambda_{n+1}})-S(\varphi\lceil\mathcal{A}_{\Lambda_{n}})}{|\Lambda_{n+1}|-|\Lambda_{n}|}$
$\displaystyle=$
$\displaystyle\frac{|\Lambda_{n+1}|}{|\Lambda_{n+1}|}\frac{S(\varphi\lceil\mathcal{A}_{\Lambda_{n+1}})}{|\Lambda_{n+1}|}-\frac{|\Lambda_{n}|}{|\Lambda_{n+1}|}\frac{S(\varphi\lceil\mathcal{A}_{\Lambda_{n}})}{|\Lambda_{n}|}$
$\displaystyle=$
$\displaystyle\frac{k^{n+2}-1}{(k-1)k^{n+1}}\frac{S(\varphi\lceil\mathcal{A}_{\Lambda_{n+1}})}{|\Lambda_{n+1}|}-\frac{k^{n+1}-1}{(k-1)k^{n}}\frac{S(\varphi\lceil\mathcal{A}_{\Lambda_{n}})}{|\Lambda_{n}|}.$

This leads to (22). In the translation-invariant case, by Corollary 4.3 we
arrive at (23). ∎

######  Example 4.5.

Let us consider the semi-infinite Cayley tree of order two $\Gamma^{+}_{2}$.
Let $\mathcal{A}_{x}=M_{2}(\mathbb{C})$ for $x\in\mathcal{A}_{V}$. By
$\sigma_{x}^{u}$, $\sigma_{y}^{u}$, $\sigma_{z}^{u}$ we denote the standard
Pauli spin matrices at site $u\in V$, i.e.

${\bf 1}\\!\\!{\rm I}^{(u)}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},\
\sigma_{x}^{(u)}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\
\sigma_{y}^{(u)}=\begin{pmatrix}0&-i\\\ i&0\end{pmatrix},\
\sigma_{z}^{(u)}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$

For every vertices $(u,(u,1),(u,2))$ we put

$\displaystyle K_{<u,(u,i)>}=\exp\\{\beta H_{u,(u,i)>}\\},\ \ i=1,2,\
\beta>0,$ (24) $\displaystyle L_{>(u,1),(u,2)<}=\exp\\{J\beta
H_{>(u,1),(u,2)<}\\},\ \ J>0,$ (25)

where

$\displaystyle H_{<u,(u,i)>}=\frac{1}{2}\big{(}{\bf 1}\\!\\!{\rm I}^{(u)}{\bf
1}\\!\\!{\rm I}^{(u,i)}+\sigma^{(u)}_{z}\sigma^{(u,i)}_{z}\big{)},$ (26)
$\displaystyle H_{>(u,1),(u,2)<}=\frac{1}{2}\big{(}{\bf 1}\\!\\!{\rm
I}^{(u,1)}{\bf 1}\\!\\!{\rm
I}^{(u,2)}+\sigma^{(u,1)}_{z}\sigma^{(u,2)}_{z}\big{)}.$ (27)

For each $n\in\mathbb{N}$ and $x\in V$, we denote

$\displaystyle A_{x,(x,1),(x,2)}=K_{<x,(x,1)>}K_{<x,(x,1)>}L_{>(x,1),(x,2)<},$
(28) $\displaystyle
K_{[m,m+1]}:=\prod_{x\in\overrightarrow{W}_{m}}A_{x,(x,1),(x,2)},\ \ 1\leq
m\leq n,$ (29)
$\displaystyle\mathbf{h}_{n}^{1/2}:=\prod_{x\in\overrightarrow{W}_{n}}(h_{x})^{1/2},\
\ \ \mathbf{h}_{n}=\mathbf{h}_{n}^{1/2}(\mathbf{h}_{n}^{1/2})^{*}$ (30)
$\displaystyle{\mathbf{K}}_{n}:=\omega_{0}^{1/2}\prod_{m=1}^{n-1}K_{[m,m+1]}\mathbf{h}_{n}^{1/2}$
(31) $\displaystyle\mathcal{W}_{n]}:={\mathbf{K}}_{n}^{*}{\mathbf{K}}_{n}.$
(32)

Here $w_{0}\in\mathcal{A}_{o,+}$ is and initial state and
$h_{x}\in\mathcal{A}_{x,+}$ is a family of positive boundary conditions.

Consider the following functional $\varphi^{(n)}_{w_{0},\mathbf{h}}$, given by

$\displaystyle\varphi^{(n)}_{w_{0},\mathbf{h}}(a)={\rm
Tr}(\mathcal{W}_{n+1]}(a\otimes{\bf 1}\\!\\!{\rm I}_{W_{n+1}})),$ (33)

for every $a\in{\mathcal{B}}_{\Lambda_{n}}$. In [27] we have shown that the
compatibility equation (10) is satisfied for the sequence
$\varphi^{(n)}_{w_{0},\mathbf{h}}$ under the following conditions:

$\displaystyle\mathrm{Tr}(\omega_{0}h_{0})=1,$ (34)
$\displaystyle\mathrm{Tr}_{x]}\big{(}{A_{x,(x,1),(x,2)}}h_{(x,1)}h_{(x,2)}A_{x,(x,1),(x,2)}^{*}\big{)}=h_{x},\
\ \textrm{for every}\ \ x\in L,$ (35)

Moreover, it has been established the existence of three quantum Markov states
$\varphi_{1},\varphi_{2}$ and $\varphi_{\alpha}$ associated with three
different translation-invariant solutions $h_{1},h_{2}$ and $h_{\alpha}$ of
(34) with

$\alpha=\frac{4}{e^{2J\beta}(e^{4\beta}+1)+2e^{2\beta}}.$

In the following we will compute the mean entropy for the Markov state
$\varphi_{\alpha}$. In [33] we found the density matrices
$\mathcal{W}_{n]}^{(\alpha)}$ of $\varphi_{\alpha}$ as follows

$\mathcal{W}_{n]}^{(\varphi)}=\exp\mathcal{H}_{n]}^{(\alpha)}$
$\mathcal{H}_{n]}^{(\alpha)}:=\sum_{x\in
W_{n-1]}}\mathcal{H}_{\alpha}^{(x,(x,1),(x,2))}$ (36)

where

$\mathcal{H}_{\alpha}^{(x,(x,1),(x,2))}:={\bf 1}\\!\\!{\rm
I}^{\otimes|\\{<_{(W_{n-1})}x\\}|}\otimes\mathcal{H}^{(x)}_{\alpha}\otimes{\bf
1}\\!\\!{\rm I}^{\otimes(|\\{>_{(W_{n-1})}x\\}|-2)},$

here

$\mathcal{H}_{\alpha}:=\ln\alpha{\bf 1}\\!\\!{\rm
I}_{M_{8}(\mathbb{C})}+2\left(\begin{array}[]{cccccccc}\beta(J+2)&&&&&&&\\\
&\beta&&&&&&\\\ &&\beta&&&\left(0\right)&&\\\ &&&\beta J&&&&\\\ &&&&\beta
J&&&\\\ &&\left(0\right)&&&\beta&&\\\ &&&&&&\beta&\\\ &&&&&&&\beta(J+2)\\\
\end{array}\right)$

and $\\{<_{(W_{n-1})}x\\}:=\\{y\in W_{n-1}\mid y<x\\}$,
$\\{>_{(W_{n-1})}x\\}:=\\{y\in W_{n-1}\mid y>x\\}$. In this notation, the
order $"x<y"$ is understood with respect to the lexicographic order applied to
the coordinates of $x$ and $y$.

It follows that

$\mathcal{W}_{n]}^{(\varphi_{\alpha})}=\mathcal{W}_{n-1]}^{(\varphi_{\alpha})}\exp\left(\sum_{x\in
W_{n-1}}\mathcal{H}_{\alpha}^{(x,(x,1),(x,2))}\right)$ (37)

The von Neuman entropy of $\varphi^{\alpha}\lceil_{\mathcal{A}_{\Lambda_{n}}}$
satisifies

$\displaystyle S(\varphi_{\alpha}\lceil_{\mathcal{A}_{\Lambda_{n}}})$
$\displaystyle=$ $\displaystyle-{\rm
Tr}\left(\mathcal{W}_{n]}^{(\varphi_{\alpha})}\log\mathcal{W}_{n]}^{(\varphi_{\alpha})}\right)$
$\displaystyle\overset{(\ref{densityn})}{=}$ $\displaystyle-{\rm
Tr}\left(\mathcal{W}_{n]}^{(\varphi_{\alpha})}\left(\log\mathcal{W}_{n-1]}^{(\varphi_{\alpha})}+\sum_{x\in
W_{n-1}}\mathcal{H}_{\alpha}^{(x,(x,1),(x,2))}\right)\right)$ $\displaystyle=$
$\displaystyle-{\rm
Tr}\left(\mathcal{W}_{n]}^{(\varphi_{\alpha})}\log\mathcal{W}_{n-1]}^{(\varphi_{\alpha})}\right)-\sum_{x\in
W_{n-1}}{\rm
Tr}\left(\mathcal{W}_{n]}^{(\varphi_{\alpha})}\mathcal{H}_{\alpha}^{(x,(x,1),(x,2))}\right)$
$\displaystyle=$
$\displaystyle-\varphi_{\alpha}\lceil_{\mathcal{A}_{\Lambda_{n}}}\left(\log\mathcal{W}_{n-1]}^{(\varphi_{\alpha})}\right)-\sum_{x\in
W_{n-1}}\varphi_{\alpha}\lceil_{\mathcal{A}_{\Lambda_{n}}}\left(\mathcal{H}_{\alpha}^{(x,(x,1),(x,2))}\right)$
$\displaystyle=$
$\displaystyle-\varphi_{\alpha}\lceil_{\mathcal{A}_{\Lambda_{n-1}}}\left(\log\mathcal{W}_{n-1]}^{(\varphi_{\alpha})}\right)-\sum_{x\in
W_{n-1}}\varphi_{\alpha}\lceil_{\mathcal{A}_{\\{x\\}\cup
S(x)}}\left(\mathcal{H}_{\alpha}^{(x,(x,1),(x,2))}\right)$

One has

$-\varphi_{\alpha}\lceil_{\mathcal{A}_{\Lambda_{n-1}}}\left(\log\mathcal{W}_{n-1]}^{(\varphi_{\alpha})}\right)=-{\rm
Tr}\left(\mathcal{W}_{n-1]}^{(\varphi_{\alpha})}\log\mathcal{W}_{n-1]}^{(\varphi_{\alpha})}\right)=S(\varphi_{\alpha}\lceil_{\mathcal{A}_{\Lambda_{n-1}}})$

and since the state $\varphi_{\alpha}$ is translation-invariant then

$\varphi_{\alpha}\lceil_{\mathcal{A}_{\\{x\\}\cup
S(x)}}\left(\mathcal{H}_{\alpha}^{(x,(x,1),(x,2))}\right)=\varphi_{\alpha}\lceil_{\mathcal{A}_{\\{x_{0}\\}\cup
S(x_{0})}}\left(\mathcal{H}_{\alpha}^{(x_{0},(x_{0},1),(x_{0},2))}\right)$

Therefore

$S(\varphi_{\alpha}\lceil_{\mathcal{A}_{\Lambda_{n}}})=S(\varphi_{\alpha}\lceil_{\mathcal{A}_{\Lambda_{n-1}}})-|W_{n-1}|\varphi_{\alpha}\lceil_{\mathcal{A}_{\\{x_{0}\\}\cup
S(x_{0})}}\left(\mathcal{H}_{\alpha}^{(x_{0},(x_{0},1),(x_{0},2))}\right).$

This leads to (21). Using (23) the mean entropy of the QMS $\varphi_{\alpha}$
is given by

$\displaystyle s(\varphi_{\alpha})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varphi_{\alpha}\lceil_{\mathcal{A}_{\\{x_{0}\\}\cup
S(x_{0})}}\left(\mathcal{H}_{\alpha}^{(x_{0},(x_{0},1),(x_{0},2))}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
Tr}\left(\mathcal{H}_{\alpha}^{(x_{0},(x_{0},1),(x_{0},2))}\exp\left(\mathcal{H}_{\alpha}^{(x_{0},(x_{0},1),(x_{0},2))}\right)\right)$
$\displaystyle=$
$\displaystyle\alpha\left((\ln\alpha+2\beta(J+2))e^{2\beta(J+2)}+2(\ln\alpha+2\beta)e^{2\beta}+(\ln\alpha+2\beta
J)e^{2\beta J}\right)$

We can compute the mean entropy of QMS $\varphi_{\alpha}$ directly from (13).
In fact,

$\displaystyle S(\varphi_{\alpha}\lceil_{\mathcal{A}_{\Lambda_{n}}})$
$\displaystyle=$ $\displaystyle-{\rm
Tr}\left(\mathcal{W}_{n]}^{(\varphi_{\alpha})}\log\mathcal{W}_{n]}^{(\varphi_{\alpha})}\right)$
$\displaystyle\overset{(\ref{densityphialpha})}{=}$
$\displaystyle-\sum_{x\in\Lambda_{n-1}}{\rm
Tr}\left(\mathcal{W}_{n]}^{(\varphi_{\alpha})}\mathcal{H}_{\alpha}^{(x,(x,1),(x,2))}\right)$
$\displaystyle=$
$\displaystyle-\sum_{x\in\Lambda_{n-1}}\varphi_{\alpha}\lceil_{\mathcal{A}_{\\{x\\}\cup
S(x)}}\left(\mathcal{H}_{\alpha}^{(x,(x,1),(x,2))}\right)$ $\displaystyle=$
$\displaystyle-|\Lambda_{n-1}|\varphi_{\alpha}\lceil_{\mathcal{A}_{\Lambda_{1}}}\left(\mathcal{H}_{\alpha}^{(x_{0},(x_{0},1),(x_{0},2))}\right)$

Then

$\displaystyle s(\varphi_{\alpha})$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\frac{S(\varphi_{\alpha}\lceil_{\mathcal{A}_{\Lambda_{n}}})}{|\Lambda_{n}|}=\lim_{n\to\infty}\frac{S(\varphi_{\alpha}\lceil_{\mathcal{A}_{\Lambda_{n}}})}{|\Lambda_{n-1}|}\frac{|\Lambda_{n-1}|}{|\Lambda_{n}|}$
$\displaystyle=$ $\displaystyle-\frac{1}{2}{\rm
Tr}\left(\mathcal{H}_{\alpha}^{(x_{0},(x_{0},1),(x_{0},2))}\exp\left(\mathcal{H}_{\alpha}^{(x_{0},(x_{0},1),(x_{0},2))}\right)\right).$

##  5 Mixing property for quantum Markov states on Cayley trees

In this section, we consider the semi-infinite Cayley tree $\Gamma^{k}_{+}$
and, it is assumed that $\mathcal{A}_{x}=M_{d}$ the $d\times d$ square
matrices over the complex field $\mathbb{C}$. Let $\varphi$ be a QMS on
$\mathcal{A}_{V}$ together with its sequence of localized transition
expectations $\mathcal{E}_{W_{n}}=\bigotimes_{x\in W_{n}}\mathcal{E}_{x}$,
here $\mathcal{E}_{x}$ is a transition expectation from
$\mathcal{A}_{\\{x\\}\cup S(x)}$ into $\mathcal{A}_{x}$. Put
$\mathcal{B}_{x}=\mathcal{E}_{x}(\mathcal{A}_{\\{x\\}\cup
S(x)})\subseteq\mathcal{A}_{x}$.

If the QMS $\varphi$ is translation-invariant, then the transition
expectations $\mathcal{E}_{x}$ can be considered as copies of a transition
expectation $\mathcal{E}$ from $M_{d}^{\otimes(k+1)}$ into $M_{d}$. In this
setting, we denote $\mathcal{B}=\mathcal{E}(M_{d}^{\otimes(k+1)})$.

Now, let $p_{1},\dots,p_{r}$ be the minimal central projections (counted with
their multiplicities) of $\mathcal{B}$, so that the reduced algebras
$p_{i}\mathcal{B}p_{i}\cong M_{d_{i}}$ for some $d_{i}\in\mathbb{N}$. Then

$\mathcal{B}=\bigoplus_{i=1}^{r}p_{i}\mathcal{B}p_{i}=\bigoplus_{i=1}^{r}M_{d_{i}}.$

Define

$E(A)=\sum_{i=1}^{r}p_{i}Ap_{i}$

Put $\mathbb{T}=\\{z\in\mathbb{C}:\quad|z|=1\\}.$

######  Definition 5.1.

A QMS $\varphi$ on $\mathcal{A}$ is said to be strongly mixing if it satisfies

$\lim_{|g|\to\infty}\varphi(a\alpha_{g}(b))=\varphi(a)\varphi(b)$

for all $a,b\in\mathcal{A}_{V}$.

Any translation-invariant QMS $\varphi$ on $\mathcal{A}_{V}$ satisfies

$\varphi\lceil_{\mathcal{A}_{V_{g}}}\circ\,\alpha_{g}=\varphi$ (38)

where $V_{g}=\alpha_{g}(V)$.

######  Theorem 5.2.

Every faithful, translation invariant QMS with localized transition
expectations on $\mathcal{A}_{V}$ is strongly mixing.

###### Proof.

Let $\varphi$ be a translation-invariant QMS on $\mathcal{A}_{V}$ together
with its sequence of localized transition expectations
$\mathcal{E}_{W_{n}}=\bigotimes_{x\in W_{n}}\mathcal{E}_{x}$ as in (6), here
$\mathcal{E}_{x}\equiv\mathcal{E}$ is a Umegaki conditional expectation from
$\mathcal{A}_{\\{x\\}\cup S(x)}$ into $\mathcal{A}_{x}$. For each $x\in W_{n}$
and $j\in\\{1,2,\dots,k\\}$, we put

$P_{j}=P_{x;j}:=\mathcal{E}_{x}\lceil_{\mathcal{A}_{x}\otimes\mathcal{B}^{(x,j)}\otimes{\bf
1}\\!\\!{\rm I}_{S(x)\setminus\\{(x,j)\\}}}.$

One can see that $P_{x;j}$ is completely positive and identity preserving from
$\mathcal{B}$ into itself. Let $\lambda\in\mathbb{T}$ and
$B\in\mathcal{B}\setminus\\{0\\}$ such that $P_{x;j}(B^{(x,j)})=\lambda
B^{(x)}$. Then for every $A^{\prime}\in\mathcal{B}$,

$\displaystyle\lambda(B^{\prime}B)^{(x)}$ $\displaystyle=$
$\displaystyle(B^{\prime(x)}\otimes{\bf 1}\\!\\!{\rm
I}_{S(x)})P_{x;j}(B^{(x,j)})$ $\displaystyle=$
$\displaystyle(B^{\prime(x)}\otimes{\bf 1}\\!\\!{\rm
I}_{S(x)})\mathcal{E}_{x}({\bf 1}\\!\\!{\rm I}\otimes B^{(x,j)})$
$\displaystyle=$ $\displaystyle\mathcal{E}_{x}(B^{\prime(x)}\otimes{\bf
1}\\!\\!{\rm I}_{S(x)\setminus\\{(x,j)\\}}\otimes B^{(x,j)})$ $\displaystyle=$
$\displaystyle\mathcal{E}_{x}({\bf 1}\\!\\!{\rm I}_{\\{x\\}\cup
S(x)\setminus}\otimes B^{(x,j)})B^{\prime(x)}\otimes{\bf 1}\\!\\!{\rm
I}_{S(x)}$ $\displaystyle=$
$\displaystyle\lambda(BB^{\prime})^{(x)}\otimes{\bf 1}\\!\\!{\rm I}_{S(x)}.$

Then $B$ belongs to the center of $\mathcal{B}$, and there exist
$\beta_{i}\in\mathbb{C}(1\leq i\leq r$) such that
$B=\sum_{i=1}^{r}\beta_{i}p_{i}$. From [4, 30, 31] there exist positive linear
functional $\rho_{ii^{\prime}}$ on $M_{i}\otimes M_{i^{\prime}}$ such that

$\mathcal{E}_{x}(p_{i}^{(x)}\otimes p_{i^{\prime}}^{(x,j)}\otimes{\bf
1}\\!\\!{\rm I}_{S(x)\setminus\\{(x,j)\\}})=\rho_{ii^{\prime}}(p_{i}\otimes
p_{i^{\prime}})p_{i}$ (39)

One has

$\displaystyle P_{x,j}(B)$ $\displaystyle=$ $\displaystyle\mathcal{E}_{x}({\bf
1}\\!\\!{\rm I}^{(x)}\otimes B^{(x,j)}\otimes{\bf 1}\\!\\!{\rm
I}_{S(x)\setminus\\{(x,j)\\}})$ $\displaystyle=$
$\displaystyle\sum_{ii^{\prime}}\mathcal{E}_{x}(p_{i}\otimes\beta_{i^{\prime}}p_{i^{\prime}}\otimes{\bf
1}\\!\\!{\rm I}_{S(x)\setminus\\{(x,j)\\}})$ $\displaystyle=$
$\displaystyle\sum_{ii^{\prime}}\beta_{i^{\prime}}\mathcal{E}_{x}(p_{i}\otimes
p_{i^{\prime}}\otimes{\bf 1}\\!\\!{\rm I}_{S(x)\setminus\\{(x,j)\\}})$
$\displaystyle\overset{(\ref{rhoii'})}{=}$
$\displaystyle\sum_{ii^{\prime}}\beta_{i^{\prime}}\pi_{ii^{\prime}}p_{i}$

where $\beta_{ii^{\prime}}=\rho_{ii^{\prime}}(p_{i}\otimes p_{i^{\prime}})>0$.
We have

$\left[\begin{array}[]{ccc}\pi_{11}&\cdots&\pi_{1r}\\\ \vdots&\ddots&\vdots\\\
\pi_{r1}&\cdots&\pi_{rr}\end{array}\right]\left[\begin{array}[]{lll}\beta_{1}\\\
\vdots\\\
\beta_{r}\end{array}\right]=\lambda\left[\begin{array}[]{lll}\beta_{1}\\\
\vdots\\\ \beta_{r}\end{array}\right]$

Since $\pi_{ii^{\prime}}>0$ for all $ii^{\prime}$ then by the Perron-Frobenius
theorem the map $P_{x,j}$ has trivial peripheral spectrum. Therefore, by [20]
the QMS $\varphi$ is strongly mixing. ∎

Now, we are going to establish an analogue of [23, Theorem 3.1].

######  Theorem 5.3.

Let $\varphi$ be a locally faithful QMS on $\mathcal{A}_{V}$. Let
$(\mathcal{H},\pi,\Omega)$ be its associated GNS triplet. The following
assertions hold true.

  * (i)

The quantum Markov state $\varphi$ on the quasi-local algebra
$\mathcal{A}_{V}$ is separating. i.e. the cyclic vector $\Omega$ is cyclic for
the algebra $\mathcal{B}:=\pi(\mathcal{A}_{V})^{{}^{\prime\prime}}$.

  * (ii)

The algebra $\mathcal{B}=\pi(\mathcal{A}_{V})^{{}^{\prime\prime}}$ is a
factor.

###### Proof.

(i). Let $(\mathcal{H},\pi,\Omega)$ be the GNS-triplet associated with the
state locally faithful state $\varphi$. Let $a\in\mathcal{A}_{\Lambda_{n}}$,
we define an operator $\hat{a}$ as

$\hat{a}\pi(b)\Omega=\pi(ba)\Omega;\quad\forall b\in\mathcal{A}_{L,loc}.$

The operator $\hat{a}$ acts on $\pi(\mathcal{A}_{L})$ and commutes with
$\pi(b)$ for every $b\in\mathcal{A}_{L,loc}$.

On the other hand, we have

$\displaystyle\langle\hat{a}\pi(b)\Omega,\hat{a}\pi(b)\Omega\rangle$
$\displaystyle=$ $\displaystyle\varphi(a^{\ast}b^{\ast}ba)$ $\displaystyle=$
$\displaystyle\varphi(E_{n+1]}(a^{\ast}b^{\ast}ba))$ $\displaystyle=$
$\displaystyle\varphi(a^{\ast}E_{n+1]}(b^{\ast}b)a)$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left((D_{\Lambda_{n+1}}a^{\ast}E_{n+1]}(b^{\ast}b)a\right)$
$\displaystyle=$
$\displaystyle\mathrm{Tr}\left(D_{\Lambda_{n+1}}^{\varphi})^{1/2}E_{n+1]}(b^{\ast}b)(D_{\Lambda_{n+1}}^{\varphi})^{1/2}\Big{|}(D_{\Lambda_{n+1}}^{\varphi})^{-1/2}a(D_{\Lambda_{n+1}}^{\varphi})^{1/2}\Big{|}^{2}\right)$
$\displaystyle\leq$
$\displaystyle\mathrm{Tr}\left((D_{\Lambda_{n+1}}^{\varphi})^{1/2}E_{n+1]}(b^{\ast}b)(D_{\Lambda_{n+1}}^{\varphi})^{1/2}\right)\Big{\|}(D_{\Lambda_{n+1}}^{\varphi})^{-1/2}a(D_{\Lambda_{n+1}}^{\varphi})^{1/2}\Big{\|}^{2}.$

and thanks to (10), one get

$\displaystyle\mathrm{Tr}\left((D_{\Lambda_{n+1}}^{\varphi})^{1/2}E_{n+1]}(b^{\ast}b)(D_{\Lambda_{n+1}}^{\varphi})^{1/2}\right)$
$\displaystyle=$
$\displaystyle\mathrm{Tr}\left(D_{\Lambda_{n+1}}^{\varphi}E_{n+1]}(b^{\ast}b)\right)$
$\displaystyle=$ $\displaystyle\varphi(E_{n+1]}(b^{\ast}b))$ $\displaystyle=$
$\displaystyle\varphi(b^{\ast}b)$ $\displaystyle=$
$\displaystyle\|\pi(b)\Omega\|^{2}.$

Thus

$\|\hat{a}\|\leq\Big{\|}(D_{\Lambda_{n+1}}^{\varphi})^{-1/2}a(D_{\Lambda_{n+1}}^{\varphi})^{1/2}\Big{\|}.$

It follows that $\hat{a}\in\pi(\mathcal{A}_{L})^{\prime}$. The set
$\\{\hat{a}\Omega=\pi(a)\Omega:a\in\mathcal{A}_{L,loc}\\}$ is dense in
$\mathcal{H}$. This means that $\Omega$ is cyclic for
$\pi(\mathcal{A}_{L})^{{}^{\prime}}$, equivalently $\Omega$ is separating for
$\pi(\mathcal{A}_{L})^{{}^{\prime\prime}}$. This proves (i).

Now according to Theorem 5.2, from [13] we arrive at (ii). ∎

## Credit Author Statement

The all authors of this paper are equally contributed for the realization of
the results.

## Declaration of Competing Interest

The authors confirm that there are no known conflicts of interest associated
with this publication and there has been no significant financial support for
this work that could have influenced its outcome.

## Data availability

The paper does not use any data.

## Acknowledgments

The authors gratefully acknowledge Qassim University, represented by the
Deanship of Scientific Research, on the financial support for this research
under the number (10173-cba-2020-1-3-I) during the academic year 1442 AH /
2020 AD.

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