# Quantum reservoir computing in finite dimensions

Rodrigo Martínez-Peña<EMAIL_ADDRESS>Instituto de Física
Interdisciplinar y Sistemas Complejos (IFISC, UIB-CSIC), Campus Universitat de
les Illes Balears, 07122 Palma de Mallorca, Spain  Juan-Pablo Ortega  juan-
<EMAIL_ADDRESS>Division of Mathematical Sciences, Nanyang
Technological University, 21 Nanyang Link, Singapore 637371

###### Abstract

Most existing results in the analysis of quantum reservoir computing (QRC)
systems with classical inputs have been obtained using the density matrix
formalism. This paper shows that alternative representations can provide
better insights when dealing with design and assessment questions. More
explicitly, system isomorphisms are established that unify the density matrix
approach to QRC with the representation in the space of observables using
Bloch vectors associated with Gell-Mann bases. It is shown that these vector
representations yield state-affine systems (SAS) previously introduced in the
classical reservoir computing literature and for which numerous theoretical
results have been established. This connection is used to show that various
statements in relation to the fading memory (FMP) and the echo state (ESP)
properties are independent of the representation, and also to shed some light
on fundamental questions in QRC theory in finite dimensions. In particular, a
necessary and sufficient condition for the ESP and FMP to hold is formulated
using standard hypotheses, and contractive quantum channels that have
exclusively trivial semi-infinite solutions are characterized in terms of the
existence of input-independent fixed points.

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##  I Introduction

The development of noisy intermediate-scale quantum (NISQ) devices is
attracting a great deal of attention from the quantum community. Recent
advancements in fields such as quantum computation [1], quantum simulation
[2], and quantum communications [3] are just examples of the prosperous future
that awaits these technologies. Nevertheless, NISQ devices are already
demonstrating in the meantime that they can be very useful for diverse
research fields like physics, chemistry, and optimization [4], even providing
quantum advantage [1, 5, 6]. Machine learning (ML) is another example of
thriving synergies with NISQ technologies. In this context, quantum machine
learning (QML) aims to exploit the specific features of quantum mechanics to
obtain an advantage over its classical counterparts when dealing with machine
learning tasks, both with classical and quantum data [7]. There is already
positive evidence in this direction, both from a theoretical point of view in
the fault-tolerant picture [8] and in experiments [9].

The flexibility and range of action of ML and QML techniques is typically
specified by the so-called universality approximation theorems. A universal
approximation property takes place when a proposed restricted family of
functions can approximate any function in a much larger class with arbitrary
precision. There are many results of this type that are part of classical
analysis dealing with, for instance, polynomials and Fourier series, and
others that were added in the early days of ML like, for example, feed-forward
neural networks [10, 11, 12]. Further results of this type have been proved
for various ML paradigms like recurrent neural networks [13], support vector
machines [14], extreme learning machines [15, 16], or kernel methods [17, 18,
19]. Universality results have also been obtained in the QML context, such as
for one qubit algorithms [20, 21], and general quantum circuits [22, 23]. A
framework in which we are particularly interested is quantum reservoir
computing (QRC). As in classical reservoir computing (RC) [24, 25, 26], QRC
harnesses the rich dynamics of (quantum) dynamical systems to solve tasks
where memory and prediction capabilities are required. Examples of application
of these techniques are found in the prediction of chaotic time-series [27,
28, 29] and complex spatiotemporal dynamics [30, 31, 32, 33]. Since the first
work on QRC [34] many have followed (see [35, 36] for reviews). This includes
both experimental implementations [37, 38, 39, 40, 41] as well as theoretical
contributions on the universal approximation question [42, 37, 43]. The latter
was inspired by the works in the classical framework [44, 45, 46, 47], where
the discrete-time setting of RC theory fits well within the quantum dynamical
map description.

The universal approximation property in the RC context brings to the table
some conditions that dynamical systems should meet to ensure it. The most
prominent ones are the echo state property (ESP) [24] and the fading memory
property (FMP) [48]. A system has the FMP if inputs that are close in the
recent past produce outputs that are also close, independently of what
happened in the distant past. The ESP guarantees that a well-defined
input/output map can be associated with our system and amounts to an existence
and uniqueness property with respect to the input sequence that is fed into
the system. The ESP and FMP are standard requirements in many stochastic and
deterministic learning paradigms since they mathematically encode the
asymptotic decorrelation (and even independence) between physical states and
initial conditions that most physical systems exhibit as time goes by. There
are many physical mechanisms that lead to the declining relevance of a given
initial condition in a system state as the temporal distance between them
increases. For instance, two pervasive phenomena in applications in this
direction are chaos (high sensitivity to initial conditions) and dissipation.
Fading memory is an important modeling feature when using physical systems
because a reservoir system can only store a finite amount of information in
its trainable parameters [49]; this implies that information must be erased as
time goes by in order to store new input information. Under very mild
mathematical conditions, ESP and FMP are equivalent to the input-forgetting
property [50] which mathematically encodes the information removal process
that is needed to learn the newly fed one.

The ESP property has been studied in the context of QRC in different works.
Nurdin and Chen [42] provided sufficient conditions for the ESP (and FMP) to
hold in terms of the contractivity of the quantum map acting on a restricted
domain [42, 37, 51], while Tran and Nakajima also define the ESP using
contractivity of the quantum map but without considering restrictions. They
also numerically connect the ESP with the spectra of the quantum maps [32,
52]. Both approaches describe the ESP in the density matrix language, that is,
the dynamical equations are defined in the space of quantum states. However,
this description might be limited somehow, because, as we will see later on,
more information can be extracted if we choose a description in terms of
observables.

This paper aims to fill some gaps in the description of QRC with finite
dimensional systems and classical inputs. More precisely, we first unify the
quantum ESP defined in previous works [42, 37, 51, 32, 52] using the norm of
quantum maps, with the classical notion of ESP using observables, and show
that they are equivalent for finite-dimensional quantum systems. We extend in
passing these results to the FMP. More specifically, we establish explicit
system isomorphisms between the description of QRC systems using the space of
density matrices as state space and the one that uses the space of Bloch
vectors associated to Gell-Mann bases. That choice of basis, which is
customary in the study of quantum systems, happens to yield non-homogeneous
affine state dynamics (the corresponding systems are called state-affine
(SAS)) of the type introduced in [45]. The connection between QRC and SAS
systems has important consequences. Our results in that context are contained
in Proposition 3 and in Theorems 5 and 9. The proposition presents a necessary
and sufficient condition for the ESP and FMP to hold in different
representations with just a few requirements, such as the compactness of the
input space. We emphasize that this hypothesis is satisfied in most RC tasks
and when dealing with implementations of RC systems with dedicated hardware
since the experimental ranges of the physical systems involved are always
finite [26]. The theorems exhibit common situations that should be avoided in
the design of quantum channels so that fully operational QRC systems are
obtained. We shall work in an idealized framework in which observables are
obtained after an infinite number of measurements, with no statistical error.
Even in such an idealized setting, we expect that our results can contribute
to the general understanding of QRC experiments in finite dimensional systems,
as they have already been implemented in [37, 38, 40, 41].

The structure of the paper is as follows. Section II introduces the general
framework and the definitions that will be needed along the paper. Definitions
of the spaces of operators and quantum maps are included in Section II.1,
while all the RC ingredients will be presented in Section II.2, together with
some preliminary results. The main results are contained in Section III.
Section IV includes a brief discussion on some of the consequences of Theorems
5 and 9, and Section V concludes the paper.

##  II Definitions

###  II.1 Quantum definitions

We start by defining the space of quantum systems. See, for example, [53, 54]
for further details. Consider a complex Hilbert space $\mathcal{H}$. The set
of all bounded operators $\mathcal{B}(\mathcal{H})$ that act on $\mathcal{H}$
is a complex vector space under point-wise addition and scalar multiplication,
and it forms an algebra under composition. If we add the involution
$A\rightarrow A^{\dagger}$ given by the adjoint operation, then
$\mathcal{B}(\mathcal{H})$ is also a $C^{*}$-algebra with respect to the
operator norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{op}$
defined by

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|A\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{op}:=\sup_{||\psi||=1}\\{||A\psi||,\
\psi\in\mathcal{H}\\},$ (1)

where $A\in\mathcal{B}(\mathcal{H})$. Along this manuscript, we will denote
all the induced operator and matrix norms with the symbol
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$.
The predual of $\mathcal{B}(\mathcal{H})$ is the Banach space
$\mathcal{T}(\mathcal{H})$ of all trace-class operators on $\mathcal{H}$ that
have finite trace norm $||A||_{1}:=\text{tr}\sqrt{AA^{\dagger}}$. The trace
norm is a particular case (with $p=1$) of the Schatten norms defined by
$||A||_{p}:=\left(\text{tr}\left(\left(\sqrt{AA^{\dagger}}\right)^{p}\right)\right)^{1/p}$.
The space of quantum density matrices $\mathcal{S}(\mathcal{H})$ is a compact
convex subset (see Section II.2 and the argument above (15)) of the normed
vector space $\mathcal{T}(\mathcal{H})$ defined by

$\mathcal{S}(\mathcal{H})=\\{\rho\in\mathcal{T}(\mathcal{H})\ |\
\rho^{\dagger}=\rho,\ \rho\geq 0,\ \text{tr}(\rho)=1\\}.$ (2)

As $\mathcal{S}(\mathcal{H})$ is a closed subset of the Banach space
$\mathcal{T}(\mathcal{H})$, it is then a complete metric space when using the
distance induced by $\left\|\cdot\right\|_{1}$. All along this paper, we
restrict ourselves to finite-dimensional Hilbert spaces, for which the spaces
of bounded and trace class operators coincide, and we shall use the symbol
$\mathcal{B}(\mathcal{H})$ to refer to both of them. Moreover, in that case,
$\mathcal{S}(\mathcal{H})$ is a complete metric space with respect to any norm
since all the norms are equivalent. We shall reserve the symbol
$d\in{\mathbb{N}}$ for the dimension of ${\mathcal{H}}$.

We now introduce the notion of quantum channel. All definitions and further
properties of these maps can be found in [55, 56, 57, 58] and references
therein. A quantum channel is a linear map
$T:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{H})$ that is
completely positive and trace preserving (CPTP). We recall here that a trace
preserving map $T$ is the one that satisfies that
$\text{tr}(T(A))=\text{tr}(A)$ for any $A\in\mathcal{B}({\mathcal{H}})$.
Moreover, we say that $T$ is positive when it maps positive semi-definite
operators to positive semi-definite operators. Finally, completely positive
maps $T$ are those positive maps which, when extended to a larger space using
the tensor map $T\otimes I_{k}$, with $I_{k}$ the identity map in dimension
$k$, they also yield a positive map for any $k\in\mathbb{N}$. A linear map is
CPTP if and only if it is possible to find a Kraus decomposition associated to
a set of operators $\\{K_{i}\\}_{i\in X}$ such that for all
$A\in\mathcal{B}(\mathcal{H})$:

$T(A)=\sum_{i\in X}K_{i}AK^{\dagger}_{i},$ (3)

where $\sum_{i\in X}K^{\dagger}_{i}K_{i}=I$ and $X$ is an index set of
cardinality at most $d^{2}$, with $d$ the dimension of ${\mathcal{H}}$. CPTP
maps obviously leave $\mathcal{S}(\mathcal{H})$ invariant and hence induce a
restricted map
$T:\mathcal{S}(\mathcal{H})\longrightarrow\mathcal{S}(\mathcal{H})$ that we
shall denote with the same symbol and use interchangeably. Note that, unlike
$\mathcal{B}({\mathcal{H}})$, the set $\mathcal{S}({\mathcal{H}})$ is not a
vector space, and hence it is only when use the map
$T:\mathcal{B}(\mathcal{H})\longrightarrow\mathcal{B}(\mathcal{H})$ that we
can talk about matrix expressions and eigenvalues for the operator $T$. It can
be shown that any CPTP map
$T:\mathcal{S}(\mathcal{H})\longrightarrow\mathcal{S}(\mathcal{H})$ is non-
expansive in the trace norm, which means that after applying $T$ to two input
states $\rho_{1},\rho_{2}\in\mathcal{S}(\mathcal{H})$, the distance between
these density matrices is either contracted or remains equal:

$||T(\rho_{1})-T(\rho_{2})||_{1}\leq||\rho_{1}-\rho_{2}||_{1}.$ (4)

We recall that the eigenvalues of a CPTP map
$T:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{H})$ are the
complex numbers $\lambda$ that make the map $T-\lambda\,\text{id}$ non-
invertible. We will denote this set by $\text{spec}(T)$. The spectral radius
$\rho(T):=\max\\{|\lambda|\ \mid\lambda\in\text{spec}(T)\\}$ of any CPTP map
is one as a consequence of (4) and of the fact that $1$ is always an
eigenvalue. Additionally, the set $\text{spec}(T)$ is invariant under complex
conjugation (see [57]). On the other hand, since the finite-dimensional vector
space $\mathcal{B}(\mathcal{H})$ is isomorphic to $\mathbb{C}^{d^{2}}$, we can
represent $T:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{H})$ as a
$d^{2}\times d^{2}$ matrix, which we write as $\widehat{T}$. This matrix
representation can be obtained by fixing an orthonormal basis in
$\mathcal{B}(\mathcal{H})$ for the Hilbert-Schmidt inner product, that is,
$\\{B_{i}\in\mathcal{B}(\mathcal{H})\\}$ with
$\text{tr}(B_{i}^{\dagger}B_{j})=\delta_{ij}$, and by setting
$\widehat{T}_{ij}=\text{tr}(B_{i}^{\dagger}T(B_{j}))$. The spectrum of the
matrix $\widehat{T}$ is given by the roots of the characteristic polynomial
$\det(\widehat{T}-\lambda I)=0$ and, since we are working with finite
dimensions, it coincides with $\text{spec}(T)$, as well as with the set of
complex values $\lambda$ that satisfy $T(X)=\lambda X$, for some non-trivial
eigenvector $X\in\mathcal{B}(\mathcal{H})$ [58].

We are particularly interested in the eigenvectors corresponding to the
eigenvalue $\lambda=1$ and that we call fixed points, that is, they are
elements $A\in\mathcal{B}(\mathcal{H})$ such that $T(A)=A$. The set of fixed
points of a CPTP map is always non-empty in the space of density matrices.
This is a consequence of Schauder’s Fixed-Point Theorem together with the
continuity of $T$ and the compactness of $\mathcal{S}(\mathcal{H})$. We will
work most of the time with quantum channels with single fixed points
$\rho^{*}\in\mathcal{S}(\mathcal{H})$ in the space of density matrices. Such
maps are called ergodic. Ergodicity of quantum channels is equivalent to the
eigenspace of $T$ associated with the eigenvalue $\lambda=1$ in
$\mathcal{B}(\mathcal{H})$ having dimension one, and to being made out of the
complex multiples of $\rho^{*}\in\mathcal{S}(\mathcal{H})$. In that case, we
obviously have (see Corollary 2 in [57]) that $T(A)=A$ with
$A\in\mathcal{B}(\mathcal{H})$ if and only if $A=\text{tr}(A)\rho^{*}$. It is
worth mentioning that the rest of the eigenvectors of an ergodic CPTP map must
be traceless. Indeed, given an eigenvalue $\lambda\neq 1$ of the CPTP map $T$
and $A$ a corresponding eigenvector, we find that
$\text{tr}(A)=\text{tr}(T(A))=\lambda\text{tr}(A)$, which implies that
$(\lambda-1)\text{tr}(A)=0$, and hence that $\text{tr}(A)=0$. Therefore, the
spectral set of a CPTP map can be decomposed as:

$\text{spec}(T)=\\{1\\}\cup\text{spec}(T|_{\mathcal{B}_{0}(\mathcal{H})}),$
(5)

where $\mathcal{B}_{0}(\mathcal{H})\subset\mathcal{B}(\mathcal{H})$ is defined
as the vector subspace of traceless operators, where the restriction
$T|_{\mathcal{B}_{0}(\mathcal{H})}$ and its corresponding matrix
representation are obviously well defined.

Repeated applications of an ergodic CPTP map do not necessarily converge to a
fixed point (see [57] for an example). If convergence to a fixed point takes
place, we say that the CPTP map is mixing. More specifically, a CPTP map is
mixing if and only if its repeated applications converge in the trace norm,
that is,

$\lim_{n\rightarrow\infty}||T^{n}(\rho)-\rho^{*}||_{1}=0,\quad\forall\rho\in\mathcal{S}(\mathcal{H}).$
(6)

Equation (6) implies that the sequence $\\{T^{n}(\rho)\\}$ converges to
$\rho^{*}$ with respect to the trace norm, but in infinite-dimensional
situations this does not necessarily imply that convergence takes place with
respect to other norms (see Definition 5.4.1 in [59]). However, in the finite-
dimensional case mixing becomes a topological property where
$\lim_{n\rightarrow\infty}T^{n}(\rho)=\rho^{*}$ for any norm and for any
$\rho\in\mathcal{S}(\mathcal{H})$ [56].

Although all mixing maps are ergodic, the converse is not true in general.
However, for continuous-time Markovian evolution, both are equivalent and such
maps are called relaxing [57]. Another important consequence of mixing
condition is the following: a CPTP map is mixing if and only if the fixed
point $\rho^{*}$ is the only eigenvector with eigenvalue $|\lambda|=1$. Then,
it is straightforward to show that

$\max\\{|\lambda|\mid\lambda\in\text{spec}(T|_{\mathcal{B}_{0}(\mathcal{H})})\\}<1.$
(7)

We say that a CPTP map is called primitive when it is mixing and its unique
fixed point $\rho^{*}>0$ has full rank. The smallest natural number $n$ for
which $T^{n}$ sends positive semidefinite matrices to positive definite
matrices is called the index of primitivity of $T$ and is denoted by
$\omega(T)$. With that notation, we say that $T^{\omega(T)}$ is a strictly
positive map. Bounds for this number are given by the quantum version of the
Wietland inequality [60, 61].

Finally, we say that a quantum channel is strictly contractive when

$||T(\rho_{1})-T(\rho_{2})||_{1}\leq r||\rho_{1}-\rho_{2}||_{1}$ (8)

for all $\rho_{1},\rho_{2}\in\mathcal{S}(\mathcal{H})$, where $0\leq r<1$. As
it is customary in the quantum channels literature, we reserve the term
strictly contractive for the trace norm, unless a different norm is explicitly
specified. It can be shown that strictly contractive channels with a full-rank
fixed point must be primitive. Indeed, the contractivity condition, together
with Banach’s Fixed Point Theorem (using that the convex closed subset of
density matrices with the trace norm is a complete metric space) guarantees
that the channel is mixing. A mixing channel with a strictly positive fixed
point is a primitive channel. The converse holds when the primitive map
becomes strictly positive, that is, it sends positive semidefinite matrices to
positive definite ones: given a primitive channel $T$, $T^{\omega(T)}$ is
strictly contractive (see Theorem VI.3 in [61]). Contractive maps can be also
constructed by composing a strictly contractive channel with a general CPTP
map.

An important conclusion that can be drawn from all these considerations is
that strictly contractive channels, which will be relevant along this work,
can be constructed by either using a map like $T^{\omega(T)}$, where $T$ is a
primitive channel, or by composing a strictly contractive channel with any
other CPTP map, as it has been done in some examples in QRC [40, 39]. Examples
of mixing/relaxing CPTP maps with full-rank single fixed points can be found
in the Lindblad-Gorini-Kossakowski-Sudarshan equation (see [62] for a summary
of the necessary and sufficient conditions). As these maps belong to the
quantum Markov semigroup, iterative applications of the channels yield
$T_{\Delta\tau}^{\omega(T)}=T_{\omega(T)\Delta\tau}$, where $\Delta\tau$
represents the time of action of the map $T_{\Delta\tau}$. Therefore, taking
$\Delta\tau^{\prime}\geq\omega(T)\Delta\tau$ we obtain a strictly contractive
map.

###  II.2 RC definitions

We now define quatum reservoir computing (QRC) systems in the density matrix
formalism. Classical RC definitions and mathematical details can be found in,
for instance, [44]. QRC maps are determined by two equations, namely, the
state-space and the readout or observation equations. The state equation is
given by a family of CPTP maps
$T:\mathcal{B}(\mathcal{H})\times\mathbb{R}^{n}\rightarrow\mathcal{B}(\mathcal{H})$,
with $n\in\mathbb{N}$ being the number of input features, which are taken to
be real values (classical inputs). The maps $T$ and $h$ will be, most of the
time, tacitly assumed to be continuous. The output is obtained from the
readout map $h:\mathcal{B}(\mathcal{H})\rightarrow\mathbb{R}^{m}$, with
$m\in\mathbb{N}$, which maps operators in $\mathcal{B}(\mathcal{H})$ to the
Euclidean space $\mathbb{R}^{m}$. Inputs are typically bi-infinite discrete-
time sequences of the form ${\bf z}=(\dots,{\bf z}_{-1},{\bf z}_{0},{\bf
z}_{1},\dots)\in(\mathbb{R}^{n})^{\mathbb{Z}}$, and outputs
$\textbf{y}\in(\mathbb{R}^{m})^{\mathbb{Z}}$ have the same structure. A QRC
system is hence determined by the state-space transformations:

$\begin{cases}&A_{t}=T(A_{t-1},{\bf z}_{t}),\\\
&\textbf{y}_{t}=h(A_{t}),\end{cases}$ (9)

where $t\in\mathbb{Z}$ denotes the time index. Analogously, one can define the
same setting for semi-infinite discrete-time sequences:
$(\mathbb{R}^{n})^{\mathbb{Z}_{-}}=\\{{\bf z}=(\dots,{\bf z}_{-1},{\bf
z}_{0})\ |\ {\bf z}_{i}\in\mathbb{R}^{n},i\in\mathbb{Z}_{-}\\}$ for left-
infinite sequences and $(\mathbb{R}^{n})^{\mathbb{Z}_{+}}=\\{{\bf z}=({\bf
z}_{0},{\bf z}_{1},\dots)\ |\ {\bf
z}_{i}\in\mathbb{R}^{n},i\in\mathbb{Z}_{+}\\}$ for right-infinite sequences.
Similar definitions apply to $(D_{n})^{\mathbb{Z}}$,
$(D_{n})^{\mathbb{Z}_{-}}$, and $(D_{n})^{\mathbb{Z}_{+}}$ with elements in
the subset $D_{n}\subset\mathbb{R}^{n}$. We can also construct sequence spaces
$(\mathcal{B}(\mathcal{H}))^{\mathbb{Z}}$ for the space of bounded (trace-
class) operators:

$(\mathcal{B}(\mathcal{H}))^{\mathbb{Z}}=\\{\textbf{A}=(\dots,A_{-1},A_{0},A_{1},\dots)\\\
\mid A_{i}\in\mathcal{B}(\mathcal{H}),\ i\in\mathbb{Z}\\}.$

Analogous definitions for
$(\mathcal{B}(\mathcal{H}))^{\mathbb{Z}_{-}}$,$(\mathcal{B}(\mathcal{H}))^{\mathbb{Z}_{+}}$,
$(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}}$,
$(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}_{-}}$, and
$(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}_{+}}$ follow immediately.

A natural way to construct CPTP state-space transformations is to insert the
input dependence using the Kraus decomposition that we introduced in (3), that
is,

$T(A,{\bf z})=\sum_{i\in X}K_{i}({\bf z})AK^{\dagger}_{i}({\bf z}),$ (10)

and for any input ${\bf z}\in\mathbb{R}^{n}$, the matrices $\\{K_{i}({\bf
z})\\}_{i\in X}$ satisfy that $\sum_{i\in X}K^{\dagger}_{i}({\bf z})K_{i}({\bf
z})=I$. The by-design CPTP character of the map
$T:\mathcal{B}(\mathcal{H})\times\mathbb{R}^{n}\rightarrow\mathcal{B}(\mathcal{H})$
in (10) implies that it naturally restricts to a state equation
$T:\mathcal{S}(\mathcal{H})\times\mathbb{R}^{n}\rightarrow\mathcal{S}(\mathcal{H})$
with density matrices as state space, that we shall use interchangeably in the
sequel and denote using the same symbol.

The Echo State Property (ESP). Consider the QRC system defined in (9) or its
analog for the subsets
$\mathcal{S}(\mathcal{H})\subset\mathcal{B}(\mathcal{H})$ and
$D_{n}\subset\mathbb{R}^{n}$, that is, $T:\mathcal{S}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{S}(\mathcal{H})$. Given an input sequence ${\bf
z}\in(D_{n})^{\mathbb{Z}}$, we say that
$\bm{\rho}\in(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}}$ is a solution of (9) for
the input ${\bf z}$ if the components of the sequences ${\bf z}$ and
$\bm{\rho}$ satisfy the first relation in (9) for any $t\in\mathbb{Z}$. We say
that the QRC system has the echo state property (ESP) when it has a unique
solution for each input ${\bf z}\in(D_{n})^{\mathbb{Z}}$. More explicitly, for
each ${\bf z}\in(D_{n})^{\mathbb{Z}}$, there exists a unique sequence
$\bm{\rho}\in(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}}$ such that

$\rho_{t}=T(\rho_{t-1},{\bf z}_{t}),\ \text{for all}\ t\in\mathbb{Z}.$ (11)

Filters and functionals. Let
$\mathcal{S}(\mathcal{H})\subset\mathcal{B}(\mathcal{H})$ be the space of
density matrices and let $D_{n}\subset\mathbb{R}^{n}$ be a subset in the input
space. A map of the type
$U:(D_{n})^{\mathbb{Z}}\rightarrow(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}}$ is
called a filter associated to the QRC system (9) when it satisfies that

$U({\bf z})_{t}=T\left(U({\bf z})_{t-1},{\bf z}_{t}\right),\ \mbox{for all
${\bf z}\in(D_{n})^{\mathbb{Z}}$ and $t\in\mathbb{Z}$.}$

Filters induce what we call functionals
$H:(D_{n})^{\mathbb{Z}}\rightarrow\mathcal{S}(\mathcal{H})$ via the relation
$H({\bf z})=U({\bf z})_{0}$. It is clear that a uniquely determined filter can
be associated with a QRC system that satisfies the ESP. The filter maps, in
that case, any input sequence to the unique solution of the QRC system
associated with it. A filter is called causal if it only produces outputs that
depend on present and past inputs. More formally, causality means that for any
two inputs ${\bf z},\textbf{v}\in(D_{n})^{\mathbb{Z}}$ that satisfy ${\bf
z}_{\tau}=\textbf{v}_{\tau}$ for any $\tau\leq t$, for a given
$t\in\mathbb{Z}$, we have $U({\bf z})_{t}=U(\textbf{v})_{t}$. The filter $U$
is called time-invariant if there is no explicit time dependence on the system
that determines it, that is, it commutes with the time delay operator defined
as $\mathcal{T}_{\tau}({\bf z})_{t}:={\bf z}_{t-\tau}$. Filters associated to
QRC systems of the type (9) are always causal and time-invariant (Proposition
2.1 in [44]). As noted in previous works [48, 44, 45], there is a bijection
between causal and time-invariant filters and functionals on
$(D_{n})^{\mathbb{Z}_{-}}$. Then, we can restrict our work to causal and time-
invariant filters with target and domain in spaces of left-infinite sequences.

The Fading Memory Property (FMP). Infinite product spaces can be endowed with
Banach space structures associated to the supremum norm and weighted norms.
The supremum norm for input sequences is defined as $||{\bf
z}||_{\infty}:=\sup_{t\in\mathbb{Z}}\\{||{\bf z}_{t}||\\}$, and for operator
sequences as
$||\textbf{A}||_{\infty}:=\sup_{t\in\mathbb{Z_{-}}}\\{||A_{t}||\\}$, where
$||\cdot||$ represents a given vector and matrix norm, respectively. The
symbols $l^{\infty}(\mathbb{R}^{n})$, $l_{\pm}^{\infty}(\mathbb{R}^{n})$,
$l^{\infty}(\mathcal{B}(\mathcal{H}))$ and
$l_{\pm}^{\infty}(\mathcal{B}(\mathcal{H}))$ denote the Banach spaces formed
by the elements in the corresponding infinite product spaces with a finite
supremum norm.

We now define the weighted norm. Let $w:\mathbb{N}\rightarrow(0,1]$ be a
decreasing sequence with zero limit and $w_{0}=1$. The weighted norm
$||\cdot||_{w}$ on $(\mathbb{R})^{\mathbb{Z_{-}}}$ is defined as

$||{\bf z}||_{w}:=\sup_{t\in\mathbb{Z}_{-}}\\{w_{-t}||{\bf z}_{t}||\\},$ (12)

and the space

$l^{w}_{-}(\mathbb{R}^{n})=\\{{\bf z}\in(\mathbb{R}^{n})^{\mathbb{Z_{-}}}|\
||{\bf z}||_{w}<\infty\\},$ (13)

with weighted norm $||\cdot||_{w}$ forms a Banach space (see Appendix A.2 in
[44]). In the same vein, we can define

$\begin{split}&||\textbf{A}||_{w}:=\sup_{t\in\mathbb{Z}_{-}}\\{w_{-t}||A||\\},\\\
&l^{w}_{-}(\mathcal{B}(\mathcal{H}))=\\{\textbf{A}\in(\mathcal{B}(\mathcal{H}))^{\mathbb{Z_{-}}}|\
||\textbf{A}||_{w}<\infty\\}.\end{split}$ (14)

It can be shown that $l^{w}_{-}(\mathcal{B}(\mathcal{H}))$ is a Banach space
as well.

We now turn to the space of density matrices $\mathcal{S}(\mathcal{H})$ and
recall that since for positive semidefinite matrices the trace operator is
submultiplicative (see Exercise 7.2.P26 in [59]) we can conclude that
$\operatorname{tr}(\rho^{2})\leq 1$ for all $\rho\in\mathcal{S}(\mathcal{H})$.
This observation implies that the elements in $\mathcal{S}(\mathcal{H})$ have
Frobenius norms bounded by one. Since we are in finite dimensions, this
statement holds for any other matrix norm (eventually with a bounding constant
different from one) and allows us to conclude that

$(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}_{-}}\subset
l^{\infty}_{-}(\mathcal{B}(\mathcal{H}))\subset
l^{w}_{-}(\mathcal{B}(\mathcal{H})),$ (15)

for any weighting sequence $w$. The previous boundedness consideration,
together with the closedness of $\mathcal{S}(\mathcal{H})$ in
$\mathcal{B}(\mathcal{H})$ implies that $\mathcal{S}(\mathcal{H})$ is
necessary compact. An important consequence of this fact is that the relative
topology induced by the $l^{w}_{-}(\mathcal{B}(\mathcal{H}))$ on
$(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}_{-}}$ coincides with the product
topology (see Corollary 2.7 in [44]).

Take now a subset $D_{n}\subset\mathbb{R}^{n}$ such that
$(D_{n})^{\mathbb{Z}_{-}}\subset l^{w}_{-}(\mathbb{R}^{n})$ and consider a QRC
system $T:\mathcal{S}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{S}(\mathcal{H})$ that has the ESP. We say that $T$
has the fading memory property (FMP) when the corresponding functional
$H:(D_{n})^{\mathbb{Z}_{-}}\rightarrow\mathcal{S}(\mathcal{H})$ is a
continuous map between the metric spaces
$((D_{n})^{\mathbb{Z}_{-}},||\cdot||_{w})$ and
$((\mathcal{S}(\mathcal{H}))^{\mathbb{Z}_{-}},||\cdot||_{w})$, for some
weighting sequence $w$. If $D_{n}$ is compact, once $H$ is continuous for a
given weighting sequence $w$, then it is continuous for all weighting
sequences (see [44, Theorem 2.6]).

The compactness of $\mathcal{S}(\mathcal{H})$ implies that we can apply with
straightforward modifications Theorem 3.1 in [44] to prove the following
statement.

######  Proposition 1.

Let $D_{n}\subset\mathbb{R}^{n}$ be a compact subset of $\mathbb{R}^{n}$. Let
$T:\mathcal{S}(\mathcal{H})\times D_{n}\rightarrow\mathcal{S}(\mathcal{H})$ be
a continuous QRC system such that the CPTP maps $T(\cdot,{\bf
z}):\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{H})$ are strictly
contractive for all ${\bf z}\in D_{n}$ as in (8) with a common contraction
constant $0\leq r<1$ associated to some norm in $\mathcal{B}(\mathcal{H})$
(not necessarily $\left\|\cdot\right\|_{1}$). Then, the QRC system induced by
$T$ has the ESP and the FMP.

##  III Results

Finding the expression for the QRC filter of a system that has the ESP is not
straightforward in the language of density matrices since the dependence on
the initial condition has to be addressed. More explicitly, given a general
CPTP map expressed using its Kraus decomposition, the relation (10) can be
iterated $n$-time steps into the past in order to obtain a quantum state
$\rho_{t}^{n}(\rho^{0}_{t-n})\in\mathcal{S}({\mathcal{H}})$ at time $t$ out of
an initial condition $\rho^{0}_{t-n}\in\mathcal{S}({\mathcal{H}})$ specified
at time $t-n$ via the formula:

$\footnotesize\begin{split}&\\!\\!\\!\\!\rho_{t}^{n}(\rho^{0}_{t-n})=\\\
&\sum_{i_{0},\ldots,i_{n-1}\in
X}\left(\overleftarrow{\prod}^{n-1}_{l=0}K_{i_{l}}({\bf
z}_{t-l})\right)\rho^{0}_{t-n}\left(\overrightarrow{\prod}^{n-1}_{l=0}K^{\dagger}_{i_{l}}({\bf
z}_{t-l})\right),\end{split}$ (16)

where $\overleftarrow{\prod}^{n-1}_{l=0}K_{i_{l}}({\bf
z}_{t-l})=K_{i_{0}}({\bf z}_{t})\cdots K_{i_{n-1}}({\bf z}_{t-n-1})$ and
$\overrightarrow{\prod}^{n-1}_{l=0}K^{\dagger}_{i_{l}}({\bf
z}_{t-l})=K^{\dagger}_{i_{n-1}}({\bf z}_{t-n-1})\cdots
K^{\dagger}_{i_{0}}({\bf z}_{t})$. If the QRC system has the ESP and hence a
filter
$U:(D_{n})^{\mathbb{Z}}\longrightarrow\left(\mathcal{S}({\mathcal{H}})\right)^{\mathbb{Z}}$
can be associated to it, it necessarily has to satisfy

$U({\bf z})_{t}=\lim_{n\rightarrow\infty}\rho_{t}^{n}(\rho^{0}_{t-n}),$ (17)

and this value has to be independent of the initial conditions
$\rho^{0}_{t-n}$. This fact has been explicitly shown in [42] (and in Chapter
2 of [51] in more detail) in the case of strictly contractive CPTP maps with
respect to the operator norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{p}$
associated to Schatten norms. We recall that given
$T:\mathcal{B}({\mathcal{H}})\rightarrow\mathcal{B}({\mathcal{H}})$, we define

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{p}:=\sup_{||A||_{p}=1}\\{||T(A)||_{p}\mid
A\in\mathcal{B}({\mathcal{H}})\\},$ (18)

for some $p\in[1,\infty)$ and $\left\|\cdot\right\|_{p}$ the $p$-Schatten
norm. Using this notation, the contractivity condition on the CPTP map $T$ is
stated by requiring that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T|_{\mathcal{B}_{0}(\mathcal{H})}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{p}<1$,
for some $p\in[1,\infty)$. It is obvious that this condition ensures that the
hypotheses of Proposition 1 are satisfied, which in turn implies that $T$ has
the ESP and the FMP (notice that given
$\rho_{1},\rho_{2}\in\mathcal{S}({\mathcal{H}})$ and ${\bf z}\in D_{n}$
arbitrary, the difference $T(\rho_{1},{\bf z})-T(\rho_{2},{\bf
z})\in\mathcal{B}_{0}(\mathcal{H})$ and hence the hypothesis
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T|_{\mathcal{B}_{0}(\mathcal{H})}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{p}<1$
implies that $\left\|T(\rho_{1},{\bf z})-T(\rho_{2},{\bf
z})\right\|_{p}<\left\|\rho_{1}-\rho_{2}\right\|_{p}$). Since we are in a
finite-dimensional context, it suffices to impose contractivity for just one
norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{p}$,
$p\in[1,\infty)$, in order to ensure that the limit in (17) exists and that
its value is independent of the initial condition.

The expression (16) shows that it is not possible to write down a closed-form
expression for the filter of a QRC system with the ESP when using the density
matrix representation, in which the initial dependence condition has been
eliminated. This feature is ultimately due to the linearity of the setup.
Already in the classical framework (see [45]), it has been shown that this can
be avoided by working with affine instead of purely linear systems. In the
quantum context, it can also be observed (see [37, 41]) that the non-
homogeneous state-space system given by
$\rho_{t}=(1-\epsilon)T(\rho_{t-1},{\bf z}_{t})+\epsilon\sigma$, where
$T(\rho_{t-1},{\bf z}_{t})$ is a CPTP map, $\sigma$ an arbitrary density
matrix, and $0<\epsilon<1$, defines a unique filter $U({\bf
z})_{t}=\epsilon\sigma+\epsilon\sum^{\infty}_{j=1}(1-\epsilon)^{j}T(\sigma,{\bf
z}_{t+1-j})$ in which the dependence on initial conditions has disappeared.

In the following subsections, we shall circumvent this problem by showing that
certain matrix representations of finite-dimensional QRC systems on density
matrices have a built-in non-homogeneous affine structure that makes them into
non-homogeneous state-affine systems (SAS) of the type introduced in [45].
More specifically, we shall be working with the Bloch vector representation of
quantum finite dimensional systems [63, 64] associated to a given Gell-Mann
basis. This idea is not new in QRC and it can already be seen in the seminal
work [34] or, more recently, in the Methods section of [40]. In the paragraphs
that follow, we shall explore in depth this representation, mostly in
connection with the available literature on SAS systems (Section III.2), which
will allow us later on in Section III.3 to identify various design constraints
on quantum channels.

###  III.1 Matrix representation of quantum channels

We will start by introducing the notation necessary for the matrix
representation of quantum channels. In the next section, we shall focus on a
specific choice of basis adapted to density matrices. Let
$\\{B_{i}\\}_{i\in\left\\{1,\ldots,d^{2}\right\\}}$ be an orthonormal basis
for the vector space $\mathcal{B}(\mathcal{H})$, when endowed with the
Hilbert-Schmidt inner product, that is,
$\text{tr}(B^{\dagger}_{i}B_{j})=\delta_{ij}$. Using any such basis we can
represent any operator $A\in\mathcal{B}(\mathcal{H})$ as
$A=\sum_{i=1}^{d^{2}}a_{i}B_{i}$, with $a_{i}=\text{tr}(B_{i}^{\dagger}A)$.
Analogously, we can express any linear map
$T:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{H})$ as

$T(A)=\sum^{d^{2}}_{i,j=1}\widehat{T}_{ij}a_{j}B_{i},\ \mbox{where
$\widehat{T}_{ij}=\text{tr}(B^{\dagger}_{i}T(B_{j}))$.}$ (19)

This observation implies that QRC systems $T:\mathcal{S}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{S}(\mathcal{H})$ admit an equivalent representation
as a system $\widehat{T}:V\times D_{n}\rightarrow V$, where
$V\subset\mathbb{R}^{d^{2}}$ is the subset of real Euclidean space that
contains the coordinate representations of the elements in
$\mathcal{S}(\mathcal{H})$ using the basis
$\\{B_{i}\\}_{i\in\left\\{1,\ldots,d^{2}\right\\}}$.

This statement can be formalized using the language of system morphisms (see
[65, 66] for the standard definitions and elementary facts). Consider the
state-space systems determined by the triples $({\cal X}_{i},F_{i},h_{i})$,
$i\in\left\\{1,2\right\\}$, with $F_{i}:{\cal X}_{i}\times{\cal
Z}\longrightarrow{\cal X}_{i}$ and $h_{i}:{\cal X}_{i}\longrightarrow{\cal
Y}$. A map $f:{\cal X}_{1}\longrightarrow{\cal X}_{2}$ is a morphism between
the systems $({\cal X}_{1},F_{1},h_{1})$ and $({\cal X}_{2},F_{2},h_{2})$
whenever it satisfies the following two properties:

  1. (i)

System equivariance: $f(F_{1}({\bf x}_{1},{\bf z}))=F_{2}(f({\bf x}_{1}),{\bf
z})$, for all ${\bf x}_{1}\in{\cal X}_{1}$ and ${\bf z}\in{\cal Z}$.

  2. (ii)

Readout invariance: $h_{1}({\bf x}_{1})=h_{2}(f({\bf x}_{1}))$, for all ${\bf
x}_{1}\in{\cal X}_{1}$.

When the map $f$ has an inverse $f^{-1}$ and this inverse is also a morphism
between the systems determined by $({\cal X}_{2},F_{2},h_{2})$ and $({\cal
X}_{1},F_{1},h_{1})$, we say that $f$ is a system isomorphism and the systems
$({\cal X}_{1},F_{1},h_{1})$ and $({\cal X}_{2},F_{2},h_{2})$ are isomorphic.
We note that given a system $F_{1}:{\cal X}_{1}\times{\cal
Z}\longrightarrow{\cal X}_{1},h_{1}:{\cal X}_{1}\longrightarrow{\cal Y}$ and a
bijection $f:{\cal X}_{1}\longrightarrow{\cal X}_{2}$, the map $f$ is a system
isomorphism with respect to the system $F_{2}:{\cal X}_{2}\times{\cal
Z}\longrightarrow{\cal X}_{2},h_{2}:{\cal X}_{2}\longrightarrow{\cal Y}$
defined by

$\displaystyle F_{2}({\bf x}_{2},{\bf z})$ $\displaystyle:=f(F_{1}(f^{-1}({\bf
x}_{2}),{\bf z})),$ (20) $\displaystyle h_{2}({\bf x}_{2})$
$\displaystyle:=h_{1}(f^{-1}({\bf x}_{2})).$ (21)

for all ${\bf x}_{2}\in{\cal X}_{2}$, ${\bf z}\in{\cal Z}$.

Consider now the QRC system given by the quantum channel
$T:\mathcal{S}(\mathcal{H})\times D_{n}\rightarrow\mathcal{S}(\mathcal{H})$
and the readout map $h:\mathcal{B}(\mathcal{H})\rightarrow\mathbb{R}^{m}$,
$m\in\mathbb{N}$. Using the orthonormal basis
$\mathcal{B}=\\{B_{i}\\}_{i\in\left\\{1,\ldots,d^{2}\right\\}}$ and the
discussion above define the map

$\begin{array}[]{cccc}G_{\mathcal{B}}:&\mathbb{C}^{d^{2}}&\longrightarrow&\mathcal{B}({\mathcal{H}})\\\
&\mathbf{a}&\longmapsto&\sum_{i=1}^{d^{2}}a_{i}B_{i}.\end{array}$ (22)

This map is a linear homeomorphism. Define
$V=G_{\mathcal{B}}^{-1}(\mathcal{S}(\mathcal{H}))\subset\mathbb{R}^{d^{2}}$ as
well as the map (that we denote with the same symbol)
$G_{\mathcal{B}}:V\longrightarrow\mathcal{S}(\mathcal{H})$ that we obtain by
restriction of the domain and codomain in (22). This restricted map is also a
homeomorphism when $V$ and $\mathcal{S}(\mathcal{H})$ are endowed with their
relative topologies [67, Theorem 18.2]. With all these ingredients, it is
straightforward to verify that the QRC system $(\mathcal{S}(\mathcal{H}),T,h)$
is system isomorphic to $(V,\widehat{T},\widehat{h})$ with
$\widehat{T}:V\times D_{n}\longrightarrow V$ and
$\widehat{h}:V\longrightarrow\mathbb{R}^{m}$ given by

$\displaystyle\widehat{T}(\mathbf{a},{\bf z})$
$\displaystyle:=G_{\mathcal{B}}^{-1}\left(T\left(G_{\mathcal{B}}(\mathbf{a}),{\bf
z}\right)\right),$ (23) $\displaystyle\widehat{h}(\mathbf{a})$
$\displaystyle:=h(G_{\mathcal{B}}(\mathbf{a})),$ (24)

and that the isomorphism is given by the map
$G_{\mathcal{B}}:V\longrightarrow\mathcal{S}(\mathcal{H})$. The procedure that
we just spelled out can be reproduced for any other (orthonormal) basis
$\mathcal{B}^{\prime}$ of $\mathcal{B}({\mathcal{H}})$, in which case we would
obtain another system $(V^{\prime},\widehat{T}^{\prime},\widehat{h}^{\prime})$
which is obviously isomorphic to both $(V,\widehat{T},\widehat{h})$ and
$(\mathcal{S}(\mathcal{H}),T,h)$.

The system isomorphisms that we just defined and the compactness of the state
spaces where they are defined allow us to establish important connections
between the filters that they define. The following proposition describes
those connections in detail.

######  Proposition 2.

Let $T:\mathcal{S}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{S}(\mathcal{H})$ be a quantum channel,
$h:\mathcal{S}(\mathcal{H})\longrightarrow\mathbb{R}^{m}$ a readout, and let
$\mathcal{B}=\\{B_{i}\\}_{i\in\left\\{1,\ldots,d^{2}\right\\}}$ be an
orthonormal basis for $\mathcal{B}(\mathcal{H})$. Let $\widehat{T}:V\times
D_{n}\longrightarrow V$ be the isomorphic system defined in (22) and
$\widehat{h}:V\longrightarrow\mathbb{R}^{m}$ the corresponding readout defined
in (24). Then:

  1. (i)

Given an input ${\bf z}\in({\mathbb{R}}^{n})^{\mathbb{Z}}$, a sequence
$\bm{\rho}\in\left(\mathcal{S}(\mathcal{H})\right)^{\mathbb{Z}}$ is a solution
for that input for the system determined by $T$ if and only if the sequence
${\cal
G}_{\mathcal{B}}^{-1}\left(\bm{\rho}\right)\in\left(V\right)^{\mathbb{Z}}$ is
a solution for the system associated to $\widehat{T}$. The symbol ${\cal
G}_{\mathcal{B}}=\prod_{\mathbb{Z}}G_{\mathcal{B}}:(V)^{\mathbb{Z}}\longrightarrow\left({\cal
S}({\mathcal{H}})\right)^{\mathbb{Z}}$ stands for the product map.

  2. (ii)

$T$ has the ESP if and only if $\widehat{T}$ has the ESP. In that case, the
filters $U_{T}$ and $U_{\widehat{T}}$ (respectively, $U_{T}^{h}$ and
$U_{\widehat{T}}^{\widehat{h}}$) determined by $T$ and $\widehat{T}$
(respectively, by $(T,h)$ and $(\widehat{T},\widehat{h})$) satisfy that
$U_{T}={\cal G}_{\mathcal{B}}\circ U_{\widehat{T}}$ (respectively,
$U_{T}^{h}=U_{\widehat{T}}^{\widehat{h}}$).

  3. (iii)

$U_{T}$ has the FMP if and only if $U_{\widehat{T}}$ has the FMP.

  4. (iv)

Let $V^{\prime}$ be a set homeomorphic to $V$. The relations (20)-(21)
determine in that case an isomorphic system
$\widehat{T}^{\prime}:V^{\prime}\times D_{n}\longrightarrow V^{\prime}$,
$\widehat{h}^{\prime}:V^{\prime}\rightarrow\mathbb{R}^{m}$ that has the ESP
and the FMP if and only if $\widehat{T}$ and $\widehat{h}$ have that property.

###### Proof.

Parts (i), (ii), and (iv) are a straightforward consequence of Proposition 2.2
in [44]. As to part (iii), given that by (ii) $U_{T}={\cal
G}_{\mathcal{B}}\circ U_{\widehat{T}}$, it suffices to prove that the
bijection ${\cal
G}_{\mathcal{B}}=\prod_{\mathbb{Z}}G_{\mathcal{B}}:(V)^{\mathbb{Z}}\longrightarrow\left({\cal
S}({\mathcal{H}})\right)^{\mathbb{Z}}$ is a homeomorphism when the domain and
the target are endowed with a weighted norm. In order to prove that, recall
first that, using the observation right under (15), the weighted norms in
$(V)^{\mathbb{Z}}$ and $\left({\cal S}({\mathcal{H}})\right)^{\mathbb{Z}}$
induce the product topology due to the compactness of $V$ and
$\mathcal{S}(\mathcal{H})$. This immediately implies that (Theorem 19.6 in
[67]) ${\cal G}_{\mathcal{B}}=\prod_{\mathbb{Z}}G_{\mathcal{B}}$ is continuous
due to the continuity of $G_{\mathcal{B}}$. The same argument can be
immediately applied to the inverse map ${\cal G}_{\mathcal{B}}^{-1}$, which
proves the statement. ∎

###  III.2 Non-homogenous state affine system representation

The goal of this section is to choose a specific basis in
$\mathcal{B}({\mathcal{H}})$ for the representation of the quantum channel
$T:\mathcal{B}(\mathcal{H})\times D_{n}\rightarrow\mathcal{B}(\mathcal{H})$
(equivalently, $T:\mathcal{S}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{S}(\mathcal{H})$) in which the associated system
isomorphic representation $\widehat{T}:\mathbb{C}^{d^{2}}\times
D_{n}\longrightarrow\mathbb{C}^{d^{2}}$ (equivalently, $\widehat{T}:V\times
D_{n}\longrightarrow V$) has a non-homogeneous state-affine form of the type
introduced in [45].

More specifically, we choose a generalized Gell-Mann basis [63, 64, 68]. This
is an orthonormal basis made of Hermitian operators in which, by convention,
its first element is the normalized identity, namely, $B_{1}=I/\sqrt{d}$. The
remaining $(d^{2}-1)$ traceless Hermitian operators are the generators

$\mathcal{B}_{0}=\left\\{\sigma_{i}\right\\}_{i\in\left\\{1,\ldots,d^{2}-1\right\\}}$
(25)

of the fundamental representation of the Lie algebra $\mathfrak{su}(d)$ of
SU($d$). The case $d=2$ corresponds to the case of one qubit, and the Gell-
Mann basis is made of the standard Pauli matrices. The orthonormality of the
Gell-Mann basis is guaranteed by the product property of the fundamental
representation of $\mathfrak{su}(d)$, namely,
$\sigma_{a}\sigma_{b}=\delta_{ab}I/(2d)+\sum^{d^{2}-1}_{c}f_{c}\sigma_{c}$,
where $\sigma_{a},\sigma_{b}$ are two elements of the Gell-Mann basis for
$\mathfrak{su}(d)$, and $f_{c}$ are complex coefficients. The resulting Gell-
Mann basis
$\mathcal{B}=\left\\{B_{i}\right\\}_{i\in\left\\{1,\ldots,d^{2}\right\\}}$ of
${\cal B}({\mathcal{H}})$ is hence given by $B_{1}=I/\sqrt{d}$ and
$B_{i}=\sigma_{i-1}$, $1<i\leq d^{2}$. Note that the subset
$\mathcal{B}_{0}=\left\\{B_{i}\right\\}_{i\in\left\\{2,\ldots,d^{2}\right\\}}$
is a basis for the vector subspace ${\cal B}_{0}({\mathcal{H}})\subset{\cal
B}({\mathcal{H}})$ of codimension $1$ made of traceless operators.

If our system is made of $N$ $d$-dimensional systems (qudits), we can extend
this basis to $\mathcal{B}\left({\mathcal{H}}^{\otimes^{N}}\right)$ by
tensorization. The orthonormality of the tensorized basis with respect to the
Hilbert-Schmidt inner product is preserved since $\text{tr}(A\otimes
B)=\text{tr}(A)\text{tr}(B)$ for any two $A,B\in\mathcal{B}({\mathcal{H}})$.

We now go back to the system constituted by one qudit and spell out the matrix
expression $\widehat{T}:\mathbb{C}^{d^{2}}\times
D_{n}\longrightarrow\mathbb{C}^{d^{2}}$ of a CPTP map
$T:\mathcal{B}({\mathcal{H}})\times
D_{n}\longrightarrow\mathcal{B}({\mathcal{H}})$ in the basis $\mathcal{B}$
that we just introduced, by using the prescription introduced in (19). We
first note that the choice $B_{1}=I/\sqrt{d}$ and the trace-preserving
character of $T(\cdot,{\bf z})$ for any ${\bf z}\in D_{n}$, imply that
$\widehat{T}(\cdot,{\bf z})_{11}=1$. Analogously, $\widehat{T}(\cdot,{\bf
z})_{1j}=\text{tr}(T(B_{j},{\bf z}))/\sqrt{d}=0$ for $1<j\leq d^{2}$, since
$T$ is trace-preserving. This implies that the matrix $\widehat{T}(\cdot,{\bf
z})$ can be written as

$\widehat{T}(\cdot,{\bf z})=\left(\begin{matrix}1&{\bf 0}_{d^{2}-1}\\\ q({\bf
z})&p({\bf z})\end{matrix}\right)$ (26)

where $p({\bf z})$ is the square matrix of dimension $d^{2}-1$ with complex
entries

$p({\bf z})_{ij}:=\left(\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\mathcal{B}_{0}(\mathcal{H})\right)}\right)_{ij}\\\
=\left(\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right)_{ij}=\text{tr}(B_{i}T(B_{j},{\bf
z})),$ (27)

$1<i,j\leq d^{2}$, and $q({\bf z})\in\mathbb{C}^{d^{2}-1}$ is given by $q({\bf
z})_{i}=\text{tr}(B_{i}T(I))/\sqrt{d}$, $1<i\leq d^{2}$. The symbol ${\bf
0}_{d^{2}-1}$ denotes a zero-row of length $d^{2}-1$.

Now, given that any element $\rho\in\mathcal{S}({\mathcal{H}})$ has
coordinates in the Gell-Mann basis of the form
$\left(1/\sqrt{d},\mathbf{x}^{\top}\right)^{\top}\in\mathbb{C}^{d^{2}}$ with
$\mathbf{x}\in\mathbb{C}^{d^{2}-1}$ called the Bloch vector, the matrix form
(26) implies that

$\widehat{T}(\cdot,{\bf z})\left(\begin{matrix}1/\sqrt{d}\\\
\mathbf{x}\end{matrix}\right)=\left(\begin{matrix}1/\sqrt{d}\\\ p({\bf
z})\textbf{x}+q({\bf z})\end{matrix}\right).$ (28)

This expression implies that $\widehat{T}:V\times D_{n}\longrightarrow V$
admits a system-isomorphic representation $\widehat{T}_{0}:V_{0}\times
D_{n}\longrightarrow V_{0}$ on the set
$V_{0}=\left\\{\mathbf{x}\in\mathbb{C}^{d^{2}-1}\mid\left(1/\sqrt{d},\mathbf{x}^{\top}\right)^{\top}\in
V\right\\}\subset\mathbb{C}^{d^{2}-1}$ given by

$\begin{array}[]{cccc}\widehat{T}_{0}:&V_{0}\times
D_{n}&\longrightarrow&V_{0}\\\ &(\mathbf{x},{\bf z})&\longmapsto&p({\bf
z})\textbf{x}+q({\bf z}),\end{array}$ (29)

with readout
$\widehat{h}_{0}(\mathbf{x})=\widehat{h}\left(\left(1/\sqrt{d},\mathbf{x}^{\top}\right)^{\top}\right)$.
The system isomorphism is in this case, given by the map

$\begin{array}[]{cccc}i_{0}:&V_{0}&\longrightarrow&V\\\ &\mathbf{x}\in
V_{0}&\longmapsto&\left(1/\sqrt{d},\mathbf{x}^{\top}\right)^{\top}.\end{array}$
(30)

Part (iv) of Proposition 2 guarantees that the dynamical properties of the
system $(\widehat{T},\widehat{h},V)$ (and hence those of the QRC system
$(T,h,\mathcal{S}({\mathcal{H}}))$) are equivalent to those of
$(\widehat{T}_{0},\widehat{h}_{0},V_{0})$.

The importance of this observation is that it links
$(T,h,\mathcal{S}({\mathcal{H}}))$ to the non-homogeneous state-affine system
(SAS) introduced in [45] and for which various universality properties have
been additionally proved in [46, 69]. SAS are defined as state equations that
have the form spelled out in (29). Strictly speaking, the SAS systems studied
in the above-cited references impose polynomial or trigonometric dependences
of $q$ and $p$ on the inputs, while in our situation, the prescription
introduced in (10) is capable of accommodating more general forms.

It has been shown in [45] that when such a system has the ESP, the
corresponding filter
$U_{\widehat{T}_{0}}:\left(D_{n}\right)^{\mathbb{Z}}\rightarrow\left(V_{0}\right)^{\mathbb{Z}}$
can be written as

$U_{\widehat{T}_{0}}({\bf
z})_{t}=\sum^{\infty}_{j=0}\left(\prod^{j-1}_{k=0}p({\bf
z}_{t-k})\right)q({\bf z}_{t-j}),$ (31)

where $\prod_{k=0}^{j-1}p({\bf z}_{t-k}):=p({\bf z}_{t})\cdot p({\bf
z}_{t-1})\cdots p({\bf z}_{t-j+1})$. A first sufficient condition for ESP and
FMP has been formulated in [45] by imposing that $\sigma_{\text{max}}(p({\bf
z}))<1$ for all ${\bf z}\in D_{n}$, where $\sigma_{\text{max}}$ is the maximum
singular value of matrix $p({\bf z})$. Given a matrix $A$, the maximum
singular value is equal to the 2-Schatten induced norm:
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|A\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{2}=\sigma_{\text{max}}(A)$,
where the singular values of $A$ are the square-roots of the eigenvalues of
$AA^{\dagger}$. An improved sufficient condition could be potentially found
using other matrix norms as in [70].

The next proposition uses this hint and, moreover, spells out equivalent
necessary and sufficient conditions for the ESP and FMP to hold in the three
different equivalent representations for QRC systems that we have introduced
in this section. More explicitly, the statement addresses the ESP and the FMP
for the operator representation $T:\mathcal{B}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{B}(\mathcal{H})$, the matrix representation
$\widehat{T}:\mathbb{C}^{d^{2}}\times D_{n}\longrightarrow\mathbb{C}^{d^{2}}$
associated to the Gell-Mann basis $\mathcal{B}$, and the SAS representation
$\widehat{T}_{0}:\mathbb{C}^{d^{2}-1}\times
D_{n}\longrightarrow\mathbb{C}^{d^{2}-1}$ introduced in (29).

######  Proposition 3.

Let $T:\mathcal{B}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{B}(\mathcal{H})$ be a continuous QRC system. The
following three statements are equivalent:

  1. (i)

There exists an operator norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
and $\epsilon>0$ such that

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})|_{\mathcal{B}_{0}(\mathcal{H})}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1-\epsilon,\
\mbox{for all ${\bf z}\in D_{n}$}.$ (32)

  2. (ii)

There exists a matrix norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
in the space of complex $d^{2}\times d^{2}$ matrices such that

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1-\epsilon,\
\mbox{for all ${\bf z}\in D_{n}$},$ (33)

with $\mathcal{B}_{0}$ the trace-zero elements in the basis $\mathcal{B}$
defined in (25) and $G_{\mathcal{B}}$ the isomorphism defined in (22) with
respect to the Gell-Mann basis.

  3. (iii)

There is a matrix norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
in the space of complex $(d^{2}-1)\times(d^{2}-1)$ matrices such that the SAS
representation $\widehat{T}_{0}:\mathbb{C}^{d^{2}-1}\times
D_{n}\longrightarrow\mathbb{C}^{d^{2}-1}$ introduced in (29) satisfies that

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|p({\bf
z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1-\epsilon,\ \mbox{for
all ${\bf z}\in D_{n}$},$ (34)

If any of these three equivalent statements hold and $D_{n}$ is compact, then:

  1. 1.

The isomorphic systems $T:\mathcal{S}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{S}(\mathcal{H})$, $\widehat{T}:V\times
D_{n}\rightarrow V$, and $\widehat{T}_{0}:V_{0}\times D_{n}\rightarrow V_{0}$,
have the ESP and the FMP, and hence continuous filters
$U_{T}:(D_{n})^{\mathbb{Z}}\longrightarrow\left(\mathcal{S}(\mathcal{H})\right)^{\mathbb{Z}}$,
$U_{\widehat{T}}:(D_{n})^{\mathbb{Z}}\longrightarrow V^{\mathbb{Z}}$, and
$U_{\widehat{T}_{0}}:(D_{n})^{\mathbb{Z}}\longrightarrow V_{0}^{\mathbb{Z}}$
can be associated to them.

  2. 2.

In such case, the filter $U_{\widehat{T}_{0}}$ is then given by (31).
$U_{\widehat{T}}$ is determined by $U_{\widehat{T}}={\cal I}_{0}\circ
U_{\widehat{T}_{0}}$, with ${\cal I}_{0}=\prod_{\mathbb{Z}}i_{0}$ and $i_{0}$
as in (30). Finally, $U_{T}={\cal G}_{\mathcal{B}}\circ U_{\widehat{T}}$, with
${\cal G}_{\mathcal{B}}=\prod_{\mathbb{Z}}G_{\mathcal{B}}$ the map introduced
in Proposition 2.

  3. 3.

The contraction conditions (32)-(34) are necessary for the ESP and the FMP to
hold.

###### Proof.

The equivalences between the contraction conditions (32)-(34) can be shown by
using norms in $\mathbb{C}^{d^{2}}$ and $\mathcal{B}({\mathcal{H}})$ that make
the map $G_{\mathcal{B}}$ in (22) into an isometry. More explicitly, let us
start with an operator norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
for which the maps $T(\cdot,{\bf
z}):\mathcal{B}({\mathcal{H}})\longrightarrow\mathcal{B}({\mathcal{H}})$
satisfy the condition (32). Assume that this operator norm is associated with
a given norm $\left\|\cdot\right\|_{\mathcal{B}({\mathcal{H}})}$ in
$\mathcal{B}({\mathcal{H}})$, that is,
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}=\sup_{A\neq
0}\left\\{\left\|T(A,{\bf
z})\right\|_{\mathcal{B}({\mathcal{H}})}/\left\|A\right\|_{\mathcal{B}({\mathcal{H}})}\right\\}$.
Take now the norm $\left\|\cdot\right\|_{\mathbb{C}^{d^{2}}}$ in
$\mathbb{C}^{d^{2}}$ with respect to which $G_{\mathcal{B}}$ is an isometry,
that is, set
$\left\|\mathbf{a}\right\|_{\mathbb{C}^{d^{2}}}=\left\|G_{\mathcal{B}}(\mathbf{a})\right\|_{\mathcal{B}({\mathcal{H}})}$,
for all $\mathbf{a}\in\mathbb{C}^{d^{2}}$, and denote by
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\widehat{T}(\cdot,{\bf
z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{\prime}=\sup_{\mathbf{a}\neq
0}\left\\{\left\|\widehat{T}(\mathbf{a},{\bf
z})\right\|_{\mathbb{C}^{d^{2}}}/\left\|\mathbf{a}\right\|_{\mathbb{C}^{d^{2}}}\right\\}$.
Using just these definitions, it is easy to see that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}={\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\widehat{T}(\cdot,{\bf
z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{\prime}$ and,
moreover, that

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})|_{\mathcal{B}_{0}(\mathcal{H})}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}={\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{\prime},$

which can be easily used to prove the equivalence between (32) and (33). The
equivalence between (33) and (34) follows from the fact that, as it can be
seen in (26),

$\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}=p({\bf
z}).$

The claims $1$ and $2$ in the second part of the proposition are
straightforward consequences of Propositions 1 and 2. We now prove the last
claim in point $3$ using the SAS representation (29) for the continuous QRC
system $T$ that, this time, it is assumed to have the ESP and the FMP. If we
iterate $n$ times this state equation using an arbitrary vector
$\mathbf{x}_{0}\in\mathbb{C}^{d^{2}-1}$ as an initial condition, we obtain the
vector $U^{n,\mathbf{x}_{0}}_{\widehat{T}_{0}}({\bf
z})_{t}\in\mathbb{C}^{d^{2}-1}$ defined by:

$U^{n,\mathbf{x}_{0}}_{\widehat{T}_{0}}({\bf
z})_{t}=\sum^{n-1}_{j=0}\left(\prod^{j-1}_{k=0}p({\bf z}_{t-k})\right)q({\bf
z}_{t-j})+\prod^{n-1}_{j=0}p({\bf z}_{t-j})\mathbf{x}_{0}.$

If we now assume that the system has the ESP, we necessarily have that

$U_{\widehat{T}_{0}}({\bf
z})_{t}=\lim_{n\rightarrow\infty}U^{n,\mathbf{x}_{0}}_{\widehat{T}_{0}}({\bf
z})_{t}=\lim_{n\rightarrow\infty}U^{n,\mathbf{x}_{0}^{\prime}}_{\widehat{T}_{0}}({\bf
z})_{t},$ (35)

with $U_{\widehat{T}_{0}}$ the filter in (31) and
$\mathbf{x}_{0},\mathbf{x}_{0}^{\prime}\in\mathbb{C}^{d^{2}-1}$ arbitrary
vectors. The last equality in (35) and the arbitrary character of
$\mathbf{x}_{0},\mathbf{x}_{0}^{\prime}\in\mathbb{C}^{d^{2}-1}$ imply that

$\prod^{\infty}_{j=0}p({\bf
z}_{t-j})=\lim_{n\rightarrow\infty}\prod^{n-1}_{j=0}p({\bf z}_{t-j})=0,$

necessarily. We now notice that since $T$ is a continuous map, then so is the
dependence of $p({\bf z})$ on the inputs ${\bf z}$. Moreover, since, in this
case, $D_{n}$ is assumed to be a compact set, then so is the matrix set
$\left\\{p({\bf z})\mid{\bf z}\in D_{n}\right\\}$. Now, Corollary 6.4 in [71]
guarantees the existence of matrix norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
for which (34) is satisfied, as required. ∎

######  Remark 4.

Using the characterization of the mixing property in (7), it is clear that a
necessary condition for the QRC $T$ to satisfy the contractivity hypothesis in
the previous proposition and, in passing, satisfy the ESP and the FMP, is that
all the maps $T(\cdot,{\bf z})$ are mixing for all ${\bf z}\in D_{n}$. Indeed,
since the spectral radius is a lower bound for any matrix norm (see [59,
Theorem 5.6.9]) we have that $\lambda_{\text{max}}\left(\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right)\leq{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$.
Consequently, if
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1-\epsilon$
then $\lambda_{\text{max}}\left(\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right)\leq
1-\epsilon$, which is equivalent to $\lambda_{\text{max}}\left(T(\cdot,{\bf
z})|_{\mathcal{B}_{0}(\mathcal{H})}\right)\leq 1-\epsilon$ and then all maps
$T(\cdot,{\bf z})$ are mixing by (7). We emphasize that this mixing condition
is necessary but not sufficient since, as the composition of two mixing maps
is not necessarily mixing, the existence of the limit in (17) is not
guaranteed even if each of the factor operators is mixing.

###  III.3 Some constrains on CPTP maps for QRC

The SAS representation allows us to easily characterize situations like the
one in Proposition 3 in which a QRC system has the ESP and the FMP and,
moreover, it allows us to write explicitly down the corresponding filter (31).
As we shall now see in this section, more interesting facts can be derived
from this representation having to do with design features that should be
avoided, as they produce systems with only trivial solutions. The first one
concerns unital quantum channels, that is, channels that satisfy

$T(I,{\bf z})=I\quad\mbox{for all ${\bf z}\in D_{n}$}.$

That situation is studied in the next theorem, which will be generalized in
Theorem 9 to the case of QRC systems that exhibit an input-independent fixed
point.

######  Theorem 5.

Let $T:\mathcal{B}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{B}(\mathcal{H})$ be a QRC system for which there
exists an operator norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
and $\epsilon>0$ such that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})|_{\mathcal{B}_{0}(\mathcal{H})}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1-\epsilon$,
for all ${\bf z}\in D_{n}$. Then, the corresponding filter $U_{T}$ is constant
with $U_{T}({\bf z})_{t}=I/d$ for all ${\bf z}\in(D_{n})^{\mathbb{Z}}$
(equivalently $U_{\widehat{T}_{0}}({\bf z})_{t}={\bf 0}$) if and only if $T$
is unital.

###### Proof.

We prove this statement by using the SAS representation associated to the
Gell-Mann basis $\mathcal{B}$. If $T$ is unital, that is, $T(I,{\bf z})=I$,
then the expressions (19) and (26) imply that $q({\bf
z})_{i}=\text{tr}(B_{i}T(I,{\bf z}))/\sqrt{d}=\text{tr}(B_{i})/\sqrt{d}=0$,
for all $i\in\left\\{2,\ldots,d^{2}\right\\}$. Therefore, the SAS state
equation becomes, in this case, $\textbf{x}_{t}=p({\bf
z}_{t})\textbf{x}_{t-1}$, which is a homogeneous equation whose only solution
under the contractivity hypotheses is the trivial one, that is,
$\textbf{x}_{t}={\bf 0}$ for all $t\in\mathbb{Z}$. Consequently,
$U_{\widehat{T}_{0}}={\bf 0}$. Proposition 3 implies then that $U_{T}={\cal
G}_{\mathcal{B}}\circ{\cal I}_{0}\circ U_{\widehat{T}_{0}}=I/d$.

Conversely, suppose that $U_{T}({\bf z})_{t}=I/d$ for all ${\bf
z}\in(D_{n})^{\mathbb{Z}}$. Since the filter $U_{T}$ is determined by the
recursions $U_{T}({\bf z})_{t}=T\left(U_{T}({\bf z})_{t-1},{\bf
z}_{t}\right)$, then, this relation implies that $T(I,{\bf z})=I$, for all
${\bf z}\in D_{n}$. ∎

######  Remark 6.

There is a close relation between unital quantum channels and contractivity.
Indeed, it has been shown that unital maps are contractive for all Schatten
$p$-norms, that is,
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{p}\leq 1$ (see
Theorem 2.4 in [72]) when $T(\cdot,{\bf z})$ is unital.

######  Example 7.

Consider the depolarizing channel
$\mathcal{E}:\mathcal{S}({\mathcal{H}})\longrightarrow\mathcal{S}({\mathcal{H}})$
defined by

$\mathcal{E}(\rho)=(1-\lambda)\rho+\lambda\frac{I}{d},$ (36)

where $0\leq\lambda\leq 1$ denotes the probability of finding the system at
the maximally mixed state $I/d$. If we arrange the previous equation as an
input-dependent channel, we can write the following state equation:

$\rho_{t}=\mathcal{E}(\rho_{t-1},{\bf
z}_{t})=(1-\lambda_{t})\rho_{t-1}+\lambda_{t}\frac{I}{d},$ (37)

where $\lambda_{t}:=\lambda({\bf z}_{t})$ is a function of the input at each
time step. This map is strictly contractive whenever
$r=\sup_{t\in\mathbb{Z}}(1-\lambda_{t})<1$ in which case $r$ is the
contraction rate. Let us compute the state after $n$ backwards iterations:

$\begin{split}\rho^{n}_{t}&=\left(\prod^{n-1}_{i=0}(1-\lambda_{t-i})\right)\rho_{t-n}\\\
&+\sum^{n-1}_{i=0}\left(\lambda_{t-i}\left(\prod^{i-1}_{j=0}(1-\lambda_{t-j})\right)\right)\frac{I}{d}.\end{split}$
(38)

Taking the limit $n\rightarrow\infty$, it is easy to see that the first
summand vanishes because $1-\lambda_{t-i}\leq r<1$ for all $t\in\mathbb{Z}$,
and hence all possible dependence on the initial condition disappears.
Regarding the second summand, since the map is strictly contractive, the limit
$\lim_{n\rightarrow\infty}\rho^{n}_{t}$ exists and must yield a density
matrix, which means that
$\sum^{\infty}_{i=0}\left(\lambda_{t-i}\left(\prod^{i-1}_{j=0}(1-\lambda_{t-j})\right)\right)=1$
because of the normalization. This limit hence defines the filter
$U_{\mathcal{E}}({\bf z})_{t}=I/d$, which is consistent with the conclusion of
Theorem 5 since (37) is a strictly contractive unital map.

We could have found the same result by directly using (29), for which we need
to define the extension of the depolarizing channel to the whole space of
bounded operators $\mathcal{B}(\mathcal{H})$. We define then the CPTP map
$\mathcal{E}^{\prime}:\mathcal{B}({\mathcal{H}})\longrightarrow\mathcal{B}({\mathcal{H}})$

$\mathcal{E}^{\prime}(A)=(1-\lambda)A+\lambda\text{tr}(A)\frac{I}{d}.$ (39)

The SAS representation associated with the input-dependent version of this map
is such that $q({\bf z}_{t})=0$ because the map is unital. It is also easy to
see that the system equation becomes $\textbf{x}_{t}=p({\bf
z}_{t})\textbf{x}_{t-1}=(1-\lambda_{t})\textbf{x}_{t-1}$. The filter
$U_{\widehat{\mathcal{E}}^{\prime}_{0}}$ of this QRC equation satisfies that
$U_{\widehat{\mathcal{E}}^{\prime}_{0}}({\bf z})_{t}={\bf 0}$, for all
$t\in\mathbb{Z}$.

######  Example 8.

The next case is an example of “poorly engineered” QRC system which can be
detected using Theorem 5. Indeed, we shall introduce a model of dissipation
using tuneable local losses, but Theorem 5 will discard its long-term
applicability because it is unital.

Let us define the Markovian master equation that governs the dynamics between
input injections:

$\dot{\rho}=-i[H({\bf z}_{t}),\rho]+\gamma L\rho
L^{\dagger}-\frac{\gamma}{2}\\{L^{\dagger}L,\rho\\},$ (40)

where $L$ is the jump operator and $\\{L^{\dagger}L,\rho\\}$ denotes the
anticommutator. We define the input-dependent Hamiltonian as $H({\bf
z}_{t})=h({\bf z}_{t})\sigma^{x}/2$, where $h({\bf z}_{t})$ will be an
arbitrary function of the input, and the jump operator as $L=\sigma^{z}$. This
is a single qubit under the influence of an external magnetic field in the $x$
direction of the real space (whose intensity varies between inputs) with a
local dephasing. Notice that the Hamiltonian is considered as time-independent
when integrating the dynamics since it is constant between input injections.

Going from the density matrix language to the real variable linear description
requires to find the CPTP map representation of (40). Since Markovian master
equations are CPTP linear transformations on their own, we just need to find
the Kraus decomposition that represents the dynamics of this master equation
as a map. We will follow the procedure as explained in [73] (see Section 2 in
the reference for the details of the algorithm). In particular, the Kraus
decomposition of a single qubit can be written in the following form:

$T(\rho)=\sum_{i,j}S^{(U)}_{ij}B_{i}\rho B^{\dagger}_{j},$ (41)

where $B_{i}$ are the basis elements of a single qubit in the operator space
($\\{I,\sigma^{x},\sigma^{y},\sigma^{z}\\}$) and $S^{(U)}_{ij}:=U^{\dagger}SU$
is a unitary transformation of the Choi matrix $S$. Applying (41) to the
definition of matrix $T$ in (19), we can find the map of the single qubit
observables:

$\left(\begin{matrix}1\\\ \braket{\sigma^{x}}_{t}\\\
\braket{\sigma^{y}}_{t}\\\
\braket{\sigma^{z}}_{t}\end{matrix}\right)=\left(\begin{matrix}1&0&0&0\\\
0&\widehat{T}_{22}&0&0\\\ 0&0&\widehat{T}_{33}&\widehat{T}_{34}\\\
0&0&\widehat{T}_{43}&\widehat{T}_{44}\end{matrix}\right)\left(\begin{matrix}1\\\
\braket{\sigma^{x}}_{t-1}\\\ \braket{\sigma^{y}}_{t-1}\\\
\braket{\sigma^{z}}_{t-1}\end{matrix}\right),$ (42)

where $\braket{\sigma^{a}}:=\text{tr}(\sigma^{a}\rho)$ is the expected value
of the spin projection in the $a$ direction of the real space. The expressions
for each matrix element are shown below:

$\begin{matrix}\widehat{T}_{22}=e^{-2\gamma\Delta\tau},\\\
\widehat{T}_{33}=e^{-\gamma\Delta\tau}\left(\cosh\left(\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)-\frac{\gamma}{\sqrt{\gamma^{2}-h_{t}^{2}}}\sinh\left(\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)\right),\\\
\widehat{T}_{44}=e^{-\gamma\Delta\tau}\left(\cosh\left(\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)+\frac{\gamma}{\sqrt{\gamma^{2}-h_{t}^{2}}}\sinh\left(\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)\right),\\\
\widehat{T}_{34}=\frac{h_{t}e^{-\gamma\Delta\tau}}{\sqrt{\gamma^{2}-h_{t}^{2}}}\sinh\left(\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right),\\\
\widehat{T}_{43}=-\widehat{T}_{34},\end{matrix}$ (43)

where have shortened notation by setting $h_{t}=h({\bf z}_{t})$. The
eigenvalues of matrix $\widehat{T}$ can be computed analytically:
$\lambda_{1}=1$, $\lambda_{2}=e^{-2\gamma\Delta\tau}$,
$\lambda_{3}=e^{-(\gamma+\sqrt{\gamma^{2}-h^{2}_{t}})\Delta\tau}$ and
$\lambda_{4}=e^{-(\gamma-\sqrt{\gamma^{2}-h^{2}_{t}})\Delta\tau}$. The moduli
of the eigenvalues are $|\lambda_{1}|=1$,
$|\lambda_{2}|=e^{-2\gamma\Delta\tau}<1$ and
$|\lambda_{3}|=e^{-(\gamma+\sqrt{\gamma^{2}-h^{2}_{t}})\Delta\tau}<1$.
Eigenvalue $|\lambda_{4}|=e^{-(\gamma-\sqrt{\gamma^{2}-h^{2}_{t}})\Delta\tau}$
is smaller than one if and only if $h_{t}\neq 0$. Under that condition, the
map is a mixing channel with a single fixed point. It can be checked that the
map is unital so the fixed point is the maximally mixed state:

$\rho^{*}=\left(\begin{matrix}1/2&0\\\ 0&1/2\end{matrix}\right).$ (44)

Let us prove that there exists some norm where
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|p(\textbf{z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1$
for all inputs. A straightforward induced norm to evaluate is the 2-Schatten
induced norm. The singular values of the restriction

$p(\textbf{z})=\left(\begin{matrix}\widehat{T}_{22}&0&0\\\
0&\widehat{T}_{33}&\widehat{T}_{34}\\\
0&\widehat{T}_{43}&\widehat{T}_{44}\end{matrix}\right)$ (45)

are $\sigma_{1}=e^{-2\gamma\Delta\tau}<1$,
$\sigma_{2}=e^{-\gamma\Delta\tau}\sqrt{f_{+}}$ and
$\sigma_{3}=e^{-\gamma\Delta\tau}\sqrt{f_{-}}$, where

$\begin{split}f_{\pm}&=\frac{1}{\gamma^{2}-h^{2}_{t}}\left(-h^{2}_{t}+\gamma^{2}\cosh\left(2\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)\right.\\\
&\left.\pm\gamma\sqrt{\sinh^{2}\left(\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)}\sqrt{-4h^{2}_{t}+2\gamma^{2}+2\gamma^{2}\cosh\left(2\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)}\right).\end{split}$
(46)

Figure 1: Density plot of the singular values (a) $\sigma_{2}$ and (b)
$\sigma_{3}$. The diagonal elements $h_{t}=\gamma$ are not determined because
of the denominator $1/(\gamma^{2}-h^{2}_{t})$.

Figure 1 numerically shows that for $\gamma\neq h_{t}$ and away from the axes
$h_{t}=0$ and $\gamma=0$, we find $\sigma_{2},\sigma_{3}<1$. Therefore, the
system has the ESP and the FMP. As we showed in Theorem 5, the corresponding
filter (17) is necessarily trivial and given by

$U_{T}({\bf z})_{t}=\rho^{*}=\left(\begin{matrix}1/2&0\\\
0&1/2\end{matrix}\right).$ (47)

Since the Bloch vector for this constant matrix is $(0,0,0)^{\top}$, this
shows that in the SAS representation $U_{\widehat{T}_{0}}({\bf
z})_{t}=(0,0,0)^{\top}$.

Unital quantum channels are very common in the quantum information literature
because of their practical advantages and well-known mathematical properties,
see for example [74, 75], and references therein. However, Theorem 5 discards
the possibility of relying on unital contractive channels for QRC for long
input sequences (see also Section IV for more details). The physical
explanation for this behavior stems from the fact that the fixed point of
these maps does not depend on the input. Then, after each application of the
channel, decoherence always leads to the same stationary state (the maximally
mixed state), which does not keep track of these inputs. This hinders any
possibility of storing the input information into the degrees of freedom of
the quantum system since all coherences fade out and the diagonal elements of
the density matrix become equal.

We believe that the observation that we just made is very relevant, namely
that quantum channels with input-independent fixed points become memoryless in
the long-term. This fact is proved in the next theorem that generalizes
Theorem 5.

######  Theorem 9.

Let $T:\mathcal{B}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{B}(\mathcal{H})$ be a QRC system for which there
exists an operator norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
and $\epsilon>0$ such that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})|_{\mathcal{B}_{0}(\mathcal{H})}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1-\epsilon$,
for all ${\bf z}\in D_{n}$. Then, $T$ has an input-independent fixed point
$\rho^{\ast}\in\mathcal{S}({\mathcal{H}})$, that is, $T(\rho^{*},{\bf
z})=\rho^{*}$, for all ${\bf z}\in D_{n}$, if and only if the corresponding
filter $U_{T}$ is constant, that is, $U_{T}({\bf
z})_{t}=\rho^{\ast}\in\mathcal{S}({\mathcal{H}})$, for all ${\bf
z}\in(D_{n})^{\mathbb{Z}}$ (equivalently, $U_{\widehat{T}_{0}}({\bf
z})_{t}=i_{0}^{-1}\left(G_{\mathcal{B}}^{-1}(\rho^{\ast})\right)$) .

###### Proof.

We first note that the contractivity hypothesis implies by Proposition 3 that
the system associated to $T$ has the ESP and hence has a unique solution for
each input. The hypothesis $T(\rho^{*},{\bf z})=\rho^{*}$, for all ${\bf z}\in
D_{n}$, obviously implies that the constant sequence equal to $\rho^{*}$ is a
solution for any input ${\bf z}\in(D_{n})^{\mathbb{Z}}$ and hence $U_{T}({\bf
z})_{t}=\rho^{\ast}\in\mathcal{S}({\mathcal{H}})$, for all ${\bf
z}\in(D_{n})^{\mathbb{Z}}$. Conversely, since the filter $U_{T}$ is determined
by the recursions

$U_{T}({\bf z})_{t}=T\left(U_{T}({\bf z})_{t-1},{\bf z}_{t}\right),$ (48)

then, if we always have that $U_{T}({\bf z})_{t}=\rho^{\ast}$, the relation
(48) implies that $T(\rho^{*},{\bf z})=\rho^{*}$, for all ${\bf z}\in D_{n}$.
∎

######  Remark 10.

The differences between the hypotheses in Theorems 5 and 9 on fixed points are
apparent when the QRC system is expressed using the SAS representation in
terms of the functions $q$ and $p$. Indeed, as we saw in the proof of Theorems
5, $q({\bf z})=0$ for and ${\bf z}\in D_{n}$, in that case, and the input
dependence takes place only through $p$. This is the case in Example 7.
Theorem 9 allows for an input dependence through $q$ too.

######  Example 11.

We define a Markovian master equation for the dynamics between input
injections:

$\dot{\rho}=-i[H({\bf z}_{t}),\rho]+\gamma L\rho
L^{\dagger}-\frac{\gamma}{2}\\{L^{\dagger}L,\rho\\},$ (49)

where $H({\bf z}_{t})=h({\bf z}_{t})\sigma^{z}/2$ and $L=\sigma^{-}$. This is
a single qubit under the influence of an external magnetic field in the $z$
direction with local dissipation. The matrix expression $\widehat{T}$ for the
associated system is:

$\left(\begin{matrix}1\\\ \braket{\sigma^{x}}_{t}\\\
\braket{\sigma^{y}}_{t}\\\
\braket{\sigma^{z}}_{t}\end{matrix}\right)=\left(\begin{matrix}1&0&0&0\\\
0&e^{-\frac{\gamma\Delta\tau}{2}}\cos(h_{t}\Delta\tau)&e^{-\frac{\gamma\Delta\tau}{2}}\sin(h_{t}\Delta\tau)&0\\\
0&-e^{-\frac{\gamma\Delta\tau}{2}}\sin(h_{t}\Delta\tau)&e^{-\frac{\gamma\Delta\tau}{2}}\cos(h_{t}\Delta\tau)&0\\\
e^{-\gamma\Delta\tau}-1&0&0&e^{-\gamma\Delta\tau}\end{matrix}\right)\left(\begin{matrix}1\\\
\braket{\sigma^{x}}_{t-1}\\\ \braket{\sigma^{y}}_{t-1}\\\
\braket{\sigma^{z}}_{t-1}\end{matrix}\right).$ (50)

The eigenvalues of $\widehat{T}$ can be computed analytically:
$\lambda_{1}=1$, $\lambda_{2}=e^{-\gamma\Delta\tau}$,
$\lambda_{3}=e^{-\frac{\gamma\Delta\tau}{2}-ih_{t}\Delta\tau}$ and
$\lambda_{4}=e^{-\frac{\gamma\Delta\tau}{2}+ih_{t}\Delta\tau}$. The moduli of
the eigenvalues are $|\lambda_{1}|=1$, $|\lambda_{2}|=e^{-\gamma\Delta\tau}<1$
and $|\lambda_{3}|=|\lambda_{4}|=e^{-\frac{\gamma\Delta\tau}{2}}<1$, so $T$ is
a mixing channel. In this case, the single fixed point is a pure input-
independent state with density matrix

$\rho^{*}=\left(\begin{matrix}0&0\\\ 0&1\end{matrix}\right).$ (51)

Let us see if it is true that there exists some
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|p(\textbf{z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1$
for all inputs. The singular values of the restriction

$p(\textbf{z})=\left(\begin{matrix}e^{-\frac{\gamma\Delta\tau}{2}}\cos(h_{t}\Delta\tau)&e^{-\frac{\gamma\Delta\tau}{2}}\sin(h_{t}\Delta\tau)&0\\\
-e^{-\frac{\gamma\Delta\tau}{2}}\sin(h_{t}\Delta\tau)&e^{-\frac{\gamma\Delta\tau}{2}}\cos(h_{t}\Delta\tau)&0\\\
0&0&e^{-\gamma\Delta\tau}\end{matrix}\right)$ (52)

are $\sigma_{1}=e^{-\gamma\Delta\tau}<1$ and
$\sigma_{2}=\sigma_{3}=e^{-\frac{\gamma\Delta\tau}{2}}<1$. Therefore, the
system has the ESP and FMP. As we showed in Theorem 9, the associated filter
is necessarily constant, and (17) necessarily yields the filter

$U_{T}({\bf z})_{t}=\rho^{*}=\left(\begin{matrix}0&0\\\
0&1\end{matrix}\right).$ (53)

Since the Bloch vector for this constant matrix is $(0,0,-1)^{\top}$, this
shows that in the SAS representation $U_{\widehat{T}_{0}}({\bf
z})_{t}=(0,0,-1)^{\top}$.

We can double-check the solution by explicitly computing the filter (31). Take
$p({\bf z}_{t})$ as in (52) and $q({\bf
z}_{t})^{\top}=(0,0,e^{-\gamma\Delta\tau}-1)$. Since
$\sigma_{\text{max}}(p({\bf z}_{t}))<1$, the filter in (31) exists. Using now
that $q({\bf z}_{t})$ is input-independent, we can write

$\begin{split}U_{\widehat{T}_{0}}({\bf
z})_{t}&=\sum^{\infty}_{j=0}\left(\prod^{j-1}_{k=0}p({\bf
z}_{t-k})\right)q({\bf z}_{t-j})\\\
&=\left(\sum^{\infty}_{j=0}\prod^{j-1}_{k=0}p({\bf
z}_{t-k})\right)q=(e^{-\gamma\Delta\tau}-1)M_{3},\end{split}$ (54)

where $M_{3}$ is the third column of the matrix
$M=\sum^{\infty}_{j=0}\prod^{j-1}_{k=0}p({\bf z}_{t-k})$. Notice that the
third column of the product $\prod^{j-1}_{k=0}p({\bf z}_{t-k})$ is
$(0,0,e^{-k\gamma\Delta\tau})^{\top}$. Then, the column $M_{3}$ equals

$M_{3}=\left(\begin{matrix}0\\\ 0\\\
\sum^{\infty}_{j=0}e^{-k\gamma\Delta\tau}\end{matrix}\right)=\left(\begin{matrix}0\\\
0\\\ \frac{1}{1-e^{-\gamma\Delta\tau}}\end{matrix}\right),$ (55)

and it hence follows that $U_{\widehat{T}_{0}}({\bf z})_{t}=(0,0,-1)^{\top}$.

To conclude, we show in Figure 2 a numerical example of input driving. We
choose $h(z_{t})=z_{t}\frac{h}{2}\sigma^{z}$ as the external magnetic field
function, where $h$ is a constant and $z_{t}$ is a unidimensional random
input. The input will be drawn from a random uniform distribution in the
interval $[0,1]$. As can be seen, the observables exhibit a transient time
after which they converge to the input-independent stationary state.

Figure 2: Dynamics of the spin projections $\braket{\sigma^{a}}_{t}$ for
$a=x,y,z$ when driven with a random input sequence with Hamiltonian
$H=z_{t}\frac{h}{2}\sigma^{z}$. The initial condition is the maximal coherent
state $\rho=1/2\sum^{1}_{i,j=0}\ket{i}\bra{j}$ and the system parameters are
$\Delta\tau=1$, $\gamma=1$ and $h=1$.

##  IV Discussion

Theorems 5 and 9 bring up a connection between long-term computation and noisy
intermediate-scale quantum (NISQ) devices: they have a finite time of
operation due to decoherence. Consider a model of the type

$\rho_{t}=T(\rho_{t},{\bf z}_{t})=\mathcal{E}_{\text{deco}}(\mathcal{U}({\bf
z}_{t})\rho_{t-1}\mathcal{U}^{\dagger}({\bf z}_{t})),$ (56)

where $\mathcal{E}_{\text{deco}}$ represents the decoherence produced by the
contact of the system with an external environment. This model is present in
QRC experimental works like [40, 39], where the unitary dynamics
$\mathcal{U}({\bf z}_{t})$ is given by a quantum circuit. Theorem 5 explains
what happens in the extreme case in which the quantum noise of the device is
unital. If the decoherence channel $\mathcal{E}_{\text{deco}}$ is a unital
strictly contractive map then

$T\left(\frac{I}{d},{\bf
z}_{t}\right)=\mathcal{E}_{\text{deco}}\left(\mathcal{U}({\bf
z}_{t})\frac{I}{d}\mathcal{U}^{\dagger}({\bf
z}_{t})\right)=\frac{\mathcal{E}_{\text{deco}}(I)}{d}=\frac{I}{d},$

that is, $T(\cdot,{\bf z})$ becomes a unital strictly contractive map for all
${\bf z}\in D_{n}$. Theorem 5 shows, in this case, that the filter becomes
trivial after the injection of long input sequences. Instances of unital
decoherence can be found in depolarizing channels, like in Example 7, or in
dephasing channels. A dephasing channel damps the coherences of the density
matrix but does not affect the diagonal elements.

More generally, Theorem 9 can be interpreted as a “common sense” warning: it
explicitly states that the input codification must have a measurable influence
on the attractor of the natural dynamics of the CPTP map. Otherwise, there is
no possibility of storing input information in the long run. We emphasize that
this is independent of the type of dissipation that the quantum channel
produces. We can connect the implications of this theorem with the NISQ
discussion and (56). If the input is codified in some particular coherences of
the system, one must be careful that decoherence does not destroy those matrix
elements, because then input-dependence would vanish, and the resulting filter
would become trivial.

This discussion does not imply that models like (56) are useless. Indeed, the
opposite has been proven in previous works for short-term memory tasks [40,
39]. Theorems 5 and 9 only rigorously establish something that was already
known about NISQ devices, that is, that there is a coherence time in which
they can be exploited. Equivalently, models affected by these theorems have
the ESP, but the resulting input/output dynamics becomes trivial for long
input sequences.

The coherence time limitation affects to all quantum platforms to a greater or
lesser extent. Then, either QRC proposals are subject to operate on shorter
time scales than the natural noise time scale (as done in [40, 39]), or the
QRC system is carefully design to integrate it. For example, as we will see in
Section IV.1, dephasing can be integrated as part of a QRC system that is not
affected by Theorems 5 and 9.

We could further extend this analysis to QRC models with measurements. As an
example we take the model proposed in [76]. In this reference, the quantum
measurement is applied at each time step after the input dependent CPTP map
$T$. The measurement scheme is introduced by modeling an indirect measurement
with a continuous-variable ancilla [77, 78, 79, 80, 81], producing a quantum
reservoir with stochastic dynamics. For simplicity, we will restrict ourselves
to the case of a single qubit, and we will average the quantum states over the
limit of infinite measurements, yielding an unconditional state which is led
by a deterministic CPTP quantum channel [82]. Besides, we choose, without loss
of generality, to take measurements in the $z$ direction. Under these
conditions, the CPTP map is

$\rho_{t}=M\odot T(\rho_{t-1},\textbf{z}_{t}),$ (57)

where $\odot$ represents the Hadamard or element-wise matrix product and $M$
is defined as

$M=\begin{pmatrix}1&e^{-\frac{g^{2}}{2}}\\\
e^{-\frac{g^{2}}{2}}&1\end{pmatrix}.$ (58)

The measurement strength $g$ allows us to quantify the decoherence introduced
by sharp measurements ($g\gg 1$), while for $g\ll 1$ the state is weakly
perturbed. It is straightforward to see that this model is introducing
dephasing, such that we can rewrite (57) as

$\rho_{t}=\mathcal{E}_{\text{deph}}\left(T(\rho_{t-1},\textbf{z}_{t})\right),$
(59)

where the dephasing channel $\mathcal{E}_{\text{deph}}$ is defined as

$\mathcal{E}_{\text{deph}}(\rho)=e^{-\frac{g^{2}}{2}}\rho+(1-e^{-\frac{g^{2}}{2}})\sigma^{z}\rho\sigma^{z}.$
(60)

As we explained above, unitary dynamics for the map $T$ would lead to a
memoryless reservoir in the long-term. However, one could engineer a mixing
map $T$ such that there is a competition between the attractors of maps $T$
and $\mathcal{E}_{\text{deph}}$. The final fixed point of (59) would be
somewhere between the original fixed point of $T$ and a diagonal state (which
is the shape of the fixed points of $\mathcal{E}_{\text{deph}}$).

We conclude by presenting an example of a qubit with tunable local dissipation
that fulfills all the requirements to be a “properly engineered” QRC system.
Then, we extend it with the measurement model of [76] to show that it still
constitutes a proper QRC system.

###  IV.1 A “properly engineered” QRC system

We start by introducing the model without measurements. The Markovian master
equation that governs the dynamics between input injections is:

$\dot{\rho}=-i[H({\bf z}_{t}),\rho]+\gamma L\rho
L^{\dagger}-\frac{\gamma}{2}\\{L^{\dagger}L,\rho\\},$ (61)

where $H({\bf z}_{t})=h({\bf z}_{t})\sigma^{x}/2$, and $L=\sigma^{-}$. The
corresponding matrix expression $\widehat{T}$ is

$\left(\begin{matrix}1\\\ \braket{\sigma^{x}}_{t}\\\
\braket{\sigma^{y}}_{t}\\\
\braket{\sigma^{z}}_{t}\end{matrix}\right)=\left(\begin{matrix}1&0&0&0\\\
0&\widehat{T}_{22}&0&0\\\
\widehat{T}_{31}&0&\widehat{T}_{33}&\widehat{T}_{34}\\\
\widehat{T}_{41}&0&\widehat{T}_{43}&\widehat{T}_{44}\end{matrix}\right)\left(\begin{matrix}1\\\
\braket{\sigma^{x}}_{t-1}\\\ \braket{\sigma^{y}}_{t-1}\\\
\braket{\sigma^{z}}_{t-1}\end{matrix}\right),$ (62)

where the expressions for the matrix elements are shown below:

$\begin{matrix}\widehat{T}_{22}=e^{-\frac{\gamma\Delta\tau}{2}},\\\
\widehat{T}_{33}=e^{-\frac{3\gamma\Delta\tau}{4}}\left(\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)+\frac{\gamma}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right),\\\
\widehat{T}_{44}=e^{-\frac{3\gamma\Delta\tau}{4}}\left(\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)-\frac{\gamma}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right),\\\
\widehat{T}_{34}=\frac{4h_{t}e^{-\frac{3\gamma\Delta\tau}{4}}}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right),\\\
\widehat{T}_{43}=-\widehat{T}_{34},\\\ \widehat{T}_{31}=\frac{2\gamma
h_{t}}{\gamma^{2}+2h^{2}_{t}}\left\\{-1+e^{-\frac{3\gamma\Delta\tau}{4}}\left(\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)+\frac{3\gamma}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right)\right\\},\\\
\widehat{T}_{41}=\frac{\gamma}{\gamma^{2}+2h^{2}_{t}}\left\\{-\gamma+e^{-\frac{3\gamma\Delta\tau}{4}}\left(\gamma\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)-\frac{\gamma^{2}+8h^{2}_{t}}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right)\right\\}.\end{matrix}$
(63)

Figure 3: Density plot of the singular values (a) $\sigma_{2}$ and (b)
$\sigma_{3}$. The elements $\gamma=4h_{t}$ are not determined because of the
denominator $1/(\gamma^{2}-16h^{2}_{t})$.

The eigenvalues of matrix $\hat{T}$ are: $\lambda_{1}=1$,
$\lambda_{2}=e^{-\frac{\gamma\Delta\tau}{2}}$,
$\lambda_{3}=e^{-(3\gamma^{2}+\sqrt{\gamma^{2}-16h^{2}_{t}})\frac{\Delta\tau}{4}}$
and
$\lambda_{4}=e^{-(3\gamma^{2}-\sqrt{\gamma^{2}-16h^{2}_{t}})\frac{\Delta\tau}{4}}$.
The three eigenvalues $\lambda_{2}$, $\lambda_{3}$ and $\lambda_{4}$ have
always modulus smaller than one when $\gamma\neq 0$. Then, the master equation
fulfills the conditions for having a single full-rank fixed point [62], whose
density matrix is

$\rho^{*}=\frac{1}{\gamma^{2}+2h^{2}_{t}}\left(\begin{matrix}h^{2}_{t}&i\gamma
h_{t}\\\ -i\gamma h_{t}&\gamma^{2}+h^{2}_{t}\end{matrix}\right).$ (64)

Finally, we compute the singular values of the restriction to the traceless
hyperplane. These values are $\sigma_{1}=e^{-\frac{\gamma\Delta\tau}{2}}<1$,
$\sigma_{2}=e^{-\frac{3\gamma\Delta\tau}{4}}\sqrt{f_{+}}$ and
$\sigma_{3}=e^{-\frac{3\gamma\Delta\tau}{4}}\sqrt{f_{-}}$, where

$\begin{split}f_{\pm}&=\frac{1}{\gamma^{2}-16h^{2}_{t}}\left(-16h^{2}_{t}+\gamma^{2}\cosh\left(\frac{\Delta\tau}{2}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right.\\\
&\left.\pm\gamma\sqrt{\sinh^{2}\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)}\sqrt{-64h^{2}_{t}+2\gamma^{2}+2\gamma^{2}\cosh\left(\frac{\Delta\tau}{2}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)}\right).\end{split}$
(65)

Figure 3 shows that for $\gamma\neq 4h_{t}$ and away from the axis $\gamma=0$,
we find $\sigma_{2},\sigma_{3}<1$, demonstrating the ESP and the FMP. Since
the fixed point $\rho^{*}$ is input dependent, this system exhibits the
necessary ingredients to be a competent QRC system.

As in Example 11, we show in Figure 4 a numerical example of input driving. We
choose again $h(z_{t})=z_{t}\frac{h}{2}\sigma^{x}$ as the external magnetic
field function, modifying its direction. Now, the observables
$\braket{\sigma^{z}}$ and $\braket{\sigma^{y}}$ exhibit an explicit response
to the driving, while $\braket{\sigma^{x}}$ converges to its input-independent
stationary value (which can be predicted from (62)).

Figure 4: Dynamics of the spin projections $\braket{\sigma^{a}}_{t}$ for
$a=x,y,z$ when driven with a random input sequence with Hamiltonian
$H=z_{t}\frac{h}{2}\sigma^{x}$. The initial condition is the maximal coherent
state $\rho=1/2\sum^{1}_{i,j=0}\ket{i}\bra{j}$ and the system parameters are
$\Delta\tau=1$, $\gamma=1$ and $h=1$.

Now we further extend the model to incorporate the measurement formalism
described in [76]. As the composition of CPTP maps can be described as the
product of their matrix representations [58], we just need to obtain the Kraus
operators of (60), which are $K_{0}=\sqrt{e^{-\frac{g^{2}}{2}}}I$,
$K_{1}=\sqrt{1-e^{-\frac{g^{2}}{2}}}\ket{0}\bra{0}$ and
$K_{2}=\sqrt{1-e^{-\frac{g^{2}}{2}}}\ket{1}\bra{1}$, where $\ket{0}$ and
$\ket{1}$ are the basis states in the $z$ axis. The matrix representation of
$\mathcal{E}_{\text{deph}}$ in the Pauli matrix basis is then

$\widehat{\mathcal{E}}_{\text{deph}}=\left(\begin{matrix}1&0&0&0\\\
0&e^{-\frac{g^{2}}{2}}&0&0\\\ 0&0&e^{-\frac{g^{2}}{2}}&0\\\
0&0&0&1\end{matrix}\right).$ (66)

It is straightforward to check that the maximum singular value of
$\widehat{\mathcal{E}}_{\text{deph}}$ restricted to the traceless hyperplane
is equal to one. The final matrix
$\widehat{T}^{\prime}=\widehat{\mathcal{E}}_{\text{deph}}\widehat{T}$ is

$\widehat{T}^{\prime}=\left(\begin{matrix}1&0&0&0\\\
0&e^{-\frac{g^{2}}{2}}\widehat{T}_{22}&0&0\\\
e^{-\frac{g^{2}}{2}}\widehat{T}_{31}&0&e^{-\frac{g^{2}}{2}}\widehat{T}_{33}&e^{-\frac{g^{2}}{2}}\widehat{T}_{34}\\\
\widehat{T}_{41}&0&\widehat{T}_{43}&\widehat{T}_{44}\end{matrix}\right).$ (67)

Given that the maximum singular value of $p(\textbf{z})$ is smaller than one
(for $\gamma\neq 4h_{t}$ and $\gamma\neq 0$), we find that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|p^{\prime}(\textbf{z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{2}\leq{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|p_{\text{deph}}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{2}\cdot{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|p(\textbf{z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{2}<1-\epsilon$
for some $\epsilon>0$ in the 2-Schatten norm, where $p(\textbf{z})$,
$p^{\prime}(\textbf{z})$ and $p_{\text{deph}}$ are the restrictions to the
traceless hyperplane of matrices $\widehat{T}$, $\widehat{T}^{\prime}$ and
$\widehat{\mathcal{E}}_{\text{deph}}$ respectively (given by (27)). Then, the
QRC system has the ESP and the FMP. The single fixed point of the map is given
by

$\rho^{*}=\frac{1}{\gamma^{2}+2h^{2}_{t}}\left(\begin{matrix}h^{2}_{t}-f_{1}&i\gamma
h_{t}(1-f_{2})\\\ -i\gamma
h_{t}(1-f_{2})&\gamma^{2}+h^{2}_{t}+f_{1}\end{matrix}\right),$ (68)

where

$\begin{split}&f_{1}=\frac{4\gamma
h^{2}_{t}\sinh\left(\frac{g^{2}}{4}\right)\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)}{\sqrt{\gamma^{2}-16h_{t}^{2}}\left(\cosh\left(\frac{g^{2}+3\gamma\Delta\tau}{4}\right)-\cosh\left(\frac{g^{2}}{4}\right)\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right)+\gamma\sinh\left(\frac{g^{2}}{4}\right)\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)},\\\
&f_{2}=\frac{\sinh\left(\frac{g^{2}}{4}\right)\left(e^{\frac{3\gamma\Delta\tau}{4}}-\cosh\left(\frac{g^{2}}{4}\right)\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)+\frac{\gamma}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{g^{2}}{4}\right)\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right)}{\cosh\left(\frac{g^{2}+3\gamma\Delta\tau}{4}\right)-\cosh\left(\frac{g^{2}}{4}\right)\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)+\frac{\gamma}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{g^{2}}{4}\right)\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)}.\end{split}$
(69)

This fixed point is produced by the competition between the original fixed
point in (64) ($g\rightarrow 0$) and a diagonal state ($g\rightarrow\infty$).
With this we can conclude that this engineered model, even including the
measurement protocol, leads to an operational QRC system in the long-term run.

##  V Conclusions

In this paper, we have unified the density matrix approach of previous works
in QRC with the Bloch vector representation. Moreover, we have shown that
these representations are linked by system isomorphisms and that various
results concerning the ESP and FMP are independent of the chosen
representation. We have also observed that the QRC dynamics in the Bloch
vectors representation amounts to that of a state-affine system (SAS) of the
type introduced in [45] and for which numerous theoretical results have been
established. We have capitalized on this connection to shed some light on
fundamental questions in QRC theory in finite dimensions. In particular, we
found a necessary and sufficient condition for the ESP and FMP in terms of the
existence of an induced norm that bounds the CPTP map for all inputs,
determining a guideline for its election. The necessity of this boundedness
hypothesis emerges out of the compactness of the input space, which is a
common requirement in the RC literature. If the input space is not compact,
sufficient conditions can still be found in terms of the weighting sequence
[50]. Besides, we described common situations in which QRC systems become
useless in long term runs which can be summarized by saying that quantum
channels that exhibit input-independent fixed points yield trivial
input/output dynamics.

Our work sets the grounds for further analysis and exploration of the QRC
theory. Future work can follow several paths, such as studying the connection
between spectral properties of QRC models and their performance in memory and
information processing tasks, studying infinite-dimensional quantum
reservoirs, or including generalized measurements (positive operator-valued
measures) and the effect of a finite number of measurements in the statistics
of expected values, given the effect that they imprint in the resources of QRC
algorithms [76].

##  VI Acknowledgments

We thank A. Sannia for useful discussion and inspiration for this work, and D.
Burgarth and M. Rahaman for answering our questions regarding their work and
its relation to our study. We also thank the editor and two anonymous referees
for input that has significantly improved the paper. R.M.-P. and J.-P.O.
acknowledge partial financial support from the Swiss National Science
Foundation (Grant No. 200021 175801/1). R.M.-P. acknowledges the Spanish State
Research Agency for support through the Severo Ochoa and María de Maeztu
Program for Centers and Units of Excellence in R&D (Grant No. MDM-2017-0711)
and through the QUARESC project (Projects No. 2019-109094GB-C21 and
2019-109094GB-C22/AEI/10.13039/501100011033). Part of this work was funded by
MICINN/AEI/FEDER and the University of the Balearic Islands through a
predoctoral fellowship (Grant No. MDM-2017-0711-18-1) for R.M.-P. R.M.-P. also
is grateful for the hospitality of the Division of Mathematical Sciences of
the Nanyang Technological University, where most of these results were
obtained.

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