# Concrete Quantum Channels and Algebraic Structure of Abstract Quantum
Channels

M.N.N. Namboodiri  Former Professor,Department of Mathematics, Cochin
University of Science and Technology, Kochi, India.
IMRT,Thiruvananthapuram<EMAIL_ADDRESS>
###### Abstract

This article analyzes the algebraic structure of the set of all quantum
channels and its subset consisting of quantum channels that have Holevo
representation. The regularity of these semigroups under composition of
mappings are analysed. It is also known that these sets are compact convex
sets and, therefore, rich in geometry. An attempt is made to identify
generalized invertible channels and also the idempotent channels. When
channels are of the Holevo type, these two problems are fully studied in this
article. The motivation behind this study is its applicability to the
reversibility of channel transformations and recent developments in resource-
destroying channels, which are idempotents. This is related to the coding-
encoding problem in quantum information theory. Several examples are provided,
with the main examples coming from pre-conditioner maps which assigns
preconditioners to matrices , in numerical linear algebra.Thus the known pre-
conditioner maps are viewd as a quantum- channel in finite dimentions.

††preprint: AIP

##  I Introduction

Over the past few years, there has been significant progress in the
construction and testing of preconditioners for Toeplitz and block Toeplitz
matrices using Korovkin’s classical approximation theorems through positive
linear maps. These preconditioners have been found to exhibit complete
positivity, leading to an abstract formulation of Korovkin-type theorems in a
non-commutative context. Interestingly, these preconditioner maps can be
viewed as abstract quantum channels.

In this concise article, we explore this viewpoint by examining various
related quantities such as the Kraus representation, channel capacity, and
fidelity. We also discuss the algebraic structure of the class of quantum
channels, which forms a semigroup under composition of mappings. Furthermore,
we prove that the set of quantum channels in the Holevo form constitutes a
sub-semigroup without an identity. We obtain a characterization of idempotents
in terms of the associated stochastic matrices, which is significant as the
range of idempotents remains invariant under quantum operations. Remarkably,
we observe that the preconditioner maps are idempotents belonging to one of
the aforementioned semigroups. In recent years, idempotent quantum channels
have proved to be useful in coding-encoding problems.

Completely positive linear maps between $C^{*}$-algebras have emerged as
powerful tools in non-commutative analysis, quantum mechanics, quantum
information theory, and linear algebra problems (see references such as Holevo
(2019), Davies (1976), and Kumar, Namboodiri, and Serra-Capizzano (2013)). As
mentioned in the abstract, we focus on studying the maps $P_{U}(.)$ and
$\tilde{P}_{U}$, described in the articles Serra Capizzano (1998) and Kumar,
Namboodiri, and Serra-Capizzano (2013), which assign good preconditioners to
elements in the class $M_{n}(C)$ of $n\times n$ matrices. We observe that
Lemma 2.1 on page 311 of Serra Capizzano (1998) implies that $P_{U}$ and its
general versions introduced in Kumar, Namboodiri, and Serra-Capizzano (2013)
can be viewed as abstract quantum channels. In line with this perspective, we
examine these maps in the context of quantum channels (C-Q Channels)
introduced and studied in references such as Holevo (2019), Ahiable  _et al._
(2021), and Choi (1975). Inspired by the work of Alexander Holevo and David
Kribs, we determine the channel fidelity, compute the Choi matrix Ahiable  _et
al._ (2021), and analyze other relevant quantities associated with the map
$P_{U}(.)$ as well as its general version, $\tilde{P}_{U}$. Additionally, we
provide several concrete examples of unitary matrices $U$ that define $P_{U}$.

This article is divided into five sections. Sections I and II are devoted to
the introduction and preliminaries, and a few new results of the
preconditioner maps $P_{U}$ and $\tilde{P}_{U}$ concerned. A few related
concepts, such as channel capacity and channel-fidelity, are considered in the
next couple of sections. It is proved that the channel capacity of $P_{U}$ is
infinite. Section III is mainly devoted to the algebraic structure of quantum
channels, especially in the Holevo-form. Section IV is devoted to the role of
precondioner maps $P_{U}$ and $\tilde{P}_{U}$ as resource -destroying channels
in coding-encoding problems mentioned in the introduction. Finally, in the
last section V, a few research problems for future considerations are given.
As mentioned in the abstract, idempotent quantum-channels are known to be
useful in connection with coding-encoding problems. A possiblity of such
interpretations for precoditioner map is discussed in remarks 9 below Liu, Hu,
and Lloyd (2017),Korzekwa  _et al._ (2022).
Throughout sections from I to IV, all spaces considered are of finite-
dimentions.

##  II Preliminaries

Throughout this article, we consider completely positive linear maps on
$M_{n}(C)$, the set of all matrices of order$n$ with complex entries. As
mentioned in the introduction, we concentrate on the maps considered in Serra
Capizzano (1998) and Kumar, Namboodiri, and Serra-Capizzano (2013) for
generating preconditioners in numerical linear algebra.
Preconditioners are very important tools in Numerical Linear Algebra to deal
with ill-conditioned problems that arise naturally while considering real-
world problems. In this regard, constructing appropriate preconditioners is an
equally important problem. As mentioned in the abstract, one such construction
is given in [6], page 4.

######  Definition 1.

(Preconditioners) Let $M_{n}(C)$ be the set of all matrices of order $n$ over
$C$.For a pre-assigned unitary $U\in M_{n}(C)$,let $M(U)$ denote the set of
all matrices simultaniuosly diagonalised by $U$.To be more explicit,

$M(U)=\\{A=U\Delta U^{*}:\Delta diagonal\\}.$

A corresponding pre-conditioner $P_{U}(A)$ for a given matrix $A$ is then
given by

$P_{U}(A)=U\sigma(U^{*}AU)U^{*},$

where for a given matrix $B$, $\sigma(B)$ denotes the matrix obtained by
retaining the diagonal part of $B$ and putting zero elsewhere.

######  Remark 1.

In fact ,it can be seen that $P_{U}$ is the orthogonal projection of the
$n^{2}$\- dimentional Hilbert-space $M_{n}(C)$ with respect to the inner
product $\langle A,B\rangle=Tr(AB^{*})$, onto the subspace $M(U)$.

Next we recall Kumar, Namboodiri, and Serra-Capizzano (2013) a more general
version $\tilde{P}_{U}$, which is defined as follows:

######  Definition 2.

(Kumar, Namboodiri, and Serra-Capizzano (2013)) Let $\\{P_{k};k=1,2,...,d\\}$
be a finite collection of mutually orthogonal projections such that
$\Sigma_{k=1}^{d}P_{k}=I_{n}$, where $d\leq n$.Let

$\Psi(A)=\Sigma_{k=1}^{d}P_{k}AP_{k},$ (1)

and

$\tilde{M}_{U}=\\{A\in M_{n}(C):U^{*}AU=\Psi(A)\\}.$ (2)

Recall the definition of $\tilde{P}_{U}$:

$\tilde{P}_{U}(A)=U(\Psi(U^{*}AU))U^{*},A\in M_{n}(C).$ (3)

######  Theorem 1.

Kumar, Namboodiri, and Serra-Capizzano (2013)

  * With $A,B\in M_{n}(C)$

  * [1]

$\tilde{P}_{U}(\alpha A+\beta
B)=\alpha\tilde{P}_{U}(A)+\beta\tilde{P}_{U}(B),$

  * [2]

$\tilde{P}_{U}(A^{*})=\tilde{P}_{U}(A)^{*},$

  * [3]

$tr(\tilde{P}_{U}(A))=Tr(A),$

  * [5]

$\|A-\tilde{P}_{U}(A)\|^{2}_{F}=\|A\|_{F}^{2}-\|\tilde{P}_{U}(A)\|^{2}_{F},$

where $\|.\|_{F}$ denotes the Frobinious norm of matrices.

It is clear that $P_{U}$ is a special case of $\tilde{P}_{U}$ if the
projections $\\{P_{k}:k=1,2,...d\\}$ are precisely rank one pairwise
orthogonal projections $\\{p_{k}:k=1,2,...,n\\}$. However, we treat them
differently as and when the situation arises.

######  Example 1.

Several examples of unitaries $U$ arise in concrete situations such as
discrete Fourier transforms, interpolation of functions etc. Serra Capizzano
(1998),Kumar, Namboodiri, and Serra-Capizzano (2013). We quote them here for
the sake of completeness.

Let $v=\left\\{{v_{n}}\right\\}_{n\in
N}\,\,\textrm{with}\,v_{n}=\left({v_{nj}}\right)_{j\leq n-1}$ be a sequence of
trigonometric functions on an interval $I.$ Let
$S=\left\\{{S_{n}}\right\\}_{n\in N}$ be a sequence of grids of n points on
$I,$ namely, $S_{n}=\left\\{{x_{i}^{n},i=0,1,.\,.\,.\,n-1}\right\\}$. Let us
suppose that the generalized Vandermonde matrix

${V_{n}=}\left({{v}_{{nj}}{(}x_{i}^{n}{)}}\right)_{{i;j=0}}^{{n-1}}$

is a unitary matrix. Then, algebra of the form $M_{{U_{n}}}$ is a
trigonometric algebra if $U_{n}={V_{n}}^{*}$ with $V_{n}$ a generalized
trigonometric Vandermonde matrix.

We quote examples of trigonometric algebras with the following choice of the
sequence of matrices ${U_{n}}$ and grid $S_{n}$.

  * [1]

$U_{n}=F_{n}=\left({\frac{{1}}{{\sqrt{n}}}e^{ijx_{i}^{n}}}\right)\,\,,\,\,\,\,i,j=0,1,.\,.\,.\,n-1,$

  * [2]

$S_{n}=\left\\{{x_{i}^{n}=\frac{{2i\pi}}{n},i=0,1,.\,.\,.\,n-1}\right\\}\subset
I=\left[{-\pi,\pi}\right]$

  * [3]

$U_{n}=G_{n}=\left({\sqrt{\frac{2}{n+1}}}sin(j+1)x_{i}^{n}\right)\,\,,\,\,\,\,i,j=0,1,.\,.\,.\,n-1,$

  * [4]

$S_{n}=\left\\{{x_{i}^{n}=\frac{{(i+1)\pi}}{n+1},i=0,1,.\,.\,.\,n-1}\right\\}\subset
I=\left[{0,\pi}\right]$

  * [5]

$U_{n}=H_{n}=\left(\frac{{1}}{{\sqrt{n}}}\left[{{sin(jx_{i}^{n})+cos(jx_{i}^{n})}}\right]\right)\,\,,\,\,\,\,i,j=0,1,.\,.\,.\,n-1.$

The complete- positivity of $\tilde{P}_{U}$ was observed in Kumar, Namboodiri,
and Serra-Capizzano (2013), and this aspect was used to prove Korovkin-type
theorems. As mentioned in the abstract, we highlight a different point of view
regarding $\tilde{P}_{U}$, namely the quantum -channel  property mentioned in
the introduction. To be more precise, theorem 1 above describes the conditions
of an abstract quantum channelHolevo (2019).
Next, we recall the definition of complete positivity followed by a few very
important representations of such maps.

######  Definition 3.

(Complete positivity) Let $A$ and $B$ be $C^{*}$-algebras over complex numbers
and $\Phi:A\rightarrow B$ be a linear map.Then $\Phi$ is said to be completely
positive if

$\Phi^{(n)}:A\otimes M_{n}(C)\rightarrow B\otimes M_{n}(C),$ (4)

defined as

$\Phi^{(N)}(a_{(i,j)})=(\Phi(a_{(i,j)})),$ (5)

is positivity preserving for all positive integers $n$ and $(a_{(i,j)})\in
A\otimes M_{n}(C)$.

As mentioned in the introduction, we consider only finite dimensional cases.

######  Definition 4.

Let$\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ be the Hilbert spaces associated
with the input and output systems of a channel, respectively. A channel in the
Schrodinger picture is a linear, completely positive, trace-preserving map
$\Phi$

$\Phi:\mathcal{T}(\mathcal{H}_{A})\rightarrow\mathcal{T}(\mathcal{H}_{B}),$

and dually, a channel in the Heisenberg picture is a linear, completely
positive map $\Phi^{*}$:

$\Phi^{*}:\mathcal{B}(\mathcal{H}_{A})\rightarrow\mathcal{B}(\mathcal{H}_{B})$

and dually a channel in the Heisenberg picture is a linear, completely
positive map that preserves the identity. Here $\mathcal{T}(H)$(respectively
$\mathcal{B}(\mathcal{H}))$ denotes the set of all trace class operators on a
Hilbert space $H$ (respectively bounded linear operators).

Next, a few important representations of completely positive maps, such as
Stienspring-dilation theorem, Kraus-representation, Holevo-form, are given
below.

######  Theorem 2.

Stienspring- Dilation Theorem - finite-dimensional case Assume that
$\mathcal{H}_{A},\mathcal{H}_{B}$ are finite-dimentional.Then for every
completely positive map $\Phi^{*}:B(\mathcal{H}_{A})\rightarrow
B(\mathcal{H}_{B})$ there exists a Hilbert space $\mathcal{H}_{E}$ and an
operator $V:\mathcal{H}_{A}\rightarrow\mathcal{H}_{B}\otimes\mathcal{H}_{E}$
such that

$\Phi^{*}(X)=V^{*}(X\otimes I_{E}),X\in B(\mathcal{H}_{B})$ (6)

Dually,

$\Phi^{*}(S)=TrVSV_{*},S\in\mathcal{T}(\mathcal{H}_{A}).$ (7)

The map $\Phi^{*}$ is unital, i.e., identity preserving (the map $\Phi$
preserves trace ) if and only if the operator $V$ is isometric.

Next we state the well known (Kraus- Representation)Kribs  _et al._ (2021)

######  Theorem 3.

A map $\Phi^{*}$ is completely positive if and only if it can be represented
in the form

$\Phi^{*}(X)=\Sigma_{k=1}^{d}V^{*}_{k}XV_{k},$ (8)

where $V_{k:}:\mathcal{H}_{A}\rightarrow\mathcal{H}_{B}$,or dually

$\Phi(S)=\Sigma_{k=1}^{d}V_{k}SV^{*}_{k},S\in\mathcal{T}(\mathcal{H}_{A}).$
(9)

The map $\Phi^{*}$ is unital ($\Phi$ is trace-preserving ) if and only if

$\Sigma_{k=1}^{d}V_{k}^{\star}V_{k}=I.$ (10)

This representation is not unique, but there is a representation with the
minimal number of components arising from minimal Stienspring representation!
Next, we compute the Kraus-representation of $P_{U}$, which is necessary for
the subsequent analysis of $P_{U}$ as a quantum channel.

######  Theorem 4.

The map $\tilde{P}_{U}:M_{n}(C)\rightarrow M_{n}(C)$ has Kraus-representation
given by

$\tilde{P}_{U}(A)=\Sigma_{k=1}^{d}V_{k}^{*}AV_{k},$ (11)

where $V_{k}=UP_{k}U^{*}$ , $\Psi(A)=\Sigma_{k=1}^{d}P_{k}AP_{k}$ ,the
pinching function defining $P_{U}$.

###### Proof.

The proof follows immediately by substituting the pinching $\Psi$ in the
defining equation of $\tilde{P}_{U}$ and then grouping the terms
appropriately. ∎

######  Definition 5.

(Holevo - Form) A quantum-channel $\Phi$ on $M_{n}(C)$ for which there exist
density operators $\\{S_{1},S_{2},...,S_{m}]\\}$ in $M_{n}(C)$ and a
resolution of identity $\\{X_{1},X_{2},...,X_{m}\\}$ such that

$\Phi(S)=\Sigma_{k=1}^{m}S_{k}T_{r}(SX_{k}),$ (12)

for all density operators $S$ is called the Holevo-FormKribs  _et al._ (2021)
of $\Phi$.
In addition, if there exists an orthonormal basis
$\\{e_{1},e_{2},...,e_{n}\\}$ such that $S_{k}=|{e_{k}\rangle\langle e_{k}}|$
for $k=1,2,...,n$, then such a channel is called a classical -quantum
channel(C-Q Channel).

###  II.1 Entanglement

######  Definition 6.

Let $\Phi$ be a quantumn channel on $M_{n}(C)\rightarrow M_{m}(C)$.Then $\Phi$
is called entanglement breaking  if for all $k\geq 1$ and states $X\in
M_{n}(C)\otimes M_{m}(C)$, the output state $(id_{k}\otimes\Phi)(X)$ is
separable, where $id_{k}$ is the identity map on $M_{n}(C)$.

######  Remark 2.

It is well known that $\Phi$ is entanglement breaking if and only if $\Phi$
has a following Kraus representation namely (refer Theorem 4, Horodecki, Shor,
and Ruskai (2003))

$\Phi(X)=\Sigma_{j=1}^{n}V_{j}^{*}XV_{j}$ (13)

where each $V_{j}$ is of rank 1.So among $\\{\tilde{P}_{U},P_{U}\\}$ only
$P_{U}$ is entanglement breaking.

##  III Algebraic Structure

In this section, the algebraic structure of the collection of all quantum
channels and the associated stochastic matrices Kribs  _et al._ (2021) are
considered. One of the motivation in considering the algebraic structure is
identification of idempotent Channels.Interestingly, recent articles by Kamil
Korzekwa, Zbigniew Puchała,etal.,Korzekwa  _et al._ (2022) and Zi-Wen Liu
etal.,Liu, Hu, and Lloyd (2017) are centred arround idempotent channels, in
the encoding-decoding problem in quantum communication theory. Let $QC(M_{n})$
and $HQC(M_{n})$ denote the collection of all quantum channels and the set of
all quantum channels in the Holevo-form on $M_{n}(C)$ respectively.

######  Theorem 5.

The set $QC(M_{n})$ is a semigroup with identity(i.e., monoid) and
$\\{\tilde{P}_{U},P_{U}:Uunitary\in{M_{n}}\\}$ are idempotents in $QC(M_{n})$
under composition of operators on $M_{n}(C)$.Also,the set $HQC(M_{n})$ is a
sub- semigroup of $QC(M_{n})$ without identity, containing $P_{U}$.

###### Proof.

It is straightforward to see that $QC(M_{n})$ is a semigroup with identity.
The maps $\\{\tilde{P}_{U},P_{U}:Uunitary\in{M_{n}}\\}$ are orthogonal
projections onto the subspace $\tilde{M}_{U}$ and therefore they are
idempotents.
Let $\Phi_{j}:j=1,2$ be in $HQC(M_{n})$ and

$\Phi_{1}(\rho)=\Sigma_{k}Tr(F_{k}\rho)R_{k},$ (14)

and and

$\Phi_{2}(\rho)=\Sigma_{j}Tr(H_{J}\rho)S_{J},$ (15)

be the respective Holevo-representation.Here $\rho,R_{k},S_{j}$ are density
matrices and $F_{k},H_{j}$ are POVM,$j,k$.Therefore we have

$\Phi_{1}\circ\Phi_{2}(\rho)=\Sigma_{k}Tr[F_{k}\Sigma_{j}Tr(H_{j}\rho)S_{j}]R_{k},\\\
$ (16) $=\Sigma_{k}\Sigma_{j}Tr(H_{j}\rho)Tr(F_{k}S_{j})S_{j}]R_{k},\\\ $
$=\Sigma_{k}\Sigma_{k}TrTr[F_{k}S_{j}Tr(H_{j}\rho)]R_{k},\\\ $
$=\Sigma_{j}Tr(H_{j}\rho)\Sigma_{k}Tr(F_{k}S_{j})R_{k},\\\ $
$=Tr(H_{j}\rho).\Sigma_{k}Tr(F_{k}S_{j}).$

Let $L_{J}=\Sigma_{k}Tr(F_{k}S_{j})R_{k}$.We show that $L_{j}$ is a density
matrix for each $j$.Clearly,each $L_{j}$ is positive definite.Now we ahve

$Tr(L_{j})=Tr\Sigma_{k}Tr(F_{k}S_{j})R_{k}=\Sigma_{k}Tr(F_{k}S_{j})Tr(R_{k})$
(17)

Interchanging summation, we get

$Tr(L_{j})=\Sigma_{k}Tr(F_{k}S_{j})=Tr(\Sigma_{k}F_{k})S_{j}$

But $(\Sigma_{k}F_{k}=I$ and $Tr(S_{j})=1$ ,since $\\{F_{k}:k=1,2,..,n\\}$ is
positive operator valued measure (POVM) and $S_{j}\&R_{k}$ are density
matrices..Therefore we have

$Tr(S_{j})=1.$

Thus we have $L_{j}$ is density for each $j$. Hence $\Phi_{1}\circ\Phi_{2}$
has the following Holevo representation

$\Phi_{1}\circ\Phi_{2}(\rho)=\Sigma_{j}Tr(H_{j}\rho)L_{j.}$ (18)

This completes the proof. ∎

Each element in $HQC(M_{n})$ can be associted with a stochastic matrix and
vice versa Kribs  _et al._ (2021).The following theorems examine the structure
of this association.

######  Theorem 6.

Assume that $\Phi$ has the Holevo form (6.1) and let the associated stochastic
matrix $A(\Phi)$ with $(i,j)^{th}$ entry $a_{i,j}$ be as follows:

$a_{i,j}=Tr(F_{j}R_{k})$ (19)

$i,j=1,2,..,r.$.Then the quantum channel (6.1) in the Holevo form is an
idempotent if and only if the probability vectors $(Tr(\rho F_{1}),Tr(\rho
F_{2}),...,Tr(\rho F_{r}))^{t}$ are all steady-state vectors of the
corresponding stochastic matrix (6.7) Ahiable  _et al._ (2021).

###### Proof.

It is known that the above matrix $A((\Phi))$ is a stochastic matrix Ahiable
_et al._ (2021).Now,$\Phi^{2}=\Phi$ implies that

$\Sigma_{i=1}^{n}Tr(\Phi(\rho)F_{i})R_{i}=\Phi(\rho)$ (20)

Therefore we have,

$\Sigma_{i=1}^{n}Tr(\Sigma_{j=1}^{r}Tr(\rho
F_{j})R_{j}F_{i})R_{i}=\Sigma_{k=1}^{r}Tr(\rho F_{k})R_{k}.$ (21)

i.e., we have the following identity

$\Sigma_{k=1}^{r}Tr(\rho
F_{k})R_{k}.=\Sigma_{i=1}^{r}[\Sigma_{j=1}^{r}Tr{(\rho
F_{j})Tr{(R_{j}F_{i})}}]R_{i}$ (22)

If $\\{R_{k}:k=1,2,...,r\\}$ is a linearly independent set, the above
equations to a system of linear equations:

$[\Sigma_{j=1}^{r}Tr{(\rho F_{j})Tr{(R_{j}F_{k})}}]=Tr(\rho F_{k})$ (23)

for $k=1,2,...,r$, for all density matrices $\rho$.
We can rewrite the above equation as a matrix equation as

$\begin{bmatrix}a_{1,1}\quad a_{1,2}\quad,...a_{1,r}\\\ a_{2,1}\quad
a_{2,2},\quad...,a_{2,r}\\\ .\quad.,...,.\\\ .\quad,...,.\\\ a_{r,1}\quad
a_{r,2}\quad,\quad.....a_{r,r}.\end{bmatrix}\begin{bmatrix}Tr(\rho F_{1})\\\
Tr{(\rho F_{2})}\\\ .\\\ .\\\ .\\\ Tr(\rho
F_{r}).\end{bmatrix}=\begin{bmatrix}Tr(\rho F_{1})\\\ Tr{(\rho F_{2})}\\\ .\\\
.\\\ .\\\ Tr(\rho F_{r}).\end{bmatrix}.$ (24)

∎

######  Remark 3.

It is trivial to see that when $\Phi=P_{U}$, the matrix (19) above is the
identity matrix, and the equation 24 is trivially satisfied.
Another related question is the generalized invertibility of quantum channels.
However, Theorem 6.3 above will be useful to address the question of quantum
channels in the Holevo form. Recall the definition of the generalized inverse
of an element in a semigroup.

######  Remark 4.

By Theorem 3, section 2 Horodecki, Shor, and Ruskai (2003),$\Phi\in
HQC(M_{n})$ if and only if $\Phi$ has a Kraus representation

$\Phi(T)=\Sigma_{k=1}^{d}V_{k}^{*}TV_{k},$ (25)

where each $V_{k}$ is of rank one and for all $T$. A simple, direct
computation shows that the composition of any two maps in has Holevo
form.However, the details are provided below since explicit expression of this
will be of use at a later context.
Let $\Phi_{j}:j=1,2$ be in $HQC(M_{n})$ . Without loss of generality,we may
assume that

$\Phi_{j}(T)=\Sigma_{k}^{m_{j}}V_{k,j}^{*}TV_{k,j}$ (26)

where $V_{k,j}$ has rank one for all $k$ and $j=1,2.$, for all $T\in
M_{n}(C)$. Therefore we have

$\Phi_{1}(\Phi_{2}(T))=\Sigma_{k=1}^{m_{1}}\Sigma_{l=1}^{m_{2}}V_{k,1}^{*}V_{l,2}^{*}TV_{l,2}V_{k,1},$
(27)

and it is clear that if $V_{k,1}$ is not orthogonal to $V_{l,2}$ then the
product $V_{k,1}V_{l,2}$ is of rank one.This completes the wanted cmputation.

######  Definition 7.

Let $\mathcal{S}$ be a semigroup and $a\in\mathcal{S}$.An element
$a^{\dagger}$ is called a generalised inverse of $a$ if

$aa^{\dagger}a=a.$ (28)

It is called a semi-inverse if in addition

$a^{\dagger}aa^{\dagger}=a^{\dagger}.$ (29)

Now the equations (28) and (29) above implies that $aa^{\dagger}$ and
$a^{\dagger}a$ are idempotents.If $a^{\dagger}\in HQC(M_{n})$ ,then
$aa^{\dagger}$ is an idempotent in $HQC(M_{n})$.
We investigate the relationship between the compositon of elements in
$HQC(M_{n})$ and the product of the corresponding stochastic matrices.

######  Remark 5.

By definition, the associated stochastic matrix of $\Phi_{1}\circ\Phi_{2}$ is
the matrix $a_{i,j}$ where

$a_{i,j}=Tr(H_{i}L_{j})=Tr(H_{i}L_{j})=Tr(H_{i}\Sigma_{k}Tr(F_{k}S_{j})R_{k}),$
(30) $=\Sigma_{k}Tr(F_{k}S_{j})Tr(R_{k}H_{i}).$

Next, we use equation 30 above to relate the associated stochastic matrices of
$\Phi_{1},\Phi_{2}\quad\&\quad\Phi_{1}\circ\Phi_{2}$. The summary is as
follows. For each quantum channel $\Phi$ in the Holevo form, let $[\Phi]$
denote the corresponding Stochastic matrix. Then we have the following
theorem.

######  Theorem 7.

Let $\Phi\quad\&\quad\Psi$ be quantum channels in the Holevo form. Then

$[\Psi\circ\Phi]=[\Psi][\Phi]$ (31)

where the modified matrix multiplication is as in equation 30.

###### Proof.

By equation 30, the associated stochastic matrix of $\Phi_{1}\circ\Phi_{2}$ is
the matrix $a_{i,j}$ where

$a_{i,j}=Tr(H_{i}L_{j})=Tr(H_{i}L_{j})=Tr(H_{i}\Sigma_{k}Tr(F_{k}S_{j})R_{k}),$
(32) $=\Sigma_{k}Tr(F_{k}S_{j})Tr(R_{k}H_{i}).$

Next, we use equation 30 above to relate the associated stochastic matrices of
$\Phi_{1},\Phi_{2}\quad\&\quad\Phi_{1}\circ\Phi_{2}$. ∎

######  Remark 6.

By using the well known charecterisation of idempotent stochastic matrices by
J.L.Doob Doob (1942) ,the above theorem will provide a clear picture of
idempotent quantum channels in the Holevo-form.One can also get information
about generalised invertibility by using Theorem 7 above and the
charecterisation of generalised invertibile stocahstic matrices,Theorem
2,p.152 ,J.R.Wall Wall (1975).

######  Definition 8.

Entnaglement fidelity

Let $S=|\psi_{AR}\rangle\langle\Psi_{AR}|$ be a pure state in the Hilbertspace
$\mathcal{H_{A}}\otimes\mathcal{H_{R}}$.The entanglement fidelity
$F_{e}(S,\Phi)$ is defined as follows:

$F_{e}(S,\Phi)=\langle\psi_{AR}\rangle|(\Psi\otimes
Id_{R})(|\psi_{AR}\rangle\langle\psi_{AR}|)|\Psi_{AR}\rangle,$

If the channel $\Psi(.)$ has the following Kraus representation

$\Psi(S)=\Sigma_{k=1}^{d}V_{K}SV_{K}^{*},$

then

$F_{e}(S,\Psi)=\Sigma_{k=1}^{d}|T_{r}V_{K}S|^{2}.$

######  Proposition 8.

$P_{U}$ is a classical -quantum-classical (c-q-c) channel: equivalently, it is
an entanglement-breaking- channel.

###### Proof.

Recall that $P_{U}{X}=\Sigma_{j=1}^{n}p_{j}Xp_{j}$ where
$\\{p_{j}:j=1,2,...,n\\}$ are pairwise orthogonal rank one projections with
$\Sigma_{j=1}^{n}p_{j}=I$.But $Xp_{j}=\lambda_{j}(X)p_{j}$ ,$\lambda_{j}(X)$
are scalars since $p_{j}$ is minimal for each $j$.By taking trace we find that

$\lambda_{j}(X)=T_{r}(Xp_{j}),$

for each $X$ and $j$. Hence we get,for $X\in M_{n}$

$P_{U}(X)=\Sigma_{j=1}^{n}p_{j}T_{R}(Xp_{j}).$

Therefore, $P_{U}$ is a C-Q channel, hence entanglement-breaking. ∎

Next, we examine the type of channels $\tilde{P}_{U}$, in general.

######  Theorem 9.

Among quantum channels $\tilde{P}_{U}$, only $P_{U}$ is a c-q-c channel.

###### Proof.

Let if possible $\tilde{P_{U}}$ has Holevo form,

$\tilde{P}_{U}(\rho)=\Sigma P_{k}\rho P_{k}=\Sigma_{k=1}^{r}Tr(\rho
F_{k})R_{k},$ (33)

where $F_{k}$ and $R_{k}$ are POVM (postive operator valued measures) and
density operators respectively and $\\{P_{k}:k=1,2,...,m\\}$ is a collection
of pairwise orthogonal projections such that $\Sigma_{k=1}^{m}P_{k}=I$ and
$m<n$,for every density $\rho$.Let $P_{k_{0}}$ be such that rank $P_{k_{0}}$ >
1.This is because $\tilde{P}_{U}$ is differenr from $P_{U}$. Consider a
density $\rho=x\otimes x$ $P_{k_{0}}x=x,\&\|x\|=1$ where $k_{0}$ is
arbitrarily chosen.Then we have

$\Sigma_{k=1}^{m}P_{k}x\otimes xP_{k}=\Sigma_{k=1}^{r}Tr(x\otimes
xF_{k})R_{k}.$ (34)

A straightforward computation shows that

$\Sigma_{k=1}^{m}P_{k}x\otimes xP_{k}=x\otimes x.$ (35)

∎

######  Remark 7.

Algebraically, the problem of identifying idempotents and identifying
invertible/generalized invertible elements is important. Notice that there are
no invertible elements in $HQC(M_{n})$.

##  IV Codes ,Achievable rates and Channel Capacity

In this section, the channel capacity of $P_{U}$ is found to be $\infty$. In
what follows, we rely heavily on the concepts and results given in p.p. $147$,
chapter $8$ Holevo (2019). Let us recall the definition of the classical
capacity of the quantum channel Holevo (2019) and a few preliminaries and
notations. We rstrict to finite dimentional case here.Let
$H=H_{A}=H_{B}=C^{n}$ and $\Phi:H\rightarrow H$ be a Quantum Channel and

$\Phi^{\bigotimes{n}}:B(H^{\bigotimes{n}})\rightarrow
B(H^{\bigotimes{n}}),composite,memoryless-channel.$

Notations:$\\{i=1,2,...M\\}$ denotes classical messages and correspondingly
$\\{S_{i}:i=1,2,...M\\}$ denotes density operators and $i\rightarrow S_{i}$
encodes each $i$.Now $\Phi(S_{i})$ decodes.Correspondingly the block code for
the composite channel comprises a c-q channel $i\rightarrow S_{i}^{(n)}$ that
encodes the classical messages $i$ into input states $S_{i}^{(n)}$ in the
space $H^{\bigotimes{n}}$, and a q-c channel observable $M^{(n)}$, here in the
same space, decoding the output states $\Phi^{\bigotimes{n}[S_{i}^{(n)}]}$
into classical messages $j$.The following is a diagrammatic representation of
this :

$i\rightarrow
S_{i}^{(n)}\rightarrow\Phi^{\bigotimes{n}}[S_{i}^{(n)}]\rightarrow{j}.$

######  Definition 9.

A code $(\Sigma^{(n)},M^{(n)})$ of length $n$ and size $N$ for the composite
channel $\Phi^{\bigotimes{n}}$ consists of an encoding, given by a collection
of states

$\Sigma^{(n)}=\\{S_{i}^{(n)};i=0,1,2,...,J\\}$

in $H^{\bigotimes}{n}$,and a decoding ,described by an obserable

$M^{(n)}=\\{M_{J}^{(n)};j=1,2,...,J.\\}$

in $H^{(n)}$.

######  Definition 10.

The Classical Capacity of a Quantum Channel
The classical capacity $C(\Phi)$of the quantum channel $\Phi$ is defined as
the least upper bound of the achievable rates $R$ such that

$lim_{n\rightarrow\infty}p_{e}(n2^{nR})=0$

where, for given $n$ and size $J$

$p_{e}(n,N)=min\\{p_{e}(\Sigma^{(n)},M^{(n)})\\}=max_{i=1,2,...,N}[1-P_{\Sigma
M}(i.i)].$

Here for each pair$(j,i)$the probability that the outcome is $i$ when the
input is $j$ is given by

$P_{M}(j,i)=Tr\Phi^{\bigotimes}(n)[S_{i}^{n}]M^{\bigotimes(n)}_{j}.$ (36)

######  Theorem 10.

Let $P_{U}(.)$ be as above.Then the classical capacity of the classical-
quantum-channel $P_{U}(.)$ has infinite capacity.i.e,

$C(P_{U}(.))=Log_{2}(J).$ (37)

###### Proof.

Now consider the pre-conditioner $P_{U}(.)$ where $S_{j}=p_{j}$ for each$j$
and

$P_{U}(X)=\Sigma_{j=1}^{N}p_{j}T_{r}(Xp_{j})$ (38)

Hence a code $(\Sigma^{(n)},M^{(n)})$ of length $n$ and size $N$ for the
composite channel $P^{\bigotimes n}_{U}$ is given by

$\Sigma_{(n)}=\\{p^{(n)}_{j}:j=1,2,..J\\}\&M^{(n)}=\\{p_{j}^{(n)}:j=0,1,2,...,J.\\}$
(39)

It is more or less clear that $p_{\Sigma M}(i|i)=1$ for all values of $i$ and
hence $p_{e}(n,J)=0$ for all vaues of $n$ and $J$.Thus with the above
mentioned code $P_{U}(.)$ has capasity $Log_{2}(J)$,by defenition 10 above.

$C(P_{U}(.))=Log_{2}(J).$ (40)

∎

######  Remark 8.

Equation (40) above shows that the capacity $C(P_{U}(.))\longrightarrow\infty$
logarithemically as the size $J\longrightarrow\infty$.In Example1 ,each
$U_{n}orS_{n}$ depends on the number of grid points chosen.Thus we infer that
the capacity keeps increasing logarethemically as the number of grid points
increases.

######  Remark 9.

Before considering next aspect, we recall coding-encoding techniques discribed
by Kamil Korzekwa, Zbigniew Puchała,etal.,in Korzekwa  _et al._ (2022) and Zi-
Wen Liu, Xueyuan Hu,etal.,in Liu, Hu, and Lloyd (2017) using idempotent
quantum-channels,the so called  resource destroying channels .The following
basic identities involving idempotent channels are used in the above articles;

$\Phi\circ\Delta=\Delta=\Delta\circ\Phi$ (41)

where $\Delta$ is an idempotent resource destroying channel and $\Psi$ and is
an eppropriate quantum channel that encodesKorzekwa  _et al._ (2022).
A variant of equation 37 above was considered in Liu, Hu, and Lloyd (2017) Iis
as follows;

$\Phi\circ\Delta=\Delta\circ\Phi\circ\Delta,$ (42)

and its dual form

$\Delta\circ\Phi=\Delta\circ\Phi\circ\Delta.$ (43)

A choice of $\Delta$ can be the preconditioner map $P_{U}$ and
$\Phi=\tilde{P}_{U}$ for various unitaries $U$.

######  Example 2.

Another example could be as follows.Let $S$ be a permution of
$\\{1,2,...,n\\}$ where $n$ is the dimention appearinf in $M_{n}(C)$.Define
$\Phi_{S}$ as follows;

$\Phi_{S}(T)=US\circ\sigma(U^{*}TU)U^{*},$ (44)

$T\in M_{n}(C)$.Then $\Phi_{S}$ is a quantum channel for each $S$.

It can be easily seen that $\Delta$ and $\phi_{S}$ satisfy the following
equations.

  * [1]

$\Phi\circ\Phi_{S}=\Phi_{S}=\Phi_{S}\circ\Phi,and\quad therefore$

  * [2]

$\Phi\circ\Phi_{S}\circ\Phi=\Phi_{S}.$

Observe that $\Phi_{S}$ is not an idempotent quantum-channel, unless of course
$S^{2}=I$.

##  V Infinite Dimensional Analogue of the Maps $\tilde{P}_{U}$

In what follows, we consider the infinite-dimensional versions of the results
presented in the preceding sections.

Let $H$ be an infinite-dimensional separable Hilbert space over complex
numbers, and $\tau(H)$ be the Banach $*$-algebra of all trace-class operators
equipped with the trace norm. For $A\in\tau(H)$, the trace norm of $A$ is
defined as:

$|A|_{1}=\text{Tr}(|A|),$ (45)

where $|A|=\sqrt{(AA)}$.

Quantum information theory and quantum channels in infinite dimensions are
mathematically interesting and have recently caught the attention of applied
signal analysts. The survey article Sharma and Shenoy (2022) provides useful
information in this direction. Additionally, the monograph by Brian Davies
Davies (1976) is also highly important. However, it is clear that one can
mimic the finite-dimensional theory and appropriately extend it to infinite
dimensions. First, we recall the definition of a quantum channel in infinite
dimension.

######  Definition 11.

Kholevo and Shirokov (2010). Let $H$ be a separable complex Hilbert space, and
$\tau(H)$ be the Banach $*$-algebra of all trace-class operators on $H$. Then,
a quantum channel is defined as a linear, trace-preserving map $\Phi$ on
$\tau(H)$ such that the dual map $\Phi^{*}$ on the $C^{*}$-algebra $B(H)$ is
completely positive.

######  Remark 10.

Every quantum channel $\Phi$ is continuous under the trace norm. Since its
dual map $\Phi^{*}$ is completely positive, it is continuous under the
operator norm on $B(H)$. Moreover, every linear map on a Banach space with a
continuous dual is continuous, which can be shown using a simple application
of the Hahn-Banach theorem.

Furthermore, the quantum channel $\Phi$, being continuous under the trace
norm, implies that its dual $\Phi^{*}$ is an ultra-weakly continuous (i.e.,
normal) completely positive map on $B(H)$. Therefore, there exists a sequence
${V_{k}:k=1,2,\ldots}$ in $B(H)$ such that Davies (1976), Friedland (2019):

$\Phi^{*}(T)=\sum_{k=1}^{\infty}V_{k}^{*}TV_{k},$ (46)

and hence its pre-dual map is:

$\Phi^{*}(S)=\sum_{j=1}^{\infty}V_{k}SV_{k}^{*},$ (47)

for all $S\in\tau(H)$.

Next, we introduce the infinite-dimensional versions of the preconditioner
maps $P_{U}$ and $\tilde{P}U$ as follows. We mimic the existing definition of
$P_{U}$ as well as its general version $\tilde{P}U$. Let $U$ be a unitary
operator in $B(H)$, and let ${P_{j}:j=1,2,\ldots}$ be a set of rank-one
orthogonal projections such that $\sum{j=1}^{\infty}P_{j}=I$, where $I$ is the
identity operator on $H$. For each $S\in\tau(H)$, define
$\sigma(S)=\sum{j=1}^{\infty}P_{j}SP_{j}$. Now, as before, we define a closed
subspace $M(U)$ as follows:

$M(U)={T\in\tau(H):U^{*}TU=\sigma(S)\text{ for some }S\in\tau(H)}.$ (48)

The continuity of the map $\sigma$ under the norm $|.|_{1}$ implies that
$M(U)$ is a closed subspace of the Hilbert space $\tau(H)$. We will now prove
the infinite-dimensional version of the preconditioner map $P_{U}$.

######  Theorem 11.

Let $M(U)$ be the closed subspace of $\tau(H)$ given by equation 48 above, and
let $P$ be the orthogonal projection of $\tau(H)$ onto $M(U)$. Then $P$ has
the following properties:

  1. 1.

$P(T)=U\sigma(U^{T}U)U^{*}$ for $T\in\tau(H)$, where
$\sigma(T)=\sum_{j=1}^{\infty}P_{j}TP_{j}$ for each $T\in\tau(H)$.

  2. 2.

$P$ is completely positive and trace preserving.

  3. 3.

For each $T\in\tau(H)$, $\|T-P(T)\|_{1}^{2}=\|T\|_{1}^{2}-\|P(T)\|_{1}^{2}$.

###### Proof.

  1. 1.

Clearly, $P(T)\in M(U)$ for all $T\in\tau(H)$, and the range of $P$ is $M(U)$.
We show that $P$ is self-adjoint and idempotent, which completes the proof by
uniqueness. We have:

$\displaystyle P(P(T))$ $\displaystyle=U\sigma(U^{*}P(T)U)U^{*}$
$\displaystyle=U\sigma(U^{*}U\sigma(U^{*}TU)U^{*}U)U^{*}$
$\displaystyle=U\sigma(U^{*}TU)U^{*}$ $\displaystyle=P(T).$

Thus, $P$ is idempotent.

Next, we show that $P$ is self-adjoint. For $S,T\in\tau(H)$, consider the
following:

$\displaystyle\langle P(T),S\rangle$
$\displaystyle=\operatorname{Tr}(P(T)S^{*})$
$\displaystyle=\operatorname{Tr}(U\Sigma_{j=1}^{\infty}P_{j}U^{*}TUP_{J}U^{*}S^{*})$
$\displaystyle=\operatorname{Tr}(S^{*}U\Sigma_{j=1}^{\infty}P_{j}U^{*}TUP_{j}U^{*})$
$\displaystyle=\operatorname{Tr}(U^{*}S^{*}U)\Sigma_{j=1}^{\infty}P_{j}U^{*}TUP_{j}$
$\displaystyle=\Sigma_{j=1}^{\infty}\operatorname{Tr}(P_{j}U^{*}S^{*}UP_{j}U^{*}TU)$
$\displaystyle=\operatorname{Tr}(TU\Sigma_{j=1}^{\infty}P_{j}U^{*}S^{*}UP_{j})$
$\displaystyle=\operatorname{Tr}(TP(S)^{*})$ $\displaystyle=\langle
T,P(S)\rangle.$

This shows that $P$ is self-adjoint. Hence, $P$ is the orthogonal projection
of $\tau(H)$ onto $M(U)$.

  2. 2.

It is clear that the map $P$ has the following Kraus representation:

$P(T)=\sum_{j=1}^{\infty}UP_{j}U^{*}TUP_{j}U^{*}.$

Therefore, $P$ is completely positive.

  3. 3.

It follows from the Pythagorean theorem in abstract Hilbert spaces.

∎

######  Remark 11.

We can define the infinite-dimensional version of $\tilde{P}_{U}$ as follows:

######  Definition 12.

Let ${p_{k}:k=1,2,\ldots}$ be a sequence of orthogonal and pairwise orthogonal
projections such that $\sum_{k=1}^{\infty}p_{k}=I$. Let

$\Psi(T)=\sum_{k=1}^{\infty}p_{k}Tp_{k}.$ (49)

For a unitary $U\in B(H)$, the map $\tilde{P}_{U}$ is defined as

$\tilde{P}_{U}(T)=U\Psi(U^{*}TU)U^{*}.$

for each $T\in\tau(H)$.

There are a number of concrete unitary operators in infinite dimention such as
Fourier transform on $L^{2}(R^{N})$.One can also create more examplles from
finite dimention as follows:
Let $H_{n}$ be an $n$-dimentional subspace of a seperable Hilbert space $H$
and let $U_{n}$ be a unitary operator on $H_{n}$.Define a unitary $U$ on $H$
as follows:

$U=U_{n}\oplus(I-P_{n}),$ (50)

where $P_{n}$ is the orthogonalprojection of $H$ onto $H_{n}$. We conclude
this article by noting that our future work will include the infinite-
dimensional version of the results presented in this article.In addition to
the concrete unitaries listed at the beginning,one can also consider the
Discrete Quantum Fourier Transform ,which may be more suitable for the theme
of this article.However,it is yet to analyse this aspect.

## Acknowledgements:

I am thankful to KSCSTE, Government of Kerala, for the financial support
provided through the Emeritus Scientist Fellowship, which enabled the
completion of a part of this article. This work was also presented at the ’XI
International Conference of the Georgian Mathematical Union, Aug. 2021,
Batumi, Georgia’. Additionally, I would like to express my gratitude to the
Institute of Mathematics, Research and Training (IMRT), Thiruvananthapuram,
for the weekly research lectures on topics related to the theory of
semigroups.

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