# Entanglement of bosonic systems under monitored evolution

Quancheng Liu and Klaus Ziegler  Department of Physics, Institute of
Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 52900,
Israel
Institut für Physik, Universität Augsburg, D-86135 Augsburg, Germany

###### Abstract

The evolution of non-interacting bosons in the presence of repeated projective
measurements is studied. Following the established approach, this monitored
evolution is characterized by the first detected return and the first detected
transition probabilities. We show that these quantities are directly related
to the entanglement entropy and to the entanglement spectrum of a bipartite
system. Calculations with specific values for the number of bosons, the number
of measurements and the time step between measurements reveal a sensitive and
often strongly fluctuating entanglement entropy. In particular, we demonstrate
that in the vicinity of special values for the time steps the evolution of the
entanglement entropy is either stationary or performs dynamical switching
between two or more stationary values. In the entanglement spectrum, on the
other hand, this complex behavior can be associated with level crossings,
indicating that the dominant quantum states and their entanglement respond
strongly to a change of the system parameters. We discuss briefly the role of
time averaging to remove the fluctuations of the entanglement entropy.

##  I Introduction

Repeated measurement on a quantum system has been used to determine the first
detected return (FDR) to the initial state or the first detected transition
(FDT) to a state that is different from the initial state. The idea is to
prepare the quantum system in an initial state, let it evolve unitarily and
perform at discrete times $\\{j\tau\\}_{j=1,2,...}$ projective measurements of
a given state. If the measurement is successful, which means that the given
state is detected, the experiment is stopped. If the measurement is
unsuccessful, where the given state is not detected, the experiment continues
by evolving another time step $\tau$, followed by another measurement. After
$m$ unsuccessful measurements and another unitary evolution over time $\tau$
we eventually determine the probability of the system to be in a certain state
[1]. This protocol has been applied to single-particle states to detect the
particle location on a graph [2, 3, 4, 5, 6, 7, 8].

For the evolution of a system with more than one particle the entanglement of
the quantum state is a fundamental effect. It can be characterized, for
instance, by the Rényi entanglement entropy that measures the quantum
correlations between two subsystems under a spatial bipartition [9, 10, 11,
12, 13]. Probing the entanglement entropy has become an important and popular
concept to study measurement-induced entanglement transitions, to characterize
the many-body evolution and many-body localization [14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29] and to classify the topology of
quantum systems.

In this paper we will study the Rényi entanglement entropy in a system of non-
interacting bosons, which is subject to periodically repeated projective
measurements. This enables us to study the FDR/FDT probabilities as well as
the entanglement entropy and entanglement spectrum, and to investigate the
relation between both quantities.

The focus of this paper is on physical aspects of the monitored evolution that
can also be applied to quantum computing [30]. Although quantum computing is
typically based on qubits (i.e., spin states), potentially bosonic systems
could also be used [31]. A promising example are photonic states. In
particular, we consider $N$ non-interacting bosons, distributed in two wells
which are coupled by tunneling. This can be experimentally realized as a pair
of photonic cavities that are coupled by an optical fiber [32, 33, 34, 35,
36]. The underlying Hilbert space is $N+1$-dimensional and enables us to study
the scaling behavior of the entanglement with $N$.

The structure of the paper is that in Sect. II we provide a general theory
part with the definitions of the FDR/FDT probabilities and their relations to
the reduced density matrix, the entanglement entropy and the entanglement
spectrum. More details for the calculation of the FDR/FDT probabilities are
provided in Sect. II.1. This includes an approach that connects the monitored
evolution due to repeated measurements to the unitary evolution. The reader,
who is not interested in the theoretical concepts but more in the results of
the monitored evolution, can skip this section. Then in Sect. III our approach
to the monitored evolution is applied to the tunneling of $N$ non-interacting
bosons in a double well. For this specific model the eigenvalues and spectral
weights are calculated. Finally, in Sect. IV specific examples in terms of the
parameters of the model are presented for the entanglement entropy and the
entanglement spectrum.

##  II General concept of projected measurements

The quantum system is characterized by the density operator that reads in the
presence of repeated projective measurements

$\rho^{m}(\tau)=\frac{1}{{\cal N}}e^{-iH\tau}(\Pi
e^{-iH\tau})^{m-1}|\psi_{0}\rangle\langle\psi_{0}|(e^{iH\tau}\Pi)^{m-1}e^{iH\tau}$
(1)

with the normalization ${\cal N}=Tr[e^{-iH\tau}(\Pi
e^{-iH\tau})^{m-1}|\psi_{0}\rangle\langle\psi_{0}|(e^{iH\tau}\Pi)^{m-1}e^{iH\tau}]$.
This density operator describes a quantum walk [37], where after each time
steps $\tau$ a projective measurement $\Pi$ is applied. The latter prevents
the visit of the Hilbert space after a time step $\tau$ that is orthogonal to
the $\Pi$-projected Hilbert space. Although we can choose $\Pi$ freely, we
will consider below a special choice within the Fock basis of our model, where
the orthogonal projection $\Pi$ into a $N$-dimensional subspace.

For the following discussion we consider a product space ${\cal
H}_{1}\otimes{\cal H}_{2}$ which is spanned by the basis
$\\{|n,N-n\rangle\\}_{0\leq n\leq N}$. This should be understood as a basis
for $n$ particles in ${\cal H}_{1}$ and $N-n$ particles in ${\cal H}_{2}$,
where the total number of particles $N$ is fixed. The basis states can also
contain additional quantum numbers, which are not relevant for the general
discussion. In Sect. III we will consider the special case in which the
product Hilbert space represents a double well, where $n$ bosons are in the
left and $N-n$ bosons are in the right well. Then the basis consists of Fock
states $|n,N-n\rangle\equiv|n\rangle|N-n\rangle$ without additional quantum
numbers.

Returning to the general case, in the basis $\\{|n,N-n\rangle\\}_{0\leq n\leq
N}$ the $(N+1)\times(N+1)$ density matrix reads
$\rho^{m}_{n,n-N;n^{\prime},N-n^{\prime}}=\langle
n,N-n|\rho^{m}(\tau)|n^{\prime},N-n^{\prime}\rangle$ with
$n,n^{\prime}=0,\ldots,N$. After summing over all basis states of ${\cal
H}_{2}$ the reduced density matrix ${\hat{\rho}}$ becomes an
$(N+1)\times(N+1)$ diagonal matrix with elements

${\hat{\rho}}^{m}_{nn}=\sum_{n^{\prime}=0}^{N}\langle
n,n^{\prime}|\rho^{m}(\tau)|n,n^{\prime}\rangle=\langle
n,N-n|\rho^{m}(\tau)|n,N-n\rangle$ (2) $=\frac{1}{\cal N}\langle
n,N-n|e^{-iH\tau}(\Pi
e^{-iH\tau})^{m-1}|\psi_{0}\rangle\langle\psi_{0}|(e^{iH\tau}\Pi)^{m-1}e^{iH\tau}|n,N-n\rangle.$

With the projector $P_{n}:=|n,N-n\rangle\langle n,N-n|$ the density matrix
elements can also be written as a trace expression

${\hat{\rho}}^{m}_{nn}=\frac{1}{\cal N}Tr\left[P_{n}e^{-iH\tau}(\Pi
e^{-iH\tau})^{m-1}P_{0}(e^{iH\tau}\Pi)^{m-1}e^{iH\tau}\right],$ (3)

where we have assumed $|\psi_{0}\rangle=|0,N\rangle$ for the initial state.
One should note that the right-hand side of Eq. (2) with $\Pi$ replaced by
$\Pi_{n}={\bf 1}-P_{n}$ is either the first detected return (FDR) probability
(for $n=0$) or the first detected transition (FDT) probability (for $n>0$).
Therefore, known results of the FDR/FDT probabilities [38, 39, 6] can be
directly used for the reduced density matrix through the relation

${\hat{\rho}}^{m}_{nn}=\frac{|\phi_{m;n0}|^{2}}{\sum_{n=0}^{N}|\phi_{m;n0}|^{2}}$
(4)

with the FDT amplitude for $|0,N\rangle\to|n,N-n\rangle$ ($n\neq 0$) after $m$
measurements

$\phi_{m;n0}:=\langle n,N-n|e^{-iH\tau}(\Pi_{n}e^{-iH\tau})^{m-1}|0,N\rangle$
(5)

and the corresponding FDR amplitude $\phi_{m;00}$ for
$|0,N\rangle\to|0,N\rangle$. It is crucial to note that
$\sum_{n=0}^{N}|\phi_{m;n0}|^{2}\leq 1$ for $m>1$ due to the projection $\Pi$.
Some known results for the FDR/FDT amplitudes are summarized in Sect. II.1.

With this expression for ${\hat{\rho}}^{m}_{nn}$ we can introduce the Rényi
entropy [23] as a quantitative measure for the entanglement of the two Hilbert
spaces ${\cal H}_{1}$ and ${\cal H}_{2}$

${\cal S}_{\alpha}(\tau,N,m)=\frac{1}{1-\alpha}\log_{2}{\rm
Tr}[{({\bar{\rho}}^{m})^{\alpha}(\tau)}].$ (6)

In general, $\alpha$ is a free parameter and typical values used are
$\alpha=2,3$ [23]. For the subsequent calculations we set $\alpha=2$, i.e., we
will calculate ${\cal S}_{2}(\tau,N,m)$ as the entanglement entropy (EE).

While the EE reveals a global measure for the monitored evolution, the
entanglement spectrum (ES) [40] provides a local measure of the evolution for
the transition between individual states $|0,N\rangle\to|n,N-n\rangle$. In
other words, it reveals the contribution of individual states to the entangled
state of the evolving quantum system. It is defined as the logarithm of the
reduced density matrix eigenvalues. In the present case the reduced density
matrix is already diagonal, such that

$\xi_{m;n}=-\log({\bar{\rho}}^{m}_{nn})=-2\log(|\phi_{m;n0}|)+\log(\sum_{n=0}^{N}|\phi_{m;n0}|^{2}).$
(7)

The smallest value of $\xi_{m;n}$ corresponds with the dominant transition
$|0,N\rangle\to|n,N-n\rangle$ after $m$ measurements. A crossing of the lowest
levels upon changing of the time step $\tau$ or $m$ is reminiscent of a phase
transition in classical statistical systems due to a crossing of the ground
state energies.

The FDR/FDT amplitude in Eq. (5) is written in the Fock basis. It is
convenient to express the evolution operator in the eigenbasis
$\\{|E_{k}\rangle\\}$ of the Hamiltonian as

$\phi_{m;n0}=\sum_{\\{k_{j}=0\\}}^{N}\langle\psi_{n}|E_{k_{1}}\rangle
e^{-iE_{k_{1}}\tau}\langle E_{k_{1}}|\Pi_{n}|E_{k_{2}}\rangle
e^{-iE_{k_{2}}\tau}\cdots\langle E_{k_{m-1}}|\Pi_{n}|E_{k_{m}}\rangle
e^{-iE_{k_{m}}\tau}\langle E_{k_{m}}|\psi_{0}\rangle.$ (8)

For the projector $\Pi_{n}={\bf 1}-|n,N-n\rangle\langle n,N-n|$ a typical
matrix element then reads

$\langle E_{k}|({\bf 1}-|n,N-n\rangle\langle
n,N-n|)|E_{k^{\prime}}\rangle=\delta_{kk^{\prime}}-q^{*}_{n,k}q_{n,k^{\prime}}=:({\bf
1}-Q_{n}^{*}EQ_{n})_{kk^{\prime}}$ (9)

with $q_{n,k}=\langle n,N-n|E_{k}\rangle$ and $q^{*}_{n,k}=\langle
E_{k}|n,N-n\rangle$. $E$ is the $(N+1)\times(N+1)$ matrix, whose matrix
elements are 1. Moreover, $Q_{n}$ is a diagonal matrix, consisting of the
elements $\\{q_{n,k}\\}$. Then the FDR/FDT amplitude of Eq. (5) can be written
as

$\phi_{m;n0}=\sum_{k,k^{\prime}=0}^{N}q_{n,k}D_{k}[({\bf
1}-Q_{n}^{*}EQ_{n})D]^{m-1}_{kk^{\prime}}q^{*}_{0,k^{\prime}}$ (10)

with $D_{k}=e^{-iE_{k}\tau}$. It should be noticed that
$(Q_{n}^{*})^{-1}(D-Q_{n}^{*}EQ_{n}D)^{m-1}Q_{n}^{*}$ is a function of
$Q_{n}Q_{n}^{*}$ (cf. Ref. [41]): Since $Q_{n}$, $D$ are diagonal matrices,
they commute and we get the relation

$(D-Q_{n}^{*}EDQ_{n})^{m-1}=Q_{n}^{*}(D-EDQ_{n}Q_{n}^{*})^{m-1}(Q_{n}^{*})^{-1},$
(11)

which is proved by complete induction. Moreover, we can write
$(D-Q^{*}_{n}EDQ_{n})^{m-1}=D^{-1/2}T_{n}^{m-1}D^{1/2}$ with
$T_{n}:=D^{1/2}({\bf 1}-Q^{*}_{n}EQ_{n})D^{1/2}$ and for Eq. (10)

$\phi_{m;n0}=Tr(D^{1/2}T_{n}^{m-1}D^{1/2}Q^{*}_{0}EQ_{n}).$ (12)

Thus, the matrix $T_{n}$ represents the monitored evolution under repeated
measurements, which is the analogue of the unitary evolution matrix
$U=\exp(-iH\tau)$ in the Fock basis. Its largest eigenvalues control the large
$m$ (i.e., stationary) behavior of $\phi_{m;n0}$, while the smaller
eigenvalues decay quickly. Therefore, the task is to identify those largest
eigenvalues for the given parameters of the model.

The matrix $T_{n}$ has some important properties that are useful for the
calculation of $\phi_{m;n0}$. First, since
$\sum_{k}q_{n,k}q^{*}_{n^{\prime},k}=\delta_{nn^{\prime}}$, there is a
complete set of right/left eigenvectors of the Hermitean matrix ${\bf
1}-Q^{*}_{n}EQ_{n}$ with an eigenvalue 0 and $N$ eigenvalues 1 due to

$\sum_{k^{\prime}}(\delta_{kk^{\prime}}-q^{*}_{n,k}q_{n,k^{\prime}})q^{*}_{n^{\prime},k^{\prime}}=(1-\delta_{nn^{\prime}})q^{*}_{n^{\prime},k}\
{\rm and}\ \
\sum_{k}q_{n^{\prime},k}(\delta_{kk^{\prime}}-q^{*}_{n,k}q_{n,k^{\prime}})=(1-\delta_{nn^{\prime}})q_{n^{\prime},k^{\prime}}.$
(13)

Defining the vector ${\bf q}_{n}:=(q_{n,0},q_{n,1},\ldots,q_{n,N})^{T}$, we
get $Q^{*}_{n}EQ_{n}{\bf q}^{*}_{n^{\prime}}={\bf q}_{n}\delta_{nn^{\prime}}$,
which implies for the vector ${\bf
x}:=\sum_{n^{\prime}}a_{n^{\prime}}D^{-1/2}{\bf q}^{*}_{n^{\prime}}$

$T_{n}{\bf x}=\cases{D{\bf x}&for $a_{n}=0$\cr 0&for $a_{n^{\prime}}=0$
($n^{\prime}\neq n$)\cr}.$ (14)

It is possible that $T_{n}{\bf x}=D{\bf x}$ reduces to $T_{n}{\bf
x}=e^{i\varphi}{\bf x}$ when we have $D{\bf x}=e^{i\varphi}{\bf x}$, which can
happen in the presence of degenerate $E_{j}\tau\ ({\rm mod}\ 2\pi)$ or when
some components of the vectors ${\bf q}_{n^{\prime}}$ vanish. In this case
${\bf x}$ is an eigenvector of $T_{n}$ whose eigenvalue lies on the unit
circle of the complex plane. This means that ${\bf x}$ does not decay but
accumulates only a phase. This eigenvector does not contribute to the FDR/FDT
amplitude though, since ${\bf x}$ is orthogonal to ${\bf q}_{n}$ and,
therefore, $EQ_{n}{\bf x}=0$.

Besides these special vectors, the eigenvalues of $T_{n}$ can be quite complex
in general. On the other hand, even a two-level system (i.e., $N=1$, $n=0,1$)
with energies $E_{0,1}$ is already instructive. In this case $T_{n}$ has two
eigenvalues:

$\lambda_{0}=0\ ,\ \
\lambda_{1}=(1-|q_{n,0}|^{2})e^{-2iE_{0}\tau}+|q_{n,0}|^{2}e^{-2iE_{1}\tau},$
(15)

where $|q_{n,1}|^{2}=1-|q_{n,0}|^{2}$ and
$|\lambda_{1}|^{2}=1-2(1-|q_{n,0}|^{2})|q_{n,0}|^{2}(1-\cos[2(E_{1}-E_{0})\tau])$.
Thus, the eigenvalues of $T_{n}$ are on the unit disk, one at the center and
the other one only for special values on the unit circle, namely for
$|q_{n,0}|=0,1$ and/or for $(E_{1}-E_{0})\tau=0\ ({\rm mod}\ \pi)$. This means
that, except for the special values with $|\lambda_{1}|=1$, $T_{n}^{m-1}$
decays exponentially fast. We will see subsequently that this type of behavior
exits also for larger systems. In particular, we will study a system of $N$
non-interacting bosons.

###  II.1 FDR/FDT amplitudes

Next we discuss the connection between the unitary and the monitored evolution
by a linear relation. We consider only the FDR/FDT amplitude, since this is
the building block for the other physical quantities, according to our
discussion in the previous section. The ‘first detected passage time problem’,
as discussed in Refs. [1, 4] for a single particle on a tight-binding graph,
can be directly generalized to the evolution in a general Hilbert space. The
unitary evolution of the transition $|\psi_{0}\rangle\to|\psi_{n}\rangle$ for
the time $\tau$ provides the amplitudes

$v_{m}:=\phi_{0;n0}(m\tau)=\langle\psi_{n}|e^{-iHm\tau}|\psi_{0}\rangle\ ,\ \
u_{m}:=\phi_{0;00}(m\tau)=\langle\psi_{0}|e^{-iHm\tau}|\psi_{0}\rangle.$ (16)

There exists a mapping from the unitary amplitudes in Eq. (16) to the FDR/FDT
amplitudes $\phi_{t,m}:=\phi_{m,1}=\langle\psi|\left(e^{-i\tau H}(\Pi
e^{-i\tau H}\right)^{m-1}|\psi_{0}\rangle$ with $\Pi={\bf
1}-|\psi_{0}\rangle\langle\psi_{0}|$ as (cf. App. B)

${\vec{\phi}}_{t}=({\bf 1}+\Gamma)^{-1}{\vec{v}}$ (17)

with the $n$-component vectors
${\vec{\phi}}_{t}=(\phi_{t,1},\phi_{t,2},...,\phi_{t,m})$,
${\vec{v}}=(v_{1},v_{2},...,v_{m})$ and with the triangular matrix $\Gamma$
whose elements are

$\Gamma_{ij}=\cases{u_{i-j}&$1\leq i-j\leq m-1$\cr 0&otherwise\cr}.$ (18)

In other words, the FDR/FDT amplitudes can be recursively constructed from the
unitary amplitudes in Eq. (17). This is solved by a discrete Fourier
transformation with

$\sum_{m\geq
1}z^{m}\phi_{t,m}={\hat{\phi}}_{t}(z)=\frac{{\hat{v}}(z)}{1+{\hat{u}}(z)}$
(19)

for $z$ inside the complex unit disk (i.e., $|z|<1$), and with the Fourier
transformed unitary amplitudes of Eq. (16):

${\hat{u}}(z)=z\sum_{j=0}^{N}\frac{|\langle\Psi_{0}|E_{j}\rangle|^{2}}{e^{iE_{j}\tau}-z}\
,\ \ \ {\hat{v}}(z)=z\sum_{j=0}^{N}\frac{\langle\Psi|E_{j}\rangle\langle
E_{j}|\Psi_{0}\rangle}{e^{iE_{j}\tau}-z}.$ (20)

The advantage of using a continuous function ${\hat{\phi}}_{t}(z)$ rather than
a discrete function $\phi_{t,m}$ is that analytic tools, such as integration
or perturbation theory, can be employed to exploit its properties.

Once the function ${\hat{\phi}}_{t}(z)$ is known, $\phi_{t,m}$ can be
retrieved as the residue of the Cauchy integral: Since ${\hat{\phi}}(z)$ does
not have poles inside the unit disk due to
$1+{\hat{u}}=\sum_{j}|\langle\Psi_{0}|E_{j}\rangle|^{2}/[1-z\exp(-iE_{j}\tau)]$,
the FDR/FDT amplitude reads

$\phi_{t,m}=\frac{1}{2\pi i}\int_{C}z^{-m-1}\frac{{\hat{v}}}{1+{\hat{u}}}dz$
(21)

with a contour $C$ around $z=0$ smaller than the unit circle in order to avoid
the poles of ${\hat{v}}/(1+{\hat{u}})$. The function
${\hat{\phi}}_{r}={\hat{u}}/(1+{\hat{u}})$ is a uni-modular function of the
form ${\hat{\phi}}_{r}(e^{i\omega})=e^{if(\omega)}$ with a characteristic
winding number that is equal to the dimensionality of the underlying Hilbert
space in the absence of degeneracies for $E_{j}\tau\ ({\rm mod}\ 2\pi)$ [6].
The example of $N=8$ non-interacting bosons is visualized in Fig. 1.

##  III Physical model: non-interacting bosons in a double well

Within the single mode approximation [42], the double well with $N$ bosons can
be described as a two-site Bose-Hubbard model,

$H=-J(a_{l}^{\dagger}a_{r}+a_{r}^{\dagger}a_{l})+\frac{U}{2}(n_{l}^{2}+n_{r}^{2}),$
(22)

where $a^{\dagger}_{L,R}$ ($a_{L,R}$) are the bosonic creation (annihilation)
operators in the left/right potential wells, and
$n_{l,r}=a_{l,r}^{\dagger}a_{l,r}$ are the corresponding number operators. $J$
describes the tunneling of bosons between the two wells, and $U$ represents
the particle-particle interaction, which favors energetically a symmetric
distribution of bosons in the double well when $U>0$. Using Fock states
$|n,N-n\rangle\equiv|n\rangle|N-n\rangle$ ($n=0,\cdots,N$) as a basis of the
Hilbert space, the corresponding Hamiltonian matrix has a tridiagonal
structure with

$H_{n,n^{\prime}}=\langle n,N-n|H|n^{\prime},N-n^{\prime}\rangle$ (23)
$=U[(N-n)^{2}+n^{2}]\delta_{n,n^{\prime}}-J\sqrt{n(N+1-n)}\delta_{n,n^{\prime}-1}-J\sqrt{n^{\prime}(N+1-n^{\prime})}\delta_{n,n^{\prime}+1}.$

These matrix elements represent an $(N+1)$-site tight-binding chain with
broken translational invariance, where the tunneling rate $-J\sqrt{n(N+1-n)}$
and the potential $U[(N-n)^{2}+n^{2}]$ are minimal at the center and grow
symmetrically towards the endpoints. The ‘sites’ $n$ and $n^{\prime}$ are
connected by hopping of a single particle. For the subsequent discussion we
ignore the diagonal terms by setting $U=0$, which gives us $N$ non-interacting
bosons in a double well.

For non-interacting bosons we can calculate ${\hat{\rho}}_{m;nn}$ explicitly,
since the energy levels and the spectral weight factors $\langle
n,N-n|E_{k}\rangle$ are known. This system can be realized for photons at a
beam splitter [43, 44] or in two harmonic cavities, which are connected
through an optical fiber [32, 33, 34, 35, 36]. Then the tunneling Hamiltonian
$H=-J(a^{\dagger}_{l}a_{r}+a^{\dagger}_{r}a_{l})$ for $N$ bosons has $N+1$
equidistant energy levels $E_{k}=-J(N-2k)$ ($k=0,1,...,N$) with eigenstates

$|E_{k}\rangle=\frac{2^{-N/2}}{\sqrt{k!(N-k)!}}(a_{l}^{\dagger}+a_{r}^{\dagger})^{k}(a_{l}^{\dagger}-a_{r}^{\dagger})^{N-k}|0,0\rangle\
,$ (24)

where the normalization follows directly from
$(a^{\dagger})^{l}|0\rangle=\sqrt{l!}|l\rangle$. Thus, the fastest
oscillations occur with frequency $NJ$ and the characteristic parameter for
the evolution is $J\tau$.

The knowledge of the eigenstates enables us to calculate the spectral weights
as scalar products

$q_{n,k}:=\langle n,N-n|E_{k}\rangle=\frac{2^{-N/2}}{\sqrt{k!(N-k)!}}\langle
n,N-n|(a_{l}^{\dagger}+a_{r}^{\dagger})^{k}(a_{l}^{\dagger}-a_{r}^{\dagger})^{N-k}|0,0\rangle$
(25) $=\frac{2^{-N/2}}{\sqrt{k!(N-k)!}}\langle n,N-n|\sum_{l=0}^{k}{k\choose
l}(a^{\dagger}_{l})^{l}(a^{\dagger}_{r})^{k-l}\sum_{l^{\prime}=0}^{N-k}{N-k\choose
l^{\prime}}(a^{\dagger}_{l})^{l^{\prime}}(-a^{\dagger}_{r})^{N-k-l^{\prime}}|0,0\rangle,$

and since the left and right operators commute, we obtain after reordering

$=\frac{2^{-N/2}}{\sqrt{k!(N-k)!}}\sum_{l=0}^{k}{k\choose
l}\sum_{l^{\prime}=0}^{N-k}{N-k\choose l^{\prime}}(-1)^{N-k-l^{\prime}}\langle
n,N-n|(a^{\dagger}_{l})^{l+l^{\prime}}(a^{\dagger}_{r})^{N-l-l^{\prime}}|0,0\rangle.$

Due to the orthogonality of the states the sum vanishes unless
$l+l^{\prime}=n$. Then from the $l$ summation there are two constraints for
$l^{\prime}$:

$0\leq l^{\prime}=n-l\ ,\ \ \ n-l=l^{\prime}\leq N-k,$

which are equivalent to $n+k-N\leq l\leq n$, such that

$\cases{n+k-N\leq l\leq n&for $n+k-N>0$\cr 0\leq l\leq n&for $n+k-N\leq
0$\cr}.$

This result, together with $(a^{\dagger})^{l}|0\rangle=\sqrt{l!}|l\rangle$,
implies for the scalar product

$q_{n,k}=2^{-N/2}\sqrt{{N\choose k}\Big{/}{N\choose
n}}\cases{\sum_{l=0}^{\min\\{k,n\\}}{k\choose l}{N-k\choose
n-l}(-1)^{N-k-n+l}&for $n+k\leq N$\cr\sum_{l=n+k-N}^{\min\\{k,n\\}}{k\choose
l}{N-k\choose n-l}(-1)^{N-k-n+l}&for $n+k>N$\cr}.$ (26)

The orthonormal condition of the states $|n,N-n\rangle$ and the states
$|E_{k}\rangle$, respectively, lead to

$\sum_{n=0}^{N}q_{n,k}q_{n,k^{\prime}}=\delta_{kk^{\prime}}\ \ \ {\rm and}\ \
\ \sum_{k=0}^{N}q_{n,k}q_{n^{\prime},k}=\delta_{nn^{\prime}}.$ (27)

In particular, for the special cases $n=0$ and $n=N$ we have

$q_{0,k}=2^{-N/2}\frac{\sqrt{N!}}{\sqrt{k!(N-k)!}}(-1)^{N-k}\ ,\ \ \
q_{N,k}=2^{-N/2}\frac{\sqrt{N!}}{\sqrt{k!(N-k)!}}.$ (28)

These specific expressions can be entered into the FDR/FDT amplitude
$\phi_{m;n0}$ of Eq. (12), using $D_{k}=e^{-iE_{k}\tau}=e^{iJ(N-2k)\tau}$.
Moreover, $Q_{n}$, the diagonal matrix with elements $q_{n,k}$ for fixed $n$,
is real here, which enables us to write

$\phi_{m;n0}=Tr[D(D-Q_{n}EDQ_{n})^{m-1}Q^{*}_{0}EQ_{n}]=Tr[D^{1/2}T^{m-1}D^{1/2}Q^{*}_{0}EQ_{n}].$
(29)

This gives immediately the reduced density matrix of Eq. (4), the EE of Eq.
(6) and the ES of Eq. (7). Some examples for the eigenvalues of the monitored
evolution matrix $T$ are presented in Figs. 2, 3.

Figure 1:  ${\hat{\phi}}_{r}(e^{i\omega})=e^{if(\omega)}$ for
$-\pi\leq\omega<\pi$ (vertical axis) with winding number $9$ for 8 non-
interacting bosons. The density of points is inverse to the sensitivity of the
phase $f(\omega)$ to a change of $\omega$.

a)  b)

Figure 2:  51 eigenvalues of $T$ for $N=50$ bosons. a) for $J\tau=0.7\hbar$
and b) for $J\tau=0.5\hbar$.

a)  b)
a)  b)

Figure 3:  51 eigenvalues of $T$ for $N=50$ bosons. a) for
$J\tau=(\pi/6)\hbar$, b) for $J\tau=(\pi/6+10^{-2})\hbar$ c) for
$J\tau=(\pi/7)\hbar$ and d) for $J\tau=(\pi/7+10^{-2})\hbar$.

Returning to the unitary evolution, the amplitudes $u_{m}$ and $v_{m}$ of Eq.
(16) are determined by a multiple of the frequency $J\tau$:

$u_{m}=\langle 0,N|e^{-iHm\tau}|0,N\rangle=\cos^{N}(mJ\tau),\ \ \
v_{m}=\langle N,0|e^{-iHm\tau}|0,N\rangle=(-i)^{N}\sin^{N}(mJ\tau),$ (30)

which are periodic with $m\tau J/\hbar=2\pi$ or periodic with $m\tau
J/\hbar=\pi$ for even $N$ (cf. Fig. 4). For a very short time (i.e., for
$mJ\tau\ll 1/\sqrt{N}$) we have a Gaussian decay of the Fock state
$|0,N\rangle$ as

$u_{m}=\cos^{N}(mJ\tau)\sim e^{-Nm^{2}J^{2}\tau^{2}/2},$ (31)

as also illustrated in Fig. 4. This non-exponential behavior in $\tau$
reflects the quantum Zeno effect [45].

Figure 4:  Unitary evolution: Real parts of the return amplitude $u_{1}(\tau)$
($|0,N\rangle\to|0,N\rangle$) and the transition amplitude $v_{1}(\tau)$
($|0,N\rangle\to|N,0\rangle$) as a function of the dimensionless time
$J\tau/2\hbar$ for 50 non-interacting bosons.

Finally, the Fourier transformed unitary amplitudes for
$|\psi_{0}\rangle=|0,N\rangle$ and $|\psi\rangle=|N,0\rangle$ read

${\hat{u}}(z)=2^{-N}\sum_{k=0}^{N}{{N}\choose{k}}\frac{z}{e^{iE_{k}\tau}-z}\
,\ \ \
{\hat{v}}(z)=(-2)^{-N}\sum_{k=0}^{N}{{N}\choose{k}}\frac{(-1)^{k}z}{e^{iE_{k}\tau}-z}\
.$ (32)

They can be used to calculate the FDR/FDT amplitudes as the residue of the
corresponding Cauchy integrals.

##  IV Discussion of the results

The results of the previous section will now be used to calculate the FDR/FDT
probabilities, the EE and the ES for specific realizations of the model. To
this end we note that the parameters of the bosonic system are the number of
bosons $N$, the number of measurements $m-1$ and the time steps between
measurements $\tau$. The latter always appears in the combination with the
tunneling rate $J$ as $J\tau$. This is a consequence of the fact that we have
non-interacting bosons, where tunneling is the only mechanism of the
evolution. The combination $J\tau/\hbar$ provides a dimensionless time step in
our system, which we will use subsequently.

The characteristic features of the unitary dynamics defined in Eq. (30) is
visualized in Fig. 4, which indicates a smooth variation of the amplitudes
over time for the return to the initial state $|0,N\rangle\to|0,N\rangle$ and
for the transition of all bosons to the other well
$|0,N\rangle\to|N,0\rangle$. This periodic behavior is also reflected by the
unitary evolution of the EE in Fig. 5a, which vanishes when all bosons are
either in the left or in the right well. Repeated measurements will
substantially affect this periodic behavior.

a)  b)

Figure 5:  a) The unitary evolution of $N=20$ bosons for the time
$J\tau/\hbar$ is periodic. b) The monitored evolution of the EE for a high
frequency of measurements reflects the quantum Zeno effect, where an
increasing measurement frequency reduces the EE.

a)  b)

Figure 6:  The entanglement spectrum for the 5 levels of $N=4$ bosons for
$m=1$ (a) and $m=50$ (b) measurements as a function of the time step units
$\pi J\tau/\hbar$ and $10^{-3}J\tau/\hbar$, respectively.

a)
b)

Figure 7:  Entanglement spectrum for $N=4$ centered around $J\tau=\pi/4$ (a)
and centered around $J\tau=\pi/3$ (b). The left panels present $m=50$
measurements, the right panels $m=51$ measurements. The jump of the green
level in b) causes the switching effect of the entanglement entropy between
two values, as illustrated in Fig. 8b. A unit of the time step on the abscissa
is $10^{-7}J\tau/\hbar$.

a)  b)

Figure 8:  The entanglement entropy for $N=4$ is plotted for $J\tau=\pi/4$ (a)
and $J\tau=\pi/3$ (b).

entanglement entropy: Without measurement (i.e., for $m=1$) the EE is
determined by the unitary amplitudes for all transitions
$|0,N\rangle\to|n,N-n\rangle$, which are smooth and periodic in time

$\phi_{1,n0}=e^{iNJ\tau}\sum_{k=0}^{N}e^{-2ikJ\tau}q_{n,k}q_{0,k}.$ (33)

This expression, together with Eq. (27), gives for $J\tau=0\ ({\rm mod}\
2\pi)$ and $J\tau=\pi\ ({\rm mod}\ 2\pi)$

$\phi_{1,n0}=e^{iNJ\tau}\delta_{n0},$ (34)

implying ${\cal S}_{2}(\tau,N,1)=0$. This is reflected in the plot of ${\cal
S}_{2}(\tau,N,1)$ of Fig. 5a, which indicates also a vanishing EE at
$J\tau=\pi/2\ ({\rm mod}\ 2\pi)$. The periodicity does not depend on the
number of bosons $N$, while the value of the EE increases with $N$. This is
remarkable because the eigenvalues as well as the weight $q_{n,k}$ depend
strongly on $N$. The behavior of the EE is affected by measurements, depending
on the time steps between the measurements. As visualized in Fig. 5b for
$N=20$ bosons, for a very short time step $J\tau$ between measurements the
periodic behavior of the unitary evolution disappears. The corresponding
entanglement spectra in Fig. 6 reveal that the level crossings are more
complex in case of the monitored evolution and they take place on a much
shorter time scale.

As already mentioned in the discussion of the FDR/FDT probabilities, the
unitary evolution between measurements is characterized by the phase factors
$\exp(-iJ\tau m/\hbar)$, which is periodic for $m=l$ if $J\tau/\hbar=2\pi/l$.
In other words, if the time step is a fraction of $\pi$, we expect a special
behavior of the monitored evolution, which might be reminiscent of the
periodic behavior of the unitary evolution. But how does the monitored
evolution depend on $l$? This we will analyze for $N=4$ bosons, by comparing
$l=3$ and $l=4$. Fig. 7 the four plots visualize how the ES changes from
$J\tau/\hbar=\pi/l$ at the center to $\pi/l\pm 5\cdot 10^{-5}$ at the
boundaries. The left panels represent the ES for $m=50$ measurements, the
right panels the ES for $m=51$ measurements. There are level degeneracies only
at $\pi/l$, while in the narrow vicinity the levels are well separated and the
spectrum is symmetric with respect to $\pi/l$. It should be noticed though
that for $l=4$ (Fig. 7a) the lowest level is two-fold degenerated, whereas for
$l=3$ (Fig. 7b) there is not such a degeneracy. Another remarkable difference
between $l=3$ and $l=4$ consists of the level change when the number of
measurements changes from $m=50$ (left) to $m=51$ (right). While the lowest
two levels for $l=4$ are not affected by this change, there is a drastic
change for $l=3$. The latter has two low levels for $m=50$ but three low
levels for $m=51$. The qualitative difference between these two $m$ values
reflects the fact that for an odd $l$ only an odd $m$ can satisfy the
condition $mJ\tau/\hbar=2\pi$ for peridiodicity of the phase factor. To
understand the effect of this $m$-dependence on the EE, we use the definitions
of the ES and the EE in Eqs. (6) and (7) and express the EE by the levels of
the ES as

${\cal S}_{2}=-\log_{2}\left(\sum_{n=0}^{N}e^{-2\xi_{m;n}}\right),$ (35)

where the sum is reminiscent to the sum of Boltzmann weights in statistical
mechanics. Therefore, only small values of $\xi_{m;n}$ (i.e., low levels of
the ES) contribute substantially to the EE. This means that the EE changes for
$m\to m+1$ when $l=3$ but it remains unchanged for $l=4$. This is what we see
in Fig. 8. After some fluctuations for small values of $m$, the EE becomes
stationary: For $l=4$ there is just one stationary value (Fig. 8a), while for
$l=3$ the EE switches between two stationary values (Fig. 8b). The
corresponding ES for $l=3$ reflects the switching as a jump of levels between
even and odd values of $m$, as illustrated for $m=50$ and $m=51$ in Fig. 7b.
On the other hand, for $l=4$ there in no level jump (Fig. 7a).

The above analysis relies on the fractional form $J\tau/\hbar=\pi/l$. The
behavior of the EE for other values of the time step between measurements can
change drastically and may lead to a strongly fluctuating behavior of the EE.
In general, repeated measurements have two major effects on the evolution:
they destroy the periodicity (recurrence) and they lead to more level
crossings in the ES, as illustrated in Fig. 6. The origin of these effects is
that the non-interacting bosons are coupled repeatedly in time to the
measurement apparatus, which provides a similar effect as a local boson-boson
interaction, since the measurement is performed on the same quantum state at
different times. The situation can be compared with the unitary evolution of
bosons with an interaction $U\neq 0$ in the Hamiltonian (22). We previously
studied this system and found a similar behavior of a fluctuating EE and level
crossings in the ES [27]. The fluctuations were removed by averaging over time
intervals, an approach we could also apply in the present case with $U=0$. For
the monitored evolution we could also introduce random time steps between
measurements and average over their distribution [41, 46, 47, 48, 49]. As a
disadvantage of such a time averaging, though, we would not be able to detect
the switching of the EE in Fig. 8b).

##  V Conclusion

We have studied the monitored evolution of $N$ non-interacting bosons which
tunnel between two wells. The monitoring is carried out by repeated projective
measurements. The effect of these measurements is studied in terms of FDR/FDT
probabilities to determine quantitatively the monitoring. From the FDR/FDT
probabilities we have derived the reduced density matrix for one well, the EE
and the ES. This is based on the relation (4) and enabled us to evaluate the
EE and the ES directly from the FDR/FDT probabilities. It turns out that the
EE is quite sensitive to a change of model parameters; i.e., on the number of
bosons and the time step between two measurements. The rather complex behavior
of the EE indicates that a single quantity, such as the EE, is quite limited
for the characterization of the entanglement in the present system. More
details are revealed by the ES of Eq. (7). It enables us to identify the
statistical weight of each transition $|0,N\rangle\to|n,N-n\rangle$ to the
monitored evolution individually. A characteristic feature of the ES is level
crossing. Although it already appears in the unitary evolution (cf. Fig. 6a),
it becomes much more complex for the monitored evolution in Fig. 6b) Except
for the crossing points, there is always a unique lowest level, representing
the dominant transition $|0,N\rangle\to|n,N-n\rangle$. The excitation to
higher levels is important as long they are close to the lowest levels. This
effect is important when the level $\xi_{m;n}$ changes quickly with $m$. This
can happen near the special values $J\tau/\hbar=\pi/l$ ($l$ integer), as
demonstrated in Figs. 7.

Although our approach was employed only to non-interacting bosons, it is
directly applicable to interacting bosons as well. New regimes might appear
due to the competition of particle tunneling, particle-particle interaction
and the interaction with the measurement apparatus. This is an ambitious
project for the future, in which, among other aspects, the role of Hilbert-
space localization should be addressed.

Acknowledgements

We are grateful to Eli Barkai for the useful discussions. This research is
supported by the Israel Science Foundation through Grant No. 1614/21 (Q.L.).

##  Appendix A Eigenstates

With the rotated basis $\\{a_{\pm},a^{\dagger}_{\pm}\\}$:

$a^{\dagger}_{\pm}:=\frac{1}{2}(a^{\dagger}_{l}\pm a^{\dagger}_{r})\ ,\ \
a_{\pm}=\frac{1}{2}(a_{l}\pm a_{r})$

we obtain for the tunneling operator
$a^{\dagger}_{l}a_{r}+a^{\dagger}_{r}a_{l}$

$a^{\dagger}_{l}a_{r}+a^{\dagger}_{r}a_{l}=\frac{1}{2}[(a^{\dagger}_{l}+a^{\dagger}_{r})(a_{l}+a_{r})-(a^{\dagger}_{l}-a^{\dagger}_{r})(a_{l}-a_{r})]=\frac{1}{2}(a^{\dagger}_{+}a_{+}-a^{\dagger}_{-}a_{-}),$
(36)

where $(a^{\dagger}_{l}\pm a^{\dagger}_{r})(a_{l}\pm a_{r})$ are number
operators. Then we can directly show that $a_{+}$ and $a_{-}$ and their
Hermitean conjugate commute when $a_{l}$ and $a_{r}$ commute. As a
consequence, the eigenstate reads

$|E_{k}\rangle=\frac{2^{-N/2}}{\sqrt{k!(N-k)!}}(a^{\dagger}_{+})^{k}(a^{\dagger}_{-})^{N-k}|0,0\rangle$

and the application of the tunneling operator yields

$\frac{1}{2}(a^{\dagger}_{+}a_{+}-a^{\dagger}_{-}a_{-})|E_{k}\rangle=(N-2k)|E_{k}\rangle.$

##  Appendix B Expansion of the first return/transition amplitude

Introducing the new notation for the FDR/FDT amplitude

$\phi_{m+1,1}:=\langle\psi|\left(e^{-i\tau H}\Pi\right)^{m}e^{-i\tau
H}|\psi_{0}\rangle=\langle\psi|e^{-i\tau H}\left(\Pi e^{-i\tau
H}\right)^{m}|\psi_{0}\rangle\ ,\ \ \Pi={\bf
1}-|\psi_{0}\rangle\langle\psi_{0}|\ ,$ (37)

we obtain two equivalent recursion relations, namely

$\phi_{m+1,1}=\phi_{m,2}^{\prime}-u_{1}\phi_{m,1}\ \ {\rm with}\ \
\phi_{m,k}^{\prime}=\langle\psi|e^{-ik\tau H}\left(\Pi e^{-i\tau
H}\right)^{m-1}|\psi_{0}\rangle\ ,\ \
u_{k}=\langle\psi_{0}|e^{-iHk\tau}|\psi_{0}\rangle$ (38)

from the third expression in Eq. (37) and

$\phi_{m+1,1}=\phi_{m,2}-\phi_{m,1}v_{1}\ \ {\rm with}\ \
\phi_{m,k}=\langle\psi|\left(e^{-i\tau H}\Pi\right)^{m-1}e^{-ik\tau
H}|\psi_{0}\rangle$ (39)

from the second expression in Eq. (37). Then the iteration of Eq. (38) yields
with $\phi_{t,m}\equiv\phi_{m,1}$

$\phi_{t,m}=v_{m}-\sum_{j=1}^{m-1}u_{m-j}\phi_{t,j}\ ,\ \ \ \phi_{t,1}=v_{1}$
(40)

and the iteration of Eq. (39)

$\phi_{t,m}=v_{m}-\sum_{j=1}^{m-1}\phi_{t,m-j}v_{j}\ ,\ \ \ \phi_{t,1}=v_{1}\
.$ (41)

Using the vector notation
${\vec{\phi}}_{t}:=(\phi_{t,1},\phi_{t,2},...,\phi_{t,m})$,
${\vec{v}}=(v_{1},v_{2},...,v_{m})$, with the matrix $\Gamma$ of Eq. (18) and
with the matrix $\phi_{t}=(\phi_{t,m-j})$, we can write for Eqs. (40) and (41)

$({\bf 1}+\Gamma){\vec{\phi}}_{t}={\vec{v}}\ ,\ \ {\vec{\phi}}_{t}=({\bf
1}-{\bf\phi_{t}}){\vec{v}},$ (42)

which yields $({\bf 1}-{\bf\phi_{t}})({\bf 1}+\Gamma)={\bf 1}$ and Eq. (17).

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UDK 517.98

Quantum Markov Chains on the Comb graphs: Ising model

Farrukh Mukhamedov

Department of Mathematical Sciences, College of Science,

United Arab Emirates University 15551, Al-Ain,

United Arab Emirates.

e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abdessatar Souissi

Department of Accounting, College of Business Management

Qassim University, Ar Rass, Saudi Arabia and

Preparatory institute for scientific and technical studies,

Carthage University, Amilcar 1054, Tunisia

e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

Tarek Hamdi

Department of Management Information Systems, College of Business Management

Qassim University, Ar Rass, Saudi Arabia and

Laboratoire d’Analyse Mathématiques et applications LR11ES11

Université de Tunis El-Manar, Tunisia

e-mail<EMAIL_ADDRESS>

Abstract

In the present paper, we construct quantum Markov chains (QMC) over the Comb
graphs. As an application of this construction, it is proved the existence of
the disordered phase for the Ising type models (within QMC scheme) over the
Comb graphs. Moreover, it is also established that the associated QMC has
clustering property with respect to translations of the graph. We stress that
this paper is the first one where a nontrivial example of QMC over non-regular
graphs is given. Mathematics Subject Classification: 46L53, 46L60, 82B10,
81Q10.
Key words: quantum Markov chain; Ising model; Comb graph; clustering;

##  1\. Introduction

Over the past decade, motivated largely by the prospect of superefficient
algorithms, the theory of quantum Markov chains (QMC), especially in the guise
of quantum walks, has generated a huge number of works, including many
discoveries of fundamental importance [11, 19, 25, 40]. In [23] it has been
proposed a novel approach to investigate quantum cryptography problems by
means of QMC [24] where quantum effects are entirely encoded into super-
operators labelling transitions, and the nodes of its transition graph carry
only classical information and thus they are discrete. Recently, QMC have been
applied [18, 19, 20] to the investigations of so-called ”open quantum random
walks” [13, 16, 26].

On the other hand, in physics, a spacial classes of QMC, called ”Matrix
Product States” (MPS) and more generally ”Tensor Network States” [17, 39] were
used to investigate quantum phase transitions for several lattice models. This
method uses the density matrix renormalization group (DMRG) algorithm which
opened a new way of performing the renormalization procedure in 1D systems and
gave extraordinary precise results. This is done by keeping the states of
subsystems which are relevant to describe the whole wave-function, and not
those that minimize the energy on that subsystem. Those states had appeared in
the literature in many different contexts and with different names:

  * (a)

variational ansatz for the transfermatrix in the estimation of the partition
function of a classical model [27],;

  * (b)

in the AKLT model in 1D [12], where the ground state has the form of a valence
bond solid (VBS) [28] which can be exactly written as an MPS. Translational
invariant MPS in infinite chains were thoroughly studied and characterized
mathematically in full generality in [21, 22], where they are called as
finitely correlated states (FCS) (see also [1, 3] for closely related paper
where QMC approach was used).

In [6, 7, 8, 9] it has been used a QMC approach to investigate models defined
over the Cayley trees. In this path the QMC scheme is based on the
$C^{*}$-algebraic framework (see also [10]). Furthermore, in [29, 30, 31, 35,
36, 37] we have established that Gibbs measures of the Ising model with
competing (Ising) interactions (with commuting interactions) on a Cayley
trees, can be considered as QMC. Note that if the perturbation vanishes then
the model reduces to the classical Ising one which was also examined in [14]
by means of $C^{*}$-algebraic methods. Other types of models with
$XY$-interactions on the same tree have studied in [32, 33, 34]. In [38] using
the matrix product states, it has been numerically investigated the quantum
Ising model in a transverse field on the Cayley tree. However, the Cayley tree
is considered as a regular graph, therefore, it is interesting to develop QMC
scheme for non-regular graphs. One of the simplest non-regular graph is the
Comb graph which has many applications in computer sciences. Several physical
models over such graph were investigated in [15].

The main aim of the present paper is to construct QMC over the Comb graphs. We
stress that the construction essentially uses boundary conditions associated
with density amplitudes. As an application of such kind of construction, the
existence of the disordered phase of the Ising model within QMC scheme over
the Comb graph is proved. Moreover, it is also established that the associated
QMC has clustering property with respect to translations of the graph. We
point out that our paper is the first one where a construction of nontrivial
example of QMC over the non-regular graph is given. Besides, the provided
construction will allow to investigate phase transition problem for other
kinds of models over non-regular graphs.

##  2\. Comb graphs

In this section, we recall some necessary notions about the Comb graphs. Let
$\mathcal{G}=(L,E)$ be a locally finite connected graph with infinite set of
vertices. An edge $l\in E$ is associated to the pair of its endpoints
$l=<x,y>=<y,x>$ and $E$ is then identifiable to a subset of $L\times L$.

$E\subset\\{\\{x,y\\}\;:\;x,y\in L\\}.$

Recall that:

  * •

Two vertices $x$ and $y$ are called nearest neighbors, and we denote by $x\sim
y$, if there exists an edge joining them.

  * •

A collection of the elements $x\sim x_{1}\sim\dots\sim x_{d-1}\sim y>$ is
called a path from the vertex $x$ to the vertex $y$ with length $d$.

  * •

The distance $d(x,y),x,y\in V$, on the Comb graph, is the length of the
shortest path from $x$ to $y$.

  * •

The interaction domains at a given vertex $x$ is given by

(1) $N_{x}=\bigl{\\{}y\in L\quad:\quad x\sim y\bigr{\\}}$

its cardinal $|N_{x}|$ is called the valence at the site $x$.

In the sequel, we deal with the Comb graph $\mathcal{C}_{d}:=\mathbb{N}^{\rhd
d}$, which is a tree base

Figure 1. Comb graph: $\mathbb{N}\rhd\mathbb{N}$

A natural coordinate structure is associated to the Comb graph $\mathcal{C}$,
as subset of the integer lattice $\mathbb{N}^{d}$. The vertex set is then
$E=\mathbb{N}^{d}$ and the root is the origin $x^{(0)}:=(0,\cdots,0)$. Each
vertex $x$ is identified to a d-tuple of integers
$x\equiv(j_{1},\cdots,j_{d})$. For each $k\in\\{1,2,\cdots,d\\}$ consider

$e_{k}:=(0,\cdots,0,1,0,\cdots,0)$

where $1$ appears at the $k^{th}$-component. Let us set

$\displaystyle W_{n}$ $\displaystyle=$ $\displaystyle\bigg{\\{}x\in
L\,:\,d(x,x^{(0)})=n\bigg{\\}}$ $\displaystyle=$
$\displaystyle\bigg{\\{}x=(j_{1},\cdots,j_{d})\in\mathbb{N}^{d}\,:\,\sum_{k=1}^{d}j_{k}=1\bigg{\\}}.$
$\Lambda_{n}=\bigcup_{k=0}^{n}W_{k},\qquad\Lambda_{[n,m]}=\bigcup_{k=n}^{m}W_{k},\
(n<m).$

Now for each $x\in W_{n}$, we define its direct successors by

$S(x):=\bigl{\\{}y\in W_{n+1}\,:\,x\sim y\bigr{\\}}$

It is obvious that for every vertex $x$ (distinct of the root $x^{(0)}$) one
has $|S(x)|=|N_{x}|-1$. In particular, if $x\in\Delta_{k}$ we get
$|S(x)|=2(d-k)+1$. Moreover, the tree structure of the considered graph yields
to

(2) $W_{n+1}=\bigcup_{x\in W_{n}}S(x)\quad\hbox{and}\quad S(x)\cap
S(y)=\varnothing,\quad\forall x\neq y.$

##  3\. Quantum Markov chains on the Comb graph

Let $\mathcal{A},\mathcal{B}$ be two unital C∗-algebras. A completely positive
(CP) identity preserving map
$\mathcal{E}:\mathcal{A}\otimes\mathcal{B}\to\mathcal{B}$ is called transition
expectation. Note the structure of completely positive maps was clarified [4]
and its general formulation in terms of density matrices was carried out in
the case of full matrix algebras. Mainly, every CP-map
$\mathcal{E}:\mathcal{A}\otimes\mathcal{B}\to\mathcal{B}$ is given by

$\mathcal{E}(x)=\sum_{j}\mathrm{Tr}_{\mathcal{B}]}\bigl{(}K_{j}^{\ast}xK_{j});\quad
x\in\mathcal{A}\otimes\mathcal{B}$

where $K_{j}\in\mathcal{A}\otimes\mathcal{B}$ and
$\mathrm{Tr}_{\mathcal{B}}:\mathcal{A}\otimes\mathcal{B}\to\mathcal{B}$ is the
partial trace. In particular, if $K\in\mathcal{A}\otimes\mathcal{B}$ is any
density, then

(3) $\mathcal{E}(x)=\mathrm{Tr}_{\mathcal{B}]}(K^{\ast}xK);\quad
x\in\mathcal{A}\otimes\mathcal{B}$

such that

(4) $\mathrm{Tr}_{\mathcal{B}]}(K^{\ast}K)=I_{\mathcal{B}}$

is a (identity preserving) transition expectation. In what follows, any
operator $K\in\mathcal{A}\otimes\mathcal{B}$ which satisfies (4) will be
called conditional density matrix.

Consider a triplet $\mathcal{C}\subseteq\mathcal{B}\subseteq\mathcal{A}$ of
C∗–algebras. A quasi-conditional expectation [2] is a completely positive
identity preserving linear map $E:\mathcal{A}\to\mathcal{B}$ such that
$E(ca)=cE(a)$, for all $a\in\mathcal{A}$, $c\in\mathcal{C}.$

To each vertex $x$, we associate an algebra of observable
$\mathcal{B}_{x}\equiv\mathcal{B}(\mathcal{H}_{x})$ for some finite
dimensional Hilbert space $\mathcal{H}_{x}$. Consider the quasi-local algebra
$\mathcal{B}_{L}:=\bigotimes_{x\in L}\mathcal{B}_{x}$ which the inductive
limit of the net

$\mathcal{B}_{\Lambda}:=\bigotimes_{x\in\Lambda}\mathcal{B}_{x}\otimes
I_{\Lambda^{c}};\quad\Lambda\subset L,\,|\Lambda|<\infty.$

Then

$\mathcal{B}_{L}=\overline{\mathcal{B}_{L,loc}}^{C^{\ast}}$

where

$\mathcal{B}_{L,loc}=\bigcup_{\Lambda\subset_{fin}L}\mathcal{B}_{V}.$

For the sake of simplicity, we will denote

$\mathcal{B}_{[m,n]}:=\mathcal{B}_{\Lambda_{[m,n]}}.$

Starting from any quasi-conditional expectation
$E_{n}:\mathcal{B}_{[0,n+1]}\to\mathcal{B}_{[0,n]}$ with respect to the
triplet
$\mathcal{B}_{[0,n-1]}\subseteq\mathcal{B}_{[0,n-1]}\subseteq\mathcal{B}_{[0,n]}$
one can obtain a transition expectation from $\mathcal{B}_{[n,n+1]}$ into
$\mathcal{B}_{W_{n}}$ by the mere restriction

$\mathcal{E}_{[n,n+1]}:=E_{n}\Big{|}_{\mathcal{B}_{[n,n+1]}}.$

Conversely, every transition expectation
$\mathcal{E}_{[n,n+1]}:\mathcal{B}_{[n,n+1]}\to B_{W_{n}}$ is extendable to a
quasi-conditional expectation $E_{n}$ with respect to the above triplet in the
following way

(5) $E_{n}:=id_{\mathcal{B}_{n-1]}}\otimes\mathcal{E}_{[n,n+1]}.$

######  Definition 3.1.

A state $\varphi$ on $\mathcal{B}_{L}$ is called quantum Markov chain (QMC)
with respect to a triplet
$\big{(}\rho_{0},\\{\mathcal{E}_{[n,n+1]}\\},\\{h_{n}\\}\big{)}$, where
$\rho_{0}$ is a state on $\mathcal{B}_{x^{(0)}}$,
$\mathcal{E}_{[n,n+1]}:\mathcal{B}_{[n,n+1]}\to\mathcal{B}_{W_{n}}$ is a
transition expectation and $\\{h_{n}\\}\subset\mathcal{B}_{W_{n},+}$ is a
sequence of boundary conditions) if

(6) $\varphi=\lim_{n\to+\infty}\rho_{0}\circ E_{0}\circ E_{1}\circ\cdots\circ
E_{n}\circ{\mathbf{h}}_{n+1}$

where ${\mathbf{h}}_{k}(\cdot)=h_{k}^{1/2}(\cdot)h_{k}^{1/2}$. Here we have
used the notation (5).

Thanks to (5), we can immediately verify that, the QMC $\varphi$ is evaluated
at $a=a_{0}\otimes a_{1}\otimes\dots\otimes a_{W_{n}}\in\mathcal{B}_{[0,n]}$,
$a_{j}\in\mathcal{B}_{W_{j}}$, by

$\varphi(a)=\lim_{n\to+\infty}\rho_{0}\Bigl{(}\mathcal{E}_{[0,1]}\Bigl{(}a_{W_{0}}\otimes\mathcal{E}_{[1,2]}\Bigl{(}a_{1}\otimes\cdots\otimes\mathcal{E}_{[n,n+1]}\Bigl{(}h^{1/2}_{n+1}a_{n}h^{1/2}_{n+1}\otimes
I\Bigr{)}\Bigr{)}\Bigr{)}\Bigr{)}$

which highlights the quantum Markov chain structure.

We notice that a more general definition of QMC has been formulated in [5,
10].

##  4\. Construction of QMC on $\mathbb{N}^{\rhd d}$

This section is devoted to a construction of quantum Markov chains associated
with nearest-neighbors interactions on the algebra $\mathcal{B}_{L}$.

Let $K_{[n,n+1]}\in\mathcal{B}_{[n,n+1]}$ be a conditional density matrix and
let $\mathcal{E}_{[n,n+1]}:\mathcal{B}_{[n,n+1]}\to\mathcal{B}_{W_{n}}$ its
associated transition expectation given by

(7)
$\mathcal{E}_{[n,n+1]}(a)=\mathrm{Tr}_{\mathcal{B}_{W_{n}}]}\bigl{(}K_{[n,n+1]}^{\ast}aK_{[n,n+1]}\bigr{)}$

Note that (2) leads to the identity

$\mathcal{B}_{[n,n+1]}\equiv\bigotimes_{x\in W_{n}}\mathcal{B}_{x\cup S(x)}$

A transition expectation
$\mathcal{E}_{[n,n+1]}:\mathcal{B}_{[n,n+1]}\to\mathcal{B}_{W_{n}}$ is said to
be localized if it can be decomposed as follows

(8) $\mathcal{E}_{[n,n+1]}=\bigotimes_{x\in W_{n}}\mathcal{E}_{x}$

for some transition expectation $\mathcal{E}_{x}:\mathcal{B}_{\\{x\\}\cup
S(x)}\to\mathcal{B}_{x}$ for each vertex $x\in W_{n}$.

Note that the partial trace $\mathrm{Tr}_{\mathcal{B}_{W_{n}}]}$ is by
construction localized

$\mathrm{Tr}_{\mathcal{B}_{W_{n}}]}=\bigotimes_{x\in
W_{n}}\mathrm{Tr}_{\mathcal{B}_{\\{x\\}\cup S(x)}]}$

One can see that the localized transition expectation $\mathcal{E}_{[n,n+1]}$
highlights the fine structure of the considered graph. This property played an
essential role in the study of phase transitions for quantum Markov chains on
the Cayley tree [37].

######  Definition 4.1.

A conditional density operator $K_{[n,n+1]}\in\mathcal{B}_{[n,n+1]}$ is said
to be localized if

(9) $K_{[n,n+1]}=\bigotimes_{x\in W_{n}}K_{\\{x\\}\cup S(x)}$

where $K_{\\{x\\}\cup S(x)}\in\mathcal{B}_{\\{x\\}\cup S(x)}$ for every $x\in
W_{n}$.

We immediately can prove the following fact.

######  Lemma 4.2.

If the conditional density matrix $K_{[n,n+1]}=\bigotimes_{x\in
W_{n}}K_{\\{x\\}\cup S(x)}$ is localized then the associated quasi-conditional
expectation $\mathcal{E}_{[n,n+1]}$ is localized in the sense of (8) with

(10)
$\mathcal{E}_{x}(\cdot)=\mathrm{Tr}_{\mathcal{B}_{x}}\Bigl{(}K_{\\{x\\}\cup
S(x)}^{\ast}\cdot K_{\\{x\\}\cup S(x)}\Bigr{)}.$

for each $x\in W_{n}$.

In the sequel, the pair $\\{\rho_{0},h=(h_{n})\\}$ denotes a boundary
condition such that $\rho_{0}$ is a faithful positive linear functional on the
algebra $\mathcal{B}_{(x_{0})}$ and $h_{n}\in\mathcal{B}_{W_{n}}^{+}$ for each
$n$.

Let $\varphi^{(\rho_{0},h)}_{n}$ be a linear functional on
$\mathcal{B}_{[0,n]}$ given by

(11)
$\varphi^{(\rho_{0},h)}_{n}(a)=\rho_{0}\bigl{(}E_{0}\bigl{(}E_{1}\bigl{(}\cdots\bigl{(}E_{n}\bigl{(}h^{1/2}_{n+1}ah^{1/2}_{n+1}\bigr{)}\bigr{)}\bigr{)}\bigr{)}\bigr{)}.$

Observe that for each $a\in\mathcal{B}_{[0,n]}$, one has

$E_{n}\bigl{(}h_{n+1}^{1/2}(a\otimes I)^{\ast}(a\otimes
I)h_{n+1}^{1/2}\bigr{)}\geq 0$

and since the maps $\mathcal{E}_{[j,j+1]}$ are completely positive and the
functional $\rho_{0}$ is a positive functional then the functional
$\varphi^{(\rho_{0},h)}_{n}$ is positive.

The sequence $\\{\varphi^{(\rho_{0},h)}_{n}\\}_{n}$ satisfies the
compatibility condition if

(12)
$\varphi_{n+1}^{(\rho_{0},h)}\Big{|}_{\mathcal{B}_{[0,n]}}=\varphi^{(\rho_{0},h)}_{n},\quad\
\textrm{for all}\ n.$

Clearly, the compatibility condition ensures the existence of the weak-limit

$\lim_{n\to\infty}\varphi^{(\rho_{0},h)}_{n}.$

######  Lemma 4.3.

For the same notations as above, if

(13) $\mathcal{E}_{[n,n+1]}(h_{n+1})=h_{n}\quad;\quad\forall n\in\mathbb{N}$

then the sequence $\\{\varphi^{(\rho_{0},h)}_{n}\\}_{n}$ given by (11)
satisfies the compatibility condition (12).

###### Proof.

Let $a\in\mathcal{B}_{\Lambda_{n}}$. Due to $[h_{n+i},a]=0$, ($i=1,2$) we find

$\displaystyle\varphi_{n+1}^{(\rho_{0},h)}(a)$ $\displaystyle=$
$\displaystyle\rho_{0}\bigl{(}\mathcal{E}_{[0,1]}\bigl{(}\mathcal{E}_{[1,2]}\bigl{(}\cdots\mathcal{E}_{[n,n+1]}\bigl{(}\mathcal{E}_{[n+1,n+2]}\bigl{(}h_{n+2}a\otimes
I\bigr{)}\bigr{)}\cdots\bigr{)}\bigr{)}\bigr{)}\bigr{)},$ $\displaystyle=$
$\displaystyle\rho_{0}\bigl{(}\mathcal{E}_{[0,1]}\bigl{(}\mathcal{E}_{[1,2]}\bigl{(}\cdots\mathcal{E}_{[n,n+1]}\bigl{(}\mathcal{E}_{[n+1,n+2]}\bigl{(}h_{n+2}\bigr{)}a\otimes
I\bigr{)}\cdots\bigr{)}\bigr{)}\bigr{)}\bigr{)},$ $\displaystyle=$
$\displaystyle\rho_{0}\bigl{(}\mathcal{E}_{[0,1]}\bigl{(}\mathcal{E}_{[1,2]}\bigl{(}\cdots\mathcal{E}_{[n,n+1]}\bigl{(}h_{n+1}a\otimes
I\bigr{)}\cdots I\bigr{)}\cdots\bigr{)}\bigr{)}\bigr{)}\bigr{)},$
$\displaystyle=$
$\displaystyle\rho_{0}\bigl{(}\mathcal{E}_{[0,1]}\bigl{(}\mathcal{E}_{[1,2]}\bigl{(}\cdots\mathcal{E}_{[n,n+1]}\bigl{(}h^{1/2}_{n+1}ah^{1/2}_{n+1}\otimes
I\bigr{)}\cdots I\bigr{)}\cdots\bigr{)}\bigr{)}\bigr{)}\bigr{)},$
$\displaystyle=$ $\displaystyle\varphi_{n}^{(\rho_{0},h)}(a).$

This completes the proof. ∎

######  Theorem 4.4.

Let $K_{\\{x\\}\cup S(x)}\in\mathcal{B}_{\\{x\\}\cup S(x)}$ be given as above.
Assume that the boundary condition $(\rho_{o},h=(h_{u})_{u\in L})$ satisfies
the following conditions

(14) $\rho_{o}(h_{o})=1$ (15)
$\mathrm{Tr}_{\mathcal{B}_{x}]}\Bigl{(}K_{\\{x\\}\cup
S(x)}^{\ast}I^{(x)}\otimes\bigotimes_{y\in S(x)}h^{(y)}K_{\\{x\\}\cup
S(x)}\Bigr{)}=h^{(x)}$

then the sequence $\\{\varphi^{(\rho_{0},h)}_{n}\\}$ satisfies the
compatibility condition (12). Moreover, there exists a quantum Markov chain.

###### Proof.

From Lemma 4.2, the maps $\mathcal{E}_{[n,n+1]}$ are completely positive and
localized in the sense of (8). One finds

$\displaystyle\mathcal{E}_{[n,n+1]}(h_{n+1})$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{\mathcal{B}_{W_{n}}]}\bigl{(}K_{[n,n+1]}^{\ast}h_{n}K_{[n,n+1]}\bigr{)},$
$\displaystyle=$ $\displaystyle\bigotimes_{x\in W_{n}}\Bigl{(}K_{\\{x\\}\cup
S(x)}^{\ast}I^{(x)}\otimes\bigotimes_{y\in S(x)}h^{(y)}K_{\\{x\\}\cup
S(x)}\Bigr{)},$ $\displaystyle=$ $\displaystyle\bigotimes_{x\in
W_{n}}h^{(x)},$ $\displaystyle=$ $\displaystyle h_{n}.$

Hence, by virtue of Lemma 4.3, we get the desired assertion, which completes
the proof. ∎

##  5\. Quantum Markov chains associated with Ising type models on the Comb
graph $\mathbb{N}\rhd_{0}\mathbb{N}$

In this and the forthcoming sections, we restrict ourselves to a semi-infinite
Comb graph $\mathbb{N}\rhd_{0}\mathbb{N}$ with distinguished vertex
$x^{(0)}=(0,0)$. In what follows, as usually, by $L$ we denote the set of all
vertices and $E$ is the set of edges of $\mathbb{N}\rhd_{0}\mathbb{N}$,
respectively. A simple coordinate structure is naturally associated to the
$\mathbb{N}\rhd_{0}\mathbb{N}$. Each vertex $x$ is identified with a pair of
non-negative integers $x\equiv(k,l)$. Here $k$ is the component of $x$ on the
horizontal axis and $l$ on the vertical one.

We enumerate elements of $W_{n}$ in the following way

(16)
$x_{W_{n}}^{(0)}=(n,0)\quad\,x_{W_{n}}^{(1)}=(n-1,1)\quad\,\cdots\quad\,x_{W_{n}}^{(n)}=(0,n)$

and we write

$\overrightarrow{W}_{n}:=\bigl{\\{}x_{W_{n}}^{(0)},x_{W_{n}}^{(1)},\cdots,x_{W_{n}}^{(n)}\bigr{\\}}.$

Accordingly, we distinguish two different types of vertices: ones having two
direct successors and others having only one. Define

(17) $L_{1}=\\{x\in L\,:\,|S(x)|=1\\}$ (18) $L_{2}=\\{x\in
L\,:\,|S(x)|=2\\}=\\{x_{W_{n}}^{(0)}\ :\ n\in\mathbb{N}\\}$

here as before, $S(x)$ denotes the direct successors of $x$.

It is clear that $L=L_{1}\cup L_{2}$. For $u\in L_{2}$ one has

$S(u)=\\{u+e_{1},\,u+e_{2}\\}$

whereas, elements of $L_{1}$ have a form $v=(k,l)$ with $l\geq 1$ and

$S(v)=\\{v+e_{2}\\}$

where $e_{1}=(1,0)$ and $e_{2}=(0,1)$.

Denote

(19) ${\bf 1}\\!\\!{\rm I}^{(u)}=\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right),\ \,\ \sigma_{z}^{(u)}=\left(\begin{array}[]{cc}1&0\\\
0&-1\\\ \end{array}\right).$

In what follows, we consider the same $C^{*}$-algebra $\mathcal{B}_{L}$ but
with $\mathcal{B}_{u}=M_{2}(\mathbb{C})$ for all $u\in L$.

Let $h^{(u)}=\left(\begin{array}[]{cc}h^{(u)}_{11}&h^{(u)}_{12}\\\
h^{(u)}_{21}&h^{(u)}_{22}\\\ \end{array}\right)\in M_{2}(\mathbb{C})^{+}$. The
family $\\{h^{(u)},\,u\in L\\}$ is said to be homogeneous (or translation
invariant) boundary conditions if $h^{(u)}=h^{(x^{0})}$ for each $u\in L$. We
are interested in the resolution of the equations

(20) $\mathrm{Tr}\left(K_{\\{u\\}\cup\overrightarrow{S}(u)}^{\ast}{\bf
1}\\!\\!{\rm I}^{(u)}\otimes
h_{S(u)}K_{\\{u\\}\cup\overrightarrow{S}(u)}\right)=h^{(u)},$

where

$\left\\{\begin{array}[]{cc}K_{\\{u\\}\cup
S(u)}\in\mathcal{B}_{u}\otimes\mathcal{B}_{u+e_{2}};\quad
h_{S(u)}=h^{(u)}\otimes h^{(u+e_{2})}&\hbox{if $u\in L_{1}$;}\\\ \\\
K_{\\{u\\}\cup
S(u)}\in\mathcal{B}_{u}\otimes\mathcal{B}_{u+e_{1}}\otimes\mathcal{B}_{u+e_{2}};\quad
h_{S(u)}=h^{(u)}\otimes h^{(u+e_{1})}\otimes h^{(u+e_{2})}&\hbox{if $u\in
L_{2}$.}\\\ \end{array}\right.$

In this paper, we restrict ourselves to the description of translation-
invariant solutions of (15). Therefore, we always assume that: $h^{x}=h$ for
all $x\in L$, here

$h=\left(\begin{array}[]{cc}h_{11}&h_{12}\\\ h_{21}&h_{22}\\\
\end{array}\right).$

###  5.1. An Ising model on vertices with degree two

In this subsection, we investigate a concrete Ising type model associated on
vertices $v\in L_{1}$ and their successors.

Define

(21) $\tilde{K}_{<v,v+e_{2}>}=\cos(\beta){\bf 1}\\!\\!{\rm I}^{(v)}\otimes{\bf
1}\\!\\!{\rm
I}^{(v+e_{2})}-\sin(\beta)\sigma_{z}^{(v)}\otimes\sigma_{z}^{(v+e_{2})}$

Put

$K_{\\{v\\}\cup S(u)}=\tilde{K}_{<v,v+e_{2}>}.$

Then (20) reduces to

(22) $\mathrm{Tr}\left(\tilde{K}_{<u,u+e_{2}>}{\bf 1}\\!\\!{\rm
I}^{(u)}\otimes h^{(u+e_{2})}\tilde{K}_{<u,u+e_{2}>}\right)=h^{(u)}$

One can calculate

$\displaystyle K_{\\{u\\}\cup S(u)}{\bf 1}\\!\\!{\rm I}^{(u)}\otimes
h^{(u+e_{2})}K_{\\{u\\}\cup S(u)}^{\ast}$ $\displaystyle=\cos^{(2)}(\beta){\bf
1}\\!\\!{\rm I}^{(u)}\otimes
h^{(u+e_{2})}-\cos(\beta)\sin(\beta)\sigma_{z}^{(u)}\otimes
h^{(u+e_{2})}\sigma_{z}^{(u+e_{2})}$
$\displaystyle-\cos(\beta)\sin(\beta)\sigma_{z}^{(u)}\otimes\sigma_{z}^{(u+e_{2})}h^{(u+e_{2})}+\sin^{2}(\beta){\bf
1}\\!\\!{\rm I}^{(u)}\otimes\sigma_{z}h^{(u+e_{2})}\sigma_{z}.$

Hence, (22) is equivalent to

$\left\\{\begin{array}[]{ll}\left(\frac{h^{(u+e_{2})}_{11}+h^{(u+e_{2})}_{22}}{2}\right)-\sin(2\beta)\left(\frac{h^{(u+e_{2})}_{11}-h^{(u+e_{2})}_{22}}{2}\right)=h^{(u)}_{11};\\\
\\\
\left(\frac{h^{(u+e_{2})}_{11}+h^{(u+e_{2})}_{22}}{2}\right)+\sin(2\beta)\left(\frac{h^{(u+e_{2})}_{11}-h^{(u+e_{2})}_{22}}{2}\right)=h^{(u)}_{22};\\\
\\\ h^{(u)}_{12}=h^{(u)}_{21}=0.\end{array}\right.$

According to the positivity of $h$, a simple calculation leads to the solution

(23) $h^{(u)}=a{\bf 1}\\!\\!{\rm I}^{(u)},\quad a>0.$

###  5.2. An Ising model on vertices with degree three

In this subsection, we consider another type of Ising model on triples
$(v,v+e_{1},v+e_{2})$ for each $v\in L_{2}$.

Define nearest neighbors interactions by

(24) $K_{<v,(v+e_{i})>}=\exp\\{\beta H_{<v,(v+e_{i})>}\\},\ \ i=1,2,\
\beta>0,$

where

(25) $\displaystyle H_{<v,(v,i)>}=\frac{1}{2}\big{(}{\bf 1}\\!\\!{\rm
I}^{v)}{\bf 1}\\!\\!{\rm
I}^{(v+e_{i})}+\sigma_{z}^{(v)}\sigma_{z}^{(v+e_{i})}\big{)},$

and the one level nearest neighbours interaction between $v+e_{1}$ and
$v+e_{2}$ by

(26) $L_{>v+e_{1},v+e_{2}<}=\exp\\{J\beta H_{>v+e_{1},v+e_{2}<}\\},\ \ J>0,$

where

(27) $H_{>v+e_{1},v+e_{2}<}=\frac{1}{2}\big{(}{\bf 1}\\!\\!{\rm
I}^{v+e_{1}}{\bf 1}\\!\\!{\rm
I}^{v+e_{2}}+\sigma_{z}^{v+e_{1}}\sigma_{z}^{v+e_{2}}\big{)}.$

The constant $J>0$ is known in the physics literature as coupling constant.

The defined model is called an Ising model with competing interactions per
vertices $(v,(v+e_{1}),(v+e_{2}))$.

One can check that

$\displaystyle H_{<u,u+e_{i}>}^{m}=H_{<u,u+e_{i}>}=\frac{1}{2}\big{(}{\bf
1}\\!\\!{\rm I}^{(u)}{\bf 1}\\!\\!{\rm
I}^{(u+e_{i})}+\sigma_{z}^{(u)}\sigma_{z}^{(u+e_{i})}\big{)},$ $\displaystyle
H_{>u+e_{1},u+e_{2}<}^{m}=H_{>u+e_{1},u+e_{2}<}=\frac{1}{2}\big{(}{\bf
1}\\!\\!{\rm I}^{(u+e_{1})}{\bf 1}\\!\\!{\rm
I}^{(u+e_{2})}+\sigma_{z}^{(u+e_{1})}\sigma_{z}^{(u+e_{2})}\big{)}.$

Therefore,

(28) $\displaystyle K_{<u,u+e_{i}>}=K_{0}{\bf 1}\\!\\!{\rm I}^{(u)}{\bf
1}\\!\\!{\rm I}^{(u+e_{i})}+K_{3}\sigma_{z}^{(u)}\sigma_{z}^{(u+e_{i})},$ (29)
$\displaystyle L_{>u+e_{1},u+e_{2}<}=R_{0}{\bf 1}\\!\\!{\rm I}^{(u+e_{1})}{\bf
1}\\!\\!{\rm
I}^{(u+e_{3})}+R_{3}\sigma_{z}^{(u+e_{1})}\sigma_{z}^{(u+e_{2})},$

here

$\displaystyle K_{0}=\frac{\exp{\beta}+1}{2},\ \ \
K_{3}=\frac{\exp{\beta}-1}{2},$ $\displaystyle
R_{0}=\frac{\exp{(J\beta)}+1}{2},\ \ \ R_{3}=\frac{\exp{(J\beta)}-1}{2}.$

For each $u\in L_{2}$, we define

(30) $K_{\\{v\\}\cup
S(v)}=K_{<v,v+e_{1}>}K_{<v,v+e_{2}>}L_{>v+e_{1},v+e_{2}<}.$

Hence, (20) reduces to

(31) $\mathrm{Tr}_{u]}\left(K_{\\{v\\}\cup S(v)}^{*}{\bf 1}\\!\\!{\rm
I}^{(u)}\otimes h^{(u+e_{1})}\otimes h^{(u+e_{2})}K_{\\{v\\}\cup
S(v)}\right)=h^{(u)}.$

From (28) and (29), it follows that

(32) $\displaystyle K_{\\{v\\}\cup S(v)}$ $\displaystyle=$ $\displaystyle
A{\bf 1}\\!\\!{\rm I}^{(v)}\otimes{\bf 1}\\!\\!{\rm I}^{(v+e_{1})}\otimes{\bf
1}\\!\\!{\rm
I}^{(v+e_{2})}+B\sigma_{z}^{(v)}\otimes\sigma_{z}^{(v+e_{1})}\otimes{\bf
1}\\!\\!{\rm I}^{(v+e_{2})}$ $\displaystyle+B\sigma_{z}^{(v)}\otimes{\bf
1}\\!\\!{\rm I}^{(v+e_{1})}\otimes\sigma_{z}^{(v+e_{2})}+C{\bf 1}\\!\\!{\rm
I}^{(v)}\otimes\sigma_{z}^{(v+e_{1})}\otimes\sigma_{z}^{(v+e_{2})},$

where

(33)
$\left\\{\begin{array}[]{ll}A=K_{0}^{2}R_{0}+K_{3}^{2}.R_{3}=\frac{1}{4}[\exp{(J+2)\beta}+\exp{J\beta}+2\exp{\beta}],\\\
\\\ B=K_{0}K_{3}(R_{0}+R_{3})=\frac{1}{4}\exp{J\beta}[\exp{2\beta}-1],\\\ \\\
C=K_{0}^{2}R_{3}+K_{3}^{2}R_{0}=\frac{1}{4}[\exp{(J+2)\beta}+\exp{J\beta}-2\exp{\beta}].\\\
\end{array}\right.$

Then

$\displaystyle K_{\\{v\\}\cup S(v)}^{\ast}\times[{\bf 1}\\!\\!{\rm
I}^{(v)}\otimes h^{(v+e_{1})}\otimes h^{(v+e_{2})}]\times K_{\\{v\\}\cup
S(v)}$ $\displaystyle=$ $\displaystyle\big{[}A^{2}{\bf 1}\\!\\!{\rm I}\otimes
h\otimes h+B^{2}{\bf 1}\\!\\!{\rm I}\otimes\sigma_{z}h\sigma_{z}\otimes
h+B^{2}{\bf 1}\\!\\!{\rm I}\otimes h\otimes\sigma_{z}h\sigma_{z}+C^{2}{\bf
1}\\!\\!{\rm
I}\otimes\sigma_{z}h\sigma_{z}\otimes\sigma_{z}h\sigma_{z}\big{]}$
$\displaystyle+$ $\displaystyle\big{[}AC{\bf 1}\\!\\!{\rm I}\otimes
h\sigma_{z}\otimes h\sigma_{z}+AC{\bf 1}\\!\\!{\rm
I}\otimes\sigma_{z}h\otimes\sigma_{z}h+B^{2}{\bf 1}\\!\\!{\rm
I}\otimes\sigma_{z}h\otimes h\sigma_{z}+B^{2}{\bf 1}\\!\\!{\rm I}\otimes
h\sigma_{z}\otimes\sigma_{z}h\big{]}$ $\displaystyle+$
$\displaystyle\big{[}AB\sigma_{z}\otimes h\sigma_{z}\otimes
h+AB\sigma_{z}\otimes h\otimes h\sigma_{z}+AB\sigma_{z}\otimes
h\otimes\sigma_{z}h+AB\sigma_{z}\otimes\sigma_{z}h\otimes h\big{]}$
$\displaystyle+$
$\displaystyle\big{[}BC\sigma_{z}\otimes\sigma_{z}h\sigma_{z}\otimes
h\sigma_{z}+BC\sigma_{z}\otimes
h\sigma_{z}\otimes\sigma_{z}h\sigma_{z}+BC\sigma_{z}\otimes\sigma_{z}h\sigma_{z}\otimes\sigma_{z}h+BC\sigma_{z}\otimes\sigma_{z}h\otimes\sigma_{z}h\sigma_{z}\big{]}.$

Therefore, due to the last equality, we rewrite (31) as follows

(34) $\displaystyle h^{(v)}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{x]}K_{\\{v\\}\cup S(v)}^{*}\left({\bf 1}\\!\\!{\rm
I}^{(v)}\otimes h\otimes h\right)K_{\\{v\\}\cup S(v)}$ $\displaystyle=$
$\displaystyle\left(\tau_{1}{\mathrm{Tr}(h)}^{2}+\tau_{2}{\mathrm{Tr}(\sigma_{z}h)}^{2}\right){\bf
1}\\!\\!{\rm
I}^{(v)}+\tau_{3}\mathrm{Tr}(h)\mathrm{Tr}(\sigma_{z}h)\sigma_{z}^{(v)},$

where $\theta=\exp(2\beta)>0$ and

(35)
$\left\\{\begin{array}[]{ll}\tau_{1}:=A^{2}+2B^{2}+C^{2}=\frac{1}{4}[\theta^{J}(\theta^{2}+1)+2\theta],\\\
\\\ \tau_{2}:=2(AC+B^{2})=\frac{1}{4}[\theta^{J}(\theta^{2}+1)-2\theta],\\\
\\\ \tau_{3}:=4B(A+C)=\frac{1}{2}\theta^{J}(\theta^{2}-1).\end{array}\right.$

From

$\mathrm{Tr}(h)=\frac{h_{11}+h_{22}}{2}\quad;\quad\mathrm{Tr}(\sigma_{z}h)=\frac{h_{11}-h_{22}}{2}$

the equation (34) reduces to

(36)
$\left\\{\begin{array}[]{lll}\mathrm{Tr}(h)=\tau_{1}\mathrm{Tr}(h)^{2}+\tau_{2}\mathrm{Tr}(\sigma_{z}h)^{2},\\\
\\\
\mathrm{Tr}(\sigma_{z}h)=\tau_{3}\mathrm{Tr}(h)\mathrm{Tr}(\sigma_{z}h),\\\
\\\ h_{21}=0,\quad h_{12}=0.\\\ \end{array}\right.$

Since, we are interested in transition-invariant solutions, it is convenient
to find those one with $h_{1,1}=h_{2,2}$. So, (36) is reduced to

$h_{11}=h_{22}=\frac{1}{\tau_{1}}.$

Hence,

(37) $h^{(u)}=\frac{1}{\tau_{1}}{\bf 1}\\!\\!{\rm I}^{(u)};\quad\forall u\in
L_{2}.$

Now we are ready to formulate a main result of this section.

######  Theorem 5.1.

For the Ising (ZZ) coupling model (21) and the Ising model with competing
interactions (24), (26), $\beta>0,J>0$ on the Comb graph
$\mathbb{N}\rhd_{0}\mathbb{N}$, there exists a quantum Markov chain $\varphi$
with homogeneous boundaries. Moreover, it can be written on the local algebra
$\mathcal{B}_{L,loc}$ by:

(38)
$\varphi(a)=\alpha^{n}\mathrm{Tr}\bigg{(}a\prod_{i=0}^{n-1}K_{[i,i+1]}K_{[i,i+1]}^{*}\bigg{)},\
\ \forall a\in B_{[0,n]}.$

We notice that the QMC $\varphi$ is called disordered phase of the Ising
model.

###### Proof.

According to the Ising type models considered in the subsection 5.1 and the
subsection 5.2, the equation (20) admits a unique translation-invariant
solution

(39) $h^{(u)}_{\alpha}=\left(\begin{array}[]{cc}\alpha&0\\\ 0&\alpha\\\
\end{array}\right)$

where $\alpha=\frac{1}{\tau_{1}}$. The initial functional $\rho_{0}$ can be
chosen of the form

$\rho_{0}(a)=\tau_{1}\mathrm{Tr}(a)$

so that $\rho_{0}(h^{x^{(0)}})=\mathrm{Tr}(\omega_{0}h^{x^{(0)}})=1$, where
$\omega_{0}=\tau_{1}{\bf 1}\\!\\!{\rm I}^{x^{(0)}}$
Let $n\in\mathbb{N},a\in B_{[0,n]}$, according (39) and from Lemma 4.3, one
finds

$\displaystyle\varphi(a)$ $\displaystyle=$
$\displaystyle\varphi_{n}^{(\rho_{0},h)}(a)$ $\displaystyle=$
$\displaystyle\rho_{0}\Bigl{(}\mathcal{E}_{[0,1]}\Bigl{(}a_{W_{0}}\otimes\mathcal{E}_{[1,2]}\Bigl{(}a_{1}\otimes\cdots\otimes\mathcal{E}_{[n-1,n+1]}\Bigl{(}h_{n}a\Bigr{)}\Bigr{)}\Bigr{)}\Bigr{)}$
$\displaystyle=$
$\displaystyle\mathrm{Tr}\bigl{(}\omega_{0}\mathrm{Tr}_{\mathcal{B}_{W_{0}]}}\Bigl{(}K_{[0,1]}^{*}\mathrm{Tr}_{\mathcal{B}_{W_{1}]}}\Bigl{(}K_{[1,2]}^{*}\cdots\otimes\mathrm{Tr}_{\mathcal{B}_{W_{n-1}]}}\Bigl{(}K_{[n-1,n]}^{*}h_{n}aK_{[n-1,n]}\Bigr{)}\cdots
K_{[1,2]}\Bigr{)}K_{[0,1]}\Bigr{)}\Bigr{)}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\bigg{(}\omega_{0}h_{n}\bigg{(}\prod_{m=0}^{n-1}K_{[m,m+1]}^{*}aK_{[m,m+1]}\bigg{)}\bigg{)}$
$\displaystyle=$
$\displaystyle\alpha^{|W_{n}|-1}\mathrm{Tr}\bigg{(}a\prod_{m=0}^{n-1}K_{[m,m+1]}K_{[m,m+1]}^{*}\bigg{)}$
$\displaystyle=$
$\displaystyle\alpha^{n}\mathrm{Tr}\bigg{(}a\prod_{i=0}^{n-1}K_{[i,i+1]}K_{[i,i+1]}^{*}\bigg{)}.$

This completes the proof. ∎

##  6\. Clustering property

The Comb graph $\mathbb{N}\rhd_{0}\mathbb{N}=(V,E)$ is invariant under the
action of the semi-group $\mathcal{G}^{+}$ of translations of the form

(40) $J_{n}:u=(k,l)\in\mathbb{N}\times\mathbb{N}\mapsto u+ne_{1}=(k+n,l).$

For each $g=(n,0)\in L_{2}$ we denote

$\tau_{g}(u):=J_{n}(u).$

In these notations, we have $\mathcal{G}^{+}:=\\{\tau_{g}\ :\ g\in L_{3}\\}$.

######  Lemma 6.1.

Assume that $n_{0}\in\mathbb{N}$. Let $a\in\mathcal{B}_{[0,n_{0}]}$ and
$b\in\mathcal{B}_{x^{(0)}}$. Then

(41) $\lim_{n\to+\infty}\varphi(aJ_{n}(b))=\varphi(a)\varphi(b),$

where $J_{n}(b)=b^{(x_{W_{n}}^{(0)})}$.

###### Proof.

For large enough $n$, one gets $ab_{n}=a\otimes b_{n}$, $n>n_{0}$. So,

$\displaystyle\varphi(a\otimes b_{n})$ $\displaystyle=$
$\displaystyle\varphi_{n}^{(\rho_{0},h_{\alpha})}(a\otimes b_{n})$
$\displaystyle=$
$\displaystyle\rho_{0}\circ\mathcal{E}_{[0,1]}\circ\cdots\circ\mathcal{E}_{[n_{0},n_{0}+1]}(a\otimes\mathcal{E}_{[n_{0}+1,n_{0}+2]}({\bf
1}\\!\\!{\rm I}_{W_{n_{0}+1}}\otimes\cdots$
$\displaystyle\otimes\mathcal{E}_{[n-1,n]}\circ\hat{\mathcal{E}}_{[n,n+1]}(b_{n}\otimes{\bf
1}\\!\\!{\rm I}_{W_{n+1}}).$

Here, the boundary condition is taken, as before,
$h^{(u)}=h_{\alpha}=\alpha{\bf 1}\\!\\!{\rm I}$ is a common fixed points of
the systems (22) and (31) and the initial state
$\rho(\cdot)=\mathrm{Tr}(\omega_{0}\,\cdot),\,\omega_{0}=\frac{1}{\alpha}{\bf
1}\\!\\!{\rm I}$ with $\alpha=\frac{4}{\theta^{J}(\theta^{2}+1)+2\theta}$.

Then, one finds

$\displaystyle\hat{\mathcal{E}}_{[n,n+1]}(b_{n}\otimes{\bf 1}\\!\\!{\rm
I}_{W_{n+1}})=\mathrm{Tr}_{n]}\left(K_{[n,n+1]}^{\ast}\mathbf{h}_{n+1}^{1/2}(b\otimes{\bf
1}\\!\\!{\rm I}_{W_{n+1}})\mathbf{h}_{n+1}^{1/2}K_{[n,n+1]}\right)$
$\displaystyle=$ $\displaystyle\bigotimes_{u\in
W_{n}}\mathrm{Tr}_{u]}\left(K_{\\{u\\}\cup S(u)}^{\ast}\bigotimes_{v\in
S(u)}(h^{(v)})^{1/2}(b\otimes{\bf 1}\\!\\!{\rm I}_{W_{n+1}})\bigotimes_{v\in
S(u)}(h^{(v)})^{1/2}K_{[n,n+1]}\right)$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{x_{W_{n}}^{(0)}]}\left(K_{\\{x_{W_{n}}^{(0)}\\}\cup
S(x_{W_{n}}^{(0)})}^{\ast}b^{(x_{W_{n}}^{(0)})}\otimes
h_{S(x_{W_{n}}^{(0)})}K_{\\{x_{W_{n}}^{(0)}\\}\cup
S(x_{W_{n}}^{(0)})}\right)\otimes$ $\displaystyle\bigotimes_{u\in
W_{n}\setminus\\{x_{W_{n}}^{(0)}\\}}\mathrm{Tr}_{u]}\left(K_{\\{u\\}\cup
S(u)}^{\ast}\bigotimes_{v\in S(u)}h^{(v)}K_{[n,n+1]}\right).$

By (15)

$\mathrm{Tr}_{u]}\left(K_{\\{u\\}\cup S(u)}\bigotimes_{v\in
S(u)}h^{(v)}K_{[n,n+1]}^{\ast}\right)=h^{(u)}$

and

$\mathrm{Tr}_{x_{W_{n}}^{(0)}]}\left(K_{\\{x_{W_{n}}^{(0)}\\}\cup
S(x_{W_{n}}^{(0)})}^{\ast}b^{(x_{W_{n}}^{(0)})}\left({\bf 1}\\!\\!{\rm
I}^{(x_{W_{n}}^{(0)})}\otimes
h_{S(x_{W_{n}}^{(0)})}\right)K_{\\{x_{W_{n}}^{(0)}\\}\cup
S(x_{W_{n}}^{(0)})}\right)\\\ $ (42)
$=\alpha^{2}\left[(A^{2}+C^{2})b^{(x_{W_{n}}^{(0)})}+2B^{2}\sigma_{z}b^{(x_{W_{n}}^{(0)})}\sigma_{z}\right].$

Then

$\mathcal{E}_{[n-1,n]}({\bf 1}\\!\\!{\rm
I}_{W_{n-1}}\otimes\hat{\mathcal{E}}(b^{(x_{W_{n}}^{(0)})}\otimes{\bf
1}\\!\\!{\rm I}))=\alpha^{2}(A^{2}+C^{2})\mathcal{E}_{[n-1,n]}\left({\bf
1}\\!\\!{\rm I}_{W_{n-1}}\otimes b^{(x_{W_{n}}^{(0)})}\otimes\bigotimes_{u\in
W_{n}\setminus\\{x_{W_{n}}^{(0)}\\}}h^{(u)}\right)$
$+2\alpha^{2}B^{2}\mathcal{E}_{[n-1,n]}\left({\bf 1}\\!\\!{\rm
I}_{W_{n-1}}\otimes\sigma_{z}b^{(x_{W_{n}}^{(0)})}\sigma_{z}\otimes\bigotimes_{u\in
W_{n}\setminus\\{x_{W_{n}}^{(0)}\\}}h^{(u)}\right).$

A small calculation leads to

$\mathrm{Tr}_{x_{W_{n-1}}^{(0)}]}\left(K_{\\{x_{W_{n-1}}^{(0)}\\}\cup
S(x_{W_{n-1}}^{(0)})}{\bf 1}\\!\\!{\rm I}^{(x_{W_{n-1}}^{(0)})}\otimes
b^{(x_{W_{n-1}}^{(0)}+e_{1})}\otimes
h^{(x_{W_{n-1}}^{(0)}+e_{2})}K_{\\{x_{W_{n-1}}^{(0)}\\}\cup
S(x_{W_{n-1}}^{(0)})}^{*}\right)$
$=\alpha\left((A^{2}+2B+C^{2})\mathrm{Tr}(b){\bf 1}\\!\\!{\rm
I}^{x_{W_{n-1}}^{(0)}}+2B(A+C)\mathrm{Tr}(\sigma_{z}b)\sigma_{z}^{x_{W_{n-1}}^{(0)}}\right)$
$=\mathrm{Tr}_{x_{W_{n-1}}^{(0)}]}\left(K_{\\{x_{W_{n-1}}^{(0)}\\}\cup
S(x_{W_{n-1}}^{(0)})}{\bf 1}\\!\\!{\rm
I}^{(x_{W_{n-1}}^{(0)})}\otimes(\sigma_{z}b\sigma_{z})^{(x_{W_{n-1}}^{(0)}+e_{1})}\otimes
h^{(x_{W_{n-1}}^{(0)}+e_{2})}K_{\\{x_{W_{n-1}}^{(0)}\\}\cup
S(x_{W_{n-1}}^{(0)})}^{*}\right).$

This implies

$\mathcal{E}_{[n-1,n]}({\bf 1}\\!\\!{\rm
I}_{W_{n-1}}\otimes\hat{\mathcal{E}}(b^{(x_{W_{n}}^{(0)})}\otimes{\bf
1}\\!\\!{\rm
I}))=\mathrm{Tr}(b)h_{n-1}+\mathrm{Tr}(\sigma_{z}b)(\frac{\tau_{3}}{4\tau_{1}})\sigma_{z}^{x_{W_{n-1}}^{(0)}}h_{n-1}.$

On the other hand, using (42) one gets

$\varphi(b^{(o)})=\rho_{0}(\mathcal{E}_{[0,1]}(b^{(o)}\otimes
h_{1}))=\mathrm{Tr}(b)$

Since for each $k\in\mathbb{N}$ we obtain

$\mathrm{Tr}_{x_{W_{k}}^{(0)}]}\left(K_{\\{x_{W_{k}}^{(0)}\\}\cup
S(x_{W_{k}}^{(0)})}{\bf 1}\\!\\!{\rm
I}^{(x_{W_{k}}^{(0)})}\otimes\sigma_{z}^{(x_{W_{k}}^{(0)}+e_{1})}\otimes
h^{(x_{W_{k}}^{(0)}+e_{2})}K_{\\{x_{W_{k}}^{(0)}\\}\cup
S(x_{W_{k}}^{(0)})}^{*}\right)$
$=\alpha(A+B)C\sigma_{z}^{(x_{W_{k}}^{(0)})}=\frac{\tau_{3}}{4\tau_{1}}\sigma_{z}^{(x_{W_{k}}^{(0)})}.$

and taking into account (15), a simple iteration leads to

$\mathcal{E}_{[k,k+1]}({\bf 1}\\!\\!{\rm
I}\otimes\cdots\mathcal{E}_{[n-1,n]}({\bf 1}\\!\\!{\rm
I}\otimes\hat{\mathcal{E}}_{[n,n+1]}(b^{x_{W_{n}}^{(1)}})=\varphi(b)h_{k}+\mathrm{Tr}(\sigma_{z}b)\left(\frac{\tau_{3}}{4\tau_{1}}\right)^{n-k}(\sigma_{z}^{(x_{W_{k}}^{(1)})}\otimes{\bf
1}\\!\\!{\rm I}_{W_{k}\setminus\\{x_{W_{k}}^{(1)}\\}})h_{k}.$

In particular, by taking $k=n_{0}+1,n_{0}+2$, one gets

$\varphi_{n}(aJ_{n}(b))=\varphi(b)\rho_{0}(\mathcal{E}_{[0,1]}(a_{W_{0}}\otimes\mathcal{E}_{[1,2]}(a_{W_{1}}\cdots\mathcal{E}_{[n_{0}-1,n_{0}]}(a_{W_{n_{0}-1}}\otimes\mathcal{E}_{[n_{0},n_{0}+1]}(a_{\Lambda}\otimes
h_{n_{0}+1}))))$
$+\mathrm{Tr}(\sigma_{z}b)\left(\frac{\tau_{3}}{4\tau_{1}}\right)^{n-n_{0}-2}\rho_{0}(\mathcal{E}_{[0,1]}(a_{W_{0}}\otimes\mathcal{E}_{[1,2]}(a_{W_{1}}\cdots\mathcal{E}_{[n_{0},n_{0}+1]}(a_{\Lambda}\otimes\mathcal{E}_{[n_{0}+1,n_{0}]}(\sigma_{z}^{(x_{W_{n_{0}+1}})}\otimes
h_{n_{0}+2})))).$

Thus

(43)
$\varphi_{n}(aJ_{n}(b))=\varphi(b)\varphi_{n_{0}}(a)+\mathrm{Tr}(\sigma_{z}b)\left(\frac{\tau_{3}}{4\tau_{1}}\right)^{n-n_{0}-2}\varphi_{n_{0}+1}(a\otimes\sigma_{z}^{(x_{W_{n_{0}+1}})}).$

From (35) it follows that

$\frac{\tau_{3}}{4\tau_{1}}=\frac{\theta^{J}(\theta^{2}-1)}{2(\theta^{J}(\theta^{2}+1)+2\theta)}\in[0,\frac{1}{2}];\quad\theta=\exp(2\beta)>1,J>0.$

Now taking the limit $n\to+\infty$ in (43) we arrive at (41). ∎

The main result of this section is the following.

######  Theorem 6.2.

Let $\varphi$ be a QMC associated with the Ising with (ZZ) coupling model on
the comb graph $\mathbb{N}\rhd_{0}\mathbb{N}$. Then

(44) $\lim_{|g|\to+\infty}\varphi(a\tau_{g}(b))=\varphi(a)\varphi(b)$

for all $a,b\in\mathcal{B}$.

###### Proof.

Let $a,b\in\mathcal{B}_{L,loc}$. There exist $n,0,m_{0}\in\mathbb{N}$ such
that $a=a_{n_{0}}\in\mathcal{B}_{[0,n_{0}]}$ and
$b=b_{m_{0}}\in\mathcal{B}_{[0,m_{0}]}$. Without lost of generality, we may
assume that $b$ is localized in the following form

$b=\bigotimes_{u\in[0,m_{0}]}b_{u}=\bigotimes_{k=0}^{m_{0}}b_{W_{k}};\quad
b_{W_{k}}=\bigotimes_{u\in W_{k}}b_{u}\in\mathcal{B}_{W_{k}}$

One can see that

$J_{n}(b)=\bigotimes_{u\in[0,m_{0}]}b_{u}^{(u+ne_{1})}=\bigotimes_{v\in[n,n+m_{0}]}\tilde{b}_{v}\in\mathcal{B}_{[n,n+m_{0}]}$

where

$\tilde{b}_{v}=\left\\{\begin{array}[]{ll}b_{v-ne_{1}},&\hbox{if
$v-ne_{1}\in\Lambda_{m_{0}}$;}\\\ 1,&\hbox{otherwise.}\end{array}\right.$

Define

(45) $F_{J_{n}(W_{k})}(a):=\bigotimes_{v\in
J_{n}(W_{k})}\mathrm{Tr}_{v]}\left(K_{\\{v\\}\cup S(v)}aK^{*}_{\\{v\\}\cup
S(v)}\right);\quad a\in\mathcal{B}_{[k,k+1]}$

Taking into account (15), one finds

$\hat{\mathcal{E}}_{[n+m_{0},n+m_{0}+1]}(\tilde{b}_{W_{n+m_{0}}})))=\left(\bigotimes_{v\in
J_{n}(W_{m_{0}})}\mathrm{Tr}_{v]}\left(K_{\\{v\\}\cup
S(v)}b_{v}h_{S(v)}K^{*}_{\\{v\\}\cup
S(v)}\right)\right)\otimes\left(\bigotimes_{v\in W_{n+m_{0}}\setminus
J_{n}(W_{m_{0}})}h_{v}\right)$
$=\alpha^{|W_{m_{0}+1}|}F_{J_{n}(W_{m_{0}})}(\tilde{b}_{J_{n}(W_{m_{0}})})\otimes\left(\bigotimes_{v\in
W_{n+m_{0}}\setminus J_{n}(W_{m_{0}})}h_{v}\right).$

According to the structure of the comb graph $\mathbb{N}\rhd_{0}\mathbb{N}$
one has

$J_{n}(W_{k+1})=\bigcup_{v\in J_{n}(W_{k})}S(v).$

Then

$\displaystyle{\mathcal{E}}_{[n+m_{0}-1,n+m_{0}]}(\tilde{b}_{W_{n+m_{0}-1}}\otimes\hat{\mathcal{E}}_{[n+m_{0},n+m_{0}+1]}(\tilde{b}_{W_{n+m_{0}}}))))=$
$\displaystyle=\alpha^{|W_{m_{0}+1}|}\bigotimes_{u\in
J_{n}(W_{m_{0}-1})}\mathrm{Tr}_{u]}\left(K_{\\{u\\}\cup
S(u)}\tilde{b}_{u}\otimes
F_{J_{n}(W_{m_{0}})}(\tilde{b}_{J_{n}(W_{m_{0}})})K_{\\{u\\}\cup S(u)}\right)$
$\displaystyle\otimes\bigotimes_{w\in W_{n+m_{0}-1}\setminus
J_{n}(W_{m_{0}-1})}h_{w},$
$\displaystyle=\alpha^{|W_{m_{0}+1}|}F_{J_{n}(W_{m_{0}-1})}\left(\tilde{b}_{J_{n}(W_{m_{0}-1})}\otimes
F_{J_{n}(W_{m_{0}})}(\tilde{b}_{J_{n}(W_{m_{0}})})\right)\otimes\bigotimes_{w\in
W_{n+m_{0}-1}\setminus J_{n}(W_{m_{0}-1})}h_{w}.$

An iteration leads to

$\mathcal{E}_{[n,n+1]}(\tilde{b}_{W_{n}}\otimes\cdots\mathcal{E}_{[n+m_{0}-1,n+m_{0}]}(\tilde{b}_{W_{n+m_{0}-1}}\otimes\hat{\mathcal{E}}_{[n+m_{0},n+m_{0}]}(\tilde{b}_{n+m_{0}})))$
$=\alpha^{|W_{m_{0}+1}|}F_{J_{n}(W_{0})}\left(\tilde{b}_{J_{n}(W_{0})}\otimes\cdots
F_{J_{n}(W_{m_{0}-1})}\left(\tilde{b}_{J_{n}(W_{m_{0}-1})}\otimes
F_{J_{n}(W_{m_{0}})}(\tilde{b}_{J_{n}(W_{m_{0}})})\right)\right)\otimes\bigotimes_{w\in
W_{n}\setminus J_{n}(W_{0})}h_{w}.$

Since $J_{n}(W_{0})=\\{x_{W_{n}}^{(0)}\\}$ and $h_{w}=\alpha{\bf 1}\\!\\!{\rm
I}$ for each $w$ then

$\displaystyle\mathcal{E}_{[n,n+1]}(\tilde{b}_{W_{n}}\otimes\cdots\mathcal{E}_{[n+m_{0}-1,n+m_{0}]}(\tilde{b}_{W_{n+m_{0}-1}}\otimes\hat{\mathcal{E}}_{[n+m_{0},n+m_{0}]}(\tilde{b}_{n+m_{0}})))$
$\displaystyle=\alpha^{|W_{m_{0}+1}|+|W_{n}|-1}F_{x_{W_{n}}^{(0)}}\left(\tilde{b}_{J_{n}(W_{0})}\otimes\cdots
F_{J_{n}(W_{m_{0}-1})}\left(\tilde{b}_{J_{n}(W_{m_{0}-1})}\otimes
F_{J_{n}(W_{m_{0}})}(\tilde{b}_{J_{n}(W_{m_{0}})})\right)\right)\otimes{\bf
1}\\!\\!{\rm I}_{W_{n}\setminus\\{x_{W_{n}}^{(0)}\\}},$
$\displaystyle=b_{n}\otimes h_{n+1}$

where

$g_{n}:=\alpha^{|m_{0}|}F_{x_{W_{n}}^{(0)}}\left(\tilde{b}_{J_{n}(W_{0})}\otimes\cdots
F_{J_{n}(W_{m_{0}-1})}\left(\tilde{b}_{J_{n}(W_{m_{0}-1})}\otimes
F_{J_{n}(W_{m_{0}})}(\tilde{b}_{J_{n}(W_{m_{0}})})\right)\right)\in\mathcal{B}_{x_{W_{n}}^{(0)}}.$

One can easily check that $b_{n}=\alpha^{|m_{0}|}J_{n}(b)$ with

$g=F_{W_{0}}\left(b_{W_{0}}\otimes\cdots
F_{W_{m_{0}-1}}\left(b_{W_{m_{0}-1}}\otimes
F_{W_{m_{0}}}(b_{W_{m_{0}}})\right)\right)\in\mathcal{B}_{W_{0}}.$

Then

$\displaystyle\varphi(a_{n_{0}}\otimes
J_{n}(b_{m_{0}})=\rho_{0}(\mathcal{E}_{[0,1]}(a_{W_{0}}\cdots\mathcal{E}_{[n_{0},n_{0}+1]}(a_{W_{n_{0}}}\otimes\mathcal{E}_{[n_{0},n_{0}+1]}({\bf
1}\\!\\!{\rm I}_{W_{n_{0}+1}}\otimes\cdots$
$\displaystyle\mathcal{E}_{[n,n+1]}(\tilde{b}_{W_{n}}\otimes\cdots\mathcal{E}_{[n+m_{0}-1,n+m_{0}]}(\tilde{b}_{W_{n+m_{0}-1}}\otimes\hat{\mathcal{E}}_{[n+m_{0},n+m_{0}]}(\tilde{b}_{n+m_{0}})))))))$
$\displaystyle=\rho_{0}(\mathcal{E}_{[0,1]}(a_{W_{0}}\cdots\mathcal{E}_{[n_{0},n_{0}+1]}(a_{W_{n_{0}}}\otimes\mathcal{E}_{[n_{0},n_{0}+1]}({\bf
1}\\!\\!{\rm I}_{W_{n_{0}+1}}\otimes\cdots\mathcal{E}_{[n,n+1]}(g_{n}\otimes
h_{n+1})))))$ $\displaystyle=\alpha^{|m_{0}|}\varphi_{n}(a_{n_{0}}J_{n}(g))$

By Lemma 6.1, one finds

$\lim_{n\to+\infty}\varphi_{n}(a_{n_{0}}J_{n}(g))=\varphi(a_{n_{0}})\varphi(g).$

Due to $F_{W_{k}}=\mathcal{E}_{[k,k+1]}$ for $0\leq k\leq m_{0}$, we obtain

$\alpha^{|m_{0}|}\varphi(g)=\varphi(b_{m_{0}}).$

Therefore,

$\lim_{n\to+\infty}\varphi(a_{n_{0}}J_{n}(b_{m_{0}}))=\varphi(a_{n_{0}})\varphi(b_{m_{0}})$

which completes the proof. ∎

## Acknowledgments

The authors gratefully acknowledge Qassim University, represented by the
Deanship of Scientific Research, on the financial support for this research
under the number (cba-2019-2-2-I-5400) during the academic year 1440 AH / 2019
AD.

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