# Decay of quantum conditional mutual information for purely generated
finitely correlated states

Pavel Svetlichnyy  T.A.B. Kennedy  School of Physics, Georgia Institute of
Technology, Atlanta, GA, 30332-0430, USA.

(August 27, 2024)

###### Abstract

The connection between quantum state recovery and quantum conditional mutual
information (QCMI) is studied for the class of purely generated finitely
correlated states (pgFCS) of one-dimensional quantum spin chains. For a
tripartition of the chain into two subsystems separated by a buffer region, it
is shown that a pgFCS is an approximate quantum Markov chain, and stronger,
may be approximated by a quantum Markov chain in trace distance, with an error
exponentially small in the buffer size. This implies that, (1) a locally
corrupted state can be approximately recovered by action of a quantum channel
on the buffer system, and (2) QCMI is exponentially small in the size of the
buffer region. Bounds on the exponential decay rate of QCMI and examples of
quantum recovery channels are presented.

##  I Introduction

In this paper we investigate properties of quantum conditional mutual
information (QCMI) and quantum state recovery for the class of many-particle
states known alternatively as purely generated finitely correlated states
(pgFCS) or uniform matrix product states.Fannes  _et al._ (1992, 1994);
Klümper  _et al._ (1993) These states were introduced in the early 1990’s and
used to describe the ground states of translationally invariant one-
dimensional spin chains, including the well-known model of Affleck, Kennedy
Lieb and Tasaki (AKLT) as a special case.Affleck  _et al._ (1988)
Independently, the matrix product states were introduced by Klümper et al
Klümper  _et al._ (1993) and separately shown by Vidal Vidal (2003) to be an
efficient representation of slightly entangled states, such as the ground
states of local gapped Hamiltonians.Perez-Garcia  _et al._ (2007) Proposals
for the controlled experimental preparation of such states have been reported
in the literature.Schön  _et al._ (2007); Zhou  _et al._ (2021) The larger
class of finitely correlated states (FCS) and the closely related matrix
product density operators (MPDO),Verstraete  _et al._ (2004) can be used to
describe mixed states of systems with local Hamiltonians and finite range
interactions, such as finite temperature Gibbs states.Kuwahara  _et al._
(2020); Kato and Brandão (2019)

Positivity of QCMI is one of the cornerstones of quantum information theory,
being equivalent to the strong subadditivity of quantum entropy.Wilde (2017);
Lieb and Ruskai (1973) In the theory of quantum state recovery,Petz (1986,
1988); Hayden  _et al._ (2004); Fawzi and Renner (2015); Sutter  _et al._
(2016); Junge  _et al._ (2018); Flammia  _et al._ (2017) viewed as a
generalized quantum error correction problem, the states for which recovery is
exact are known as quantum Markov chains. These states are intimately
connected with the QCMI as is clear in the following setting. Consider a
quantum system (a collection of spatially separated spins) subdivided into
three parts, denoted $A$, $B$ and $C,$ see Figure 1a. The QCMI, denoted
$I(A:C|B)$ indicates in broad terms the quantum information mutually shared by
subsystems $A$ and $C$ given specific information about the state of subsystem
$B$. Two essential and non-trivial properties of $I(A:C|B)$ are that it is
non-negative $I(A:C|B)\geq 0,$ as a direct consequence of the strong
subadditivity of quantum entropy, Lieb and Ruskai (1973) and that the equality
condition $I(A:C|B)=0$ defines the class of quantum states with density
operator $\rho_{ABC}$, which are precisely the quantum Markov chains. In this
setting the problem of quantum state recovery, in either exact or approximate
form, may be approached by investigating when $I(A:C|B)$ is zero or very
small, conditions which require investigation of quantum states described as
either exact or approximate quantum Markov chains. This specific quantum
information-theoretic task gives QCMI a compelling physical interpretation.

We study quantum state recovery for a one-dimensional quantum system $ABC$ for
which the density operator $\rho_{ABC}$ is the reduced density operator of a
pgFCS, respectively, uniform matrix product state. From here on we will
principally use the former appellation and refer the reader to the dictionary
between these two languages compiled in Appendix A. As we discuss further
below, $\rho_{ABC}$ is an approximate quantum Markov chain and hence can be
approximately recovered.

The pgFCS is a subset of the class of FCS, some of whose properties we review
in Section III.3. The FCS can efficiently describe systems with limited
entanglement and their preparation by using quantum circuits of limited
depth.Kato and Brandão (2019); Brandão and Kastoryano (2019) It was recently
shown that FCS may be used to approximate Gibbs states of systems described by
one-dimensional local Hamiltonians.Kato and Brandão (2019); Kuwahara  _et al._
(2020) (In Refs. Guth Jarkovský  _et al._ , 2020; Kuwahara  _et al._ , 2021 it
was shown that the same class of states can be approximated by matrix product
operators (MPO). It is not immediately clear that the resulting MPO are FCS or
MPDO. The latter are called _locally purifiable_ in the sense of Ref.
Verstraete  _et al._ , 2004.) Conversely, it is conjectured that a generic FCS
approximates the Gibbs state of some local gapped Hamiltonian.Chen  _et al._
(2021) While Gibbs states are of broad physical interest, their sampling has
been employed in a quantum algorithm proposed to speed up the computation of
semi-definite programs.Brandao and Svore (2017); Brandão  _et al._ (2019); van
Apeldoorn  _et al._ (2020)

The QCMI is defined for a system $ABC$ with the Hilbert space
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$ by
Watrous (2018)

$I(A:C|B):=S(\rho_{AB})+S(\rho_{BC})-S(\rho_{ABC})-S(\rho_{B}),$ (1)

where $S(\rho)=-\mathrm{tr}(\rho\log\rho)$ is the von Neumann entropy, and
omission of a subscript implies taking a partial trace, e.g.,
$\rho_{AB}=\mbox{tr}_{C}\rho_{ABC}$. As noted earlier, QCMI is non-negative,
$I(A:C|B)\geq 0,$ as an immediate consequence of the strong subadditivity of
von-Neumann entropy. Lieb and Ruskai (1973) In Refs. Petz, 1986, 1988 it was
established that the condition $I(A:C|B)=0$ ($\rho_{ABC}$ is a quantum Markov
chain) is equivalent to the existence of a quantum channel, a completely
positive trace-preserving map, Stinespring (1955); Watrous (2018)
$\mathcal{R}_{B\rightarrow BC}$, which recovers $\rho_{ABC}$ from $\rho_{AB}$
exactly, i.e., $\mathcal{R}_{B\rightarrow BC}(\rho_{AB})=\rho_{ABC}$. From
this property follows the particular structure of quantum Markov chains which
we summarize in Theorem III.5. Hayden  _et al._ (2004) The recovery map is not
unique, and we can always explicitly construct the Petz recovery map, which is
defined (on the support of $\rho_{B}$) by Petz (1986, 1988); Hayden  _et al._
(2004)

$\mathcal{P}_{B\rightarrow
BC}(X):=\rho_{BC}^{\frac{1}{2}}(\rho_{B}^{-\frac{1}{2}}X\rho_{B}^{-\frac{1}{2}}\otimes\mathbb{1}_{C})\rho_{BC}^{\frac{1}{2}},$
(2)

where $\rho_{B}^{-\frac{1}{2}}$ is a pseudo-inverse of
$\rho_{B}^{\frac{1}{2}}$. In the case when a recovery quantum channel does not
restore the state exactly, the _recovery error_ is quantified by the trace
distance,

$\epsilon:=\|\rho_{ABC}-\mathcal{R}_{B\rightarrow BC}(\rho_{AB})\|_{1},$

where $\|X\|_{1}=\mathrm{tr}(\sqrt{X^{\dagger}X})$. The recovery error, for a
candidate recovery channel $\mathcal{R}_{B\rightarrow BC}$, bounds QCMI from
above via an Alicki-Fannes type inequality Alicki and Fannes (2004); Winter
(2016) (see Appendix B),

$\displaystyle I(A:C|B)$
$\displaystyle\leq\frac{1}{2}\epsilon\ln\dim\mathcal{H}_{A}+(1+\frac{1}{2}\epsilon)h\left(\frac{\epsilon}{2+\epsilon}\right),$
(3)

where $h(p)=-p\ln p-(1-p)\ln(1-p)$ is the binary entropy. This bound was used
to show that a Gibbs state of a 1D local Hamiltonian is an approximate quantum
Markov chain.Kato and Brandão (2019) By constructing a specific recovery
channel, for which the recovery error is subexponentially small in the size of
the region $B$, it was shown that the QCMI is also subexponentially small. In
this paper we similarly construct an approximate recovery channel for pgFCS,
and bound QCMI from above using (3).

Theoretical developments regarding the reversibility of quantum channels
obtained in a series of works Refs. Fawzi and Renner, 2015; Sutter  _et al._ ,
2016; Junge  _et al._ , 2018, led to the discovery of a recovery channel
$\mathcal{R}^{u}_{B\rightarrow BC}$, called the universal recovery map, which
may be used to bound QCMI from below,

$\frac{1}{4\ln 2}\|\rho_{ABC}-\mathcal{R}^{u}_{B\rightarrow
BC}(\rho_{AB})\|_{1}^{2}\leq I(A:C|B).$ (4)

The universal recovery map, which is a generalization of the Petz recovery map
(2), is given explicitly in Ref. Junge  _et al._ , 2018 and, as in the case of
the Petz recovery map, depends only on the marginal $\rho_{BC}$. The
inequality (4) implies that states with small QCMI are guaranteed to have a
small recovery error with respect to the universal recovery map. In Ref.
Kuwahara  _et al._ , 2020, it was shown for local Hamiltonian systems of
arbitrary dimension, and high enough temperature, that QCMI for Gibbs states
decays exponentially with the width of region $B$, separating regions $A$ and
$C$. By (4) this implies good recovery by the universal recovery map. (It is
conjectured, Kato and Brandão (2019) that in one dimension the exponential
decay of QCMI holds for any temperature.) Similar conclusions were found for
finite temperature Gibbs states of free fermions, free bosons, conformal field
theories, and holographic models.Swingle and McGreevy (2016)

Figure 1: The figure shows lattices of spins partitioned into subsystems $A$,
$B$, and $C$, which may be analyzed with the help of QCMI. In (a) we show a
typical partition of a two dimensional spin lattice with the distance $d(A,C)$
shown. In (b) and (c) subsystem $B$ is disconnected and connected,
respectively, and the distance $d(A,C)$ is again shown. In this paper we focus
on the partition (c).

A more physical picture of the recovery map is as an experimental repair
process, with an accuracy bounded by how rapidly QCMI decreases as a function
of the distance $d(A,C)$ shown in Figure 1.Flammia  _et al._ (2017) For
example, for the lattice of spins shown in Figure 1(a), suppose that quantum
state corruption or erasure has happened in the region $C$. It is then natural
to partition the lattice by means of a buffer zone $B$ surrounding $C$.Swingle
and McGreevy (2016); Brandão and Kastoryano (2019) Qualitatively, quantum
states with rapidly decaying QCMI may be recovered accurately on erased
regions with small buffer zones $B$.Swingle and McGreevy (2016); Brandão and
Kastoryano (2019); Kato and Brandão (2019) In one dimension the analysis can
be reduced to the partition shown in Figure 1(c),Kato and Brandão (2019) which
we focus on in this paper. We briefly consider the setting of Figure 1(b) in
Appendix C.

In the language of quantum memories, QCMI is connected to the task of storing
quantum information in the presence of quantum error correction,Flammia  _et
al._ (2017) and to state preparation by (circuits of) local quantum
channels.Brandão and Kastoryano (2019); Swingle and McGreevy (2016); Kato and
Brandão (2019) Let us assume that quantum channels can be applied
experimentally over regions with size limited by the linear dimension $l$.
Conceptually, a quantum state may be prepared on a much larger region by first
constructing reduced states on a grid of disconnected regions and then
patching them together by the application of a second layer of quantum
channels. The deviation of the constructed state from the target state will
depend on the value of QCMI for the partitions with $d(A,C)=O(l)$. An example
of Gibbs state construction by the application of layers of universal recovery
channels is given in Ref. Brandão and Kastoryano, 2019. A simpler, one-
dimensional example, in which the circuit consists of two layers of quantum
channels, is presented in Ref. Kato and Brandão, 2019. Note that in the latter
reference the quantum channels proposed are not universal recovery channels.
In this work we also avoid use of the universal recovery maps, relying instead
on the explicit structure of pgFCS.

Our main result, a bound on the recovery error and the related decay of QCMI,
is summarized in Theorem II.1 below. The latter can be compared to and
contrasted with the behavior of quantum mutual information (QMI), defined by
$I(A:C)=S(\rho_{A})+S(\rho_{C})-S(\rho_{AC})$. The QMI has the property that
it bounds the quantum correlations between the regions $A$ and $C$ via the
quantum Pinsker’s inequality.Hiai  _et al._ (1981) Those pgFCS which
correspond to injective MPSPerez-Garcia  _et al._ (2007) exhibit exponential
decay of both QMIWolf  _et al._ (2008) and correlations.Fannes  _et al._
(1992); Brandão and Horodecki (2015) A theoretical bound on QMI can be
obtained more straightforwardly than a similar bound on QCMI, since in the
former case the separating region $B$ is traced out and information about its
state is lost. We note that for pgFCS corresponding to non-injective MPS QMI
does not necessarily converge to zero with the growth of region $B$ (see
Appendix D for examples). By contrast, we show that for any pgFCS the QCMI
converges to zero.

The remainder of this paper is organized around the proof of Theorem II.1,
which we state in Section II. In Section III we establish our notation and
conventions, and provide relevant background and theorems on FCS, pgFCS,
quantum channels and quantum Markov chains, which we use throughout the paper.
In Section IV we prove Theorem II.1 starting with the simplest case of pgFCS
(corresponding to the injective MPS), and proceed to more general cases. While
the proof does not require consideration of specific recovery channels, we
give examples of the latter in Section V. In Sections III, IV, and V we rely
on additional technical results, presented in a sequence of appendices. We
summarize our conclusions in Section VI.

##  II Main result

In this paper we consider a pgFCS density operator, denoted by $\rho_{ABC},$
for the tripartite system $ABC$ depicted in Figure 1(c). We construct a QMC
approximating $\rho_{ABC}$ and a recovery map for which both the trace
distance error and the recovery error are exponentially small in the size of
the separating region $B$. Using inequality (3), we conclude that QCMI decays
exponentially. The methods we use are built upon those of Refs. Fannes  _et
al._ , 1992; Perez-Garcia  _et al._ , 2007; Wolf  _et al._ , 2008; Brandão and
Horodecki, 2015, which we augment with the continuity theorem for
Stinespring’s dilation,Kretschmann  _et al._ (2008) cited in Section III.2 as
Theorem III.1.

We now state our principle result,

######  Theorem II.1.

Let $\rho_{ABC}$ be the reduced density operator for a pgFCS on a finite
contiguous one-dimensional region $ABC$, where the subsystem $B$ separates the
subsystems $A$ and $C$. Then, provided that the region $B$ is large enough,

  1. 1.

There exists a QMC, denoted $\tilde{\rho}_{ABC}$, and constants $q>0$, $K>0$,
such that

$\|\rho_{ABC}-\tilde{\rho}_{ABC}\|_{1}\leq\frac{1}{2}Ke^{-q|B|}.$

  2. 2.

There exists a quantum channel $\mathcal{R}_{B\rightarrow
BC}:\mathcal{B}(\mathcal{H}_{B})\rightarrow\mathcal{B}(\mathcal{H}_{B}\otimes\mathcal{H}_{C})$,
such that

$\|\rho_{ABC}-(\mathrm{id}_{A}\otimes\mathcal{R}_{B\rightarrow
BC})\left(\rho_{AB}\right)\|_{1}\leq Ke^{-q|B|}.$

  3. 3.

Furthermore, the QCMI satisfies $I(A:C|B)\leq\tilde{K}e^{-q|B|}$, where
$\tilde{K}$ depends linearly on the sizes of the regions $A$ and $B$.

Remarks:

  1. 1.

We will show that $q=-\frac{1}{2}\log\nu$, where $\nu_{\mathrm{gap}}<\nu<1$
with $\nu_{\mathrm{gap}}:=\max_{|\nu_{i}|<1}\\{|\nu_{i}|\\}$, where $\nu_{i}$
are the eigenvalues of the quantum channel $\mathcal{E}$ induced by the pgFCS
under consideration (see Section III.3). The value of $\nu$ may be chosen
arbitrarily close to $\nu_{\mathrm{gap}}$, with compensating increase in
$n_{0}$, defined in the Remark 2(b) immediately below. See Lemma III.3 proved
in Appendix E for details.

  2. 2.

Let $d_{s}$ be the dimension of the Hilbert space $\mathcal{H}_{s}$ of a
single spin. Our proof requires that size of the region $B$, denoted by $|B|$,
is large enough that $\dim\mathcal{H}_{B}=d_{s}^{|B|}\geq d_{M}^{2}$, where
$d_{M}$ is the dimension of the _memory space_ , defined for FCS in Section
III.3. This demand is quite mild, since $\dim\mathcal{H}_{B}$ grows
exponentially with $|B|$.

  3. 3.

We may write a bound that does not require a choice of the parameter $\nu$,
however this bound will have a logarithmic correction in the exponent,

$\|\rho_{ABC}-(\mathrm{id}_{A}\otimes\mathcal{R}_{B\rightarrow
BC})\left(\rho_{AB}\right)\|_{1}\leq
K^{\prime}e^{-q^{\prime}|B|+u^{\prime}\log|B|},$

where $q^{\prime}=-\frac{1}{2}\log\nu_{\mathrm{gap}}$, and $K^{\prime}$ and
$u^{\prime}$ are constants. See Lemma E.2 for details. A similar modification
may be done to the bound on QCMI.

  4. 4.

For the setup of Figure 1(b) we have a result similar to Theorem II.1 with an
additional factor of $2$ in the pre-exponent and
$d(A,C)=\min\\{|B_{1}|,|B_{2}|\\}$ replacing $|B|$. In this case we have to
demand that both $B_{1}$ and $B_{2}$ are sufficiently large, i.e.,
$\dim\mathcal{H}_{B_{1}},\dim\mathcal{H}_{B_{2}}\geq d_{M}^{2}$. We prove this
result in Appendix C.

  5. 5.

In Theorem II.1, the points $2$ and $3$ are consequences of the stronger
statement $1$: there exist approximate QMC’s, which cannot be approximated by
(exact) QMC’s. Sutter (2018); Christandl  _et al._ (2012); Ibinson  _et al._
(2008)

##  III Preliminaries

In this section we take the opportunity to introduce some relevant background
on quantum channels, finitely correlated states and quantum Markov chains, and
to compile and restate several important results from the literature in our
notation. These are needed for the proof of Theorem II.1 in Section IV.

###  III.1 Notation and conventions

All Hilbert spaces considered in this paper are finite-dimensional and are
denoted, up to subscripts, by either $\mathcal{H}$ or $\mathcal{K}$. In
particular, we denote the Hilbert space of a single spin by $\mathcal{H}_{s}$,
and denote its dimension by $d_{s}:=\dim\mathcal{H}_{s}$. We define
$\mathcal{B}(\mathcal{H})$ to be the space of linear operators acting on
$\mathcal{H}$; the space $\mathcal{B}(\mathcal{H})$ is isomorphic to the space
of $\dim\mathcal{H}\times\dim\mathcal{H}$ complex matrices. The set of density
operators on the Hilbert space $\mathcal{H}$ is denoted by
$\mathcal{D}(\mathcal{H})$. The $n$-fold composition of a quantum channel, for
instance $\mathcal{E}$, is denoted by $\mathcal{E}^{n}$.

To lighten the notation we often omit the identity operator/map in tensor
products of operators/maps. For example, for a quantum channel
$\Phi:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{H}_{s})\otimes\mathcal{B}(\mathcal{H})$,
the quantum channel
$\Phi^{2}\equiv\Phi\circ\Phi:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{H}_{s})\otimes\mathcal{B}(\mathcal{H}_{s})\otimes\mathcal{B}(\mathcal{H})$
denotes the map $X\mapsto(\mathrm{id}_{s}\otimes\Phi)(\Phi(X))$, where
$\mathrm{id}_{s}$ is the identity map on $\mathcal{B}(\mathcal{H}_{s})$.
Analogously, if $U:\mathcal{H}\rightarrow\mathcal{K}_{2}\otimes\mathcal{H}$
and $W:\mathcal{H}\rightarrow\mathcal{K}_{1}\otimes\mathcal{H}$ are
isometries, then
$UW:\mathcal{H}\rightarrow\mathcal{K}_{1}\otimes\mathcal{K}_{2}\otimes\mathcal{H}$
denotes the isometry $(\mathbb{1}_{\mathcal{K}_{1}}\otimes U)W$, which we can
also express in terms of an orthonormal basis of $\mathcal{H}$ as

$UW:=\sum_{i,j,k=1}^{\dim\mathcal{H}}\langle k|W|j\rangle\otimes\langle
i|U|k\rangle\otimes|i\rangle\langle j|.$ (5)

For system $ABC$ the notation $\rho_{AB}$ means the partial trace of
$\rho_{ABC}$ over the Hilbert space of system $C$, i.e.,
$\rho_{AB}:=\mbox{tr}_{C}\rho_{ABC}$. We denote the number of spins in the
region $A$ by $|A|$. The support of an operator
$O:\mathcal{K}\rightarrow\mathcal{H}$, i.e., the closure of the set
$\left\\{|\psi\rangle\in\mathcal{K}\;|\;O|\psi\rangle\neq 0\right\\}$, is
denoted by $\mathrm{supp}(O)$.

###  III.2 Quantum channels

A quantum channel $\mathcal{M}$ is a completely positive trace-preserving map
$\mathcal{M}:\mathcal{B}(\mathcal{K}_{in})\rightarrow\mathcal{B}(\mathcal{K}_{out})$.
The complete positivity means that for operators in the space
$\mathcal{B}(\mathcal{K}_{R}\otimes\mathcal{K}_{in})$, where $\mathcal{K}_{R}$
is a reference Hilbert space of arbitrary dimension, the map
$\mathrm{id}_{R}\otimes\mathcal{M}$ is positive, i.e., for any
$\dim\mathcal{K}_{R}\in\mathbb{N}$ and for any
$\mathcal{B}(\mathcal{K}_{R}\otimes\mathcal{K}_{in})\ni X\geq 0$,
$(\mathrm{id}_{R}\otimes\mathcal{M})(X)\geq 0$. For finite-dimensional
$\mathcal{K}_{in}$, as considered here, it suffices to demand that
$\mathrm{id}_{R}\otimes\mathcal{M}$ is positive for the case
$\dim\mathcal{K}_{R}=\dim\mathcal{K}_{in}$. The trace-preservation implies
that $\mathrm{tr}\mathcal{M}(X)=\mathrm{tr}X$ for any
$X\in\mathcal{B}(\mathcal{K}_{in})$. For $p,q\in[1,\infty)$ we define the
$p\rightarrow q$ norm $\|\mathcal{M}\|_{p\rightarrow
q}:=\sup_{X\in\mathcal{B}(\mathcal{K}_{in})}{\|\mathcal{M}(X)\|_{q}}/{\|X\|_{p}}$,
where $\|\cdot\|_{r}$ is the Schatten $r$-norm. Watrous (2018)

By Stinespring’s dilation theorem, Stinespring (1955) in the finite-
dimensional setting, a quantum channel
$\mathcal{M}:\mathcal{B}(\mathcal{K}_{in})\rightarrow\mathcal{B}(\mathcal{K}_{out})$
may be represented in the form Watrous (2018)

$\mathcal{M}(X)=\mathrm{tr}_{E}\left(WXW^{\dagger}\right),$ (6)

where $W:\mathcal{K}_{in}\rightarrow\mathcal{K}_{E}\otimes\mathcal{K}_{out}$
is an isometry, $W^{\dagger}W=\mathbb{1}_{in}$, called the dilating isometry,
and the Hilbert space $\mathcal{K}_{E}$ is called the dilation space (or
environment). The pair $\\{\mathcal{K}_{E},W\\}$ is referred to as a dilation
of $\mathcal{M}$. For brevity, we will write that $W$ dilates the quantum
channel $\mathcal{M}$. We note that
$\dim\mathcal{K}_{E}\leq\dim\mathcal{K}_{in}\dim\mathcal{K}_{out}$.Watrous
(2018) The isometric representation is not unique,Kretschmann  _et al._ (2008)
and, if $W:\mathcal{K}_{in}\rightarrow\mathcal{K}_{E}\otimes\mathcal{K}_{out}$
and
$W^{\prime}:\mathcal{K}_{in}\rightarrow\mathcal{K}_{E^{\prime}}\otimes\mathcal{K}_{out}$
are different dilations with
$\dim\mathcal{K}_{E^{\prime}}\geq\dim\mathcal{K}_{E}$, then $W^{\prime}=UW$
for some isometry $U:\mathcal{K}_{E}\rightarrow\mathcal{K}_{E^{\prime}}$,
$U^{\dagger}U=\mathbb{1}_{E}$. In the case of
$\dim\mathcal{K}_{E}=\dim\mathcal{K}_{E^{\prime}}$, the isometry $U$ is
unitary.

The space of all quantum channels
$\mathcal{M}:\mathcal{B}(\mathcal{K}_{in})\rightarrow\mathcal{B}(\mathcal{K}_{out})$
with fixed domain and codomain can be endowed with the so-called _diamond
norm_ ,Kitaev  _et al._ (2002) also called _completely bounded norm_ ,
$\|\mathcal{M}\|_{\diamond}:=\sup_{\dim{\mathcal{K}_{R}}}\|\mathrm{id}_{R}\otimes\mathcal{M}\|_{1\rightarrow
1}$. In the finite-dimensional case the supremum is achieved for
$\dim\mathcal{K}_{R}=\dim\mathcal{K}_{in}$.Kitaev  _et al._ (2002) The quantum
channels are continuous with respect to the diamond norm, a result that is
expressed by the Continuity Theorem for Stinespring’s Representation,

######  Theorem III.1 (Ref. Kretschmann  _et al._ , 2008).

Let $\mathcal{K}_{in}$ and $\mathcal{K}_{out}$ be finite-dimensional Hilbert
spaces, and suppose that
$\mathcal{M}_{1},\mathcal{M}_{2}:\mathcal{B}(\mathcal{K}_{in})\rightarrow\mathcal{B}(\mathcal{K}_{out})$
are quantum channels with Stinespring isometries
$W_{1},W_{2}:\mathcal{K}_{in}\rightarrow\mathcal{K}_{E}\otimes\mathcal{K}_{out}$
and a common dilation space $\mathcal{K}_{E}$. We then have

$\inf_{U_{E}}\|W_{1}-U_{E}W_{2}\|^{2}\leq\|\mathcal{M}_{1}-\mathcal{M}_{2}\|_{\diamond}\leq
2\inf_{U_{E}}\|W_{1}-U_{E}W_{2}\|,$ (7)

where the minimization is with respect to all unitary
$U_{E}\in\mathcal{B}(\mathcal{K}_{E})$.

We denoted by $\|W\|$ the operator norm of $W$. In the case
$\mathcal{M}_{1}=\mathcal{M}_{2}$, we obtain $W_{1}=U_{E}W_{2}$ for some
unitary $U_{E}$, which implies that the dilating isometry is defined up to a
unitary on the dilation space, consistent with (6).

For a quantum channel $\mathcal{M}$, regarded as a linear map on the
$\dim\mathcal{K}_{in}^{2}$-dimensional vector space
$\mathcal{B}(\mathcal{K}_{in})$, we define the spectrum in the usual way:
$X\in\mathcal{B}(\mathcal{K}_{in})$ is an eigenvector with the eigenvalue
$\nu$, if $X$ is non-zero and $\mathcal{M}(X)=\nu X$. Each eigenvalue
$\nu_{i}$ ($i=1,\cdot\cdot\cdot,\dim^{2}\mathcal{K}_{in}$) of $\mathcal{M}$
satisfies $|\nu_{i}|\leq 1$, and there always exists at least one fixed point
with the eigenvalue equal to $1$.Evans and Høegh-Krohn (1978) Hence the
spectral radius of a quantum channel is equal to $1$. The set of eigenvalues
with $|\nu_{i}|=1$ is known as the _peripheral spectrum_.Fannes  _et al._
(1992) When the peripheral spectrum is a singleton set, consisting of the
single eigenvalue $1$, it is said to be trivial.

We now employ the spectral decomposition to construct a quantum channel
$\tilde{\mathcal{E}}$ from a given quantum channel $\mathcal{E}$.Szehr  _et
al._ (2015) This will be useful because that the $n$-fold composition
$\tilde{\mathcal{E}}^{n}$ is an excellent approximation to $\mathcal{E}^{n}$
in the limit $n\rightarrow\infty$.Szehr  _et al._ (2015) A dilating isometry
associated to $\tilde{\mathcal{E}}^{n}$ possesses convenient properties, and
we will take advantage of these.

As shown in Appendix F, the space of all maps
$\mathcal{B}(\mathcal{K})\rightarrow\mathcal{B}(\mathcal{K})$ is isomorphic to
$\mathcal{B}(\mathcal{K}\otimes\mathcal{K})$.

######  Definition 1.

Let $\mathcal{K}$ be a finite-dimensional Hilbert space, and
$\mathcal{E}:\mathcal{B}(\mathcal{K})\rightarrow\mathcal{B}(\mathcal{K})$ be a
map. Let $E\in\mathcal{B}(\mathcal{K}\otimes\mathcal{K})$ be the operator
isomorphic to $\mathcal{E}$, $E:=\mathrm{vec}(\mathcal{E})$. Let $E$ have the
Jordan decomposition $E=\sum_{i}(\nu_{i}P_{\nu_{i}}+N_{\nu_{i}})$, with the
$P_{\nu_{i}}$ orthogonal projectors onto the subspace of the eigenvalue
$\nu_{i}$, and the $N_{\nu_{i}}$ nilpotent operators. Define
$P_{\mathcal{E}}:=\sum_{|\nu_{i}|=1}P_{\nu_{i}}$, the projector onto the
peripheral spectrum of $\mathcal{E}$, and
$\tilde{E}:=P_{\mathcal{E}}E=\sum_{|\nu_{i}|=1}(\nu_{i}P_{\nu_{i}}+N_{\nu_{i}})$.
Then we define the map
$\tilde{\mathcal{E}}:=\mathrm{vec^{-1}}(\tilde{E})=\mathrm{vec^{-1}}(P_{\mathcal{E}}\cdot\mathrm{vec}(\mathcal{E}))$.

For $\mathcal{E}$ and $\tilde{\mathcal{E}}$ as given in Definition 1 we have,

######  Lemma III.2 (Ref. Szehr  _et al._ , 2015).

If $\mathcal{E}$ is a quantum channel, then $\tilde{\mathcal{E}}$ is a quantum
channel.

###### Proof.

One may argue (see the proof of Proposition 3.3 in Ref. Fannes  _et al._ ,
1992, for example) that the nilpotents $N_{\nu_{i}}=0$ for the peripheral
eigenvalues $\nu_{i}$, satisfying $|\nu_{i}|=1$. Moreover, the peripheral
eigenvalues form a cyclic group under multiplication, hence there exists
$m\in\mathbb{N}$, such that $\tilde{E}^{m}=\tilde{E}$, and
$\tilde{\mathcal{E}}^{m}=\tilde{\mathcal{E}}$. Then for any $n\in\mathbb{N}$,
$\tilde{\mathcal{E}}^{n(m-1)+1}=\tilde{\mathcal{E}}$. As Lemma III.3 below
assures, the sequence
$\\{\mathcal{E}^{n}-\tilde{\mathcal{E}}^{n}\\}_{n\in\mathbb{N}}$ converges to
zero in 2-2 norm (and in diamond norm, since all the spaces are finite-
dimensional), hence the subsequence
$\\{\mathcal{E}^{n(m-1)+1}-\tilde{\mathcal{E}}^{n(m-1)+1}\\}_{n\in\mathbb{N}}$
converges to zero as well. Then
$\\{\mathcal{E}^{n(m-1)+1}\\}_{n\in\mathbb{N}}$ converges to
$\tilde{\mathcal{E}}$. For any $X\in\mathcal{B}(\mathcal{K})$,
$\mathrm{tr}(\mathcal{E}^{n(m-1)+1}(X))=\mathrm{tr}(X)$, hence
$\mathrm{tr}\tilde{\mathcal{E}}(X)=\mathrm{tr}(X)$, thus $\tilde{\mathcal{E}}$
is trace-preserving. To show that $\tilde{\mathcal{E}}$ is completely
positive, it suffices to prove that for any $X\geq 0$ in
$\mathcal{B}(\mathcal{K}\otimes\mathcal{K})$, the condition
$(\mathrm{id}\otimes\tilde{\mathcal{E}})(X)\geq 0$ is satisfied. Select $X\geq
0$ in $\mathcal{B}(\mathcal{K}\otimes\mathcal{K})$, and note that
$(\mathrm{id}\otimes\mathcal{E}^{n(m-1)+1})(X)\geq 0$. The sequence
$\\{(\mathrm{id}\otimes\mathcal{E}^{n(m-1)+1})(X)\\}_{n\in\mathbb{N}}$
converges to $(\mathrm{id}\otimes\tilde{\mathcal{E}})(X)$. Since the set of
positive semidefinite operators on a finite-dimensional Hilbert space is
closed, then
$(\mathrm{id}\otimes\tilde{\mathcal{E}})(X)=\lim_{n\rightarrow\infty}(\mathrm{id}\otimes\mathcal{E}^{n(m-1)+1})(X)\geq
0$, completing the proof. ∎

The following lemma gathers results on convergence from Refs. Fannes  _et al._
, 1992; Szehr  _et al._ , 2015. For the sake of completeness, we provide its
proof in our notation in Appendix E.

######  Lemma III.3 (Theorem III.2 of Ref. Szehr  _et al._ , 2015; Ref. Fannes
_et al._ , 1992).

Let $\mathcal{E}:\mathcal{B}(\mathcal{K})\rightarrow\mathcal{B}(\mathcal{K})$
be a map with the spectral radius $1$, and let $\tilde{\mathcal{E}}$ be the
map obtained from $\mathcal{E}$ as described in Definition 1. Then for any
$\nu\in\mathbb{R}$, such that $\nu_{\mathrm{gap}}<\nu<1$, with
$\nu_{\mathrm{gap}}$ defined in Theorem II.1, there exists the constant $c>0$,
depending on $\nu$, such that

$\|\mathcal{E}^{n}-\tilde{\mathcal{E}}^{n}\|_{2\rightarrow 2}\leq c\nu^{n}.$

###  III.3 Finitely correlated states

FCS are a special class of translationally invariant quantum states on a chain
of identical finite dimensional quantum systems (spins). The structure of FCS
was characterized in Ref. Fannes  _et al._ , 1992, and is summarized here in
the form convenient for our purposes. We refer the reader to the original
papers Refs. Fannes  _et al._ , 1992, 1994 for the unabridged treatment of
FCS.

A FCS can be described by the pair of a full-rank density operator
$\sigma\in\mathcal{B}(\mathcal{H}_{M})$ and a quantum channel
$\Phi:\mathcal{B}(\mathcal{H}_{M})\rightarrow\mathcal{B}(\mathcal{H}_{s})\otimes\mathcal{B}(\mathcal{H}_{M})$,
satisfying a compatibility condition $\mathrm{tr}_{s}\Phi(\sigma)=\sigma$.
Fannes  _et al._ (1992) Here $\mathcal{H}_{M}$ and $\mathcal{H}_{s}$ are
Hilbert spaces with dimensions $d_{M}$ and $d_{s}$, respectively. The space
$\mathcal{H}_{M}$ is referred to as the _memory space_ , and $\mathcal{H}_{s}$
is the Hilbert space of a single spin. The FCS reduced density operator for a
continuous region $R$ with $|R|$ spins, is generated by $\Phi$ and $\sigma$ as

$\rho_{R}=\mbox{tr}_{M}(\Phi^{|R|}(\sigma)).$ (8)

Here $\mathrm{tr}_{M}$ is a partial trace over the Hilbert space
$\mathcal{H}_{M}$; it should not be confused with the trace over the spins in
the chain outside the region $R$. Colloquially, each $\Phi$ generates a single
spin, and a composition of $\Phi$ generates consecutive spins in the chain.
Note that the pair of $\Phi$ and $\sigma$ generating $\rho_{R}$ is not
necessarily unique, and using the results of Refs. Fannes  _et al._ , 1992,
1994 we may choose the most convenient representation for our purposes. In
this subsection we list all the relevant properties of the representation we
choose.

For the FCS $(\Phi,\sigma)$, we define the induced quantum channel
$\mathcal{E}:\mathcal{B}(\mathcal{H}_{M})\rightarrow\mathcal{B}(\mathcal{H}_{M})$
by

$\mathcal{E}(X)=\mathrm{tr}_{s}\Phi(X),$ (9)

for which $\sigma$ is a fixed point, $\mathcal{E}(\sigma)=\sigma$.

####  III.3.1 Ergodic FCS

We consider a subcollection of FCS, called _ergodic_ FCS. By definition, an
ergodic state is extremal in the convex set of translationally invariant
states. However, in the context of this paper, two other equivalent
definitions (see Proposition 3.1 of Ref. Fannes  _et al._ , 1992 and Lemma 4.1
of Ref. Evans and Høegh-Krohn, 1978) will be more useful: (i) the state for
which the eigenvalue $1$ of $\mathcal{E}$ is non-degenerate; (ii) the state
for which $\mathcal{E}$ is _irreducible_ in the sense that there does not
exist a non-trivial projector $\tilde{\Pi}$, such that
$\tilde{\Pi}\mathcal{B}(\mathcal{H}_{M})\tilde{\Pi}$ is invariant under
$\mathcal{E}$. The importance of ergodic states lies in the fact that any FCS
can be decomposed into a convex sum of ergodic FCS, which are also referred to
as _ergodic components_ (Corollary 3.2 of Ref. Fannes  _et al._ , 1992). We
collect together various observations that lead to this conclusion in the
proposition below. The item 1 follows from the Theorem 3.1 of Ref. Evans and
Høegh-Krohn, 1978 applied to $\mathcal{E}$ and $\sigma$. The items 2 and 3 are
obtained using the same reasoning as in the proof of Proposition 3.3 of Ref.
Fannes  _et al._ , 1992. The item 4 follows from Lemma 4.1 of Ref. Evans and
Høegh-Krohn, 1978 and item 2.

######  Proposition 1.

Without loss of generality, we may assume that a FCS is generated by a pair
$(\Phi,\sigma)$, with the induced quantum channel $\mathcal{E}$, for which the
following properties hold

  1. 1.

There exist $J$ orthogonal projectors
$\tilde{\Pi}_{\alpha}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{M}$,
$\sum_{\alpha=1}^{J}\tilde{\Pi}_{\alpha}=\mathbb{1}_{M}$, such that
$\mathcal{E}(\tilde{\Pi}_{\alpha}\mathcal{B}(\mathcal{H}_{M})\tilde{\Pi}_{\alpha})\subseteq\tilde{\Pi}_{\alpha}\mathcal{B}(\mathcal{H}_{M})\tilde{\Pi}_{\alpha}$.
The restriction of $\mathcal{E}$ to
$\tilde{\Pi}_{\alpha}\mathcal{B}(\mathcal{H}_{M})\tilde{\Pi}_{\alpha}$ is
irreducible.

  2. 2.

The density operator $\sigma$ is block-diagonal with respect to the
decomposition
$\mathcal{H}_{M}=\bigoplus_{\alpha=1}^{J}\tilde{\Pi}_{\alpha}\mathcal{H}_{M}$,
i.e.,
$\sigma=\bigoplus_{\alpha=1}^{J}\tilde{\Pi}_{\alpha}\sigma\tilde{\Pi}_{\alpha}$.

  3. 3.

$\Phi(\tilde{\Pi}_{\alpha}\mathcal{B}(\mathcal{H}_{M})\tilde{\Pi}_{\alpha})\subseteq\mathcal{B}(\mathcal{H}_{s})\otimes\tilde{\Pi}_{\alpha}\mathcal{B}(\mathcal{H}_{M})\tilde{\Pi}_{\alpha}$.

  4. 4.

The restriction of $\mathcal{E}$ to
$\tilde{\Pi}_{\alpha}\mathcal{B}(\mathcal{H}_{M})\tilde{\Pi}_{\alpha}$ has a
non-degenerate eigenvalue equal to 1, corresponding to the fixed point
$\tilde{\Pi}_{\alpha}\sigma\tilde{\Pi}_{\alpha}$.

The decomposition of FCS into ergodic components follows from items 2 and 3 of
Proposition 1,

$\rho_{R}=\sum_{\alpha=1}^{J}\lambda_{\alpha}\rho^{\alpha}_{R},$ (10)

where $\rho^{\alpha}_{R}=\mathrm{tr}_{M}\Phi^{|R|}(\sigma_{\alpha})$,
$\sigma_{\alpha}=\tilde{\Pi}_{\alpha}\sigma\tilde{\Pi}_{\alpha}/\mathrm{tr}(\tilde{\Pi}_{\alpha}\sigma\tilde{\Pi}_{\alpha})$,
and
$\lambda_{\alpha}=\mathrm{tr}(\tilde{\Pi}_{\alpha}\sigma\tilde{\Pi}_{\alpha})$.
We observe that each ergodic component is manifestly a FCS. One can think of
ergodic components being, in the sense of items 2 and 3, independent of each
other.

The structure of ergodic components can be analyzed further. We summarize
their properties in the proposition below, which is a restatement of
Proposition 3.3 of Ref. Fannes  _et al._ , 1992 in the language of density
operators

######  Proposition 2 (Proposition 3.3, Ref. Fannes  _et al._ , 1992).

For an ergodic FCS, generated by $(\Phi,\sigma)$, with the induced quantum
channel $\mathcal{E}$, we may assume that the following properties hold

  1. 1.

The peripheral spectrum of $\mathcal{E}$ consists of $p\in\mathbb{N}$ non-
degenerate eigenvalues $\\{\exp(\frac{2\pi
i}{p}k)\\}\;|\;k=0,\cdot\cdot\cdot,{p-1}\\}$. Here $p$ is referred to as the
_period_ of the state.

  2. 2.

There exist $p$ orthogonal projectors
$\Pi_{k}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{M}$,
$\sum_{k=0}^{p-1}\Pi_{k}=\mathbb{1}_{M}$, such that
$\mathcal{E}(\Pi_{k}\mathcal{B}(\mathcal{H}_{M})\Pi_{k})\subseteq\Pi_{k+1}\mathcal{B}(\mathcal{H}_{M})\Pi_{k+1}$
with the convention $\Pi_{p}:=\Pi_{0}$, which leads to
$\mathcal{E}^{p}(\Pi_{k}\mathcal{B}(\mathcal{H}_{M})\Pi_{k})\subseteq\Pi_{k}\mathcal{B}(\mathcal{H}_{M})\Pi_{k}$.

  3. 3.

The density operator $\sigma$ is block-diagonal with respect to the
decomposition $\mathcal{H}_{M}=\bigoplus_{k=0}^{p-1}\Pi_{k}\mathcal{H}_{M}$,
i.e., $\sigma=\bigoplus_{k=0}^{p-1}\Pi_{k}\sigma\Pi_{k}$. Moreover
$\mathcal{E}(\Pi_{k}\sigma\Pi_{k})=\Pi_{k+1}\sigma\Pi_{k+1}$, which leads to
$\mathrm{tr}(\Pi_{k}\sigma\Pi_{k})={1}/{p}$ for any $k$.

  4. 4.

The restriction of $\mathcal{E}^{p}$ to
$\Pi_{k}\mathcal{B}(\mathcal{H}_{M})\Pi_{k}$ has a trivial peripheral spectrum
with $\Pi_{k}\sigma\Pi_{k}$ the fixed point.

We observe similarities between the statements of Proposition 1 and
Proposition 2. In particular, item 1 of Proposition 1 implies that the algebra
of operators on the memory space, $\mathcal{B}(\mathcal{H}_{M})$, contains $J$
orthogonal subalgebras invariant under $\mathcal{E}$. By item 2 of Proposition
2, each of these subalgebras is supported on the memory space
$\tilde{\Pi}_{\alpha}\mathcal{H}_{M}$ of a corresponding ergodic component,
and further contains $p$ orthogonal subalgebras that are cyclically permuted
by $\mathcal{E}$.

####  III.3.2 Purely Generated FCS

A subcollection of FCS, called _purely generated FCS_ , is characterized by a
pure quantum channel $\Phi,$ of the form $\Phi(X)=VXV^{\dagger}$, where
$V:\mathcal{H}_{M}\rightarrow\mathcal{H}_{s}\otimes\mathcal{H}_{M}$ is an
isometry.Fannes  _et al._ (1992, 1994) We will say that pgFCS is induced, or
generated, by the pair $(V,\sigma)$. Equivalently these states are
characterized by vanishing mean entropy, i.e., for a continuous region $R$ of
$|R|$ spins, $\lim_{|R|\rightarrow+\infty}S(\rho_{R})/|R|=0$. Fannes  _et al._
(1994)

In the language of MPS, a pgFCS generated by $(V,\sigma)$ corresponds to the
reduced density operator of the properly defined translationally invariant
limit of the MPS $|\Psi\rangle$ as $n\rightarrow+\infty$Perez-Garcia  _et al._
(2007) (see Appendix A for the diagrammatic illustration),

$\displaystyle|\Psi\rangle=\sum_{s_{-n},\dots,s_{n}=1}^{d_{s}}$
$\displaystyle\left(L\right|M^{s_{-n}}M^{s_{-n+1}}\cdot\cdot\cdot
M^{s_{n}}\left|R\right)$
$\displaystyle\times|s_{-n}\rangle\otimes|s_{-n+1}\rangle\otimes\cdot\cdot\cdot\otimes|s_{n}\rangle,$

where $M^{s}$ is a $d_{M}\times d_{M}$ complex matrix, defined by its
elements,

$M^{s}_{ij}:=(\langle s|\otimes\langle i|)V|j\rangle,$

and $|L),|R)\in\mathbb{C}^{d_{M}}$ are boundary factors. In the limit of the
infinite chain $\mathrm{tr}_{M}$ and $\sigma$ play a similar role as the left
and right boundary factors $|L)$ and $|R)$, respectively.

For the case of pgFCS the induced channel $\mathcal{E}$ defined in (9) becomes

$\mathcal{E}(X)=\mbox{tr}_{s}\left(VXV^{\dagger}\right).$ (11)

To formulate the study of recoverability and QCMI we subdivide a finite region
$R$ into continuous adjacent subregions $A$, $B$, and $C$, as discussed in the
introduction. A pgFCS defined on this region may be expressed as

$\rho_{ABC}=\mbox{tr}_{M}\left(V_{C}V_{B}V_{A}\sigma
V_{A}^{\dagger}V_{B}^{\dagger}V_{C}^{\dagger}\right).$ (12)

The isometry
$V_{A}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{A}\otimes\mathcal{H}_{M}$,
generating all spins in the region $A$, is the $|A|$-fold product of $V$,
$V_{A}:=VV\cdot\cdot\cdot V$, in the sense of (5). The isometries $V_{B}$ and
$V_{C}$ generate all spins in the regions $B$ and $C$, respectively, and are
defined in an analogous manner. The products of these isometries are also
defined by (5).

Note that while ergodic FCS and pgFCS form distinct subsets of FCS, their
intersection, the ergodic pgFCS, is non-empty and plays an important role in
this study. A pgFCS can be decomposed into a convex sum (10) of ergodic pgFCS
$\rho^{\alpha}_{ABC}$, as shown in Lemma III.4 below,Fannes  _et al._ (1992,
1994)

$\rho_{ABC}=\sum_{\alpha=1}^{J}\lambda_{\alpha}\rho^{\alpha}_{ABC}.$ (13)

######  Lemma III.4.

A FCS is a pgFCS if and only if all its ergodic components are pgFCS.

###### Proof.

As was shown in Ref. Fannes  _et al._ , 1994, a state is a pgFCS if and only
if it has vanishing mean entropy. Note the well-known bounds on the convexity
of the quantum entropy,

$\sum_{\alpha=1}^{J}\lambda_{\alpha}S(\rho^{\alpha}_{ABC})\leq
S(\rho_{ABC})\leq\sum_{\alpha=1}^{J}\lambda_{\alpha}S(\rho^{\alpha}_{ABC})-\sum_{\alpha=1}^{J}\lambda_{\alpha}\ln\lambda_{\alpha}.$

Since

$\lim_{|ABC|\rightarrow+\infty}\sum_{\alpha=1}^{J}\lambda_{\alpha}\ln\lambda_{\alpha}/|ABC|=0,$

then

$\lim_{|ABC|\rightarrow+\infty}S(\rho_{ABC})/|ABC|=\sum_{\alpha=1}^{J}\lambda_{\alpha}\lim_{|ABC|\rightarrow+\infty}S(\rho^{\alpha}_{ABC})/|ABC|.$

Hence $\lim_{|ABC|\rightarrow+\infty}S(\rho_{ABC})/|ABC|=0$ if and only if
$\lim_{|ABC|\rightarrow+\infty}S(\rho^{\alpha}_{ABC})/|ABC|=0$ for each
$\alpha$, which implies that every ergodic component is a pgFCS. ∎

Since each ergodic component is a pgFCS, we may express $\rho_{ABC}^{\alpha}$
in the form

$\rho^{\alpha}_{ABC}=\mbox{tr}_{M_{\alpha}}\left(V_{C}^{\alpha}V_{B}^{\alpha}V_{A}^{\alpha}\sigma_{\alpha}V_{A}^{\alpha\dagger}V_{B}^{\alpha\dagger}V_{C}^{\alpha\dagger}\right),$
(14)

where all the objects are defined as in (12), and each $\sigma_{\alpha}$ is
defined on a distinct memory space $\mathcal{H}_{M_{\alpha}}$,
$\alpha=1,\cdot\cdot\cdot,J$. Naturally, we denote the elementary isometry
inducing the ergodic component by $V_{\alpha}$, i.e., $V_{A}^{\alpha}$ is the
$|A|$-fold product $V_{A}^{\alpha}:=V_{\alpha}V_{\alpha}\cdot\cdot\cdot
V_{\alpha}$ in the sense of (5). Since $\rho^{\alpha}_{ABC}$ is ergodic, the
corresponding quantum channel
$\mathcal{E}_{\alpha}:\mathcal{B}(\mathcal{H}_{M_{\alpha}})\rightarrow\mathcal{B}(\mathcal{H}_{M_{\alpha}})$,
defined as in (11),
$\mathcal{E}_{\alpha}(X)=\mbox{tr}_{s}(V_{\alpha}XV_{\alpha}^{\dagger})$, and
$\sigma_{\alpha}$ have the properties listed in Proposition 2. Now we rewrite
Proposition 1 for the case of pgFCS, making a connection between $(V,\sigma)$
generating the pgFCS $\rho_{ABC}$ and the collection of
$(V_{\alpha},\sigma_{\alpha})$, $\alpha=1,\cdot\cdot\cdot,J$, generating its
ergodic pgFCS components $\rho^{\alpha}_{ABC}$. We add an additional property,
item 5, whose interpretation is that ergodic components are not proportional
to each other; in Appendix G we show that we may assume, without loss of
generality, that this property holds.

######  Proposition 3 (Ref. Fannes  _et al._ , 1992).

Without loss of generality, we may assume that a pgFCS is generated by a pair
$(V,\sigma)$, with the induced quantum channel $\mathcal{E}$, and the
following properties hold

  1. 1.

$\mathcal{H}_{M}=\bigoplus_{\alpha=1}^{J}\mathcal{H}_{M_{\alpha}}$, where $J$
is the number of ergodic components in $\rho_{ABC}$, and
$\mathcal{H}_{M_{\alpha}}:=\tilde{\Pi}_{\alpha}\mathcal{H}_{M}$.

  2. 2.

$V=\sum_{\alpha=1}^{J}V_{\alpha}\tilde{\Pi}_{\alpha}$, where
$\tilde{\Pi}_{\alpha}$, ${\alpha=1,\cdot\cdot\cdot,J}$ is the orthogonal
projector onto $\mathcal{H}_{M_{\alpha}}$, and
$V_{\alpha}:\mathcal{H}_{M_{\alpha}}\rightarrow\mathcal{H}_{s}\otimes\mathcal{H}_{M_{\alpha}}$
is an isometry.

  3. 3.

$\sigma=\bigoplus_{\alpha=1}^{J}\lambda_{\alpha}\sigma_{\alpha}$, where
$\sigma_{\alpha}={\tilde{\Pi}_{\alpha}\sigma\tilde{\Pi}_{\alpha}}/{\mathrm{tr}(\tilde{\Pi}_{\alpha}\sigma\tilde{\Pi}_{\alpha})}$,
$\lambda_{\alpha}=\mathrm{tr}(\tilde{\Pi}_{\alpha}\sigma\tilde{\Pi}_{\alpha})$.

  4. 4.

$(V_{\alpha},\sigma_{\alpha})$ generates an ergodic pgFCS in the form (14),
with the induced quantum channel
$\mathcal{E}_{\alpha}(X)\equiv\mathcal{E}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\alpha})=\mbox{tr}_{s}(V_{\alpha}XV^{\dagger}_{\alpha})$,
for which Proposition 2 applies.

  5. 5.

For $\alpha\neq\beta$ there is no unitary
$U:\mathcal{H}_{M}\rightarrow\mathcal{H}_{M}$ and $\phi\in\mathbb{R}$, such
that $V_{\alpha}=e^{i\phi}UV_{\beta}U^{\dagger}$.

Remarks:

  1. 1.

Proposition 3 contains all the pgFCS properties that we use in this paper.

  2. 2.

The decomposition (13) into ergodic components is not necessarily the finest
one that can be performed on a pgFCS – each ergodic component may be further
decomposed into a sum of ergodic pgFCS components of period $1$. Fannes  _et
al._ (1992) The resulting components may, however, involve adjustment of the
memory Hilbert space, and blocking of spins – the procedure under which
several consecutive spins are treated as one. We prefer to deal directly with
the decomposition of a pgFCS into ergodic pgFCS components of arbitrary
period. In Section IV it will be convenient to consider an ergodic pgFCS of
period $1$ first, in order to establish the arguments in our proof.

###  III.4 Quantum Markov chains

Quantum Markov chains are defined as those states for which QCMI vanishes,
$I(A:C|B):=S(\rho_{AB})+S(\rho_{BC})-S(\rho_{ABC})-S(\rho_{B})=0$, Hayden  _et
al._ (2004) and are fully characterized by the following theorem,

######  Theorem III.5 (Ref. Hayden  _et al._ , 2004).

Let
$\rho_{ABC}\in\mathcal{D}(\mathcal{K}_{A}\otimes\mathcal{K}_{B}\otimes\mathcal{K}_{C})$.
The following three statements are equivalent:

  1. 1.

$\rho_{ABC}$ is a quantum Markov chain, _i.e._ , $I(A:C|B)=0$.

  2. 2.

There exists a quantum channel $\mathcal{R}_{B\rightarrow
BC}:\mathcal{B}(\mathcal{K}_{B})\rightarrow\mathcal{B}(\mathcal{K}_{B}\otimes\mathcal{K}_{C})$,
such that $\rho_{ABC}=\mathcal{R}_{B\rightarrow BC}(\rho_{AB})$. On
$\mathcal{B}(\mathrm{supp}(\rho_{B}))$ this channel can be taken to be the
_Petz recovery map_ $\mathcal{P}_{B\rightarrow
BC}(X):=\rho_{BC}^{\frac{1}{2}}\rho_{B}^{-\frac{1}{2}}X\rho_{B}^{-\frac{1}{2}}\rho_{BC}^{\frac{1}{2}},$
although this choice is not unique.

  3. 3.

There is a decomposition
$\mathrm{supp}(\rho_{B})\cong\bigoplus_{k=1}^{k_{\mathrm{max}}}\mathcal{K}_{b^{l}_{k}}\otimes\mathcal{K}_{b^{r}_{k}}$,
_i.e._ , there exists a unitary isomorphism
$I:\mathrm{supp}(\rho_{B})\rightarrow\bigoplus_{k=1}^{k_{\mathrm{max}}}\mathcal{K}_{b^{l}_{k}}\otimes\mathcal{K}_{b^{r}_{k}}$,
such that
$I\rho_{ABC}I^{\dagger}=\bigoplus_{k=1}^{k_{\mathrm{max}}}\lambda_{k}\rho_{Ab^{l}_{k}}\otimes\rho_{b^{r}_{k}C}$,
where
$\rho_{Ab^{l}_{k}}\in\mathcal{D}(\mathcal{K}_{A}\otimes\mathcal{K}_{b^{l}_{k}})$,
$\rho_{b^{r}_{k}C}\in\mathcal{D}(\mathcal{K}_{b^{r}_{k}}\otimes\mathcal{K}_{C})$,
$\lambda_{k}>0$ and $\sum_{k=1}^{k_{\mathrm{max}}}\lambda_{k}=1$.

For a pgFCS $\rho_{ABC}$ we will construct an approximating state
$\tilde{\rho}_{ABC}$, and show that it is a quantum Markov chain by proving
that it satisfies the property $3$, whence properties $1$ and $2$ as well.

The theorem has a useful corollary, whose proof is omitted, which we will
employ in the next section,

######  Corollary III.5.1.

If $\mathcal{K}_{B}=\mathcal{K}_{B_{1}}\oplus\mathcal{K}_{B_{2}},$ and
$\rho_{AB_{1}C}\in\mathcal{D}(\mathcal{K}_{A}\otimes\mathcal{K}_{B_{1}}\otimes\mathcal{K}_{C})$
and
$\rho_{AB_{2}C}\in\mathcal{D}(\mathcal{K}_{A}\otimes\mathcal{K}_{B_{2}}\otimes\mathcal{K}_{C})$
are quantum Markov chains, then
$\lambda\rho_{AB_{1}C}+(1-\lambda)\rho_{AB_{2}C}$, where $0\leq\lambda\leq 1$,
is a quantum Markov chain on
$\mathcal{D}(\mathcal{K}_{A}\otimes\mathcal{K}_{B}\otimes\mathcal{K}_{C})$.

##  IV Proof of theorem II.1

In this section we prove the main result, Theorem II.1. The proof relies on
two steps: (i) approximating pgFCS $\rho_{ABC}$ by another state
$\tilde{\rho}_{ABC}$ and (ii) the proof that $\tilde{\rho}_{ABC}$ is a quantum
Markov chain. Conveniently, the approximation step is the same for any pgFCS,
and we outline it immediately below. The proof that the constructed
$\tilde{\rho}_{ABC}$ is a quantum Markov chain, proceeds in several steps,
which we develop in subsequent subsections. We first consider the simplest
case of an ergodic pgFCS of period $1$, before generalizing to arbitrary
period $p.$ Finally, by taking convex sums, we deal with a general case of
pgFCS.

We now deal with the approximation step, assuming that $\tilde{\rho}_{ABC}$ is
a quantum Markov chain. This is sufficient to prove the Theorem II.1.

Any pgFCS has the form (12),

$\rho_{ABC}=\mbox{tr}_{M}\left(V_{C}V_{B}V_{A}\sigma
V_{A}^{\dagger}V_{B}^{\dagger}V_{C}^{\dagger}\right).$

We introduce an isometry
$\tilde{V}_{B}:\mathcal{H}_{A}\otimes\mathcal{H}_{M}\rightarrow\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{M}$
which acts trivially on the space $\mathcal{H}_{A}$, and use it to construct
the state

$\tilde{\rho}_{ABC}=\mbox{tr}_{M}\left(V_{C}\tilde{V}_{B}V_{A}\sigma
V_{A}^{\dagger}\tilde{V}_{B}^{\dagger}V_{C}^{\dagger}\right).$ (15)

Note that the isometry $\tilde{V}_{B}$ generates all the spins in the region
$B$, but unlike $V_{B}$, it is not a concatenation of isometric factors
generating one spin at a time. It is a key step to choose $\tilde{V}_{B}$ to
be of a particular form that ensures the state $\tilde{\rho}_{ABC}$ is
simultaneously close (in trace norm distance) to $\rho_{ABC}$ and is an exact
quantum Markov chain.

The trace distance $\|\rho_{ABC}-\tilde{\rho}_{ABC}\|_{1}$ is bounded by twice
the error in approximating $V_{B}$ by $\tilde{V}_{B}$ in operator norm,

$\displaystyle\|\rho_{ABC}-\tilde{\rho}_{ABC}\|_{1}$
$\displaystyle=\|\mbox{tr}_{M}\left(V_{C}V_{B}V_{A}\sigma
V_{A}^{\dagger}V_{B}^{\dagger}V_{C}^{\dagger}\right)$ (16)
$\displaystyle\quad-\mbox{tr}_{M}\left(V_{C}\tilde{V}_{B}V_{A}\sigma
V_{A}^{\dagger}\tilde{V}_{B}^{\dagger}V_{C}^{\dagger}\right)\|_{1}$
$\displaystyle\leq\|V_{B}V_{A}\sigma
V_{A}^{\dagger}V_{B}^{\dagger}-\tilde{V}_{B}V_{A}\sigma
V_{A}^{\dagger}\tilde{V}_{B}^{\dagger}\|_{1}$ $\displaystyle\leq
2\|V_{B}-\tilde{V}_{B}\|,$

where in the last line we have used the inequalities
$\|XY\|_{1}\leq\|X\|\|Y\|_{1}$, $\|XZX^{\dagger}-YZY^{\dagger}\|_{1}\leq
2\|X-Y\|\|Z\|_{1}$, and the fact that $\|V_{A}\sigma V_{A}^{\dagger}\|_{1}=1$.

For $\tilde{\rho}_{ABC}$ a quantum Markov chain, according to condition 2 of
Theorem III.5 there exists a recovery channel $\mathcal{R}_{B\rightarrow BC}$,
such that $\mathcal{R}_{B\rightarrow
BC}(\tilde{\rho}_{AB})=\tilde{\rho}_{ABC}$. We may select
$\mathcal{R}_{B\rightarrow BC}$ to be the Petz recovery map, defined in
statement 2 of Theorem III.5, or one of the alternatives that we discuss in
Section V. If we use such an $\mathcal{R}_{B\rightarrow BC}$ to approximately
recover $\rho_{ABC}$ from $\rho_{AB}$, then the recovery error is given by

$\displaystyle\|\rho_{ABC}-\mathcal{R}_{B\rightarrow
BC}\left(\rho_{AB}\right)\|_{1}=\|\rho_{ABC}-\mathcal{R}_{B\rightarrow
BC}\circ\mathrm{tr}_{C}\left(\rho_{ABC}\right)\|_{1}$
$\displaystyle=\|\rho_{ABC}-\tilde{\rho}_{ABC}+\mathcal{R}_{B\rightarrow
BC}\circ\mathrm{tr}_{C}\left(\tilde{\rho}_{ABC}-\rho_{ABC}\right)\|_{1}$
$\displaystyle\leq\|\rho_{ABC}-\tilde{\rho}_{ABC}\|_{1}+\|\mathcal{R}_{B\rightarrow
BC}\circ\mathrm{tr}_{C}\left(\rho_{ABC}-\tilde{\rho}_{ABC}\right)\|_{1}$
$\displaystyle\leq 2\|\rho_{ABC}-\tilde{\rho}_{ABC}\|_{1}$ (17)

where we have used the quantum Markov property, the triangle inequality, and
the property that the trace norm is non-increasing under actions of quantum
channels.

Combining (16) and (IV), we obtain a bound on the recovery error,

$\|\rho_{ABC}-\mathcal{R}_{B\rightarrow BC}\left(\rho_{AB}\right)\|_{1}\leq
4\|V_{B}-\tilde{V}_{B}\|.$ (18)

Our goal is to choose the isometry $\tilde{V}_{B}$ so that
$\|V_{B}-\tilde{V}_{B}\|$ is exponentially small in the size of the region
$B$. Consider the $|B|$-fold composition of $\mathcal{E}$, defined in (11),
$\mathcal{E}^{|B|}$. Since $\mathcal{E}$ is dilated by the elementary isometry
$V$, then the $|B|$-fold composition of $V$, that is $V_{B}$, dilates
$\mathcal{E}^{|B|}$. From $\mathcal{E}$ we construct the quantum channel
$\tilde{\mathcal{E}}$ via Definition 1, and then approximate
$\mathcal{E}^{|B|}$ by $\tilde{\mathcal{E}}^{|B|}$, the $|B|$-fold composition
of $\tilde{\mathcal{E}}$. As discussed in Section III.2, the maximum dimension
of the dilating space of $\tilde{\mathcal{E}}^{|B|}$ is equal to
$\dim\mathcal{B}(\mathcal{H}_{M})=d_{M}^{2}$. We will assume that
$\dim\mathcal{H}_{B}\geq d_{M}^{2}$, so that the dilation space of
$\tilde{\mathcal{E}}^{|B|}$ can be embedded into $\mathcal{H}_{B}$. Under this
condition, we define $\tilde{V}_{B}$ to be an isometry dilating
$\tilde{\mathcal{E}}^{|B|}$. Note that any isometry of the form
$U_{B}\tilde{V}_{B}$, where $U_{B}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{B}$
is unitary, will also be a dilating isometry for $\tilde{\mathcal{E}}^{|B|}$.
We will use this freedom of choice to minimize $\|V_{B}-\tilde{V}_{B}\|$.

By the continuity of Stinespring’s dilation, Kretschmann  _et al._ (2008) we
can relate the operator norm distance between $V_{B}$ and $\tilde{V}_{B}$ to
the distance between channels $\mathcal{E}^{|B|}$ and
$\mathcal{\tilde{E}}^{|B|}$ through the inequalities

$\inf_{U_{B}}\|V_{B}-U_{B}\tilde{V}_{B}\|^{2}\leq\|\mathcal{E}^{|B|}-\tilde{\mathcal{E}}^{|B|}\|_{\diamond}\leq
2\inf_{U_{B}}\|V_{B}-U_{B}\tilde{V}_{B}\|,$ (19)

where the infimum is taken over the set of all unitaries
$U_{B}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{B}.$ (Recall that the diamond
norm is defined by
$\|\mathcal{E}^{|B|}-\tilde{\mathcal{E}}^{|B|}\|_{\diamond}:=\sup_{n\in\mathbb{N}}\|\mbox{id}_{n}\otimes(\mathcal{E}^{|B|}-\tilde{\mathcal{E}}^{|B|})\|_{1\rightarrow
1}$.Kitaev  _et al._ (2002)) As the dimension of $\mathcal{H}_{B}$ is finite,
the infimum is attained for some $U_{B}$, which we take to be the identity,
$U_{B}=\mathbb{1}_{B}$, so that our choice of $\tilde{V}_{B}$ is optimal.
Using the first inequality in (19), we bound the distance between the
isometries as

$\|V_{B}-\tilde{V}_{B}\|\leq\sqrt{\|\mathcal{E}^{|B|}-\tilde{\mathcal{E}}^{|B|}\|_{\diamond}}.$
(20)

To bound the diamond norm
$\|\mathcal{E}^{|B|}-\tilde{\mathcal{E}}^{|B|}\|_{\diamond}$ we take advantage
of the $2\rightarrow 2$ norm, which is easier to estimate. (The $2\rightarrow
2$ norm is defined by $\|\mathcal{E}\|_{2\rightarrow
2}:=\sup_{\|X\|_{2}=1}\|\mathcal{E}(X)\|_{2}$, where the 2-norm, or Hilbert-
Schmidt norm, is given by $\|X\|_{2}:=\sqrt{\mbox{tr}(X^{\dagger}X)}$.) Since
the Hilbert space $\mathcal{H}_{M}$ is finite dimensional, the supremum
$\sup_{n\in\mathbb{N}}\|\mbox{id}_{n}\otimes(\mathcal{E}^{|B|}-\tilde{\mathcal{E}}^{|B|})\|_{1\rightarrow
1}$ is achieved when $n=d_{M}$,Kitaev  _et al._ (2002) hence
$\|\mathcal{E}^{|B|}-\tilde{\mathcal{E}}^{|B|}\|_{\diamond}=\|\mbox{id}_{d_{M}}\otimes(\mathcal{E}^{|B|}-\tilde{\mathcal{E}}^{|B|})\|_{1\rightarrow
1}$. Then, using the relations $\|X\|_{1}\leq\mathrm{rank}(X)^{1/2}\|X\|_{2}$
and
$\|\mbox{id}_{n}\otimes(\mathcal{E}^{|B|}-\tilde{\mathcal{E}}^{|B|})\|_{2\rightarrow
2}=\|\mathcal{E}^{|B|}-\tilde{\mathcal{E}}^{|B|}\|_{2\rightarrow 2}$
(tensoring with the identity map does not change the $2\rightarrow 2$ norm),
and bounding the rank with the space dimension $\mathrm{rank}(X)\leq
d_{M}^{2}$, we obtain the well-known estimate

$\|\mathcal{E}^{|B|}-\tilde{\mathcal{E}}^{|B|}\|_{\diamond}\leq
d_{M}\|\mathcal{E}^{|B|}-\tilde{\mathcal{E}}^{|B|}\|_{2\rightarrow 2}.$ (21)

From Lemma III.3 applied to the quantum channels $\mathcal{E}$ and
$\tilde{\mathcal{E}}$ it follows that,

$\|\mathcal{E}^{|B|}-\tilde{\mathcal{E}}^{|B|}\|_{2\rightarrow 2}\leq
c\nu^{|B|},$ (22)

with $c>0$ and $\nu_{\mathrm{gap}}<\nu<1$, where
$\nu_{\mathrm{gap}}:=\max_{|\nu_{i}|<1}\\{|\nu_{i}|\\}$, and $\nu_{i}$ are the
eigenvalues of $\mathcal{E}$. Combining together (20), (21), and (22), we
obtain

$\|V_{B}-\tilde{V}_{B}\|\leq\sqrt{d_{M}c}\;\nu^{|B|/2},$ (23)

and, from (18),

$\displaystyle\|\rho_{ABC}-\tilde{\rho}_{ABC}\|_{1}$ $\displaystyle\leq
2\sqrt{d_{M}c}\ \ \nu^{|B|/2}$ (24)
$\displaystyle\|\rho_{ABC}-\mathcal{R}_{B\rightarrow
BC}\left(\rho_{AB}\right)\|_{1}$ $\displaystyle\leq 4\sqrt{d_{M}c}\ \
\nu^{|B|/2}.$

Observe that statements $1$ and $2$ of Theorem II.1 hold with the constants
$K=4\sqrt{d_{M}c}$ and $q=\ln\left(\nu^{-1}\right)/2$. The bound (24) is
meaningful while $4\sqrt{d_{M}c}\nu^{|B|/2}\leq 2$, since the trace distance
between any two states cannot exceed $2$. The statement $3$ of the theorem
follows from the relation (36) in Appendix B with
$\tilde{K}=\sqrt{d_{M}c}\left(2|A|\ln
d_{s}+2-2\ln(2\sqrt{d_{M}c}\nu^{{|B|}/{2}})\right)$. Notice that
$\ln(2\sqrt{d_{M}c}\nu^{{|B|}/{2}})\leq 0$ when the bound (24) is useful. This
completes the proof.

It remains to prove that $\tilde{\rho}_{ABC}$ given in (15), and defined in
terms of the isometry $\tilde{V}_{B}$, is a quantum Markov chain.

###  IV.1 Ergodic pgFCS of period 1

We first prove that $\tilde{\rho}_{ABC}$ is a quantum Markov chain for an
ergodic pgFCS with $p=1.$ Since $\mathcal{E}$ has a trivial peripheral
spectrum and $\sigma$ is its fixed point, then $\tilde{\mathcal{E}}$
constructed according to Definition 1 sends any input into the state
proportional to $\sigma$, $\tilde{\mathcal{E}}(X)=\mathrm{tr}(X)\sigma$. It
follows that for any $n\in\mathbb{N}$,
$\tilde{\mathcal{E}}^{n}=\tilde{\mathcal{E}}$, in particular
$\tilde{\mathcal{E}}^{|B|}=\tilde{\mathcal{E}}$. We observe that the isometry
$\tilde{V}^{\prime}_{B}=\sum_{i,j=1}^{d_{M}}\sqrt{\sigma_{i}}|\xi_{ij}\rangle\otimes|i\rangle\langle
j|$ is a dilation of $\tilde{\mathcal{E}}^{|B|}=\tilde{\mathcal{E}}$, where we
have introduced an arbitrary orthonormal set of vectors
$\\{|\xi_{ij}\rangle\in\mathcal{H}_{B}\;|\;i,j=1\cdot\cdot\cdot d_{M}\\}$. The
vectors $|i\rangle\in\mathcal{H}_{M}$ and $\sigma_{i}>0$ are the eigenvectors
and corresponding eigenvalues of $\sigma$, respectively, i.e.,
$\sigma=\sum_{i=1}^{d_{M}}\sigma_{i}|i\rangle\langle i|$. Indeed,

$\displaystyle\mathrm{tr}_{B}(\tilde{V}^{\prime}_{B}X\tilde{V}^{\prime\dagger}_{B})$
$\displaystyle=\sum_{i,j,i^{\prime},j^{\prime}}\sqrt{\sigma_{i}\sigma_{i^{\prime}}}\langle\xi_{i^{\prime}j^{\prime}}|\xi_{ij}\rangle\langle
j|X|j^{\prime}\rangle\,|i\rangle\langle i^{\prime}|$
$\displaystyle=\sum_{j=1}^{d_{M}}\langle
j|X|j\rangle\,\sum_{i=1}^{d_{M}}\sigma_{i}|i\rangle\langle i|$
$\displaystyle=\mbox{tr}(X)\sigma.$

Since $\tilde{V}^{\prime}_{B}$ dilates $\tilde{\mathcal{E}}$, which is also
dilated by $\tilde{V}_{B}$, there is unitary
$U_{B}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{B}$, such that
$\tilde{V}_{B}=U_{B}\tilde{V}^{\prime}_{B}$, and we observe that this amounts
to a unitary change of the orthonormal basis $\\{|\xi_{ij}\rangle\\}$. Thus,
considering the freedom of choice of the basis vectors $|\xi_{ij}\rangle$, we
can assume without loss of generality that
$\tilde{V}_{B}=\tilde{V}^{\prime}_{B}$,

$\tilde{V}_{B}=\sum_{i,j=1}^{d_{M}}\sqrt{\sigma_{i}}|\xi_{ij}\rangle\otimes|i\rangle\langle
j|.$ (25)

Substituting (25) into (15), $\tilde{\rho}_{ABC}$ may be written explicitly,

$\displaystyle\tilde{\rho}_{ABC}=\sum_{i,i^{\prime},j,j^{\prime}}\langle
j|V_{A}\sigma V_{A}^{\dagger}|j^{\prime}\rangle$
$\displaystyle\otimes\sqrt{\sigma_{i}}|\xi_{ij}\rangle\langle\xi_{i^{\prime}j^{\prime}}|\sqrt{\sigma_{i^{\prime}}}$
(26) $\displaystyle\otimes\mbox{tr}_{M}\left(V_{C}|i\rangle\langle
i^{\prime}|V_{C}^{\dagger}\right).$

We will show by direct computation that $\tilde{\rho}_{ABC}$ satisfies the
condition 3 of Theorem III.5 and hence is a quantum Markov chain. We denote
$\mathcal{H}_{b}:=\mathrm{span}\\{|\xi_{ij}\rangle\;|\;i,j=1,\cdot\cdot\cdot,d_{M}\rangle\\}$,
and note that since their dimensions are equal, $\mathcal{H}_{b}$ and
$\mathcal{H}_{M}\otimes\mathcal{H}_{M}$ are isomorphic. We construct the
unitary isomorphism
$I:\mathcal{H}_{b}\rightarrow\mathcal{H}_{M}\otimes\mathcal{H}_{M}$, defined
by $I|\xi_{ij}\rangle=|j\rangle\otimes|i\rangle$. Under this isomorphism, a
straightforward calculation gives,

$I\tilde{\rho}_{ABC}I^{\dagger}=V_{A}\sigma
V^{\dagger}_{A}\otimes\mbox{tr}_{M}\left(V_{C}\sigma^{\frac{1}{2}}|+\rangle\langle+|\sigma^{\frac{1}{2}}V^{\dagger}_{C}\right),$

where
$|+\rangle=\sum_{i=1}^{d_{M}}|i\rangle\otimes|i\rangle\in\mathcal{H}_{M}\otimes\mathcal{H}_{M}$
is a non-normalized, maximally entangled state vector, and the partial trace
$\mathrm{tr}_{M}$ is over the second factor. If we write
$\mathcal{H}_{b}=\mathcal{H}_{b^{l}}\otimes\mathcal{H}_{b^{r}}$, where
$\mathcal{H}_{b^{l}}$ and $\mathcal{H}_{b^{r}}$ are isomorphic to
$\mathcal{H}_{M}$, we observe that $\tilde{\rho}_{ABC}$ satisfies condition 3
of Theorem III.5 with

$\displaystyle\mathrm{supp}(\rho_{B})\cong\mathcal{H}_{b}=\mathcal{H}_{b^{l}}\otimes\mathcal{H}_{b^{r}},$
$\displaystyle
I\tilde{\rho}_{ABC}I^{\dagger}=\tilde{\rho}_{Ab^{l}}\otimes\tilde{\rho}_{b^{r}C},$
$\displaystyle\tilde{\rho}_{Ab^{l}}:=V_{A}\sigma V^{\dagger}_{A},$
$\displaystyle\tilde{\rho}_{b^{r}C}:=\mbox{tr}_{M}(V_{C}\sigma^{\frac{1}{2}}|+\rangle\langle+|\sigma^{\frac{1}{2}}V^{\dagger}_{C}),$

and is thus a quantum Markov chain.

###  IV.2 Ergodic pgFCS of period $p$

For an ergodic pgFCS with $p\geq 1$, Proposition 2 states that the peripheral
spectrum of $\mathcal{E}$ contains $p$ eigenvalues, and thus for $p>1$ the
proof that $\tilde{\rho}_{ABC}$ is a quantum Markov chain must be modified.

The properties of $\mathcal{E}$ listed in Proposition 2 allow us to prove
Lemma IV.1 below, stating that (assuming $\dim\mathcal{H}_{B}\geq d_{M}^{2}$)
$\tilde{V}_{B}$ can be decomposed into
$\tilde{V}_{B}=\sum_{k=0}^{p-1}\tilde{V}^{k}_{B}$, where each
$\tilde{V}_{B}^{k}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{B}\otimes\mathcal{H}_{M}$
is a partial isometry with
$\mathrm{supp}(\tilde{V}_{B}^{k})=\Pi_{k}\mathcal{H}_{M}$, and
$\tilde{V}_{B}^{k\dagger}\tilde{V}_{B}^{k}=\Pi_{k}$. The range of
$\tilde{V}^{k}_{B}$ satisfies

$\mathrm{Range}(\tilde{V}_{B}^{k})\subseteq\mathcal{H}_{B_{k}}\otimes\Pi_{k+|B|}\mathcal{H}_{M}\subseteq\mathcal{H}_{B}\otimes\mathcal{H}_{M},$

where the subspaces $\mathcal{H}_{B_{k}}\subseteq\mathcal{H}_{B}$ are mutually
orthogonal. Below, each $\tilde{V}^{k}_{B}$ will be seen to have the form (28)
similar to (25). The operator $\sigma$ is block diagonal with respect to the
set of all projectors $\Pi_{k},$ $k=0,\cdot\cdot\cdot,p-1.$

Anticipating the validity of Lemma IV.1, below, we may express
$\tilde{\rho}_{ABC}$ in the form

$\displaystyle\tilde{\rho}_{ABC}=\mbox{tr}_{M}\left(V_{C}\left(\sum_{k_{1}=0}^{p-1}\tilde{V}^{k_{1}}_{B}\right)V_{A}\sigma
V_{A}^{\dagger}\left(\sum_{k_{2}=0}^{p-1}\tilde{V}^{k_{2}\dagger}_{B}\right)V_{C}^{\dagger}\right)$
$\displaystyle=\mbox{tr}_{M}\left(V_{C}\left(\sum_{k_{1}=0}^{p-1}\tilde{V}^{k_{1}}_{B}\right)\left(\sum_{k=0}^{p-1}\Pi_{k}V_{A}\sigma
V_{A}^{\dagger}\Pi_{k}\right)\left(\sum_{k_{2}=0}^{p-1}\tilde{V}^{k_{2}\dagger}_{B}\right)V_{C}^{\dagger}\right)$
(27)
$\displaystyle=\bigoplus_{k=0}^{p-1}\mbox{tr}_{M}\left(V_{C}\tilde{V}^{k}_{B}V_{A}\sigma
V_{A}^{\dagger}\tilde{V}_{B}^{k\dagger}V_{C}^{\dagger}\right),$

where in the first line we substitute the decomposition of $\tilde{V}_{B}$
into partial isometries and make the replacement $V_{A}\sigma
V^{\dagger}_{A}=\sum_{k=1}^{p}\Pi_{k}V_{A}\sigma V^{\dagger}_{A}\Pi_{k}$ (see
Appendix H for proof), and
$\tilde{V}^{k^{\prime}}_{B}\Pi_{k}=\delta_{kk^{\prime}}\tilde{V}^{k}_{B}$
collapses the expression to a single sum in the second line. Since the ranges
of $\tilde{V}^{k}_{B}$ are mutually orthogonal, the summands corresponding to
different $k$ are supported on orthogonal subspaces, so that the sum is
direct.

Each term in the direct sum can be expressed as in (26), and by the arguments
of Section IV.1 is shown to be (up to normalization) a quantum Markov chain.
Hence by Corollary III.5.1, $\tilde{\rho}_{ABC}$ is also a quantum Markov
chain. Thus we can use the bound for the recovery error (24), and the
statement of Theorem II.1 holds with the same constants $q$, $K$ and
$\tilde{K}$ as in Section IV.1.

We now prove that $\tilde{V}_{B}$ has the required decomposition.

######  Lemma IV.1.

For an ergodic pgFCS of period $p$, the isometry
$\tilde{V}_{B}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{B}\otimes\mathcal{H}_{M}$
defined in (15) can be constructed as the sum
$\tilde{V}_{B}=\sum_{k=0}^{p-1}\tilde{V}_{B}^{k}$, where

  1. 1.

$\tilde{V}_{B}^{k}$ is a partial isometry,
$\tilde{V}_{B}^{k\dagger}\tilde{V}_{B}^{k}=\Pi_{k}$, and
$\tilde{V}^{k}_{B}\Pi_{k^{\prime}}=\delta_{kk^{\prime}}\tilde{V}_{B}^{k}$.

  2. 2.

The range of $\tilde{V}_{B}^{k}$ satisfies
$\mathrm{Range}(\tilde{V}_{B}^{k})\subseteq\mathcal{H}_{B_{k}}\otimes\Pi_{k+|B|}\mathcal{H}_{M}\subseteq\mathcal{H}_{B}\otimes\mathcal{H}_{M}$.
The subspaces $\mathcal{H}_{B_{k}}$ are mutually orthogonal.

  3. 3.

The density operator $p\,\mathrm{tr}_{M}(V_{C}\tilde{V}_{B}^{k}V_{A}\sigma
V^{\dagger}_{A}\tilde{V}_{B}^{k\dagger}V_{C}^{\dagger})$ is a quantum Markov
chain.

Remarks:

  1. 1.

Explicitly, each $\tilde{V}_{B}^{k}$, $k=0,\cdot\cdot\cdot,p-1$, has the form

$\tilde{V}_{B}^{k}=\sqrt{p}\sum_{(i,j)\in\mathcal{O}_{k}}\sqrt{\sigma_{i}}|\xi_{ij}\rangle\otimes|i\rangle\langle
j|,$ (28)

where
$\\{|\xi_{ij}\rangle\in\mathcal{H}_{B}\;|\;i,j=1,\cdot\cdot\cdot,d_{M}\\}$ is
a set of orthonormal vectors,
$\langle\xi_{i^{\prime}j^{\prime}}|\xi_{ij}\rangle=\delta_{ii^{\prime}}\delta_{jj^{\prime}}$,
the vectors $|i\rangle,|j\rangle\in\mathcal{H}_{M}$ and the eigenvalues
$\sigma_{i}>0$ are defined by the spectral decomposition
$\sigma=\sum_{i=1}^{d_{M}}\sigma_{i}|i\rangle\langle i|$, and
$\mathcal{O}_{k}:=\\{(i,j)\;|\;|i\rangle\in\Pi_{k+|B|}\mathcal{H}_{M},|j\rangle\in\Pi_{k}\mathcal{H}_{M}\\}$.
Notice that if $p>1$, then the set $\bigcup_{k=0}^{p-1}\mathcal{O}_{k}$ does
not contain all pairs $(i,j)$.

  2. 2.

The subspaces $\mathcal{H}_{B_{k}}$ are defined as
$\mathcal{H}_{B_{k}}:=\mathrm{span}\\{|\xi_{ij}\rangle\in\mathcal{H}_{B}\;|\;(i,j)\in\mathcal{O}_{k}\\}$.
It follows that the subspaces $\mathcal{H}_{B_{k}}$ are mutually orthogonal,
and implies that for $X\in\mathcal{B}(\mathcal{H}_{M})$ and $k\neq
k^{\prime}$, the sum
$\tilde{V}^{k}_{B}X\tilde{V}^{k\dagger}_{B}+\tilde{V}^{k^{\prime}}_{B}X\tilde{V}^{k^{\prime}\dagger}_{B}$
is direct, i.e.,
$\tilde{V}^{k}_{B}X\tilde{V}^{k\dagger}_{B}\in\mathcal{B}(\mathcal{H}_{B_{k}})\otimes\mathcal{B}(\mathcal{H}_{M})$,
$\tilde{V}^{k^{\prime}}_{B}X\tilde{V}^{k^{\prime}\dagger}_{B}\in\mathcal{B}(\mathcal{H}_{B_{k^{\prime}}})\otimes\mathcal{B}(\mathcal{H}_{M})$
and
$\tilde{V}^{k}_{B}X\tilde{V}^{k\dagger}_{B}+\tilde{V}^{k^{\prime}}_{B}X\tilde{V}^{k^{\prime}\dagger}_{B}\in\left(\mathcal{B}(\mathcal{H}_{B_{k}})\oplus\mathcal{B}(\mathcal{H}_{B_{k^{\prime}}})\right)\otimes\mathcal{B}(\mathcal{H}_{M})$.

###### Proof.

As we show in Appendix I, the quantum channel $\tilde{\mathcal{E}}$, obtained
via Definition 1 from $\mathcal{E}$ with the properties listed in Proposition
2, has the form

$\tilde{\mathcal{E}}(X)=p\sum_{k=0}^{p-1}\mbox{tr}\left(\Pi_{k}X\Pi_{k}\right)\Pi_{k+1}\sigma\Pi_{k+1}.$
(29)

In $\Pi_{k+k^{\prime}}$ the addition in the subscript is interpreted modulo
$p$. Then,

$\tilde{\mathcal{E}}^{|B|}(X)=p\sum_{k=0}^{p-1}\mbox{tr}\left(\Pi_{k}X\Pi_{k}\right)\Pi_{k+|B|}\sigma\Pi_{k+|B|}.$

Notice that for each $k$ and $X\in\mathcal{B}(\mathcal{H}_{M})$ the map,
$X\mapsto p\mbox{tr}\left(\Pi_{k}X\Pi_{k}\right)\Pi_{k+|B|}\sigma\Pi_{k+|B|}$,
while not trace-preserving, is completely positive, and has an associated
dilation
$\tilde{V}_{B}^{k}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{B}\otimes\mathcal{H}_{M}$,
defined in (28).

It is easy to check that $\tilde{V}_{B}^{k}$ is a partial isometry,

$\tilde{V}_{B}^{k\dagger}\tilde{V}_{B}^{k}=\sum_{|j\rangle\in\Pi_{k}\mathcal{H}_{M}}|j\rangle\langle
j|=\Pi_{k}.$

We define the subspace
$\mathcal{H}_{B_{k}}:=\mathrm{span}\\{|\xi_{ij}\rangle\in\mathcal{H}_{B}\;|\;(i,j)\in\mathcal{O}_{k}\\}$
that has the dimension
$\dim\mathcal{H}_{B_{k}}=\mathrm{rank}(\Pi_{k})\mathrm{rank}(\Pi_{k+|B|})$.

Now we prove that the assumption $\dim\mathcal{H}_{B}\geq d_{M}^{2}$ is
consistent with mutual orthogonality of the subspaces $\mathcal{H}_{B_{k}}$.
For $\mathcal{H}_{B}$ to contain $\bigoplus_{k=0}^{p-1}\mathcal{H}_{B_{k}}$,
we require $\sum_{k=0}^{p-1}\dim\mathcal{H}_{B_{k}}\leq\dim\mathcal{H}_{B}$,
and since
$\sum_{k=0}^{p-1}\mathrm{rank}(\Pi_{k})\mathrm{rank}(\Pi_{k+|B|})\leq(\sum_{k=0}^{p-1}\mathrm{rank}(\Pi_{k}))^{2}=d^{2}_{M}$,
this condition is satisfied.

The channel $\tilde{\mathcal{E}}^{|B|}$ has the associated dilation isometry
$\tilde{V}^{\prime}_{B}:=\sum_{k=0}^{p-1}\tilde{V}^{k}_{B}$, since

$\tilde{V}_{B}^{{}^{\prime}\dagger}\tilde{V}^{\prime}_{B}=\sum_{k_{1},k_{2}=0}^{p-1}\tilde{V}^{k_{1}\dagger}_{B}\tilde{V}^{k_{2}}_{B}=\sum_{k=0}^{p-1}\tilde{V}^{k\dagger}_{B}\tilde{V}^{k}_{B}=\sum_{k=0}^{p-1}\Pi_{k}=\mathbb{1}_{B},$

and

$\displaystyle\mathrm{tr}_{B}(\tilde{V}^{\prime}_{B}X\tilde{V}^{{}^{\prime}\dagger}_{B})$
$\displaystyle=\sum_{k_{1},k_{2}=0}^{p-1}\mathrm{tr}_{B}(\tilde{V}^{k_{1}}_{B}X\tilde{V}^{k_{2}\dagger}_{B})$
$\displaystyle=\sum_{k=0}^{p-1}\mathrm{tr}_{B}(\tilde{V}^{k}_{B}X\tilde{V}^{k\dagger}_{B})$
$\displaystyle=p\sum_{k=0}^{p-1}\mbox{tr}\left(\Pi_{k}X\Pi_{k}\right)\Pi_{k+|B|}\sigma\Pi_{k+|B|}$
$\displaystyle=\tilde{\mathcal{E}}^{|B|}(X).$

We have used mutual orthogonality of $\mathcal{H}_{B_{k}}$ in both
calculations. Since both
$\tilde{V}^{\prime}_{B},\tilde{V}_{B}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{B}\otimes\mathcal{H}_{M}$
dilate $\tilde{\mathcal{E}}^{B}$, there exists unitary
$U_{B}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{B}$, such that
$\tilde{V}_{B}=U_{B}\tilde{V}^{\prime}_{B}$. Using the same reasoning as in
Section IV.1 we can, without loss of generality, take
$\tilde{V}^{\prime}_{B}=\tilde{V}_{B}$,

$\tilde{V}_{B}=\sum_{k=0}^{p-1}\tilde{V}_{B}^{k}=\sqrt{p}\sum_{k=0}^{p-1}\sum_{(i,j)\in\mathcal{O}_{k}}\sqrt{\sigma_{i}}|\xi_{ij}\rangle\otimes|i\rangle\langle
j|.$ (30)

Notice that the case $p=1$ corresponds to a single projector
$\Pi_{0}=\mathbb{1}_{M}$, so that (29) reduces to
$\tilde{\mathcal{E}}^{|B|}(X)=\mathrm{tr}(X)\sigma$, and (30) reduces to (25).

Comparing (28) to (25), we realize that the we can use the same arguments as
in Section IV.1 to prove that
$p\mathrm{tr}_{M}(V_{C}\tilde{V}_{B}^{k}V_{A}\sigma
V_{A}\tilde{V}_{B}^{k\dagger}V_{C}^{\dagger})$ is a quantum Markov chain. The
normalization constant comes from the observation that

$\displaystyle\mathrm{tr}(V_{C}\tilde{V}^{k}_{B}V_{A}\sigma
V^{\dagger}_{A}\tilde{V}_{B}^{k\dagger}V_{C}^{\dagger})$
$\displaystyle=\mathrm{tr}(\Pi_{k}V_{A}\sigma V^{\dagger}_{A}\Pi_{k})$
$\displaystyle=\mathrm{tr}\left(\Pi_{k}\mathcal{E}^{|A|}(\sigma)\Pi_{k}\right)$
$\displaystyle=\mathrm{tr}\left(\Pi_{k}\mathcal{E}^{|A|}\left(\bigoplus_{k^{\prime}=0}^{p-1}\Pi_{k^{\prime}}\sigma\Pi_{k^{\prime}}\right)\Pi_{k}\right)$
$\displaystyle=\mathrm{tr}(\Pi_{k}\bigoplus_{k^{\prime}=0}^{p-1}\Pi_{k^{\prime}+|A|}\sigma\Pi_{k^{\prime}+|A|}\Pi_{k})$
$\displaystyle=\mathrm{tr}(\Pi_{k}\sigma\Pi_{k})=\frac{1}{p}.$

∎

###  IV.3 The general case of pgFCS

We generalize the proof to the case of a convex sum of ergodic pgFCS
considered in the previous subsection. The properties of pgFCS are given in
Proposition 3, whose notation and definitions we employ below.

We will prove that $\tilde{\rho}_{ABC}$, as defined in (15), is a quantum
Markov chain by anticipating the Lemma IV.2, which states that the isometry
$\tilde{V}_{B}$ can be decomposed as a sum of partial isometries corresponding
to ergodic components,
$\tilde{V}_{B}=\sum_{\alpha=1}^{J}\tilde{V}^{\alpha}_{B}$, where
$\tilde{V}^{\alpha}_{B}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{B}\otimes\mathcal{H}_{M}$
is a partial isometry with $\tilde{\Pi}_{\alpha}\mathcal{H}_{M}$ the
orthogonal complement of its kernel, and where
$\mathrm{Range}(\tilde{V}_{B}^{\alpha})=\mathcal{H}_{B_{\alpha}}\otimes\tilde{\Pi}_{\alpha}\mathcal{H}_{M}$,
with subspaces $\mathcal{H}_{B_{\alpha}}\subseteq\mathcal{H}_{B}$ being
mutually orthogonal. The projectors $\tilde{\Pi}_{\alpha}$ are defined in
Proposition 3, and correspond to different ergodic components of $\rho_{ABC}$.
Moreover, $\tilde{V}^{\alpha}_{B}$ are such that

$\tilde{\rho}^{\alpha}_{ABC}:=\mbox{tr}_{M}\left(V^{\alpha}_{C}\tilde{V}^{\alpha}_{B}V^{\alpha}_{A}\sigma_{\alpha}V_{A}^{\alpha\dagger}\tilde{V}_{B}^{\alpha\dagger}V_{C}^{\alpha\dagger}\right)$

separately approximates the corresponding $\rho^{\alpha}_{ABC}$, defined in
Proposition 3. Then $\tilde{\rho}_{ABC}$ may be decomposed,

$\displaystyle\tilde{\rho}_{ABC}$
$\displaystyle=\mbox{tr}_{M}\left(V_{C}\left(\sum_{\alpha=1}^{J}\tilde{V}^{\alpha}_{B}\right)V_{A}\sigma
V_{A}^{\dagger}\left(\sum_{\beta=1}^{J}\tilde{V}^{\beta\dagger}_{B}\right)V_{C}^{\dagger}\right)$
$\displaystyle=\bigoplus_{\alpha=1}^{J}\lambda_{\alpha}\mbox{tr}_{M_{\alpha}}\left(V^{\alpha}_{C}\tilde{V}^{\alpha}_{B}V^{\alpha}_{A}\sigma_{\alpha}V_{A}^{\alpha\dagger}\tilde{V}^{\alpha\dagger}_{B}V_{C}^{\alpha\dagger}\right)$
$\displaystyle=\bigoplus_{\alpha=1}^{J}\lambda_{\alpha}\tilde{\rho}^{\alpha}_{ABC},$

where we have used
$V_{C}\tilde{V}^{\alpha}_{B}=V_{C}^{\alpha}\tilde{V}^{\alpha}_{B}$ and
$\tilde{V}^{\alpha}_{B}V_{A}\sigma
V^{\dagger}_{A}\tilde{V}_{B}^{\beta}=\delta_{\alpha\beta}\lambda_{\alpha}\tilde{V}^{\alpha}_{B}V^{\alpha}_{A}\sigma_{\alpha}V^{\alpha\dagger}_{A}\tilde{V}_{B}^{\beta}$.
By the proof in Section IV.2, each $\tilde{\rho}^{\alpha}_{ABC}$ is a quantum
Markov chain, and by Corollary III.5.1 $\tilde{\rho}_{ABC}$ is a quantum
Markov chain. This is sufficient to complete the proof.

The following lemma shows that $\tilde{V}_{B}$ indeed possesses the requisite
decomposition. We repeatedly use Proposition 3 in the proof.

######  Lemma IV.2.

For a pgFCS, the isometry
$\tilde{V}_{B}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{B}\otimes\mathcal{H}_{M}$
defined in (15) can be constructed as the sum
$\tilde{V}_{B}=\sum_{\alpha=1}^{J}\tilde{V}_{B}^{\alpha}$, where $J$ is the
number of ergodic components, and

  1. 1.

$\tilde{V}_{B}^{\alpha}$ is a partial isometry,
$\tilde{V}_{B}^{\alpha\dagger}\tilde{V}_{B}^{\alpha}=\tilde{\Pi}_{\alpha}$,
and
$\tilde{V}^{\alpha}_{B}\tilde{\Pi}_{\beta}=\delta_{\alpha\beta}\tilde{V}_{B}^{\alpha}$.

  2. 2.

$\mathrm{Range}(\tilde{V}_{B}^{\alpha})=\mathcal{H}_{B_{\alpha}}\otimes\tilde{\Pi}_{\alpha}\mathcal{H}_{M}\subseteq\mathcal{H}_{B}\otimes\mathcal{H}_{M}$.
The subspaces $\mathcal{H}_{B_{\alpha}}$ are mutually orthogonal.

  3. 3.

$\tilde{\rho}^{\alpha}_{ABC}=\mathrm{tr}_{M}(V^{\alpha}_{C}\tilde{V}_{B}^{\alpha}V^{\alpha}_{A}\sigma
V^{\alpha\dagger}_{A}\tilde{V}_{B}^{\alpha\dagger}V_{C}^{\alpha\dagger})$ is a
quantum Markov chain.

###### Proof.

We express $\mathcal{E}$ as

$\displaystyle\mathcal{E}(X)=\sum_{\alpha,\beta=1}^{J}\mathcal{E}\left(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\beta}\right)=\sum_{\alpha,\beta=1}^{J}\mbox{tr}_{s}\left(V_{\alpha}\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\beta}V_{\beta}^{\dagger}\right),$
(31)

where we have used $V\tilde{\Pi}_{\alpha}=V_{\alpha}\tilde{\Pi}_{\alpha}$.
Since the range of $V_{\alpha}$ is
$\mathcal{H}_{s}\otimes\tilde{\Pi}_{\alpha}\mathcal{H}_{M}$, then
$\mbox{tr}_{s}\left(V_{\alpha}\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\beta}V_{\beta}^{\dagger}\right)\in\tilde{\Pi}_{\alpha}\mathcal{B}(\mathcal{H}_{M})\tilde{\Pi}_{\beta}$.
We decompose $\tilde{\mathcal{E}}$ similarly,

$\tilde{\mathcal{E}}(X)=\sum_{\alpha,\beta=1}^{J}\tilde{\mathcal{E}}\left(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\beta}\right).$

We now show that the last equation reduces to the direct sum,
$\tilde{\mathcal{E}}(X)=\bigoplus_{\alpha=1}^{J}\tilde{\mathcal{E}}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\alpha})$.
We observe that the map,
$X\mapsto\tilde{\mathcal{E}}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\beta})$, is
obtained via Definition 1 from the map
$X\mapsto\mbox{tr}_{s}(V_{\alpha}\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\beta}V_{\beta}^{\dagger})$.
Recall that for $\alpha\neq\beta$ there is no unitary
$U:\mathcal{H}_{M}\rightarrow\mathcal{H}_{M}$ and $\phi\in\mathbb{R}$, such
that $V_{\alpha}=e^{i\phi}UV_{\beta}U^{\dagger}$, which implies by Lemma E.1
that the eigenvalues of
$X\mapsto\mbox{tr}_{s}(V_{\alpha}\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\beta}V_{\beta}^{\dagger})$
have magnitudes less than $1$. Then, following the construction in Definition
1, the map
$X\mapsto\mbox{tr}_{s}(V_{\alpha}\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\beta}V_{\beta}^{\dagger})$
is identically zero,
$\tilde{\mathcal{E}}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\beta})=0$ for
$\alpha\neq\beta$. Thus,
$\tilde{\mathcal{E}}(X)=\sum_{\alpha=1}^{J}\tilde{\mathcal{E}}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\alpha})$.
Since
$\mathcal{E}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\alpha})\in\tilde{\Pi}_{\alpha}\mathcal{B}(\mathcal{H}_{M})\tilde{\Pi}_{\alpha}$,
from Definition 1 follows that
$\tilde{\mathcal{E}}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\alpha})\in\tilde{\Pi}_{\alpha}\mathcal{B}(\mathcal{H}_{M})\tilde{\Pi}_{\alpha}$.
Since $\tilde{\Pi}_{\alpha}$ are orthonormal projectors, the sum is direct,
$\tilde{\mathcal{E}}(X)=\bigoplus_{\alpha=1}^{J}\tilde{\mathcal{E}}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\alpha})$,
and

$\tilde{\mathcal{E}}^{|B|}(X)=\bigoplus_{\alpha=1}^{J}\tilde{\mathcal{E}}^{|B|}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\alpha}).$

We define
$\tilde{V}_{B}^{\alpha}:\mathcal{H}_{M_{\alpha}}\rightarrow\mathcal{H}_{B_{\alpha}}\otimes\mathcal{H}_{M_{\alpha}}$
to be an isometric dilation of
$X\mapsto\tilde{\mathcal{E}}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\alpha})$. Here
$\mathcal{H}_{M_{\alpha}}:=\tilde{\Pi}_{\alpha}\mathcal{H}_{M}$, and
$\mathcal{H}_{B_{\alpha}}$ is the dilation space embedded in $\mathcal{H}_{B}$
of dimension
$\dim\mathcal{H}_{B_{\alpha}}\leq(\dim\mathcal{H}_{M_{\alpha}})^{2}=d^{2}_{M_{\alpha}}$.
It immediately follows that
$\tilde{V}_{B}^{\alpha\dagger}\tilde{V}_{B}^{\alpha}=\tilde{\Pi}_{\alpha}$,
proving the statement 1. From property 4 of Proposition 3 we know that
$(V_{\alpha},\sigma_{\alpha})$ generates an ergodic pgFCS with the induced
quantum channel
$X\mapsto\mathcal{E}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\alpha})$. As mentioned
before,
$X\mapsto\tilde{\mathcal{E}}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\alpha})$ is
exactly the map obtained via Definition 1 from the map
$X\mapsto\mathcal{E}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\alpha})$. Then,
separately for each $\alpha$, we can repeat the reasoning of Section IV.2,
showing that each $\tilde{V}_{B}^{\alpha}$, the dilating isometry of
$X\mapsto\tilde{\mathcal{E}}^{|B|}(\tilde{\Pi}_{\alpha}X\tilde{\Pi}_{\alpha})$,
possesses the properties which ensure that $\tilde{\rho}^{\alpha}_{ABC}$ is a
quantum Markov chain, proving the statement 3.

Now we show that we can safely choose the subspaces $\mathcal{H}_{B_{\alpha}}$
to be mutually orthogonal. Since
$\mathcal{H}_{M}=\bigoplus_{\alpha=1}^{J}\mathcal{H}_{M_{\alpha}}$, and since
we assume $\dim\mathcal{H}_{B}\geq d_{M}^{2}$, then
$\dim\mathcal{H}_{B}\geq(\sum_{\alpha=1}^{J}d_{M_{\alpha}})^{2}\geq\sum_{\alpha=1}^{J}d^{2}_{M_{\alpha}}\geq\sum_{\alpha=1}^{J}\dim\mathcal{H}_{B_{\alpha}}$,
which implies that the subspaces $\mathcal{H}_{B_{\alpha}}$ may be chosen to
be orthogonal, proving the statement 2. Then we can construct the isometry
$\tilde{V}^{\prime}_{B}=\sum_{\alpha=1}^{J}\tilde{V}^{\alpha}_{B}$. This
isometry clearly dilates $\tilde{\mathcal{E}}^{|B|}$, which is also dilated by
$\tilde{V}_{B}$. By the same reasoning as in Section IV.1 and Section IV.2 we
may take $\tilde{V}^{\prime}_{B}=\tilde{V}_{B}$,

$\tilde{V}_{B}=\sum_{\alpha=1}^{J}\tilde{V}^{\alpha}_{B}.$

Thus, $\tilde{V}_{B}$ possesses all the required properties. ∎

##  V Recovery maps

In this section we consider the recovery maps that can be used to restore
$\tilde{\rho}_{ABC}$, defined in (15), from $\tilde{\rho}_{AB}$. The obvious
choice is the Petz recovery map, defined on
$\mathcal{B}(\mathrm{supp}(\tilde{\rho}_{B}))$ in (2),
$\mathcal{P}_{B\rightarrow
BC}(X)=\tilde{\rho}_{BC}^{\frac{1}{2}}\tilde{\rho}^{-\frac{1}{2}}_{B}X\tilde{\rho}^{-\frac{1}{2}}_{B}\tilde{\rho}_{BC}^{\frac{1}{2}}$,
and which can be extended by a trivial embedding
$\mathrm{supp}(\tilde{\rho}_{B})^{\perp}\rightarrow\mathrm{supp}(\tilde{\rho}_{B})^{\perp}\otimes\mathcal{H}_{C}$
to the map $\mathcal{P}_{B\rightarrow
BC}:\mathcal{B}(\mathcal{H}_{B})\rightarrow\mathcal{B}(\mathcal{H}_{B})\otimes\mathcal{B}(\mathcal{H}_{C})$.Petz
(1986, 1988); Hayden  _et al._ (2004) The Petz recovery map can be defined for
any state that is a quantum Markov chain.

The Petz recovery map is, however, not the only map that can exactly
reconstruct a quantum Markov chain $\tilde{\rho}_{ABC}$. From Theorem III.5,
we note that there is another obvious choice of the recovery channel, which
relies on the isomorphism
$I:\mathrm{supp}(\tilde{\rho}_{B})\rightarrow\bigoplus_{k=1}^{k_{\mathrm{max}}}\mathcal{H}_{b^{l}_{k}}\otimes\mathcal{H}_{b^{r}_{k}}$
defined in Theorem III.5, and which provides the decomposition
$I\tilde{\rho}_{ABC}I^{\dagger}=\bigoplus_{k=1}^{k_{\mathrm{max}}}\lambda_{k}\tilde{\rho}_{Ab^{l}_{k}}\otimes\tilde{\rho}_{b^{r}_{k}C}$.
We gave an example of such an isomorphism in Section IV.1 for the case of an
ergodic pgFCS of period $1$. Using $I$, for a general quantum Markov chain, we
can construct the map $\mathcal{R}_{B\rightarrow
BC}:\mathcal{B}(\mathcal{H}_{B})\rightarrow\mathcal{B}(\mathcal{H}_{B}\otimes\mathcal{H}_{C})$,

$\mathcal{R}_{B\rightarrow
BC}(X):=I^{\dagger}\sum_{k=1}^{k_{\mathrm{max}}}\mathrm{tr}_{b^{r}_{k}}\left(\mathbb{P}_{k}(IXI^{\dagger})\mathbb{P}_{k}\right)\otimes\tilde{\rho}_{b^{r}_{k}C}I,$

where $\mathbb{P}_{k}$ is a projector onto
$\mathcal{H}_{b^{l}_{k}}\otimes\mathcal{H}_{b^{r}_{k}}$. We verify that this
map is indeed a recovery channel by direct computation,

$\displaystyle\mathcal{R}_{B\rightarrow
BC}(\tilde{\rho}_{AB})=I^{\dagger}\sum_{k=1}^{k_{\mathrm{max}}}\mathrm{tr}_{b^{r}_{k}}\left(\mathbb{P}_{k}(I\tilde{\rho}_{AB}I^{\dagger})\mathbb{P}_{k}\right)\otimes\tilde{\rho}_{b^{r}_{k}C}I$
$\displaystyle=I^{\dagger}\sum_{k=1}^{k_{\mathrm{max}}}\mathrm{tr}_{b^{r}_{k}}(\mathbb{P}_{k}\left(\bigoplus_{k^{\prime}=1}^{k_{\mathrm{max}}}\lambda_{k^{\prime}}\tilde{\rho}_{Ab^{l}_{k^{\prime}}}\otimes\tilde{\rho}_{b^{r}_{k^{\prime}}}\right)\mathbb{P}_{k})\otimes\tilde{\rho}_{b^{r}_{k}C}I$
$\displaystyle=I^{\dagger}\bigoplus_{k^{\prime}=1}^{k_{\mathrm{max}}}\lambda_{k^{\prime}}\mathrm{tr}_{b^{r}_{k^{\prime}}}(\tilde{\rho}_{Ab^{l}_{k^{\prime}}}\otimes\tilde{\rho}_{b^{r}_{k^{\prime}}})\otimes\rho_{b^{r}_{k}C}I$
$\displaystyle=I^{\dagger}\bigoplus_{k^{\prime}=1}^{k_{\mathrm{max}}}\lambda_{k^{\prime}}\tilde{\rho}_{Ab^{l}_{k^{\prime}}}\otimes\tilde{\rho}_{b^{r}_{k^{\prime}}C}I$
$\displaystyle=\tilde{\rho}_{ABC},$

where we have used
$I\tilde{\rho}_{AB}I^{\dagger}=I\mathrm{tr}_{C}\tilde{\rho}_{ABC}I^{\dagger}=\mathrm{tr}_{C}(I\tilde{\rho}_{ABC}I^{\dagger})$
(recall that $I$ acts non-trivially only on $\mathcal{H}_{B}$), the structure
of the quantum Markov chain from Theorem III.5 in the second line, and
$\mathbb{P}_{k}\rho_{Ab^{l}_{k^{\prime}}}\otimes\rho_{b^{r}_{k^{\prime}}C}\mathbb{P}_{k}=\delta_{{k^{\prime}}k}\rho_{Ab^{l}_{k^{\prime}}}\otimes\rho_{b^{r}_{k^{\prime}}C}$
in the third line.

Unlike the Petz recovery map that is defined in terms of the reduced density
operators $\tilde{\rho}_{BC}$ and $\tilde{\rho}_{B}$, the recovery map
$\mathcal{R}_{B\rightarrow BC}$ requires knowledge of the isomorphism $I$, the
structure of which is not obvious in the case of a general quantum Markov
chain. In the case of a quantum Markov chain approximating a pgFCS, and
possessing the form (15), we can construct the isomorphism $I$. For simplicity
of argument, we illustrate this claim using the case of ergodic pgFCS of
period $1$. Recall that $\tilde{\rho}_{ABC}$ has the form (15), and
$\tilde{V}_{B}$ has the form (25),
$\tilde{V}_{B}=\sum_{i,j=1}^{d_{M}}\sqrt{\sigma_{i}}|\xi_{ij}\rangle\otimes|i\rangle\langle
j|$, where $|i\rangle$ and $|j\rangle$ are the eigenvectors of $\sigma$, and
$|\xi_{ij}\rangle$ is an orthonormal basis set. This allows us to express
$\tilde{\rho}_{ABC}$ as in (26) and define $I$ in terms of the orthonormal
vectors $|\xi_{ij}\rangle$,
$I=\sum_{i,j=1}^{d_{M}}(|i\rangle\otimes|j\rangle)\langle\xi_{ij}|$. The
required set of orthonormal vectors
$\\{|\xi_{ij}\rangle\in\mathcal{H}_{B}\;|\;i,j=1,\cdot\cdot\cdot,d_{M}\\}$ is
the solution of the minimization problem for the norm
$\|V_{B}-\tilde{V}_{B}\|$ over unitary change of $d_{M}^{2}$ basis vectors in
$\mathcal{H}_{B}$.

Interestingly, we can construct another recovery channel, that has a slightly
less optimal bound in terms of the pre-exponential factor, but for which the
vectors $|\xi_{ij}\rangle$ can be constructed more explicitly. Again we use
the case of ergodic pgFCS of period $1$ for illustration.

For $\tilde{V}_{B}$ in the form (25), _any_ choice of the $|\xi_{ij}\rangle$
induces an exact quantum Markov chain $\tilde{\rho}_{ABC}$ of the form (15). A
particular choice of $|\xi_{ij}\rangle$ influences the bound on the recovery
error. Expressing

$\displaystyle\|V_{B}-\tilde{V}_{B}\|$
$\displaystyle=\|\sum_{i,j=1}^{d_{M}^{2}}(\langle
i|V_{B}|j\rangle-\sqrt{\sigma_{i}}|\xi_{ij}\rangle)\otimes|i\rangle\langle
j|\|.$

we observe that any choice of $|\xi_{ij}\rangle$ close to
$\sigma^{-\frac{1}{2}}_{i}\langle i|V_{B}|j\rangle$ is expected to be a good
one. Let us construct the positive semidefinite matrix

$Q_{i^{\prime}j^{\prime};ij}:=\langle
j^{\prime}|V^{\dagger}_{B}\left(\mathbb{1}_{B}\otimes|i^{\prime}\rangle\langle
i|\right)V_{B}|j\rangle,$ (32)

which is a Gramian matrix for the set of vectors $\\{\langle
i|V_{B}|j\rangle\;|\;i,j=1,\cdot\cdot\cdot,d_{M}\\}$. For large $|B|$, the
vectors in the latter set are almost orthogonal to each other, and $\|\langle
i|V_{B}|j\rangle\|$ is close to $\sigma_{i}^{1/2}$. To prove this, notice that
the map

$X\mapsto V^{\dagger}_{B}(\mathbb{1}_{B}\otimes X)V_{B}$

with $X\in\mathcal{B}(\mathcal{H}_{M})$ is adjoint (with respect to the
Hilbert-Schmidt inner product) to the quantum channel
$\mathcal{E}^{|B|}(X)=\mathrm{tr}_{B}(V_{B}XV^{\dagger}_{B})$, which converges
to $\tilde{\mathcal{E}}(X)=\mathrm{tr}(X)\sigma$ in the limit
$|B|\rightarrow\infty$. Hence the map $X\mapsto
V^{\dagger}_{B}(\mathbb{1}_{B}\otimes X)V_{B}$ converges to the adjoint of
$\tilde{\mathcal{E}}$, the map $X\mapsto\mathrm{tr}(\sigma X)\mathbb{1}_{M}$.
Thus, $Q_{i^{\prime}j^{\prime};ij}$ converges to $\langle
j^{\prime}|\mbox{tr}_{M}\left(\sigma|i^{\prime}\rangle\langle
i|\right)\mathbb{1}_{M}V_{B}|j\rangle=\sigma_{i}\delta_{ii^{\prime}}\delta_{jj^{\prime}}$.

Then we define $|\xi_{ij}\rangle$ as

$|\xi_{ij}\rangle=\sum_{m,n=1}^{d_{M}}(Q^{-\frac{1}{2}})_{mn;ij}\langle
m|V_{B}|n\rangle.$ (33)

(As we show in Appendix J, $Q$ is guaranteed to be invertible, if the region
$B$ is large enough). We verify that the set of $|\xi_{ij}\rangle$ is
orthonormal,

$\displaystyle\langle\xi_{i^{\prime}j^{\prime}}|\xi_{ij}\rangle$
$\displaystyle=\sum_{\mathclap{m,n,m^{\prime},n^{\prime}}}(Q^{-\frac{1}{2}})_{i^{\prime}j^{\prime};m^{\prime}n^{\prime}}\langle
n^{\prime}|V^{\dagger}_{B}|m^{\prime}\rangle\langle
m|V_{B}|n\rangle(Q^{-\frac{1}{2}})_{mn;ij}$
$\displaystyle=\sum_{\mathclap{m,n,m^{\prime},n^{\prime}}}(Q^{-\frac{1}{2}})_{i^{\prime}j^{\prime};m^{\prime}n^{\prime}}Q_{m^{\prime}n^{\prime};mn}(Q^{-\frac{1}{2}})_{mn;ij}$
$\displaystyle=\delta_{ii^{\prime}}\delta_{jj^{\prime}}.$

In Appendix J we show that for this choice of $|\xi_{ij}\rangle$,

$\|V_{B}-\tilde{V}_{B}\|\leq\sqrt{2(1+2^{-\frac{11}{2}})c}\;d_{M}\nu^{\frac{|B|}{2}},$
(34)

with $c$ and $\nu$ defined in Lemma III.3, leading to the recovery error

$\|\mathcal{R}_{B\rightarrow BC}(\rho_{AB})-\rho_{ABC}\|_{1}\leq
4\sqrt{2(1+2^{-\frac{11}{2}})c}\;d_{M}\nu^{\frac{|B|}{2}},$ (35)

which is worse than the recovery error (24) by the factor
$\sqrt{2(1+2^{-\frac{11}{2}})d_{M}}$.

In Appendix J we present a similar construction for the general case of pgFCS,
which is based on the same idea, but is more delicate.

##  VI Conclusion

Quantum conditional mutual information (QCMI) is non-negative by virtue of the
strong subadditivity of von Neumann entropy. We have studied the QCMI for a
homogeneous quantum spin chain described by a purely generated finitely
correlated state (pgFCS) $\rho_{ABC}$. Explicitly, by separating the chain
regions A and C by a domain B, and provided the size $|B|$ of region B is
large enough, we show there exists a quantum Markov chain which differs from
$\rho_{ABC}$ in trace distance by an error exponentially small in $|B|$. It
follows that $\rho_{ABC}$ may be recovered, within the stated error, from the
reduced density operator $\rho_{AB}$ by a quantum channel acting only on the
domain B. As a consequence this implies that the QCMI, denoted $I(A:C|B),$ is
also exponentially small in $|B|.$ We have presented quantum channels that, in
principle, can perform the state recovery. Besides the obvious choice of the
Petz recovery map, we give two other examples derived from the particular
structure of the pgFCS. We have carried out numerical experiments on the decay
of QCMI, the results of which are consistent with the bound presented in
Theorem II.1. The details will be reported elsewhere.

A natural next step would be to prove the exponential decay of QCMI for
generic FCS, as conjectured in Ref. Chen  _et al._ , 2021. Both the results
presented here and in Ref. Chen  _et al._ , 2021 point in this direction. As
discussed further in Appendix K, the pgFCS considered here and the class of
states considered in Ref. Chen  _et al._ , 2021 are in general distinct.

Generalization of our result to generic FCS seems to require an approach
different from the one we used in this paper. It remains an open question
whether a FCS exhibiting an exponential decay of QCMI is necessarily close to
a quantum Markov chain in trace distance.

###### Acknowledgements.

The authors would like to thank Shivan Mittal for helpful discussions and the
referee for helpful suggestions.

## Author declarations

### Conflict of interests

The authors have no conflicts to disclose.

## Data Availability Statement

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

## Appendix A

We present in the form of Table 1 the dictionary for conversion from the
language of pgFCS defined by the pair $(V,\sigma)$ (see Section III.3) to the
language of matrix product states (in diagrammatic notation).

pgFCS notation | Diagrammatic notation
---|---
$\left(\langle s|\otimes\langle i|\right)V|j\rangle$ |  M$j$$i$$s$
$V^{\dagger}V=\mathbb{1}_{M}$ |  $M$$M^{\dagger}$$=$
$\langle i|\sigma|j\rangle$ |  $\sigma$$i$$j$
$\mathrm{tr}_{M}(V\sigma V^{\dagger})=\mathcal{E}(\sigma)=\sigma$ |  $M$$M^{\dagger}$$\sigma$$=$$\sigma$
$\rho_{ABC}=\mbox{tr}_{M}\left(V_{C}V_{B}V_{A}\sigma V_{A}^{\dagger}V_{B}^{\dagger}V_{C}^{\dagger}\right)$ |  $M$$M$$M$ $M$$M$ $M$ $M$ $M^{\dagger}$$M^{\dagger}$$M^{\dagger}$$M^{\dagger}$$M^{\dagger}$$M^{\dagger}$$M^{\dagger}$ $\sigma$$A$$B$$C$
Table 1: The dictionary for conversion from the language of pgFCS defined by
the pair $(V,\sigma)$ to the language of matrix product states (in
diagrammatic notation)

## Appendix B

In this appendix we derive the right-hand side of the inequality (3). In Refs.
Fawzi and Renner, 2015; Sutter  _et al._ , 2016; Junge  _et al._ , 2018; Kim,
2017, the bound is based on the Alicki-Fannes inequality, Alicki and Fannes
(2004) which is the extension of the Fannes inequality. Fannes (1973) Here we
employ an improved version of the Alicki-Fannes inequality from Ref. Winter,
2016 (we refer the reader to this paper for further references on the topic).

In order to apply the results of Ref. Winter, 2016, we first express the QCMI
in terms of _quantum relative entropy_ ,
$D(\rho||\sigma):=\mathrm{tr}\rho(\ln\rho-\ln\sigma)$,

$\displaystyle I(A:C|B)$
$\displaystyle=S(\rho_{BC})-S(\rho_{ABC})+S(\rho_{AB})-S(\rho_{B})$
$\displaystyle=D(\rho_{ABC}||\rho_{BC})-D(\rho_{AB}||\rho_{B}).$

By the monotonicity of quantum relative entropy under the action of quantum
channels, Lindblad (1975); Watrous (2018) $D(\mathcal{R}_{B\rightarrow
BC}(\rho_{AB})||\mathcal{R}_{B\rightarrow BC}(\rho_{B}))\leq
D(\rho_{AB}||\rho_{B})$, thus we can estimate

$I(A:C|B)\leq D(\rho_{ABC}||\rho_{BC})-D\left(\mathcal{R}_{B\rightarrow
BC}(\rho_{AB})||\mathcal{R}_{B\rightarrow BC}(\rho_{B})\right).$

To make the result of Ref. Winter, 2016 directly applicable, we recast the
above inequality in terms of _conditional entropy_ ,
$S(R_{1}|R_{2})_{\rho}:=S(\rho_{R_{1}R_{2}})-S(\rho_{R_{2}})$ (we adopt the
notation of Ref. Winter, 2016). We notice that
$D(\rho_{ABC}||\rho_{BC})=-S(A|BC)_{\rho}$ and $D(\mathcal{R}_{B\rightarrow
BC}(\rho_{AB})||\mathcal{R}_{B\rightarrow
BC}(\rho_{B}))=-S(A|BC)_{\rho^{\prime}}$, where
$\rho_{ABC}^{\prime}=\mathcal{R}_{B\rightarrow BC}(\rho_{AB})$. Then

$I(A:C|B)\leq S(A|BC)_{\rho^{\prime}}-S(A|BC)_{\rho}.$

We apply Lemma 2 of Ref. Winter, 2016 to obtain the bound

$I(A:C|B)\leq\epsilon^{\prime}\ln\dim\mathcal{H}_{A}+(1+\epsilon^{\prime})h\left(\frac{\epsilon^{\prime}}{1+\epsilon^{\prime}}\right),$

where $h(p)=-p\ln p-(1-p)\ln(1-p)$ is the binary entropy and
$\epsilon^{\prime}=\frac{1}{2}\|\rho_{ABC}-\mathcal{R}_{B\rightarrow
BC}(\rho_{AB})\|_{1}$. This bound, unlike the one that follows from the
Alicki-Fannes inequality, is applicable for any $0\leq\epsilon^{\prime}\leq
1$. Notice that we expressed the inequality (3) in terms of
$\epsilon=\|\rho_{ABC}-\mathcal{R}_{B\rightarrow
BC}(\rho_{AB})\|_{1}=2\epsilon^{\prime}$. Reverting to our notation,

$I(A:C|B)\leq\frac{1}{2}\epsilon|A|\ln
d_{s}+(1+\frac{1}{2}\epsilon)h\left(\frac{\epsilon}{2+\epsilon}\right),$

as presented in (3), where we have substituted
$\dim\mathcal{H}_{A}=d_{s}^{|A|}$. We can also derive the simplified bound by
estimating

$\displaystyle(1+\epsilon^{\prime})h\left(\frac{\epsilon^{\prime}}{1+\epsilon^{\prime}}\right)$
$\displaystyle=-\epsilon^{\prime}\ln\left(\frac{\epsilon^{\prime}}{1+\epsilon^{\prime}}\right)+\ln(1+\epsilon^{\prime})$
$\displaystyle=-\epsilon^{\prime}\ln\epsilon^{\prime}+(1+\epsilon^{\prime})\ln(1+\epsilon^{\prime})$
$\displaystyle\leq\epsilon^{\prime}(2-\ln\epsilon^{\prime}),$

where we have used $0\leq\epsilon^{\prime}\leq 1$ and
$\ln(1+\epsilon^{\prime})\leq\epsilon^{\prime}$. This leads to the bound for
the QCMI,

$I(A:C|B)\leq\epsilon\left(\frac{1}{2}|A|\ln
d_{s}+1-\frac{1}{2}\ln\frac{\epsilon}{2}\right),$ (36)

where $0<\epsilon<2$, which we use in the proof of Theorem II.1.

## Appendix C

In this appendix we extend Theorem II.1 for the configuration of the regions
$A$, $B$, and $C$ depicted in Figure 1(b). In this case $\rho_{ABC}$ has the
form

$\rho_{ABC}=\mathrm{tr}_{M}\left(V_{A_{2}}V_{B_{2}}V_{C}V_{B_{1}}V_{A_{1}}\sigma
V_{A_{1}}^{\dagger}V_{B_{1}}^{\dagger}V^{\dagger}_{C}V^{\dagger}_{B_{2}}V^{\dagger}_{A_{2}}\right).$

We introduce approximating isometries
$\tilde{V}_{B_{1}}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{B_{1}}\otimes\mathcal{H}_{M}$
and
$\tilde{V}_{B_{2}}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{B_{2}}\otimes\mathcal{H}_{M}$
in exactly the same way as in Section IV, which we use to construct the state

$\tilde{\rho}_{ABC}=\mathrm{tr}_{M}\left(V_{A_{2}}\tilde{V}_{B_{2}}V_{C}\tilde{V}_{B_{1}}V_{A_{1}}\sigma
V_{A_{1}}^{\dagger}\tilde{V}_{B_{1}}^{\dagger}V^{\dagger}_{C}\tilde{V}^{\dagger}_{B_{2}}V^{\dagger}_{A_{2}}\right).$
(37)

Then, repeating the calculations in (IV) and (16) and using the triangle
inequality, we bound the recovery error for the recovery channel
$\mathcal{R}_{B\rightarrow
BC}:\mathcal{B}(\mathcal{H}_{B})\rightarrow\mathcal{B}(\mathcal{H}_{B}\otimes\mathcal{H}_{C})$,
for which $\mathcal{R}_{B\rightarrow
BC}(\tilde{\rho}_{AB})=\tilde{\rho}_{ABC}$, as

$\|\rho_{ABC}-\mathcal{R}_{B\rightarrow BC}\left(\rho_{AB}\right)\|_{1}\leq
4\|V_{B_{1}}-\tilde{V}_{B_{1}}\|+4\|V_{B_{2}}-\tilde{V}_{B_{2}}\|.$

For both terms we apply the bound (23), which leads to

$\displaystyle\|\rho_{ABC}-\mathcal{R}_{B\rightarrow
BC}\left(\rho_{AB}\right)\|_{1}$ $\displaystyle\leq 4\sqrt{d_{M}c}\ \
(\nu^{|B_{1}|/2}+\nu^{|B_{2}|/2})$ $\displaystyle\leq
8\sqrt{d_{M}c}\nu^{d(A,C)/2},$

where $d(A,C)=\min\\{|B_{1}|,|B_{2}|\\}$.

We can prove that $\tilde{\rho_{ABC}}$ is a quantum Markov chain using the
same reasoning as in Sections IV.1, IV.2, and IV.3. We present the proof only
for the case of ergodic pgFCS of period 1. The extensions of the argument to
the general case of pgFCS are analogous to the ones of Sections IV.2 and IV.3.

Both $\tilde{V}_{B_{1}}$ and $\tilde{V}_{B_{2}}$ have the form (25),

$\displaystyle\tilde{V}_{B_{1}}$
$\displaystyle=\sum_{i,j=1}^{d_{M}}\sqrt{\sigma_{i}}|\xi_{ij}\rangle\otimes|i\rangle\langle
j|,$ (38) $\displaystyle\tilde{V}_{B_{2}}$
$\displaystyle=\sum_{m,n=1}^{d_{M}}\sqrt{\sigma_{m}}|\zeta_{mn}\rangle\otimes|m\rangle\langle
n|.$

The vectors $|\xi_{ij}\rangle$ and $|\zeta_{mn}\rangle$ define the subspaces
$\mathcal{H}_{B_{1}}\supseteq\mathcal{H}_{b_{1}}:=\mathrm{span}\\{|\xi_{ij}\rangle\in\mathcal{H}_{B_{1}}\;|\;i,j=1,\cdot\cdot\cdot,d_{M}\\}$
and
$\mathcal{H}_{B_{2}}\supseteq\mathcal{H}_{b_{2}}:=\mathrm{span}\\{|\zeta_{mn}\rangle\in\mathcal{H}_{B_{1}}\;|\;m,n=1,\cdot\cdot\cdot,d_{M}\\}$.
We observe that we should demand $\dim\mathcal{H}_{B_{1}}\geq d_{M}^{2}$ and
$\dim\mathcal{H}_{B_{2}}\geq d_{M}^{2}$ for the construction to be possible.
Substituting (38) into (37), we explicitly express

$\displaystyle\tilde{\rho}_{ABC}=\sum_{all\,indices=1}^{d_{M}}$
$\displaystyle\langle j|V_{A_{1}}\sigma
V_{A_{1}}^{\dagger}|j^{\prime}\rangle\otimes\sqrt{\sigma_{i}}|\xi_{ij}\rangle|\xi_{i^{\prime}j^{\prime}}\rangle\sqrt{\sigma_{i^{\prime}}}\otimes\langle
n|V_{C}|i\rangle\rangle
i^{\prime}|V^{\dagger}_{C}|n^{\prime}\rangle\otimes\sqrt{\sigma_{m}}|\zeta_{mn}\rangle\langle\zeta_{m^{\prime}n^{\prime}}|\sqrt{\sigma_{m^{\prime}}}\otimes\mathrm{tr}_{M}(V_{A_{2}}|m\rangle\langle
m^{\prime}|V^{\dagger}_{A_{2}}).$

We introduce the isomorphism
$I:\mathcal{H}_{b_{1}}\otimes\mathcal{H}_{b_{2}}\rightarrow\mathcal{H}_{M}\otimes\mathcal{H}_{M}\otimes\mathcal{H}_{M}\otimes\mathcal{H}_{M}$,
defined by
$I|\xi_{ij}\rangle\otimes|\zeta_{mn}\rangle=|j\rangle\otimes|i\rangle\otimes|n\rangle\otimes|m\rangle$,
which maps $\tilde{\rho}_{ABC}$ to

$\displaystyle I\tilde{\rho}_{ABC}I^{\dagger}=V_{A_{1}}\sigma
V_{A_{1}}^{\dagger}$ $\displaystyle\otimes
V_{C}\sqrt{\sigma}|+\rangle\langle+|\sqrt{\sigma}V^{\dagger}_{C}$
$\displaystyle\otimes\mathrm{tr}_{M}\left(V_{A_{2}}\sqrt{\sigma}|+\rangle\langle+|\sqrt{\sigma}V^{\dagger}_{A_{2}}\right),$

where
$|+\rangle=\sum_{i=1}^{d_{M}}|i\rangle\otimes|i\rangle\in\mathcal{H}_{M}\otimes\mathcal{H}_{M}$.
By Proposition III.5 this state is a quantum Markov chain.

## Appendix D

In this appendix we construct explicit examples of pgFCS, for which QMI
converges to a finite limit as the size of the region $B$ is increased,
$\lim_{|B|\rightarrow+\infty}I(A:C)>0$. By contrast, Theorem II.1 proves that
QCMI converges to zero for any pgFCS.

To construct the examples we use two isometries. The first one is
$V_{1}:\mathbb{C}^{2}\rightarrow\mathbb{C}^{3}\otimes\mathbb{C}^{2}$, defined
by

$\displaystyle V_{1}=$
$\displaystyle\frac{1}{\sqrt{2}}|1\rangle\otimes|-\rangle\langle+|-\frac{1}{\sqrt{2}}|0\rangle\otimes|+\rangle\langle+|$
(39)
$\displaystyle+\frac{1}{\sqrt{2}}|0\rangle\otimes|-\rangle\langle-|-\frac{1}{\sqrt{2}}|-1\rangle\otimes|+\rangle\langle-|,$

where $\\{|-1\rangle,|0\rangle,|1\rangle\\}$ is an orthonormal basis in
$\mathbb{C}^{3}$, and $\\{|-\rangle,|+\rangle\\}$ is an orthonormal basis in
$\mathbb{C}^{2}$. This isometry is of the type defined in Ref. Fannes  _et
al._ , 1992, that generates the ground state of the generalized AKLT model. We
define
$\mathcal{E}_{1}:\mathcal{B}(\mathbb{C}^{2})\rightarrow\mathcal{B}(\mathbb{C}^{2})$
by $\mathcal{E}_{1}(X)=\mathrm{tr}_{3}(V_{1}XV_{1}^{\dagger})$, where
$\mathrm{tr}_{3}$ is the partial trace over $\mathbb{C}^{3}$. For the current
discussion it is important that the pair $(V_{1},\mathbb{1}_{2}/2)$ generates
an ergodic pgFCS of period 1. Hence, $\tilde{\mathcal{E}}_{1}$, obtained from
$\mathcal{E}_{1}$ using Definition 1 satisfies
$\tilde{\mathcal{E}}_{1}(X)=\mathrm{tr}(X)\mathbb{1}_{2}/2$. We obtain the
second isometry,
$V_{2}:\mathbb{C}^{2}\rightarrow\mathbb{C}^{3}\otimes\mathbb{C}^{2}$, from
$V_{1}$ by performing the cyclic permutation
$|-1\rangle\mapsto|0\rangle\mapsto|1\rangle\mapsto|-1\rangle$,

$\displaystyle V_{2}=$
$\displaystyle\frac{1}{\sqrt{2}}|-1\rangle\otimes|-\rangle\langle+|-\frac{1}{\sqrt{2}}|1\rangle\otimes|+\rangle\langle+|$
(40)
$\displaystyle+\frac{1}{\sqrt{2}}|1\rangle\otimes|-\rangle\langle-|-\frac{1}{\sqrt{2}}|0\rangle\otimes|+\rangle\langle-|.$

We observe
$\mathrm{tr}_{3}(V_{2}XV_{2}^{\dagger})=\mathrm{tr}_{3}(V_{1}\mathbb{1}_{2}V_{1}^{\dagger})=\mathcal{E}_{1}(X)$
for any $X\in\mathcal{B}(\mathcal{H}_{M})$.

Our first example is a pgFCS with two ergodic components. For a Hilbert space
of the spin $\mathcal{H}_{s}:=\mathbb{C}^{3}$ and a memory Hilbert space
$\mathcal{H}_{M}:=\mathbb{C}^{4}\cong\mathbb{C}^{2}\oplus\mathbb{C}^{2}$, the
quantum state is generated by

$V=\begin{pmatrix}V_{1}&0\\\
0&V_{2}\end{pmatrix},\quad\sigma=\frac{1}{4}\begin{pmatrix}\mathbb{1}_{2}&0\\\
0&\mathbb{1}_{2}\end{pmatrix}.$

It is obvious that this state by construction has two ergodic components (see
Proposition 3). We choose the regions $A$ and $C$ to each consist of a single
spin, i.e., $\mathcal{H}_{A},\mathcal{H}_{C}=\mathbb{C}^{2}$. In the limit
$|B|\rightarrow+\infty$,

$\displaystyle\tilde{\rho}_{AC}$
$\displaystyle:=\lim_{|B|\rightarrow+\infty}\mathrm{tr}_{M}(V\mathrm{tr}_{B}(V_{B}V\sigma
V^{\dagger}V_{B}^{\dagger})V^{\dagger})$
$\displaystyle=\mathrm{tr}_{M}(V\tilde{\mathcal{E}}(V\sigma
V^{\dagger})V^{\dagger})$
$\displaystyle=\frac{1}{8}\mathrm{tr}_{2}(V_{1}V^{\dagger}_{1})\otimes\mathrm{tr}_{2}(V_{1}V^{\dagger}_{1})+\frac{1}{8}\mathrm{tr}_{2}(V_{2}V^{\dagger}_{2})\otimes\mathrm{tr}_{2}(V_{2}V^{\dagger}_{2}),$

where $\mathrm{tr}_{2}$ is the partial trace over $\mathbb{C}^{2}$. In the
second line $\tilde{\mathcal{E}}$ is the channel obtained from
$\mathcal{E}:\mathbb{C}^{4}\rightarrow\mathbb{C}^{4}$,
$\mathcal{E}(X)=\mathrm{tr}_{3}(VXV^{\dagger})$, via Definition 1, which acts
as

$\displaystyle\tilde{\mathcal{E}}\begin{pmatrix}X_{11}&X_{12}\\\
X_{21}&X_{22}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\tilde{\mathcal{E}}_{1}(X_{11})&0\\\
0&\tilde{\mathcal{E}}_{1}(X_{22})\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\frac{1}{2}\mathrm{tr}_{2}(X_{11})\mathbb{1}_{2}&0\\\
0&\frac{1}{2}\mathrm{tr}_{2}(X_{22})\mathbb{1}_{2}\end{pmatrix}.$

Then

$\displaystyle\tilde{\rho}_{A}\otimes\tilde{\rho}_{C}=\rho_{A}\otimes\rho_{C}=\frac{1}{16}$
$\displaystyle\left(\mathrm{tr}_{2}(V_{1}V^{\dagger}_{1})+\mathrm{tr}_{2}(V_{2}V^{\dagger}_{2})\right)$
$\displaystyle\otimes\left(\mathrm{tr}_{2}(V_{1}V^{\dagger}_{1})+\mathrm{tr}_{2}(V_{2}V^{\dagger}_{2})\right).$

Using the explicit expressions (39) and (40), we calculate

$\displaystyle\tilde{\rho}_{AC}=$
$\displaystyle\frac{1}{32}\begin{pmatrix}1&0&0\\\ 0&2&0\\\
0&0&1\end{pmatrix}\otimes\begin{pmatrix}1&0&0\\\ 0&2&0\\\ 0&0&1\end{pmatrix}$
$\displaystyle+\frac{1}{32}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&2\end{pmatrix}\otimes\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&2\end{pmatrix},$
$\displaystyle\tilde{\rho}_{A}\otimes\tilde{\rho}_{C}=$
$\displaystyle\frac{1}{64}\begin{pmatrix}2&0&0\\\ 0&3&0\\\
0&0&3\end{pmatrix}\otimes\begin{pmatrix}2&0&0\\\ 0&3&0\\\ 0&0&3\end{pmatrix}.$

We observe that
$\tilde{\rho}_{AC}\neq\tilde{\rho}_{A}\otimes\tilde{\rho}_{C}$, which implies
that $I(A:C)\neq 0$. We verify this by explicitly calculating $I(A:C)={17}\ln
2/{16}-9\ln 3/8+5\ln 5/16\approx 0.0035$.

Now we construct an ergodic pgFCS of period $2$ for which QMI does not
converge to zero. We use the same setup as above, changing only the elementary
isometry generating the state to

$V=\begin{pmatrix}0&V_{1}\\\ V_{2}&0\end{pmatrix},$

where $V_{1}$ and $V_{2}$ are the same isometries (39) and (40), respectively.
One can check that the only eigenvalues of absolute value $1$ of
$\mathcal{E}(X)=\mathrm{tr}_{3}(VXV^{\dagger})$ are $1$ and $-1$,
corresponding to $\left(\begin{smallmatrix}\mathbb{1}_{2}/4&0\\\
0&\mathbb{1}_{2}/4\end{smallmatrix}\right)$ and
$\left(\begin{smallmatrix}\mathbb{1}_{2}/4&0\\\
0&-\mathbb{1}_{2}/4\end{smallmatrix}\right)$, respectively. Hence the
generated state is indeed an ergodic pgFCS with period 2. To determine the
asymptotic behavior of $\rho_{AC}$ we explicitly express it in terms of
$V_{1}$ and $V_{2}$,

$\rho_{ABC}=\begin{cases}\begin{aligned}
&\frac{1}{4}\mathrm{tr}_{2}(V_{1}V_{2}\cdot\cdot\cdot
V_{2}V_{1}\frac{\mathbb{1}_{2}}{2}V^{\dagger}_{1}V^{\dagger}_{2}\cdot\cdot\cdot
V^{\dagger}_{2}V^{\dagger}_{1})\\\
&+\frac{1}{4}\mathrm{tr}_{2}(V_{2}V_{1}\cdot\cdot\cdot
V_{1}V_{2}\frac{\mathbb{1}_{2}}{2}V^{\dagger}_{2}V^{\dagger}_{1}\cdot\cdot\cdot
V^{\dagger}_{1}V^{\dagger}_{2}),\\\ &\quad\quad\quad\text{if}\ \
|B|\,\mbox{mod}\,2\equiv 1,\end{aligned}\\\ \begin{aligned}
&\frac{1}{4}\mathrm{tr}_{2}(V_{2}V_{1}\cdot\cdot\cdot
V_{2}V_{1}\frac{\mathbb{1}_{2}}{2}V^{\dagger}_{1}V^{\dagger}_{2}\cdot\cdot\cdot
V^{\dagger}_{1}V^{\dagger}_{2})\\\
&+\frac{1}{4}\mathrm{tr}_{2}(V_{1}V_{2}\cdot\cdot\cdot
V_{1}V_{2}\frac{\mathbb{1}_{2}}{2}V^{\dagger}_{2}V^{\dagger}_{1}\cdot\cdot\cdot
V^{\dagger}_{2}V^{\dagger}_{1}),\\\ &\quad\quad\quad\text{if}\ \
|B|\,\mbox{mod}\,2\equiv 0.\end{aligned}\end{cases}$

Since $V_{2}=U_{s}V_{1}$ for a unitary
$U_{s}:\mathbb{C}^{3}\rightarrow\mathbb{C}^{3}$, then
$\mathrm{tr}_{3}(V_{1}XV_{1}^{\dagger})=\mathrm{tr}_{3}(V_{2}XV^{\dagger}_{2})$
for any $X\in\mathcal{B}(\mathcal{H}_{M})$. Notice that for any choice of the
indices $\\{i_{k}\in\\{1,2\\}\;|\;k=1,2,\cdot\cdot\cdot,|B|\\}$, we have
$\lim_{|B|\rightarrow+\infty}\mathrm{tr}_{B}({V_{i_{|B|}}}\cdot\cdot\cdot
V_{i_{2}}V_{i_{1}}XV_{i_{1}}^{\dagger}V_{i_{2}}^{\dagger}\cdot\cdot\cdot
V^{\dagger}_{i_{|B|}})=\lim_{|B|\rightarrow+\infty}\mathcal{E}_{1}^{|B|}(X)=\tilde{\mathcal{E}}_{1}^{|B|}(X)=\mathrm{tr}(X)\mathbb{1}_{2}/2$.
This implies that in the asymptotic limit $|B|\rightarrow+\infty$, the state
$\rho_{AC}$ oscillates between two states,

$\displaystyle\tilde{\rho}^{(1)}_{AC}$
$\displaystyle=\frac{1}{8}\mathrm{tr}_{2}(V_{1}V^{\dagger}_{1})\otimes\mathrm{tr}_{2}(V_{1}V^{\dagger}_{1})+\frac{1}{8}\mathrm{tr}_{2}(V_{2}V^{\dagger}_{2})\otimes\mathrm{tr}_{2}(V_{2}V^{\dagger}_{2}),$
$\displaystyle\tilde{\rho}^{(2)}_{AC}$
$\displaystyle=\frac{1}{8}\mathrm{tr}_{2}(V_{1}V^{\dagger}_{1})\otimes\mathrm{tr}_{2}(V_{2}V^{\dagger}_{2})+\frac{1}{8}\mathrm{tr}_{2}(V_{2}V^{\dagger}_{2})\otimes\mathrm{tr}_{2}(V_{1}V^{\dagger}_{1}).$

For both of these states,

$\displaystyle\tilde{\rho}^{(1)}_{A}\otimes\tilde{\rho}^{(1)}_{C}=\tilde{\rho}^{(2)}_{A}\otimes\tilde{\rho}^{(2)}_{C}$
$\displaystyle=\frac{1}{16}\left(\mathrm{tr}_{2}(V_{1}V^{\dagger}_{1})+\mathrm{tr}_{2}(V_{2}V^{\dagger}_{2})\right)\otimes\left(\mathrm{tr}_{2}(V_{1}V^{\dagger}_{1})+\mathrm{tr}_{2}(V_{2}V^{\dagger}_{2})\right).$

Using the explicit expressions (39) and (40), we obtain

$\displaystyle\tilde{\rho}^{(1)}_{AC}=\frac{1}{32}\begin{pmatrix}1&0&0\\\
0&2&0\\\ 0&0&1\end{pmatrix}\otimes\begin{pmatrix}1&0&0\\\ 0&2&0\\\
0&0&1\end{pmatrix}$
$\displaystyle\quad\quad\quad+\frac{1}{32}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&2\end{pmatrix}\otimes\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&2\end{pmatrix},$
$\displaystyle\tilde{\rho}^{(2)}_{AC}=\frac{1}{32}\begin{pmatrix}1&0&0\\\
0&2&0\\\ 0&0&1\end{pmatrix}\otimes\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&2\end{pmatrix}$
$\displaystyle\quad\quad\quad+\frac{1}{32}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&2\end{pmatrix}\otimes\begin{pmatrix}1&0&0\\\ 0&2&0\\\ 0&0&1\end{pmatrix},$
$\displaystyle\tilde{\rho}^{(1)}_{A}\otimes\tilde{\rho}^{(1)}_{C}=\tilde{\rho}^{(2)}_{A}\otimes\tilde{\rho}^{(2)}_{C}=\frac{1}{64}\begin{pmatrix}2&0&0\\\
0&3&0\\\ 0&0&3\end{pmatrix}\otimes\begin{pmatrix}2&0&0\\\ 0&3&0\\\
0&0&3\end{pmatrix}.$

An explicit calculation shows that $\tilde{\rho}^{(1)}_{AC}$ and
$\tilde{\rho}^{(2)}_{AC}$ have the same QMI, calculated in the previous
example, $I(A:C)={17}\ln 2/{16}-9\ln 3/8+5\ln 5/16\approx 0.0035$. Thus, while
for this ergodic pgFCS with period 2 QMI actually converges, it still does not
converge to zero.

## Appendix E

In this appendix we present a pair of lemmata for completeness.

The first lemma is a restatement of the Lemma A.2 from Ref.Cirac  _et al._ ,
2017 in a slightly different language.

######  Lemma E.1 (Lemma A.2 of Ref. Cirac  _et al._ , 2017).

Let isometries
$V_{1}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{s}\otimes\mathcal{H}_{M}$ and
$V_{2}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{s}\otimes\mathcal{H}_{M}$ be
such that full-rank density operators
$\sigma_{1},\sigma_{2}\in\mathcal{B}(\mathcal{H}_{M})$ are the only fixed
points of the quantum channels
$X\mapsto\mathrm{tr}_{s}(V_{1}XV^{\dagger}_{1})$ and
$X\mapsto\mathrm{tr}_{s}(V_{2}XV^{\dagger}_{2})$,
$X\in\mathcal{B}(\mathcal{H}_{M})$, respectively, and that these quantum
channels have no other eigenvalues of magnitude 1. Then the eigenvalues
$\nu_{i}$ of the map $X\mapsto\mathrm{tr}_{s}(V_{1}XV^{\dagger}_{2})$ are such
that $|\nu_{i}|\leq 1.$ Furthermore, if there exists the eigenvalue $\nu_{i}$
of magnitude $1$, then $V_{1}=e^{-i\phi}(\mathbb{1}_{s}\otimes
W^{\dagger})V_{2}W$ for some $\phi\in\mathbb{R}$ and unitary
$W:\mathcal{H}_{M}\rightarrow\mathcal{H}_{M}$.

###### Proof.

Let $\nu_{i}$ be an eigenvalue, so that for some $X$:

$\mbox{tr}_{s}(V_{1}XV^{\dagger}_{2})=\nu_{i}X.$

Using the fact that $\sigma_{1}$ is invertible, consider:

$\displaystyle\left|\nu_{i}\mbox{tr}(X^{\dagger}\sigma_{1}^{-1}X)\right|^{2}=\left|\mbox{tr}(X^{\dagger}\sigma_{1}^{-1}\mbox{tr}_{s}(V_{1}XV^{\dagger}_{2}))\right|^{2}$
(41) $\displaystyle=\left|\mbox{tr}((\mathbb{1}_{s}\otimes
X^{\dagger}\sigma_{1}^{-1})V_{1}{\sigma_{1}}^{1/2}\sigma_{1}^{-1/2}XV^{\dagger}_{2})\right|^{2}$
$\displaystyle\leq\mbox{tr}(X^{\dagger}\sigma_{1}^{-1}\mbox{tr}_{s}(V_{1}{\sigma_{1}}V^{\dagger}_{1})\sigma_{1}^{-1}X)\mbox{tr}(V_{2}X^{\dagger}\sigma_{1}^{-1}XV^{\dagger}_{2})$
$\displaystyle\leq\mbox{tr}(X^{\dagger}\sigma_{1}^{-1}X)^{2},$

where we have used the Cauchy–Schwarz inequality for the Hilbert-Schmidt inner
product
$|\mathrm{tr}(A^{\dagger}B)|^{2}\leq\mathrm{tr}(A^{\dagger}A)\mathrm{tr}(B^{\dagger}B)$
to get from the second line to the third. Thus $|\nu_{i}|\leq 1$. Moreover
$|\nu_{i}|=1$ only in the case of equality in the Cauchy–Schwarz inequality,
which implies:

$\sigma_{1}^{-1/2}XV^{\dagger}_{2}=e^{i\phi}{\sigma_{1}}^{1/2}V^{\dagger}_{1}(\mathbb{1}_{s}\otimes\sigma_{1}^{-1}X),$

where $\phi\in\mathbb{R}$. Using that $\sigma_{1}^{-1/2}$ is invertible and
denoting $W=\sigma_{1}^{-1}X$, we get:

$WV^{\dagger}_{2}=e^{i\phi}V^{\dagger}_{1}(\mathbb{1}_{s}\otimes W).$

Using that $V_{2}^{\dagger}V_{2}=\mathbb{1}_{M}$, we obtain:

$WW^{\dagger}=V_{1}^{\dagger}(\mathbb{1}_{s}\otimes WW^{\dagger})V_{1}.$

This means that $WW^{\dagger}$ is the eigenvector with eigenvalue $1$ of the
map $X\mapsto V_{1}^{\dagger}(\mathbb{1}_{s}\otimes X)V_{1}$. The latter map
is adjoint to the quantum channel
$X\mapsto\mathrm{tr}_{s}(V_{1}XV_{1}^{\dagger}),$ with respect to the Hilbert-
Schmidt inner product, therefore it has the same spectrum. Hence the adjoint
map has only one eigenvector with eigenvalue $1$. Since $V_{1}$ is an
isometry,
$V_{1}^{\dagger}(\mathbb{1}_{s}\otimes\mathbb{1}_{M})V_{1}=\mathbb{1}_{M}$,
the identity operator $\mathbb{1}_{M}$ is this eigenvector, and
$W^{\dagger}W=\mathbb{1}_{M}$. Replacing
$\mathrm{tr}(X^{\dagger}\sigma_{1}^{-1}X)$ with
$\mathrm{tr}(X\sigma_{2}^{-1}X^{\dagger})$ in (41) leads through the same
reasoning to $WW^{\dagger}=\mathbb{1}_{M}$. Thus $W$ is unitary and

$V_{2}=e^{-i\phi}(\mathbb{1}_{s}\otimes W^{\dagger})V_{1}W.$

∎

The second lemma is a restatement of the result used in Ref.Fannes  _et al._ ,
1992, the formal proof of which can be found in Ref.Szehr  _et al._ , 2015. We
present the proof using our notation.

######  Lemma E.2 (Theorem III.2 of Ref. Szehr  _et al._ , 2015, Ref. Fannes
_et al._ , 1992).

Let $\mathcal{E}:\mathcal{B}(\mathcal{K})\rightarrow\mathcal{B}(\mathcal{K})$
be a map with the spectral radius $1$, and let $\tilde{\mathcal{E}}$ be the
map obtained from $\mathcal{E}$ as described in Definition 1. Then for any
$\nu\in\mathbb{R}$, such that $\nu_{\mathrm{gap}}<\nu<1$, where
$\nu_{\mathrm{gap}}:=\max_{|\nu_{i}|<1}|\nu_{i}|$, there exist the constant
$c>0$, depending on $\nu$, such that

$\|\mathcal{E}^{n}-\tilde{\mathcal{E}}^{n}\|_{2\rightarrow 2}\leq c\nu^{n}.$

###### Proof.

Following the procedure described in Appendix F, we represent the maps
$\mathcal{E}$ and $\tilde{\mathcal{E}}$ with the operators $E$ and
$\tilde{E}$, respectively, that inherit the spectra of the maps.

As is shown in Appendix F,

$\|\mathcal{E}^{n}-\tilde{\mathcal{E}}^{n}\|_{2\rightarrow
2}=\|E^{n}-\tilde{E}^{n}\|.$ (42)

Then to prove the statement of the lemma we can estimate
$\|E^{n}-\tilde{E}^{n}\|$.

We make the Jordan decomposition of $E$,

$E=\sum_{i}(\nu_{i}P_{\nu_{i}}+N_{\nu_{i}}),$

where $\nu_{i}$ is an eigenvalue, $P_{\nu_{i}}$ is the projector onto the
subspace corresponding to the eigenvalue $\nu_{i}$, and $N_{\nu_{i}}$ is a
nilpotent operator with index $(K_{i}+1)$, $N_{\nu_{i}}^{K_{i}+1}=0$. The
Jordan decomposition for $\tilde{E}$ can be obtained by setting all $\nu_{i}$,
except for the ones of magnitude 1, to 0,

$\tilde{E}=\sum_{|\nu_{i}|=1}(\nu_{i}P_{\nu_{i}}+N_{\nu_{i}}),$

From the properties of the operators $P_{\nu_{i}}$ and $N_{\nu_{i}}$ in the
Jordan decomposition, $P_{\nu_{i}}P_{\nu_{j}}=\delta_{ij}P_{\nu_{i}}$,
$P_{\nu_{i}}N_{\nu_{j}}=N_{\nu_{j}}P_{\nu_{i}}=\delta_{ij}N_{\nu_{i}}$, and
$N_{\nu_{i}}N_{\nu_{k}}=\delta_{ik}N_{\nu_{i}}^{2}$, it follows that we can
express $E^{n}$ as

$E^{n}=\sum_{i}({\nu_{i}}P_{\nu_{i}}+N_{\nu_{i}})^{n}.$

For all peripheral eigenvalues $\nu_{i}$, _i.e._ such that $|\nu_{i}|=1$,
$N_{\nu_{i}}=0$ (see, for example, the proof of Proposition 3.1 in Ref.Fannes
_et al._ , 1992). Thus, we can write the expression for $E^{n}$ as

$E^{n}=\sum_{|\nu_{i}|=1}\nu^{n}_{i}P_{\nu_{i}}+\sum_{|{\nu_{i}}|<1}({\nu_{i}}P_{\nu_{i}}+N_{\nu_{i}})^{n}.$

It immediately follows that

$\tilde{E}^{n}=\sum_{|\nu_{i}|=1}\nu^{n}_{i}P_{\nu_{i}}.$

Then we can estimate

$\displaystyle\|E^{n}-\tilde{E}^{n}\|^{2}=\|\sum_{|\nu_{i}|<1}({\nu_{i}}P_{\nu_{i}}+N_{\nu_{i}})^{n}\|^{2}$
(43)
$\displaystyle=\sup_{\||\psi\rangle\|=1}\langle\psi|\sum_{|\nu_{i}|<1}(\nu_{i}^{*}P_{\nu_{i}}+N_{\nu_{i}}^{\dagger})^{n}({\nu_{i}}P_{\nu_{i}}+N_{\nu_{i}})^{n}|\psi\rangle$
$\displaystyle=\sum_{|\nu_{i}|<1}\sup_{\||\psi\rangle\|=1}\langle\psi|P_{\nu_{i}}(\nu_{i}^{*}P_{\nu_{i}}+N_{\nu_{i}}^{\dagger})^{n}({\nu_{i}}P_{\nu_{i}}+N_{\nu_{i}})^{n}P_{\nu_{i}}|\psi\rangle$
$\displaystyle\leq\sum_{|\nu_{i}|<1}\|(\nu_{i}P_{\nu_{i}}+N_{\nu_{i}})^{n}\|^{2}\sup_{\||\psi\rangle\|=1}\langle\psi|P_{\nu_{i}}^{2}|\psi\rangle$
$\displaystyle\leq\max_{0<|\nu_{i}|<1}\|(\nu_{i}P_{\nu_{i}}+N_{\nu_{i}})^{n}\|^{2}\sup_{\||\psi\rangle\|=1}\langle\psi|\sum_{|\nu_{i}|<1}P_{\nu_{i}}|\psi\rangle$
$\displaystyle\leq\max_{0<|\nu_{i}|<1}\|(\nu_{i}P_{\nu_{i}}+N_{\nu_{i}})^{n}\|^{2}\sup_{\||\psi\rangle\|=1}\langle\psi|\sum_{\nu_{i}}P_{\nu_{i}}|\psi\rangle$
$\displaystyle\leq\max_{0<|\nu_{i}|<1}\|(\nu_{i}P_{\nu_{i}}+N_{\nu_{i}})^{n}\|^{2}.$

We have used the definition of the operator norm and orthogonality of the
terms $\nu_{i}P_{\nu_{i}}+N_{\nu_{i}}$ to get the second line. The third line
follows from the identity
$\nu_{i}P_{\nu_{i}}+N_{\nu_{i}}=(\nu_{i}P_{\nu_{i}}+N_{\nu_{i}})P_{\nu_{i}},$
and the bound
$\langle\psi|A^{\dagger}A|\psi\rangle=\|A|\psi\rangle\|^{2}\leq\|A\|^{2}\langle\psi|\psi\rangle$
is used in the fourth line. The sixth and seventh lines follow from properties
of the orthogonal projectors
$\sum_{|\nu_{i}|<1}P_{\nu_{i}}\leq\sum_{\nu_{i}}P_{\nu_{i}}=\mathbb{1}.$

Now, using the fact that $N_{\nu_{i}}$ is nilpotent with the index $K_{i}+1$,
we estimate

$\displaystyle\|(\nu_{i}P_{\nu_{i}}+N_{\nu_{i}})^{n}\|$
$\displaystyle=\|P_{\nu_{i}}\sum_{k=0}^{K_{i}}\binom{n}{k}\nu_{i}^{n-k}N_{\nu_{i}}^{k}\|$
$\displaystyle\leq|\nu_{i}|^{n}\sum_{k=0}^{K_{i}}\binom{n}{k}\frac{\|N_{\nu_{i}}\|^{k}}{|\nu_{i}|^{k}}$
$\displaystyle\leq|\nu_{i}|^{n}\sum_{k=0}^{K_{i}}\frac{n^{k}}{k!}\frac{\|N_{\nu_{i}}\|^{k}}{|\nu_{i}|^{k}}$
$\displaystyle\leq|\nu_{i}|^{n}\sum_{k=0}^{K_{i}}\binom{K_{i}}{k}\left(\frac{n\|N_{\nu_{i}}\|}{|\nu_{i}|}\right)^{k}$
$\displaystyle=|\nu_{i}|^{n}\left(\frac{n\|N_{\nu_{i}}\|}{|\nu_{i}|}+1\right)^{K_{i}},$

where we have used the triangle inequality and the inequality
$\binom{n}{k}\leq\frac{n^{k}}{k!}\leq\binom{K_{i}}{k}n^{k}$ for $k\leq K_{i}$.
Denoting
$c_{1}:=\max_{0<|\nu_{i}|<1}\left\\{\|N_{\nu_{i}}\|/|\nu_{i}|)\right\\}$, and
$K_{\mathrm{max}}=\max_{0<|\nu_{i}|<1}K_{i}$, we bound

$\displaystyle\max_{0<|\nu_{i}|<1}\|(\nu_{i}P_{\nu_{i}}+N_{\nu_{i}})^{n}\|\leq\nu_{\mathrm{gap}}^{n}\left(c_{1}n+1\right)^{K_{\mathrm{max}}}.$
(44)

Combining (42), (43), and (44) together,

$\|\mathcal{E}^{n}-\tilde{\mathcal{E}}^{n}\|_{2\rightarrow
2}\leq\nu_{\mathrm{gap}}^{n}\left(c_{1}n+1\right)^{K_{\mathrm{max}}},$

manifesting an almost exponential decay. Using a slightly weaker bound, we can
make the decay to be strictly exponential. We choose some
$\nu_{\mathrm{gap}}<\nu<1$, expand

$\nu_{\mathrm{gap}}^{n}\left(c_{1}n+1\right)^{K_{\mathrm{max}}}=\left(\frac{\nu_{\mathrm{gap}}}{\nu}\right)^{n}\left(c_{1}n+1\right)^{K_{\mathrm{max}}}\nu^{n},$

and notice that first two factors can be bounded by a constant $c$, depending
on $\nu$

$\left(\frac{\nu_{\mathrm{gap}}}{\nu}\right)^{n}\left(c_{1}n+1\right)^{K_{\mathrm{max}}}\leq
c,$

which leads to the stated estimate,

$\|\mathcal{E}^{n}-\tilde{\mathcal{E}}^{n}\|_{2\rightarrow 2}\leq c\nu^{n}.$

∎

## Appendix F

In this appendix we review the matrix representation of a quantum
channel.Watrous (2018)

The map
$\mathcal{E}:\mathcal{B}(\mathcal{H}_{M})\rightarrow\mathcal{B}(\mathcal{H}_{M})$
can be represented by the operator
$E:\mathcal{H}_{M}\otimes\mathcal{H}_{M}\rightarrow\mathcal{H}_{M}\otimes\mathcal{H}_{M}$
through the process of vectorization.Watrous (2018) For the chosen orthonormal
basis $\\{|i\rangle:i=1\cdot\cdot\cdot d_{M}\\}$ of $\mathcal{H}_{M}$ the
invertible map,
$\mathrm{vec}:\mathcal{B}(\mathcal{H}_{M})\rightarrow\mathcal{H}_{M}\otimes\mathcal{H}_{M}$,
is defined through its action on the operators $|i\rangle\langle j|$, which
constitute an orthonormal basis of $\mathcal{B}(\mathcal{H}_{M})$ (with
respect to the Hilbert-Schmidt inner product),

$\mbox{vec}(|i\rangle\langle j|)=|i\rangle\otimes|j\rangle,\ \
i,j=1\cdot\cdot\cdot d_{M}.$ (45)

The inner product on the Hilbert space $\mathcal{H}_{M}\otimes\mathcal{H}_{M}$
may then be related to the Hilbert-Schmidt inner product on the Hilbert space
$\mathcal{B}(\mathcal{H}_{M})$, by

$\langle\mbox{vec}(X)|\mbox{vec}(Y)\rangle_{\mathcal{H}_{M}\otimes\mathcal{H}_{M}}=\mbox{tr}(X^{\dagger}Y)=\langle
X,Y\rangle_{\mathcal{B}(\mathcal{H}_{M})},$ (46)

for $X,Y\in\mathcal{B}(\mathcal{H}_{M}).$ It follows immediately that

$\|X\|_{2}=\|\mathrm{vec}(X)\|,$ (47)

where
$\||\Psi\rangle\|=\langle\Psi|\Psi\rangle^{\frac{1}{2}}_{\mathcal{H}_{M}\otimes\mathcal{H}_{M}}$
is the norm of the vector.

Define the operator
$E:\mathcal{H}_{M}\otimes\mathcal{H}_{M}\rightarrow\mathcal{H}_{M}\otimes\mathcal{H}_{M}$
by

$E:=\mbox{vec}\circ\mathcal{E}\circ\mbox{vec}^{-1},$ (48)

leading to the explicit form of the matrix elements of $E$,

$\langle m|\otimes\langle n|E|i\rangle\otimes|j\rangle=\langle
m|\mathcal{E}(|i\rangle\langle j|)|n\rangle,\quad i,j,m,n=1\cdot\cdot\cdot
d_{M}.$

The relations (46) and (47) imply that the $2\rightarrow 2$ norm of the map
$\mathcal{E}$ can be expressed through the operator norm of $E$,

$\displaystyle\|\mathcal{E}\|_{2\rightarrow 2}$
$\displaystyle:=\sup_{\|X\|_{2}=1}\|\mathcal{E}(X)\|_{2}$
$\displaystyle=\sup_{\|\mbox{vec}(X)\|=1}\|\mbox{vec}\circ\mathcal{E}\circ\mbox{vec}^{-1}\circ\mbox{vec}(X)\|$
$\displaystyle=\sup_{\||x\rangle\|=1}\|E|x\rangle\|$ $\displaystyle=\|E\|.$

The spectra of the map $\mathcal{E}$ and the operator $E$ coincide: if
$\mathcal{E}(X)=\lambda X$ for some $X\in\mathcal{B}(\mathcal{H}_{M})$, then
$E\mathrm{vec}(X)=\lambda\cdot\mathrm{vec}(X)$. Conversely, if
$E|\psi\rangle=\lambda|\psi\rangle$ for some
$|\psi\rangle\in\mathcal{H}_{M}\otimes\mathcal{H}_{M}$, then
$\mathcal{E}\left(\mathrm{vec}^{-1}(|\psi\rangle)\right)=\lambda\cdot\mathrm{vec}^{-1}(|\psi\rangle)$.

## Appendix G

In this appendix we show that without loss of generality, we can assume that
the decomposition (13) of a pgFCS into the sum of ergodic components does not
contain terms that are essentially equal in the sense described below.

Let us suppose there exists $\phi\in\mathbb{R}$ and a unitary
$W:\mathcal{H}_{M}\rightarrow\mathcal{H}_{M}$, such that the isometries
$V_{\alpha}$ and $V_{\beta}$ satisfy

$V_{\beta}=e^{-i\phi}(\mathbb{1}_{s}\otimes W^{\dagger})V_{\alpha}W.$ (49)

This condition defines an equivalence relation and the isometries $V_{\alpha}$
and $V_{\beta}$ are then said to be equivalent.

In this appendix we show that the general convex sum representation of the FCS
may be written as

$\rho_{ABC}=\sum_{\alpha=1}^{J}\lambda_{\alpha}\rho_{ABC}^{\alpha},$ (50)

where the $\rho_{ABC}^{\alpha}$ define a collection of $J$ ergodic pgFCS
constructed as in (12), with a collection of inequivalent isometries
$V_{\alpha}$ and full-rank density operators $\sigma_{\alpha}.$

Suppose the converse is true, then

$\displaystyle\rho^{\beta}_{ABC}$
$\displaystyle=\mbox{tr}_{M}\left(V^{\beta}_{C}V^{\beta}_{B}V^{\beta}_{A}\sigma_{\beta}V_{A}^{\beta\dagger}V_{B}^{\beta\dagger}V_{C}^{\beta\dagger}\right)$
(51) $\displaystyle=\mbox{tr}_{M}\left((\mathbb{1}_{ABC}\otimes
W^{\dagger})V^{\alpha}_{C}V^{\alpha}_{B}V^{\alpha}_{A}W\sigma_{\beta}W^{\dagger}\times\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.V_{A}^{\alpha\dagger}V_{B}^{\alpha\dagger}V_{C}^{\alpha\dagger}(\mathbb{1}_{ABC}\otimes
W)\right)$
$\displaystyle=\mbox{tr}_{M}\left(V^{\alpha}_{C}V^{\alpha}_{B}V^{\alpha}_{A}W\sigma_{\beta}W^{\dagger}V_{A}^{\alpha\dagger}V_{B}^{\alpha\dagger}V_{C}^{\alpha\dagger}\right).$

Now we show that the density operator $W\sigma_{\beta}W^{\dagger}$ is a fixed
point of the quantum channel
$X\mapsto\mathrm{tr}_{s}(V_{\alpha}XV^{\dagger}_{\alpha})$,

$\displaystyle\mbox{tr}_{s}(V_{\alpha}W\sigma_{\beta}W^{\dagger}V^{\dagger}_{\alpha})$
$\displaystyle=W\mbox{tr}_{s}\left((\mathbb{1}\otimes
W^{\dagger})V_{\alpha}W\sigma_{\beta}W^{\dagger}\times\right.$
$\displaystyle\qquad\qquad\qquad\left.V^{\dagger}_{\alpha}(\mathbb{1}\otimes
W)\right)W^{\dagger}$
$\displaystyle=W\mbox{tr}_{s}(V_{\beta}\sigma_{\beta}V^{\dagger}_{\beta}))W^{\dagger}$
$\displaystyle=W\sigma_{\beta}W^{\dagger},$

where we have used the fact that $\sigma_{\beta}$ is a fixed point of the map
$X\mapsto\mathrm{tr}_{s}(V_{\beta}XV^{\dagger}_{\beta})$. Thus,
$W\sigma_{\beta}W^{\dagger}$ is the fixed point of the quantum channel
$X\mapsto\mathrm{tr}_{s}(V_{\alpha}XV^{\dagger}_{\alpha})$. Since
$\rho_{ABC}^{\alpha}$ is ergodic, the density operator $\sigma_{\alpha}$ is
the only fixed point of this map. Then
$W\sigma_{\beta}W^{\dagger}=\sigma_{\alpha}$. Combining this fact with (51),
we obtain that $\rho^{\alpha}_{ABC}=\rho^{\beta}_{ABC}$. Thus,
$\lambda_{\alpha}\rho^{\alpha}_{ABC}+\lambda_{\beta}\rho^{\beta}_{ABC}=(\lambda_{\alpha}+\lambda_{\beta})\rho^{\alpha}_{ABC}$.
We can combine in the convex sum (50) all the terms satisfying the relation
(49) together.

## Appendix H

In this appendix we present a technical result, required in the proof of
Theorem II.1 in Section IV.2,

$V_{A}\sigma
V^{\dagger}_{A}\in\mathcal{B}(\mathcal{H}_{A})\otimes\bigoplus_{k=0}^{p-1}\Pi_{k}\mathcal{B}(\mathcal{H}_{M})\Pi_{k}.$

We closely follow the reasoning of the proof of Proposition 3.3 of Ref. Fannes
_et al._ , 1992. Here
$V_{A}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{A}\otimes\mathcal{H}_{M}$ is the
isometry generating the spins in the region $A$ as in (12), and
$\sigma\in\mathcal{D}(\mathcal{H}_{M})$ and the projectors
$\\{\Pi_{k}\in\mathcal{B}(\mathcal{H}_{M})\;|\;k=0,\cdot\cdot\cdot,p-1\\}$,
$\sum_{k=0}^{p-1}\Pi_{k}=\mathbb{1}_{M}$, are defined in Proposition 2.

Let us express

$V_{A}\sigma
V_{A}^{\dagger}=\sum_{n,m=1}^{d_{s}^{|A|}}|\phi_{n}\rangle\langle\phi_{m}|\otimes\langle\phi_{n}|V_{A}\sigma
V_{A}^{\dagger}|\phi_{m}\rangle,$

where
$\\{|\phi_{n}\rangle\in\mathcal{H}_{A}|\;n=1,\cdot\cdot\cdot,d_{s}^{|A|}\\}$
is an orthonormal basis. It is clear that

$\langle\phi_{n}|V_{A}\sigma
V_{A}^{\dagger}|\phi_{m}\rangle\in\mathrm{span}\\{\langle\zeta|V_{A}\sigma
V^{\dagger}_{A}|\psi\rangle\;|\;|\zeta\rangle,|\psi\rangle\in\mathcal{H}_{A}\\}.$

Notice that $\langle\zeta|V_{A}\sigma
V^{\dagger}_{A}|\psi\rangle=\mathrm{tr}_{A}(|\psi\rangle\langle\zeta|V_{A}\sigma
V^{\dagger}_{A})$ and the vector space
$\mathrm{span}\\{|\psi\rangle\langle\zeta|\;|\;|\zeta\rangle,|\psi\rangle\in\mathcal{H}_{A}\\}=\mathcal{B}(\mathcal{H}_{A})$
can be spanned by positive semi-definite operators
$O_{A}\in\mathcal{B}(\mathcal{H}_{A})$. Hence,

$\displaystyle\mathcal{B}(\mathcal{H}_{M})=\mathrm{span}\\{\mathrm{tr}_{A}(O_{A}V_{A}\sigma
V^{\dagger}_{A})\;|\;O_{A}\geq 0,\,O_{A}\in\mathcal{B}(\mathcal{H}_{A})\\}.$

Since any $O_{A}\geq 0$ satisfies $O_{A}\leq\|O_{A}\|\mathbb{1}_{A}$, it
follows that

$\mathrm{tr}_{A}(O_{A}V_{A}\sigma
V_{A}^{\dagger})\leq\|O_{A}\|\mathrm{tr}_{A}(V_{A}\sigma
V^{\dagger}_{A})=\|O_{A}\|\sigma,$

where we have used $\mathrm{tr}_{A}(V_{A}\sigma V^{\dagger}_{A})=\sigma$. The
operators $\mathrm{tr}_{A}(O_{A}V_{A}\sigma V_{A}^{\dagger})$ and
$\|O_{A}\|\sigma$ can be ordered in this way only if
$\mathrm{tr}_{A}(O_{A}V_{A}\sigma
V_{A}^{\dagger})\in\bigoplus_{k=0}^{p-1}\Pi_{k}\mathcal{B}(\mathcal{H}_{M})\Pi_{k}$,
since
$\sigma\in\bigoplus_{k=0}^{p-1}\Pi_{k}\mathcal{B}(\mathcal{H}_{M})\Pi_{k}$.
Then

$\displaystyle\mathrm{span}\\{\mathrm{tr}_{A}(O_{A}V_{A}\sigma
V^{\dagger}_{A})\;|\;O_{A}\geq 0,\,O_{A}\in\mathcal{B}(\mathcal{H}_{A})\\}$
$\displaystyle\in\bigoplus_{r=1}^{p}\Pi_{k}\mathcal{B}(\mathcal{H}_{M})\Pi_{k},$

from which it follows that $\langle\zeta|V_{A}\sigma
V^{\dagger}_{A}|\psi\rangle\in\bigoplus_{k=0}^{p-1}\Pi_{k}\mathcal{B}(\mathcal{H}_{M})\Pi_{k}$,
and

$V_{A}\sigma
V^{\dagger}_{A}\in\mathcal{B}(\mathcal{H}_{A})\otimes\bigoplus_{k=0}^{p-1}\Pi_{k}\mathcal{B}(\mathcal{H}_{M})\Pi_{k}.$

This leads to the identity $V_{A}\sigma
V_{A}=\sum_{k=0}^{p-1}\Pi_{k}V_{A}\sigma V_{A}^{\dagger}\Pi_{k}$.

## Appendix I

In this appendix we prove that the quantum channel $\tilde{\mathcal{E}}$
obtained from $\mathcal{E}$ by Definition 1 of Section III.2, and used in
Section IV.2, takes the form

$\tilde{\mathcal{E}}(X)=p\sum_{r=0}^{p-1}\mbox{tr}\left(\Pi_{r}X\Pi_{r}\right)\Pi_{r+1}\sigma\Pi_{r+1}.$

We use the notation established in Section IV.2.

Let $E\in\mathcal{B}(\mathcal{H}_{M}\otimes\mathcal{H}_{M})$ and
$\tilde{E}\in\mathcal{B}(\mathcal{H}_{M}\otimes\mathcal{H}_{M})$ be the
operators representing $\mathcal{E}$ and $\tilde{\mathcal{E}}$ as described in
Appendix F. The maps adjoint to $\mathcal{E}$ and $\tilde{\mathcal{E}}$, with
respect to Hilbert-Schmidt inner product, are represented by the operators
$E^{\dagger}$ and $\tilde{E}^{\dagger}$, respectively. From Proposition 3.3 of
Ref. Fannes  _et al._ , 1992 we know that the peripheral spectrum of
$E^{\dagger}$ consists of $p$ non-degenerate eigenavalues $\exp(\frac{2\pi
i}{p}r)$, where $r=0,\cdot\cdot\cdot,p-1$, which correspond to the
eigenvectors $u_{r}$, which are unitary operators. Each of $u_{r}$ can be
decomposed as $u_{r}=\sum_{k=0}^{p-1}\exp(\frac{2\pi i}{p}rk)\Pi_{k}$, where
$\\{\Pi_{k}\\}_{k=0,\cdot\cdot\cdot,p-1}$ is the set of projectors defined in
Proposition 2, $\sum_{k=0}^{p-1}\Pi_{k}=\mathbb{1}_{M}$.

For brevity of notation in this appendix we denote the vectors
$\mathrm{vec}(X)\in\mathcal{H}_{M}\otimes\mathcal{H}_{M}$, corresponding to
the operators $X\in\mathcal{B}(\mathcal{H}_{M})$ through the isomorphism
$\mathrm{vec}$ defined in Appendix F, as $|X\rangle:=\mathrm{vec}(X)$. Then,