#  $C^{*}$-extreme entanglement breaking maps on operator systems

Sriram Balasubramanian∗ Department of Mathematics
IIT Madras, Chennai - 600036, India<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Neha Hotwani1 Department of Mathematics
IIT Madras, Chennai - 600036, India<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{E}$ denote the set of all unital entanglement breaking (UEB)
linear maps defined on an operator system $\mathcal{S}\subset M_{d}$ and,
mapping into $M_{n}$. As it turns out, the set $\mathcal{E}$ is not only
convex in the classical sense but also in a quantum sense, namely it is
$C^{*}$-convex. The main objective of this article is to describe the
$C^{*}$-extreme points of this set $\mathcal{E}$. By observing that every EB
map defined on the operator system $\mathcal{S}$ dilates to a positive map
with commutative range and also extends to an EB map on $M_{d}$, We show that
the $C^{*}$-extreme points of the set $\mathcal{E}$ are precisely the UEB maps
that are maximal in the sense of Arveson ([A] and [A69]) and that they are
also exactly the linear extreme points of the set $\mathcal{E}$ with
commutative range. We also determine their explicit structure, thereby
obtaining operator system generalizations of the analogous structure theorem
and the Krein-Milman type theorem given in [BDMS]. As a consequence, we show
that $C^{*}$-extreme (UEB) maps in $\mathcal{E}$ extend to $C^{*}$-extreme UEB
maps on the full algebra. Finally, we obtain an improved version of the main
result in [BDMS], which contains various characterizations of $C^{*}$-extreme
UEB maps between the algebras $M_{d}$ and $M_{n}$.

###### Key words and phrases:

Operator systems, Entanglement breaking maps, Maximal, Dilation, Extension,
Mapping cone, $C^{*}$-convexity, $C^{*}$-extreme, Extremal, Krein-Milman,
Schmidt number, EB rank, Choi rank.

###### 2020 Mathematics Subject Classification:

81P40, 47L07 (Primary); 15B48, 81R15, 81P42, 81P45 (Secondary)

∗ Supported by the grant MTR/2018/000113 from the Department of Science and
Technology (DST), Govt. of India.

1 Supported by the fellowship 0203/16(8)/2018-R&D-II from the National Board
for Higher Mathematics (NBHM), Govt. of India.

##  1\. Introduction

The notions of positivity and convexity are fundamental to Mathematical
analysis and in particular, to the theory of $C^{*}$-algebras. Among positive
maps between $C^{*}$-algebras, the ones that are completely so, are of
considerable interest. The study of completely positive maps was initiated by
Stinespring and Arveson in the seminal papers [S], [A69] and [A72]. Among
various results of significant importance in [A69], of particular interest to
us is an abstract characterization of the (linear) extreme points of the
convex set of completely positive maps between a $C^{*}$-algebra $\mathcal{A}$
and $B(H)$ for some Hilbert space $H$, in terms of the (minimal) Stinespring
dilation. As important as classical convexity is, it still has some
limitations in the non-commutative setting. Two ”non-commutative” convexity
notions that have gathered significant attention recently are matrix-convexity
and $C^{*}$-convexity, the former introduced and studied by Webster and
Winkler in [W] and [WW] and the latter by Hopenwasser, Loebl, Moore and
Paulsen in [LP] and [HMP]. Although these notions appear to be similar, they
are vastly different as was pointed out by Farenick in [F]. Farenick and
Morenz also obtained a complete characterization of $C^{*}$-extreme unital
completely positive maps between $C^{*}$-algebras in [FM93] and [FM97].
Further contributions on this and related topics can be found in [BBK], [BK].

Our main objects of focus in this article are entanglement breaking (EB) maps.
These maps have drawn considerable attention recently and are particularly
sought after in quantum information theory. In the finite dimensional setting,
an important aspect of the set of unital entanglement breaking (UEB) maps is
that it is $C^{*}$-convex. This warrants the study of $C^{*}$-extreme UEB
maps. A complete description of such maps between matrix algebras was obtained
in [BDMS]. The purpose of this article is four fold. Firstly, we obtain the
explicit structure of UEB maps defined on a finite-dimensional operator system
that are maximal with respect to the dilation order (see [A]) on the set of
UEB maps (See Theorem 1.10). As a consequence we show that every UEB map on a
finite dimensional operator system dilates to a maximal UEB map (See Theorem
1.11). Secondly, we show that UEB maps on a finite-dimensional operator system
that are maximal with respect to the dilation order on the set of UEB maps are
precisely the $C^{*}$-extreme UEB maps and that they are also exactly the
linear extreme UEB maps with commutative range (See Theorem 1.12). As a
consequence, we obtain an operator system generalization of a characterization
of $C^{*}$-extreme UEB maps as well as the Krein-Milman type theorem for UEB
maps given in [BDMS]. (See Corollary 1.14). It is to be noted that the partial
order used for obtaining the abstract characterization of $C^{*}$-extreme UEB
maps in [BDMS] is the usual partial order (and not the dilation order) on the
set of UEB maps, i.e., for UEB maps $\Phi$ and $\Psi$, $\Phi\leq\Psi$ if and
only if $\Psi-\Phi$ is a UEB map. Thirdly, we show that a $C^{*}$-extreme UEB
map defined on a finite dimensional operator system extends to a
$C^{*}$-extreme UEB map on the full algebra (See Corollary 1.13). Finally, we
obtain an improved version of the main result Theorem 5.3 in [BDMS] (See
Theorem 6.2). To help us prove the above mentioned assertions, we make use of
the following key observations namely, an EB map defined on a finite
dimensional operator system dilates to a positive map with commutative range
(See Theorem 1.7) and also has an EB extension to the full algebra (See
Theorem 1.5).

Before we explicitly state our main observations and results, we introduce
some notations and definitions. Throughout this article, $L$ and $K$ will
denote separable Complex Hilbert spaces and $B(L)$ will denote the C*-algebra
of bounded linear maps defined on $L$. An operator system $\mathcal{M}\subset
B(L)$ is a self adjoint subspace containing the identity operator $I_{L}$. A
linear map $\Phi:\mathcal{M}\to B(K)$ is said to be unital if
$\Phi(I_{L})=I_{K}$, positive if $\Phi(A)$ is a positive operator in $B(K)$
(in this case we write $\Phi(A)\succeq 0$ or $\Phi(A)\in B(K)^{+}$), whenever
$A$ is a positive operator in $\mathcal{M}$. A unital positive linear
functional on $\mathcal{M}$ is called a state. The notation $M_{k}$ will
denote the space of all $k\times k$ Complex matrices and $M_{k}(\mathcal{M})$,
the operator system of all $k\times k$ matrices with entries from
$\mathcal{M}$. For a given linear map $\Phi:\mathcal{M}\to B(K)$ and
$k\in{\mathbb{N}}$, the kth-ampliation $\Phi_{k}:M_{k}(\mathcal{M})\to
M_{k}(B(K))$ is defined by $\Phi_{k}([A_{i,j}])=[\Phi(A_{i,j})]$. Under the
identification of $M_{k}(\mathcal{M})$ with $M_{k}\otimes\mathcal{M}$, one
sees that $\Phi_{k}:=\mathtt{i}_{k}\otimes\Phi$, where $\mathtt{i}_{k}$ is the
identity operator on $M_{k}$. More specifically,
$\Phi_{k}:M_{k}\otimes\mathcal{M}\to M_{k}\otimes B(K)$ is the linear map
determined by $\Phi_{k}(X\otimes A)=X\otimes\Phi(A).$ The linear map
$\Phi:\mathcal{M}\rightarrow B(K)$ is said to be completely positive (CP) if
$\Phi_{k}$ is positive for all $k\in{\mathbb{N}}$.

Next we define the entanglement breaking property of linear maps. There are a
number of competing definitions for the notion of an entanglement breaking map
$\Phi:\mathcal{M}\to B(K)$ that agree with the usual notion in the case that
$\mathcal{M}=B(L)$ and $K$ is a finite dimensional Hilbert space. In any case,
they all reference the cone of (separable) matrices,

$M_{k}^{+}\otimes B(K)^{+}:=\left\\{\sum_{m=1}^{\ell}A_{m}\otimes
B_{m}\,:\,\ell\in{\mathbb{N}},A_{m}\in M_{k}^{+},B_{m}\in B(K)^{+}\right\\}.$

Since we are mainly interested in the case where $\mathcal{M}\subset B(L)$ is
an operator system and, $K$ and $L$ are finite dimensional Hilbert spaces,
inspired by the various notions of separability introduced in [CH], we will
say that the linear map $\Phi:\mathcal{M}\to B(K)$ is entanglement breaking
(EB) if $(\mathtt{i}_{k}\otimes\Phi)(X)\in\overline{M_{k}^{+}\otimes
B(K)^{+}}$, for all $k\in{\mathbb{N}}$ and
$X\in(M_{k}\otimes\mathcal{M})^{+}$, where the closure is with respect to the
norm topology on $B({\mathbb{C}}^{k}\otimes K)$. Evidently, when $K$ is finite
dimensional, $M_{k}^{+}\otimes B(K)^{+}$ is already norm closed. The
abbreviations UCP and UEB will be used for ”unital completely positive” and
”unital entanglement breaking” respectively. The collection of all UEB maps
mapping $\mathcal{M}$ to $B(K)$ will be denoted by UEB($\mathcal{M},B(K))$.
Finally, since we work mainly in the finite dimensional setting, we make the
following notational conventions.

######  Convention 1.1.

Throughout $E={\mathbb{C}}^{d}$ and $H={\mathbb{C}}^{n}$ and
$\mathcal{S}\subset B(E)$ is an operator system.

Given a linear map $\Phi:\mathcal{S}\rightarrow B(H)$, the dual functional
$s_{\Phi}:B(H)\otimes\mathcal{S}\to{\mathbb{C}}$ associated to it, is the
linear mapping determined by

$s_{\Phi}(X\otimes A)=\operatorname{trace}(\Phi(A)X^{t}),$

where $t$ is the transpose operator in $B(H)$ induced by a fixed orthonormal
basis of $H$. It is easily seen that this definition of $s_{\Phi}$ is
independent of the choice of the orthonormal basis and hence of the transpose
operator induced by it and that the correspondence $\Phi\mapsto s_{\Phi}$ is
bijective. Please see [St], [B] for more details.

The Choi matrix $C_{\Phi}$ associated with the linear map $\Phi:B(E)\to B(H)$
is defined as
$[\Phi(e_{i}e_{j}^{*})]_{i,j=1}^{d}=\sum_{i,j=1}^{d}e_{i}e_{j}^{*}\otimes\Phi(e_{i}e_{j}^{*})\in
B(E)\otimes B(H)$, where $\\{e_{1},\dots,e_{d}\\}$ is the standard orthonormal
basis of $E$. One of the many significant applications of the Choi-matrix is
the following well-known characterization of CP maps due to Choi.

######  Theorem 1.2.

([C, Theorem 1], [St, Theorem 4.1.8]) Let $\Phi:B(E)\rightarrow B(H)$ be a
linear map. The following statements are equivalent.

  * (i)

$\Phi$ is CP.

  * (ii)

$\Phi(X)=\sum_{k=1}^{\ell}V_{k}^{*}XV_{k}$, for some linear maps $V_{k}:H\to
E$.

  * (iii)

$C_{\Phi}\in(B(E)\otimes B(H))^{+}.$

The formula for $\Phi$ given in statement $(ii)$ above is called a Choi-Kraus
decomposition of $\Phi$ and the matrices coefficients $V_{k}$ are called Choi-
Kraus operators/coefficients. The minimum number of Choi-Kraus operators
required to represent $\Phi$ in the form of a Choi-Kraus decomposition is
known as the Choi-rank of $\Phi$. The Schmidt rank of the vector $\xi\in
E\otimes H$, denoted by $SR(\xi)$, is the smallest natural number $k$ such
that $\xi=\sum_{i=1}^{k}x_{i}\otimes y_{i}\in E\otimes H$. Given a completely
positive map $\Phi:B(E)\rightarrow B(H)$, by Theorem 1.2 the Choi matrix
$C_{\Phi}\in(B(E)\otimes B(H))^{+}$ and hence has a spectral decomposition of
the form $C_{\Phi}=\sum_{i=1}^{m}\xi_{i}\xi_{i}^{*}$, where $\xi_{i}\in
E\otimes H$. Let $\Pi_{\Phi}$ denote the set of all spectral decompositions of
$C_{\Phi}$. The Schmidt number of the Choi matrix $C_{\Phi}\in(B(E)\otimes
B(H))^{+}$ is denoted by $SN(C_{\Phi})$ and is defined as

$SN(C_{\Phi}):=\min_{\Gamma\in\Pi_{\Phi}}\left\\{\max_{1\leq j\leq
m}\left\\{SR(\xi_{j})\,\middle|\,\Gamma=\sum_{i=1}^{m}\xi_{i}\xi_{i}^{*}\in\Pi_{\Phi}\right\\}\right\\}.$

See [TH] for more details.

In this article, our main focus will be on EB maps. The following are a few
well-known characterizations of EB maps that will be used in the sequel.

######  Theorem 1.3.

([HSR, Theorem 4], [TH]) Let $\Phi:B(E)\rightarrow B(H)$ be a linear map. The
following statements are equivalent.

  1. (i)

$\Phi$ is EB.

  2. (ii)

$\Phi(X)=\sum_{j=1}^{m}\phi_{j}(X)R_{j}$, where the $\phi_{j}$’s are states
defined on $B(E)$ and the $R_{j}$’s are positive operators in $B(H)$.

  3. (iii)

$\Phi(X)=\sum_{k=1}^{\ell}V_{k}^{*}XV_{k}$, where each $V_{k}:H\to E$ is a
linear map of rank one.

  4. (iv)

$C_{\Phi}\in B(E)^{+}\otimes B(H)^{+}$. (i.e., $C_{\Phi}$ is separable.)

  5. (v)

$SN(C_{\Phi})=1$.

The formula for $\Phi$ given in statement (ii) of the above theorem is called
a Holevo form of $\Phi$. The EB-rank of $\Phi$ is defined as the minimum
number of Choi-Kraus operators of rank one, required to represent $\Phi$ as in
statement (iii) of Theorem 1.3. Please see [PPPR] for more details.

”Extremal” UEB maps are of utmost importance to us. Let $\mathcal{S}\subset
B(E)$ be an operator system. Here we mainly consider two notions of extreme
points of UEB($\mathcal{S},B(H))$. The UEB map $\Phi:\mathcal{S}\to B(H)$ is
said to be a linear extreme point of UEB($\mathcal{S},B(H))$ if

$\Phi=\sum_{i=1}^{k}t_{i}\Phi_{i}$

for some UEB maps $\Phi_{i}:\mathcal{S}\to B(H)$ and $t_{i}\in(0,1)$
satisfying $\sum_{i=1}^{k}t_{i}=1$, then $\Phi_{i}=\Phi$ for all $i$. Linear
extreme UCP maps betweeen $C^{*}$-algebras were characterized by Arveson in
[A69].

The notion of $C^{*}$-convexity and $C^{*}$-extreme points were first
introduced and studied in [LP] and [HMP] respectively. $C^{*}$-extreme UCP
maps were extensively studied in [FM93], [FM97], [FZ] and [Z]. In [BDMS], the
authors consider $C^{*}$-extreme UEB maps between matrix algebras. Along the
same lines, we define $C^{*}$-extreme points in UEB($\mathcal{S},B(H))$.

Let $\Phi_{1},\Phi_{2},\dots,\Phi_{k}\in\text{UEB}(\mathcal{S},B(H))$. A UEB
map $\Phi:\mathcal{S}\rightarrow B(H)$ is said to be a $C^{*}$-convex
combination of the UEB maps $\Phi_{i}$, if there exists $T_{1},\dots,T_{k}\in
B(H)$ such that $\sum_{i=1}^{k}T_{i}^{*}T_{i}=I_{H}$ and
$\Phi(X)=\sum_{i=1}^{k}T_{i}^{*}\Phi_{i}(X)T_{i}$ for every $X\in\mathcal{S}$.
The $T_{i}$’s will be referred to as the coefficients of this $C^{*}$-convex
combination. If the coefficients, i.e., the $T_{i}$’s are positive, then this
$C^{*}$-convex combination will be called a positive $C^{*}$-convex
combination.

A UEB map $\Phi:\mathcal{S}\to B(H)$ is a proper $C^{*}$-convex combination of
the UEB maps $\Phi_{i}$, $1\leq i\leq k$, if there exists invertible operators
$T_{i}\in B(H)$ such that $\sum_{i=1}^{k}T_{i}^{*}T_{i}=I_{H}$ and

$\Phi(X)=\sum_{i=1}^{k}T_{i}^{*}\Phi_{i}(X)T_{i},$ (1.1)

for all $X\in\mathcal{S}.$ This $C^{*}$-convex combination is trivial if each
$\Phi_{i}$ is unitarily equivalent to $\Phi;$ that is, there exist unitary
operators $U_{i}$ such that $\Phi_{i}(X)=U_{i}^{*}\Phi(X)U_{i}.$ The UEB map
$\Phi:\mathcal{S}\to B(H)$ is said to be a $C^{*}$-extreme point of
UEB($\mathcal{S},B(H))$ if, every representation of $\Phi$ as a proper
$C^{*}$-convex combination is trivial.

######  Remark 1.4.

Given our finite dimensionality assumptions, it is a standard observation that
$C^{*}$-extreme UEB maps from $\mathcal{S}\subset B(E)$ mapping into $B(H)$
are also linear extreme UEB maps. Please see [Z, Theorem 2.2.2] for a proof of
this fact for UCP maps. However, not every linear extreme UEB map is
$C^{*}$-extreme, as an example in [HSR] shows. See also example 5.7 in [BDMS].

With the above given definitions and remarks, we proceed to state our main
observations and results.

###  1.1. EB maps - Extension and Structure

A well-known extension theorem for CP maps due to Arveson says that every CP
map defined on an operator system in a $C^{*}$-algebra mapping into $B(K)$ for
some Hilbert space $K$, has a CP extension (See [P, Theorem 7.5]). Our first
main observation in this article is the following analogous extension theorem
for EB maps.

######  Theorem 1.5.

Let $\mathcal{S}\subset B(E)$ be an operator system and $\phi:\mathcal{S}\to
B(H)$ be an EB map. There exists an EB map $\Phi:B(E)\to B(H)$ such that
$\Phi\big{|}_{\mathcal{S}}=\phi$.

A proof of Theorem 1.5 is given in Section 2. As an immediate consequence of
the above theorem, one obtains the following operator system version of
Theorem 1.3.

######  Corollary 1.6.

The equivalence of statements (i), (ii) and (iii) of Theorem 1.3 holds even
for UEB maps defined on operator systems. More precisely, if
$\mathcal{S}\subset B(E)$ is an operator system and
$\Phi:\mathcal{S}\rightarrow B(H)$ is a linear map, then the following
statements are equivalent.

  1. (i)

$\Phi$ is EB.

  2. (ii)

$\Phi$ can be written in the Holevo form, i.e.,
$\Phi(X)=\sum_{j=1}^{m}\phi_{j}(X)R_{j}$, where the $\phi_{j}$’s are states
defined on the operator system $\mathcal{S}$ and the $R_{j}$’s are positive
operators in $B(H)$.

  3. (iii)

$\Phi$ has a Choi-Kraus decomposition with rank-one Choi-Kraus coefficients,
i.e., there exists linear maps $V_{k}:H\to E$ of rank one such that
$\Phi(X)=\sum_{k=1}^{\ell}V_{k}^{*}XV_{k}$, for all $X\in\mathcal{S}$.

It is a well-known result due to Stinespring ([P, Theorem 4.1]) that a UCP map
defined on a C*-algebra $\mathcal{A}$ mapping into $B(K)$ for some Hilbert
space $K$, dilates to a representation. In this article we point out a similar
structure theorem for EB maps (for our finite dimensional setting). The
following characterization of an EB map in terms of a dilation, is our second
main observation in this article.

######  Theorem 1.7.

Let $\mathcal{S}\subset B(E)$ be an operator system and
$\Phi:\mathcal{S}\rightarrow B(H)$ a linear map. The following statements are
equivalent.

  * (i)

$\Phi$ is a (unital) EB map.

  * (ii)

$\Phi(X)=V^{*}\Gamma(X)V$ for all $X\in\mathcal{S}$, where $V:H\rightarrow K$
is an isometry for some finite dimensional Hilbert space $K$ and
$\Gamma:\mathcal{S}\rightarrow B(K)$ is a (unital) positive map with
commutative range.

  * (iii)

$\Phi$ is the compression of a (unital) EB map with commutative range
contained in $B(K)$ for some finite dimensional Hilbert space $K$.

A proof of Theorem 1.7 is given in Section 3. It relies mainly on the
observation that an entanglement breaking map between matrix algebras factors
via the commutative C*-algebra $\ell^{\infty}_{k}$ for some $k$. See [KMP] and
[JKPP] for more details. We anticipate that the above structure theorem for
”EB” maps being compressions of ”positive” maps with commutative range should
hold for a much more general setting than is considered here (for instance,
for strongly entanglement breaking maps in the infinite dimensional setting
(See [LD])).

######  Remark 1.8.

The above theorem can also be deduced from the well-known fact that a Positive
Operator Valued Measure (POVM) dilates to a Projection Valued Measure (PVM)
(see [Y]), which in turn can be deduced from Naimark’s dilation Theorem ([P,
Theorem 4.6]).

###  1.2. Maximal UEB maps

In [A] and [A72], Arveson defined the notion of a maximal UCP dilation of a
given UCP map on an operator system. He also showed that maximal UCP dilations
always exist and that such maps are precisely the ones that satisfy a unique
extension property. Following [A] and [A72], here we consider the UEB analog
of maximal dilations (for our finite dimensional setting.) Let
$\mathcal{S}\subset B(E)$ be an operator system and let $\Phi:\mathcal{S}\to
B(H)$ be a UEB map. A linear map $\Psi:\mathcal{S}\to B(K)$ is said to be a
UEB dilation of $\Phi$ if $\Psi$ is a UEB map, $K$ is a separable Hilbert
space, and there exists an isometry $V:H\to K$ such that

$\Phi(X)=V^{*}\Psi(X)V,$ (1.2)

for all $X\in\mathcal{S}$. In this case we write $\Phi\leq\Psi$. Note that if
$H$ is identified with $VH$, then ”$\leq$” is a partial order on the set of
all UEB maps defined on $\mathcal{S}$. The UEB dilation $\Psi$ of $\Phi$ is
said to be trivial if

$\Psi(X)V(x)=V(\Phi(X)(x)),$ (1.3)

for all $X\in\mathcal{S}$ and $x\in H$. The UEB map $\Phi$ is said to be
maximal if every UEB dilation of $\Phi$ is trivial.

######  Remark 1.9.

The following observations are immediate from the above definition of
maximality for UEB maps.

  * (i)

Since $VV^{*}$ is the projection onto $\operatorname{Range}(V)$, it follows
that the UEB dilation $\Psi$ is trivial if and only if $VH$ is an invariant
subspace for $\Psi(X)$ for all $X\in\mathcal{S}$. Also, since $\mathcal{S}$ is
an operator system and $\Psi(X^{*})=\Psi(X)^{*}$ for all $X\in\mathcal{S},$ it
follows that the UEB dilation $\Psi$ of $\Phi$ is trivial if and only if $VH$
is a reducing subspace for $\Psi(X)$ for every $X\in\mathcal{S}$.

  * (ii)

Since $H$ is finite dimensional and $V:H\rightarrow K$ is an isometry, there
is no loss of generality in assuming $K$ to be finite dimensional. (See Lemma
4.4)

The following characterization of maximal UEB maps is one of our main results
in this article.

######  Theorem 1.10.

Let $\mathcal{S}\subset B(E)$ be an operator system and $\Phi:\mathcal{S}\to
B(H)$ be a UEB map. The following statements are equivalent.

  * $(i)$

$\Phi$ is maximal.

  * $(ii)$

$\Phi$ has the form

$\Phi(X)=\sum_{i=1}^{k}\phi_{i}(X)P_{i},$ (1.4)

for all $X\in\mathcal{S}$, where the $\phi_{i}$’s are distinct linear extremal
states on $\mathcal{S}$, $k\leq n$ and the $P_{i}$’s are mutually orthogonal
projections in $B(H)$ such that $\sum_{i=1}^{k}P_{i}=I_{H}$.

A proof of Theorem 1.10 is given in Section 4. Using Theorems 1.7 and 1.10, we
also observe the following.

######  Theorem 1.11.

Let $\mathcal{S}\subset B(E)$ be an operator system. Every UEB map
$\Phi:\mathcal{S}\to B(H)$ dilates to a maximal UEB map $\Psi:\mathcal{S}\to
B(K)$ with $dim(K)<\infty$.

A proof of Theorem 1.11 can be found in Section 4.

###  1.3. C*-extreme UEB maps

In [BDMS], the authors obtain various characterizations of $C^{*}$-extreme UEB
maps between matrix algebras. A primary objective of ours is to characterize
$C^{*}$-extreme UEB maps defined on finite dimensional operator systems. One
of our main results in this article along these lines is the following.

######  Theorem 1.12.

Let $\mathcal{S}\subset B(E)$ be an operator system and $\Phi:\mathcal{S}\to
B(H)$ a UEB map. Then the following statements are equivalent.

  * $(i)$

$\Phi$ is a maximal UEB map.

  * $(ii)$

$\Phi$ is a $C^{*}$-extreme point of UEB$(\mathcal{S},B(H))$.

  * $(iii)$

$\Phi$ is a linear extreme point of UEB$(\mathcal{S},B(H))$ with commutative
range.

A proof of Theorem 1.12 is given in Section 5. Combined with Theorem 1.10, the
equivalence $(i)\iff(ii)$ of Theorem 1.12 is an operator system generalization
of the equivalence $(i)\iff(v)$ of [BDMS, Theorem 5.3], which says that
$\Phi\in$ UEB($B(E),B(H)$) is $C^{*}$-extreme if and only if it has the form
given in equation (1.4).

After establishing a characterization as given above, it is only natural to
ask whether $C^{*}$-extreme UEB maps defined on operator systems in $B(E)$
extend to $C^{*}$-extreme UEB maps on the whole algebra. This is still only a
partially answered question for $C^{*}$-extreme UCP maps. Please see [Z] for
more details and some results on this problem. Here, as an application of
Theorems 1.10 and 1.12, we obtain the following extension result.

######  Corollary 1.13.

Let $\mathcal{S}\subset B(E)$ be an operator system and $\Phi:\mathcal{S}\to
B(H)$ be a $C^{*}$-extreme UEB map. There exists a $C^{*}$-extreme UEB map
$\Psi:B(E)\to B(H)$ such that $\Psi|_{\mathcal{S}}=\Phi$.

A proof of Corollary 1.13 is given in Section 5.

In [BDMS], a Krein-Milman type theorem was established for the compact convex
set UEB$(B(E),B(H))$. To be precise, it was shown that the set
UEB$(B(E),B(H))$ equals the $C^{*}$-convex hull of its $C^{*}$-extreme points.
As another application of our main results, we obtain the following operator
system analog of this result.

######  Corollary 1.14.

UEB$(\mathcal{S},B(H))$ equals the $C^{*}$-convex hull of its $C^{*}$-extreme
points.

A proof of Corollary 1.14 is given in Section 5.

##  2\. Extensions of EB maps

In this section we give a proof of Theorem 1.5. We first include some
necessary definitions adapted to our finite dimensional setting. Please see
[St, Chapter 5] for more details.

Let $M$ and $L$ be finite dimensional Hilbert spaces. Let $\mathcal{P}(M)$
denote the cone of all positive linear maps from $B(M)$ to $B(M)$. A closed
convex cone $\mathcal{C}\subset\mathcal{P}(M)$ is said to be a mapping cone if
for each nonzero $A\in B(M)^{+}$, there exists a $\varphi\in\mathcal{C}$ such
that $\varphi(A)\neq 0$ and

$\phi\circ\sigma\circ\psi\in\mathcal{C},$

for all $\sigma\in\mathcal{C}$ and CP maps $\phi,\psi:B(M)\to B(M)$. The
mapping cone $\mathcal{C}$ is said to be symmetric if $\phi\in\mathcal{C}$
implies both $\phi^{*}$ and $t\circ\phi\circ t$ are in $\mathcal{C}$, where
$t$ is the transpose operator in $B(M)$ induced by a fixed orthonormal basis
of $M$ and $\phi^{*}:B(M)\rightarrow B(M)$ is the adjoint of $\phi$ with
respect to the Hilbert-Schmidt inner product on $B(M)$, i.e., $\phi^{*}$ is
determined by

$\langle\phi(A),B\rangle=\langle A,\phi^{*}(B)\rangle,$

for all $A,B\in B(M)$, where $\langle
C,D\rangle:=\operatorname{trace}(D^{*}C)$.

######  Remark 2.1.

Typical examples of mapping cones that are symmetric are UCP$(B(M))$ and
EB$(B(M))$.

Suppose that $\mathcal{S}\subset B(L)$ is an operator system and
$\mathcal{C}\subset\mathcal{P}(M)$ is a mapping cone. A linear map
$\phi:\mathcal{S}\to B(M)$ is said to be $\mathcal{C}$-positive if the
corresponding dual functional $s_{\phi}:B(M)\otimes\mathcal{S}\to{\mathbb{C}}$
takes positive values on the cone

$P(\mathcal{S},\mathcal{C}):=\\{Y\in
B(M)\otimes\mathcal{S}:Y=Y^{*},(\alpha\otimes\mathtt{i})(Y)\succeq 0\text{ for
all $\alpha$ in $\mathcal{C}$}\\},$

where $\mathtt{i}$ is the identity map on $\mathcal{S}$.

######  Lemma 2.2.

Let $\mathcal{S}\subset B(E)$ be an operator system and $\phi:\mathcal{S}\to
B(H)$ be an EB map. If $\mathcal{C}$ denotes the mapping cone EB$(B(H))$, then
$\phi$ is $\mathcal{C}$-positive.

###### Proof.

Since $\phi$ is an EB map, it follows from [St09, Proposition 1(ii)] that the
dual functional $s_{\phi}$ takes positive values on the cone $K$, where

$K:=\\{X\in(B(H)\otimes\mathcal{S}):(\omega\otimes\mathtt{i})(X)\succeq
0\quad\textrm{for all states $\omega$ on $B(H)$}\\}.$

Thus it suffices to show that $P(\mathcal{S},\mathcal{C})\subseteq K$. To
prove this statement, let $\omega$ be a state defined on $B(H)$. Suppose also
that $Y=\sum_{i}A_{i}\otimes B_{i}\in P(\mathcal{S},\mathcal{C})$ and $Z\in
B(H)^{+}$ is a fixed matrix of rank one. Define the linear map
$\gamma:B(H)\rightarrow B(H)$ by $\gamma(X)=\omega(X)Z$. Observe that $\gamma$
has the Holevo form (See Theorem 1.3) and hence $\gamma\in\mathcal{C}$. Since
$(\alpha\otimes\mathtt{i})(Y)\succeq 0$ for all $\alpha\in\mathcal{C}$, it
follows that

$(\gamma\otimes\mathtt{i})(Y)=\sum_{i}\omega(A_{i})Z\otimes
B_{i}=Z\otimes\left[\sum_{i}\omega(A_{i})B_{i}\right]\succeq 0$

and therefore $\sum_{i}\omega(A_{i})B_{i}=(\omega\otimes\mathtt{i})(Y)\succeq
0$. Thus $Y\in K$ and the proof is complete. ∎

######  Lemma 2.3.

Let $\mathcal{S}\subset B(E)$ be an operator system and $\phi:\mathcal{S}\to
B(H)$ be an EB map. Given $\epsilon>0$, there exists an EB map $\Phi:B(E)\to
B(H)$ such that $\|\Phi(A)-\phi(A)\|<\epsilon\|A\|$ for all $A\in\mathcal{S}$.

###### Proof.

Let $\mathcal{C}=$ EB$(B(H))$. Since $\phi:\mathcal{S}\to B(H)$ is an EB map,
it follows from Lemma 2.2 that $\phi$ is $\mathcal{C}$-positive. By observing
that $\mathcal{C}$ is a symmetric mapping cone, it follows from [St, Theorem
5.1.13] that there exists a sequence
$\phi_{j}:=\sum_{i=1}^{r_{j}}\alpha_{i}^{(j)}\circ\psi_{i}^{(j)}\in\mathcal{C}$
with $\alpha_{i}^{(j)}\in\mathcal{C}$ and CP maps
$\psi_{i}^{(j)}:\mathcal{S}\to B(H)$ such that $\phi_{j}\to\phi$ in norm, in
$B(H)$. Note that norm topology on $B(H)$ coincides with the BW-topology due
to the finite dimensionality of $B(H)$. By the Arveson extension theorem [P,
Theorem 7.5], there exist CP maps $\Psi_{i}^{(j)}:B(E)\to B(H)$ such that
$\Psi_{i}^{(j)}\big{|}_{\mathcal{S}}=\psi_{i}^{(j)}$. Define
$\Phi_{j}:=\sum_{i=1}^{r_{j}}\alpha_{i}^{(j)}\circ\Psi_{i}^{(j)}$. It follows
that $\Phi_{j}:B(E)\to B(H)$ is an EB map for all $j$. Let $A\in\mathcal{S}$
be arbitrary. Since $\phi_{j}\to\phi$, it follows that there exists an
$j_{0}\in{\mathbb{N}}$ such that $\|\phi_{j}(A)-\phi(A)\|<\epsilon\|A\|$ for
all $j\geq j_{0}$. Thus for all $j\geq j_{0}$, one obtains

$\displaystyle\|\Phi_{j}(A)-\phi(A)\|=$
$\displaystyle\left\|\sum_{i=1}^{r_{j}}\left(\alpha_{i}^{(j)}\circ\Psi_{i}^{(j)}\right)(A)-\phi(A)\right\|$
$\displaystyle=$
$\displaystyle\left\|\sum_{i=1}^{r_{j}}\left(\alpha_{i}^{(j)}\circ\psi_{i}^{(j)}\right)(A)-\phi(A)\right\|$
$\displaystyle=$ $\displaystyle\|\phi_{j}(A)-\phi(A)\|$ $\displaystyle<$
$\displaystyle\epsilon\|A\|.$

Defining $\Phi:=\Phi_{j_{0}}$ completes the proof. ∎

######  Remark 2.4.

Recall that $E={\mathbb{C}}^{d}$. Due to the convexity of the mapping cone
$\mathcal{C}$ and the finite dimensionality of $B(E)$, Caratheodory’s theorem
[DD, Theorem 16.1.8] implies that the map $\phi_{j}$ in the above proof can be
written as $\phi_{j}=\sum_{i=1}^{r}\alpha_{i}^{(j)}\circ\Psi_{i}^{(j)}$, where
$r=(2d)^{2}+1$ (is independent of $j$).

###### Proof of Theorem 1.5.

Let $j\in{\mathbb{N}}$ be arbitrary. By Lemma 2.3, there exists an EB map
$\Phi_{j}:B(E)\to B(H)$ such that $\|\Phi_{j}(A)-\phi(A)\|<\frac{1}{j}\|A\|$
for all $A\in\mathcal{S}$. Note that $\|\Phi_{j}\|=\|\Phi_{j}(I_{E})\|$ and

$\displaystyle\|\Phi_{j}(I_{E})\|$
$\displaystyle\leq\|\Phi_{j}(I_{E})-\phi(I_{E})\|+\|\phi(I_{E})\|$
$\displaystyle<\frac{1}{j}+\|\phi(I_{E})\|.$

Thus the sequence $\\{\Phi_{j}\\}_{j\in{\mathbb{N}}}$ is bounded. Consider the
set

$\mathcal{K}:=\\{\Gamma:B(E)\rightarrow B(H)\,:\,\Gamma\text{ is EB and
}\|\Gamma\|\leq 1+\|\phi(I_{E})\|\\}.$

Observe that $\mathcal{K}$ is compact and $\\{\Phi_{j}\\}_{j\in{\mathbb{N}}}$
is a sequence in $\mathcal{K}$. By the compactness of $\mathcal{K}$, there
exists a subsequence $\\{\Phi_{j_{k}}\\}_{k\in{\mathbb{N}}}$ of
$\\{\Phi_{j}\\}_{j\in{\mathbb{N}}}$ and an EB map $\Phi\in\mathcal{K}$ such
that $\Phi_{j_{k}}\rightarrow\Phi$ as $k\rightarrow\infty$. The proof is
complete by observing that $\Phi\big{|}_{\mathcal{S}}=\phi$.

∎

######  Remark 2.5.

It is to be noted that Theorem 1.5 does not necessarily follow from the
extension theorem [St, Theorem 5.2.3]. It was remarked in [St18] that the
conclusion of [St, Theorem 5.2.3] holds only for Real operator systems and
that this is due to the fact that Krein’s Extension Theorem ([St, Theorem
A.3.1]) which is formulated only for Real spaces, was incorrectly applied to
Complex spaces, while proving [St, Theorem 5.2.3].

##  3\. Dilations of UEB maps

In this section, we prove Theorem 1.7. The proof relies on a crucial
observation from [RJP, Lemma 3.1]. We begin with the following remark.

######  Remark 3.1.

Let $\Psi:B(E)\rightarrow B(H)$ be a (not necessarily unital) positive map
with commutative range. The argument in the proof of [RJP, Lemma 3.1] can
easily be adapted to conclude that $\Psi$ is EB.

######  Lemma 3.2.

Let $K$ be a finite dimensional Hilbert space and $\mathcal{S}\subset B(E)$ be
an operator system. If $\Gamma:\mathcal{S}\rightarrow B(K)$ is a positive map
with commutative range, then $\Gamma$ is EB.

###### Proof.

By [HJRW, Corollary 2], there exists a commutative $C^{*}$-algebra
$\mathcal{B}\subset B(K)$ containing $\operatorname{Range}(\Gamma)$ and a CP
map $\Theta:B(E)\to\mathcal{B}\subset B(K)$ such that
$\Theta\big{|}_{\mathcal{S}}=\Gamma$. It follows from Remark 3.1 that $\Theta$
is EB. Since $\Gamma$ is the restriction of the EB map $\Theta$ to
$\mathcal{S}$, it is also EB. ∎

######  Lemma 3.3.

Let $\mathcal{S}\subset B(E)$ be an operator system and
$\Phi:\mathcal{S}\rightarrow B(H)$ be an EB map. There exists a commutative
$C^{*}$-algebra $\mathcal{A}$, a finite dimensional Hilbert space $K$, an
isometry $V:H\rightarrow K$, a unital $*$-algebra homomorphism
$\pi:\mathcal{A}\rightarrow B(K)$ and a positive map
$\eta:\mathcal{S}\rightarrow\mathcal{A}$ such that
$\Phi(X)=V^{*}(\pi\circ\eta)(X)V$, for all $X\in\mathcal{S}$.

###### Proof.

By Theorem 1.5, there exists an EB map $\Psi:B(E)\to B(H)$ such that
$\Psi\big{|}_{\mathcal{S}}=\Phi$. It follows from Theorem 1.3 and Corollary
1.6, $\Phi$ can be written in Holevo form, i.e.,
$\Phi(X)=\sum_{j=1}^{m}\phi_{j}(X)R_{j}$ for all $X\in\mathcal{S}$, where the
$\phi_{j}$’s are states on $\mathcal{S}$ and the $R_{j}$’s are positive
operators in $B(H)$. Without loss of generality, assume that $\|\Phi\|\leq 1$.

Case(i) - $\Phi$ is non-unital: Let $\mathcal{A}=\ell^{\infty}_{m+1}$. Define
the linear maps $\gamma:\mathcal{A}\rightarrow B(H)$ and
$\eta:\mathcal{S}\rightarrow\mathcal{A}$ by
$\gamma(x_{1},\dots,x_{m+1})=\sum_{j=1}^{m+1}x_{j}R_{j}$ and
$\eta(X)=(\phi_{1}(X),\dots,\phi_{m}(X),0)$, where
$R_{m+1}=I_{H}-\sum_{j=1}^{m}R_{j}=I_{H}-\Phi(I_{E})\in B(H)^{+}$. Note that
$\gamma$ and $\eta$ are positive maps with $\gamma$ also being unital. In
fact, by [P, Theorem 3.9], $\gamma$ is a UCP map since the domain of $\gamma$
is a commutative C*-algebra. By Stinespring’s dilation theorem [P, Theorem
4.1], there exists a finite dimensional Hilbert space $K$, an isometry
$V:H\rightarrow K$ and a unital $*$-algebra homomorphism
$\pi:\mathcal{A}\rightarrow B(K)$ such that $V^{*}\pi(\cdot)V=\gamma(\cdot)$.

Case(ii) - $\Phi$ is unital: Let $\mathcal{A}=\ell^{\infty}_{m}$. Define the
linear maps $\gamma:\mathcal{A}\rightarrow B(H)$ and
$\eta:\mathcal{S}\rightarrow\mathcal{A}$ by
$\gamma(x_{1},\dots,x_{m})=\sum_{j=1}^{m}x_{j}R_{j}$ and
$\eta(X)=(\phi_{1}(X),\dots,\phi_{m}(X))$. Note that both $\gamma$ and $\eta$
are unital positive maps. Arguing as above, one obtains a finite dimensional
Hilbert space $K$, an isometry $V:H\rightarrow K$ and a unital $*$-algebra
homomorphism $\pi:\mathcal{A}\rightarrow B(K)$ such that
$V^{*}\pi(\cdot)V=\gamma(\cdot)$.

In both cases above, observe that $\Phi=\gamma\circ\eta$. It follows that
$\Phi(X)=V^{*}(\pi\circ\eta)(X)V,$ for all $X\in\mathcal{S}$. ∎

###### Proof of Theorem 1.7.

To prove $(i)\implies(ii)$, observe that by Lemma 3.3, there exists a
commutative $C^{*}$-algebra $\mathcal{A}$, a finite dimensional Hilbert space
$K$, an isometry $V:H\rightarrow K$, a unital $*$-algebra homomorphism
$\pi:\mathcal{A}\rightarrow B(K)$ and a positive map
$\eta:\mathcal{S}\rightarrow\mathcal{A}$ such that
$\Phi(X)=V^{*}(\pi\circ\eta)(X)V$ for all $X\in\mathcal{S}$. Define
$\Gamma:=\pi\circ\eta$. Since $\pi$ is a $*$-algebra homomorphism and $\eta$
is a positive map with commutative range, it follows that $\Gamma$ is a
positive map with commutative range that dilates $\Phi$. Finally, observe that
if $\Phi$ is unital, then $\Gamma$ is too.

The implication $(ii)\implies(iii)$ follows directly from Lemma 3.2.

To prove $(iii)\implies(i)$, let $\Phi(X)=V^{*}\Gamma(X)V$ for all
$X\in\mathcal{S}$, where $V:H\to K$ is an isometry for some finite dimensional
Hilbert space $K$ and $\Gamma:\mathcal{S}\to B(K)$ is an EB map with
commutative range. It follows from Theorem 1.3 and Corollary 1.6 that
$\Gamma(X)=\sum_{j=1}^{\ell}\psi_{j}(X)R_{j}$ for some states $\psi_{j}$ on
$\mathcal{S}$ and positive operators $R_{j}$ in $B(K)$. Observe that for each
$X\in\mathcal{S}$,

$\displaystyle\Phi(X)=$ $\displaystyle
V^{*}\left(\sum_{j=1}^{\ell}\psi_{j}(X)R_{j}\right)V=\sum_{j=1}^{\ell}\psi_{j}(X)(V^{*}R_{j}V),$

which is again a map in the Holevo form and hence is EB. Finally, if $\Gamma$
is unital, then $\Phi$ is too. ∎

##  4\. Maximal UEB Dilations

In this section we prove Theorems 1.10 and 1.11. The proof of Theorem 1.10
makes use of the following Lemmas, which contain some key properties of
maximal UEB maps defined on operator systems.

######  Lemma 4.1.

Let $\mathcal{S}\subset B(E)$ be an operator system and
$\Phi:\mathcal{S}\rightarrow B(H)$ be a UEB map. If $\Phi$ has commutative
range, then

$\Phi(X)=\sum_{i=1}^{k}\phi_{i}(X)P_{i},$ (4.1)

where the $\phi_{i}$’s are distinct states on $\mathcal{S}$ and the $P_{i}$’s
are mutually orthogonal projections in $B(H)$ such that
$\sum_{i=1}^{k}P_{i}=I_{H}$.

###### Proof.

Let $\mathcal{A}=C^{*}(\operatorname{Range}(\Phi))\subset B(H)$. There exists
a unital $*$-algebra isomorphism $\pi:\mathcal{A}\to\ell_{k}^{\infty}$ for
some $k$. Consider $~{}\pi\circ\Phi:\mathcal{S}\to\ell_{k}^{\infty}$. For each
$X\in\mathcal{S}$, $\pi\circ\Phi(X)=\sum_{i=1}^{k}\lambda_{i,X}e_{i}$, where
$\lambda_{i,X}$’s are scalars (depending on $X$) and $\\{e_{1},\dots,e_{k}\\}$
is the standard basis of $\ell_{k}^{\infty}$. For $1\leq i\leq k$, define
$\phi_{i}:\mathcal{S}\to{\mathbb{C}}$ via $\phi_{i}(X)=\lambda_{i,X}$. Since
$\pi\circ\Phi$ is a unital positive map, it follows that $\phi_{i}$’s are
states on $\mathcal{S}$. Indeed

$\Phi(X)=\sum_{i=1}^{k}\phi_{i}(X)P_{i},$

where $P_{i}=\pi^{-1}(\\{e_{i}\\})$. Note that the $P_{i}$’s are mutually
orthogonal projections in $B(H)$ such that $\sum_{i=1}^{k}P_{i}=I_{H}$.
Without loss of generality, one can assume that the $\phi_{i}$’s are distinct
states on $\mathcal{S}$ because if two $\phi_{i}$’s are the same, we can sum
the corresponding projections, i.e., the $P_{i}$’s together, and obtain a
single projection. ∎

######  Lemma 4.2.

Let $\mathcal{S}\subset B(E)$ be an operator system. If $\Phi:\mathcal{S}\to
B(H)$ is a maximal UEB map, then $\operatorname{Range}(\Phi)$ is commutative.

###### Proof.

Since $\Phi$ is a UEB map, it follows from part (iii) of Theorem 1.7 that
there exists a finite dimensional Hilbert space $K$, a UEB map
$\Gamma:\mathcal{S}\to B(K)$ with commutative range and an isometry
$V:H\rightarrow K$ such that $V^{*}\Gamma(X)V=\Phi(X)$, for all
$X\in\mathcal{S}$. Since $\Phi$ is a maximal UEB map, $\Gamma$ is a trivial
UEB dilation of $\Phi$. Thus it follows from Remark 1.9 that the subspace $VH$
is an invariant subspace for $\Gamma(X)$ for all $X\in\mathcal{S}$. Since
$VV^{*}$ is the projection of $K$ onto $\operatorname{Range}(V)$, it follows
that

$\displaystyle\Phi(X)\Phi(Y)(a)$
$\displaystyle=V^{*}\Gamma(X)VV^{*}\Gamma(Y)V(a)$
$\displaystyle=V^{*}\Gamma(X)\Gamma(Y)V(a)$
$\displaystyle=V^{*}\Gamma(Y)\Gamma(X)V(a)$
$\displaystyle=V^{*}\Gamma(Y)VV^{*}\Gamma(X)V(a)$
$\displaystyle=\Phi(Y)\Phi(X)(a),\quad\text{ for all $a\in H$.}\qed$

######  Lemma 4.3.

Let $\mathcal{S}\subset B(E)$ be an operator system and let
$\Phi:\mathcal{S}\to B(H)$ be a UEB map defined by
$\Phi(X)=\sum_{i=1}^{k}\phi_{i}(X)P_{i}$, where $\phi_{i}$ are states defined
on $\mathcal{S}$ and $P_{i}\in B(H)$ are mutually orthogonal projections
satisfying $\sum_{i=1}^{k}P_{i}=I_{H}$. If $\Phi$ is a maximal UEB map, then
each $\phi_{i}$ is a linear extremal state.

###### Proof.

Fix $i\in\\{1,2,\dots,k\\}$. Suppose that $\sigma,\tau$ are states on
$\mathcal{S}$ such that

$\phi_{i}=t\sigma+(1-t)\tau,$

for some $t\in(0,1)$. It suffices to show that $\sigma=\tau$. Let
$F=\operatorname{Range}(P_{i})\subset H$ and $G=F^{\perp}\subset H$. Let
$L:=F\oplus F\oplus G$. Define $V:H\rightarrow L$ by

$V(x+y)=(\sqrt{t}x,\sqrt{(1-t)}x,y),$

for all $x\in F$ and $y\in G$. It is easily seen that $V$ is an isometry. Let
$\Psi:\mathcal{S}\to B(L)$ be defined by

$\Psi(X)(x,y,z)=\bigg{(}\sigma(X)x,\tau(X)y,\sum_{j\neq
i}\phi_{j}(X)P_{j}(z)\bigg{)},$

for all $X\in\mathcal{S},x,y\in F$ and $z\in G$. Observe that $\Psi$ is
unital. Define the coordinate projections $Q_{1},Q_{2},Q_{3}$ on $L$ by
$Q_{1}(x,y,z)=(x,0,0),Q_{2}(x,y,z)=(0,y,0)$ and $Q_{3}(x,y,z)=(0,0,z)$. One
sees that $\Psi(X)=\sigma(X)Q_{1}+\tau(X)Q_{2}+\left(\sum_{j\neq
i}\phi_{j}(X)P_{j}\right)Q_{3}$, which is in Holevo form for $\Psi$. By
Corollary 1.6, it follows that $\Psi$ is a UEB map. Observe that

$\displaystyle\langle V^{*}(u,v,w),(x+y)\rangle=$
$\displaystyle\langle(u,v,w),V(x+y)\rangle$ $\displaystyle=$
$\displaystyle\langle(u,v,w),(\sqrt{t}x,\sqrt{(1-t)}x,y)\rangle$
$\displaystyle=$ $\displaystyle\sqrt{t}\langle u,x\rangle+\sqrt{(1-t)}\langle
v,x\rangle+\langle w,y\rangle$ $\displaystyle=$
$\displaystyle\langle\sqrt{t}u+\sqrt{(1-t)}v,x\rangle+\langle w,y\rangle$
$\displaystyle=$ $\displaystyle\langle\sqrt{t}u+\sqrt{(1-t)}v+w,x+y\rangle,$

for all $u,v,x\in F$ and $w,y\in G$. Thus $V^{*}$ is defined by

$V^{*}(u,v,w):=\sqrt{t}u+\sqrt{(1-t)}v+w.$

For each $X\in\mathcal{S}$, $x\in F$ and $y\in G$, it follows that

$\displaystyle V^{*}\Psi(X)V(x+y)$
$\displaystyle=V^{*}\Psi(X)(\sqrt{t}x,\sqrt{(1-t)}x,y)$
$\displaystyle=V^{*}\bigg{(}\sqrt{t}\sigma(X)x,\sqrt{(1-t)}\tau(X)x,\sum_{j\neq
i}\phi_{j}(X)P_{j}(y)\bigg{)}$
$\displaystyle=t\sigma(X)x+(1-t)\tau(X)x+\sum_{j\neq i}\phi_{j}(X)P_{j}(y)$
$\displaystyle=\phi_{i}(X)x+\sum_{j\neq i}\phi_{j}(X)P_{j}(y)$
$\displaystyle=\sum_{j=1}^{k}\phi_{j}(X)P_{j}(x+y)$
$\displaystyle=\Phi(X)(x+y).$

Hence $\Psi$ is a UEB dilation of $\Phi$. By hypothesis, $\Psi(X)VH\subset VH$
for all $X\in\mathcal{S}$. Choosing $0\neq x\in F$, it follows that

$\displaystyle\big{(}\sqrt{t}\sigma(X)x,\sqrt{(1-t)}\tau(X)x,0\big{)}$
$\displaystyle=\Psi(X)V(x)$ $\displaystyle=V(z+w)=(\sqrt{t}z,\sqrt{(1-t)}z,w)$

for some $z\in F$ and $w\in G$. Hence $\sigma(X)x=\tau(X)x$ for each
$X\in\mathcal{S}$. This in turn implies that $\sigma=\tau$ and the proof is
complete. ∎

######  Lemma 4.4.

Let $\mathcal{S}\subset B(E)$ be an operator system and let
$\Phi:\mathcal{S}\to B(H)$ be a UEB map. If every UEB dilation
$\Psi:\mathcal{S}\to B(L)$ of $\Phi$ is trivial whenever $L$ is finite
dimensional, then $\Phi$ is a maximal UEB map.

###### Proof.

Let $\Psi:\mathcal{S}\to B(K)$ be a UEB dilation of $\Phi$ where $K$ is an
infinite dimensional separable Hilbert space. Let the isometry $V:H\rightarrow
K$ be such that $V^{*}\Psi(X)V=\Phi(X)$ for all $X\in\mathcal{S}$. Let
$\\{e_{1},\dots,e_{n}\\}$ denote an orthonormal basis for $VH$. Extend it to
an orthonormal basis $\\{e_{i}\,:\,i\in{\mathbb{N}}\\}$ for the Hilbert space
$K$. For each $m\geq n$, let
$K_{m}:=\operatorname{span}\\{e_{1},\dots,e_{m}\\}$, $W_{m}:K_{m}\rightarrow
K$ denote the inclusion map and $P_{m}$ denote the orthogonal projection of
$K$ onto $K_{m}$. Observe that $P_{m}=W_{m}^{*}$. Define $V_{m}=P_{m}\circ V$
and $\Psi_{m}:\mathcal{S}\to B(K_{m})$ to be the compression of $\Psi$ to
$K_{m}$, i.e.,
$\Psi_{m}(X)=P_{m}\Psi(X)\big{|}_{K_{m}}=W_{m}^{*}\Psi(X)W_{m}$. Let
$k\in{\mathbb{N}}$ and $Z\in(M_{k}\otimes\mathcal{S})^{+}$. Since $\Psi$ is an
EB map, $(\mathtt{i}_{k}\otimes\Psi)(Z)\in\overline{M_{k}^{+}\otimes
B(K_{m})^{+}}$. It follows that
$(\mathtt{i}_{k}\otimes\Psi_{m})(Z)=(\mathtt{i}_{k}\otimes
W_{m})^{*}((\mathtt{i}_{k}\otimes\Psi)(Z))(\mathtt{i}_{k}\otimes
W_{m})\in\overline{M_{k}^{+}\otimes B(K_{m})^{+}}=M_{k}^{+}\otimes
B(K_{m})^{+}.$ Thus $\Psi_{m}:\mathcal{S}\to B(K_{m})$ is a UEB map. Moreover,
$V_{m}$ is an isometry and $V_{m}^{*}\Psi_{m}(X)V_{m}=\Phi(X)$ for all
$X\in\mathcal{S}$, i.e., $\Psi_{m}$ is a UEB dilation of $\Phi$. Let $a\in H$
and $X\in\mathcal{S}$. Take $V_{m}=P_{m}\circ V$, where $V$ is defined as
above. Since $K_{m}$ is finite dimensional, by hypothesis, it follows that
$\Psi_{m}$ is a trivial UEB dilation of $\Phi$. By Remark 1.9, one gets that
$\Psi_{m}(X)V_{m}H\subset V_{m}H=VH$. Finally, for $x\in H$,

$\displaystyle\Psi(X)(Vx)$
$\displaystyle=\lim_{m\rightarrow\infty}P_{m}\Psi(X)P_{m}(Vx)=\lim_{m\rightarrow\infty}W_{m}^{*}\Psi(X)W_{m}P_{m}(Vx)$
$\displaystyle=\lim_{m\rightarrow\infty}\Psi_{m}(X)V_{m}(x)\in VH.\qed$

######  Remark 4.5.

It is to be noted that the definition of an EB map adopted here in terms of
separability in the norm-closure sense, is not the usual way it is defined in
the literature. If one works with a weak*-continuous CP map (or equivalently,
a normal CP map), say $\Phi:B(L)\rightarrow B(M)$, then there is a way of
defining an EB map that is along the lines of the usual definition namely:
$\Phi:B(L)\rightarrow B(M)$ is EB if for every $k\in{\mathbb{N}}$,
$\Psi:\mathfrak{T}(M)\rightarrow\mathfrak{T}(L)$ maps positive matrices in
$M_{k}\otimes\mathfrak{T}(M)$ to separable matrices in
$M_{k}\otimes\mathfrak{T}(L)$, where $\mathfrak{T}(L)$ denotes the space of
trace-class operators on the Hilbert space $L$, and $\Psi$ is the unique map
whose (Banach space) adjoint $\Psi^{*}$ equals $\Phi$. This certainly is a
generalization of the usual definition of an EB map from the finite to the
infinite dimensional setting. But since here we mainly work on unital maps
defined only on operator systems, defining the ”entanglement breaking”
property using this type of duality becomes a challenge.

We would like to emphasize that the above remark is only relevant when either
of the Hilbert spaces $L$ or $M$ is infinite dimensional and so does not
impact the results here.

###### Proof of Theorem 1.10.

(i)$\implies$ (ii): Since $\Phi$ is a maximal UEB map, it follows from Lemma
4.2 that $\Phi$ has commutative range. By applying Lemma 4.1, one gets that
$\Phi(X)=\sum_{i=1}^{k}\phi_{i}(X)P_{i}$, where the $\phi_{i}$’s are distinct
states defined on $\mathcal{S}$ and $P_{i}\in B(H)$ are mutually orthogonal
projections satisfying $\sum_{i=1}^{k}P_{i}=I_{H}$. That the $\phi_{i}$’s are
linear extremal states on $\mathcal{S}$ follows from an application of Lemma
4.3.

To prove the implication (ii) $\implies$ (i), let $\Psi$ be an arbitrary UEB
dilation of $\Phi;$ that is, $\Psi:\mathcal{S}\to B(K)$ is a UEB map such that
there exists an isometry $V:H\to K$ satisfying $\Phi(X)=V^{*}\Psi(X)V,$ for
all $X\in\mathcal{S}$. By Lemma 4.4, one can assume that $K$ is finite
dimensional. It suffices to show that $\Psi(X)Vx\in VH$ for all $x\in H$ and
$X\in\mathcal{S}$. If $x=0$, then there is nothing to prove. Suppose that
$x\neq 0$. We first consider the case $x\in\operatorname{Range}(P_{j})$ for
some $j$. For such an $x$, it follows that $\Phi(X)x=\phi_{j}(X)x$ and hence
$V^{*}\Psi(X)Vx=\phi_{j}(X)x$.

Let a Holevo form (see Theorem 1.3 and Corollary 1.6) for $\Psi$ be given by
$\Psi(X)=\sum_{i=1}^{k}\psi_{i}(X)R_{i}$, where the $\psi_{i}$’s are distinct
states and the $R_{i}$’s are positive matrices with
$\sum_{i=1}^{k}R_{i}=I_{K}$. Observe that

$\displaystyle\phi_{j}(X)\|x\|^{2}=\langle\phi_{j}(X)x,x\rangle$
$\displaystyle=\langle\Phi(X)x,x\rangle=\left\langle\left(\sum_{i=1}^{k}V^{*}\psi_{i}(X)R_{i}V\right)x,x\right\rangle$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{k}\psi_{i}(X)\left\langle
R_{i}Vx,Vx\right\rangle.$ (4.2)

Thus, $\phi_{j}(X)=\sum_{i=1}^{k}\alpha_{i}\psi_{i}(X)$, where
$\alpha_{i}=\frac{1}{\|x\|^{2}}\langle R_{i}Vx,Vx\rangle\geq 0$. Note that
$\alpha_{i}$’s are independent of $X$. Also, since $V$ is an isometry and
$\sum_{i=1}^{k}R_{i}=I_{K}$, it follows that

$\sum_{i=1}^{k}\alpha_{i}=\frac{1}{\|x\|^{2}}\sum_{i=1}^{k}\langle
R_{i}Vx,Vx\rangle=\frac{1}{\|x\|^{2}}\left\langle\left(\sum_{i=1}^{k}R_{i}\right)Vx,Vx\right\rangle=1.$
(4.3)

Combining equations (4) and (4.3), it follows that $\phi_{j}$ is a convex
combination of the $\psi_{i}$’s. Since the $\psi_{i}$’s are distinct and
$\phi_{j}$ is a linear extremal state, it follows that there exists an index
$i_{0}$ such that $\alpha_{i_{0}}=1$, $\phi_{j}=\psi_{i_{0}}$ and
$\alpha_{i}=\frac{1}{\|x\|^{2}}\langle R_{i}Vx,Vx\rangle=0$ for all $i\neq
i_{0}$. Since the $R_{i}$’s are positive, it follows that $R_{i}Vx=0$ for all
$i\neq i_{0}$, implying $Vx=\sum_{i=1}^{k}R_{i}Vx=R_{i_{0}}Vx$. Thus

$\displaystyle\Psi(X)Vx=$
$\displaystyle\psi_{i_{0}}(X)R_{i_{0}}Vx=\psi_{i_{0}}(X)Vx\in VH.$

Now using the fact that the $P_{i}$’s are mutually orthogonal projections
satisfying $\sum_{i=1}^{k}P_{i}=I_{H}$ and
$\oplus_{i=1}^{k}\operatorname{Range}(P_{i})=H$, it follows that $\Psi(X)Vx\in
V(H)$ for all $x\in H$, and the proof is complete. ∎

Our next task is to prove Theorem 1.11, before which we prove the following
crucial observation concerning a Holevo form of a UEB map.

######  Lemma 4.6.

Let $\mathcal{S}\subset B(E)$ be an operator system. If
$\Phi:\mathcal{S}\rightarrow B(H)$ is a UEB map, then
$\Phi(X)=\sum_{j=1}^{k}\phi_{j}(X)S_{j}$ for each $X\in\mathcal{S}$, where
$\phi_{j}$ are distinct linear extremal states on $\mathcal{S}$ and $S_{j}$
are positive operators in $B(H)$ satisfying $\sum_{j=1}^{k}S_{j}=I_{H}$.

###### Proof.

Let a Holevo form for $\Phi$ be given by

$\Phi(X)=\sum_{i=1}^{r}\phi_{i}(X)R_{i},$

where the $\phi_{i}$’s are states on $\mathcal{S}$ and the $R_{i}$’s are
positive operators in $B(H)$ such that $\sum_{i=1}^{r}R_{i}=I_{H}$. Observe
that the set of all states on $\mathcal{S}$ is a compact and convex subset of
the dual of $\mathcal{S}$. A well known consequence of Caratheodory’s theorem
(See [DD, Theorem 16.1.8] and [DD, Corollary 16.1.9]) and the Krein-Milman
theorem is that in a finite dimensional topological vector space, a compact
convex set $\mathcal{C}$ equals the convex hull of its extreme points. Using
this fact here, each $\phi_{i}$ can be written as a convex combination of
linear extremal states, i.e.,

$\phi_{i}(X)=t_{i,1}\varphi_{i,1}(X)+\dots+t_{i,\ell_{i}}\varphi_{i,\ell_{i}}(X),$

where $t_{i,j}\in[0,1]$ with $\sum_{j=1}^{\ell_{i}}t_{i,j}=1$ and the
$\varphi_{i,j}$’s are linear extremal states on $\mathcal{S}$. It follows that

$\displaystyle\Phi(X)$
$\displaystyle=\sum_{i=1}^{r}\sum_{j=1}^{\ell_{i}}t_{i,j}\varphi_{i,j}(X)R_{i}=\sum_{i=1}^{r}\sum_{j=1}^{\ell_{i}}\varphi_{i,j}(X)(t_{i,j}R_{i}),$

for all $X\in\mathcal{S}$. Note that by combining suitable terms in the above
sum, it can be rewritten in the desired form. This completes the proof. ∎

###### Proof of Theorem 1.11.

Using Theorem 1.5 and the Holevo form (see Theorem 1.3 and Corollary 1.6) for
EB maps, we have

$\Phi(X)=\sum_{i=1}^{r}\phi_{i}(X)R_{i},$ (4.4)

for all $X\in\mathcal{S}$, where the $\phi_{i}$’s are distinct states on
$\mathcal{S}$ and the $R_{i}$’s are positive operators in $B(H)$ such that
$\sum_{i=1}^{r}R_{i}=I_{H}$. By Lemma 4.6, there is no loss of generality in
assuming that the $\phi_{i}$’s in equation (4.4) are distinct linear extremal
states on $\mathcal{S}$. By Lemma 3.3, there exists a finite dimensional
Hilbert space $K$, an isometry $V:H\rightarrow K$, $r\in{\mathbb{N}}$, a
unital $*$-algebra homomorphism $\pi:\ell^{\infty}_{r}\rightarrow B(K)$ and a
positive map $\eta:\mathcal{S}\rightarrow\ell^{\infty}_{r}$ such that

$\Phi(X)=V^{*}(\pi\circ\eta)(X)V,$

for all $X\in\mathcal{S}$. As observed in the proof of Lemma 3.3, the
unitality of $\Phi$ implies the unitality of $\eta$. Note that
$\pi(x_{1},\dots,x_{r})=\sum_{i=1}^{r}x_{i}\pi(e_{i})$, where
$\\{e_{1},\dots,e_{r}\\}$ is the standard basis of $\ell_{r}^{\infty}$. Also
$(\pi\circ\eta)(X)=\pi\big{(}\sum_{i=1}^{r}\phi_{i}(X)e_{i}\big{)}=\sum_{i=1}^{r}\phi_{i}(X)\pi(e_{i})$,
for all $X\in\mathcal{S}$. Since $\pi$ is a unital $*$-algebra homomorphism,
it follows that the $\pi(e_{i})$’s are mutually orthogonal projections and
$\sum_{i=1}^{r}\pi(e_{i})=I_{K}$. Define $\Psi:=\pi\circ\eta$. It follows from
Theorem 1.10 that $\Psi$ is maximal. This completes the proof. ∎

##  5\. $C^{*}$-extreme UEB maps on Operator Systems

In this section we prove Theorem 1.12 and Corollary 1.13. The proofs use
techniques from [FM93], [FM97], [FZ] and [Z]. We also need the following
Lemmas, the first of which contains an equivalent definition of a
$C^{*}$-extreme UEB map and the second one contains a description of
UEB$(\mathcal{S},B(H))$ in terms of $C^{*}$-convex combinations.

######  Lemma 5.1.

Let $\mathcal{S}\subset B(E)$ be an operator system and
$\Phi:\mathcal{S}\rightarrow B(H)$ be a UEB map. The following statements are
equivalent.

  * (i)

$\Phi$ is a $C^{*}$-extreme point of UEB$(\mathcal{S},B(H))$.

  * (ii)

If $\Phi=\sum_{i=1}^{2}T_{i}^{*}\Phi_{i}T_{i}$, for some invertible operators
$T_{1},T_{2}\in B(H)$ satisfying $T_{1}^{*}T_{1}+T_{2}^{*}T_{2}=I_{H}$, then
there exist unitaries $U_{1},U_{2}\in B(H)$ such that
$\Phi_{i}(X)=U_{i}^{*}\Phi(X)U_{i}$, $i=1,2$.

Lemma 5.1 is essentially the operator system version of [BDMS, Proposition
3.2]. The proof given there works equally well for the operator system setting
too.

######  Lemma 5.2.

Let $\mathcal{S}\subset B(E)$ be an operator system, $\mathcal{K}$ denote the
set

$\displaystyle\\{\Psi:\mathcal{S}\rightarrow B(H)$
$\displaystyle:\Psi(X)=g(X)I_{H},\,g\text{ is a linear extremal state on
}\mathcal{S}\\},$

$\mathcal{E}$ denote the $C^{*}$-convex hull of $\mathcal{K}$, i.e., the set
of all $C^{*}$-convex combinations of elements of $\mathcal{K}$ and
$\mathcal{E}_{+}$ denote the set of all positive $C^{*}$-convex combinations
of elements of $\mathcal{K}$. The following statements hold.

  1. (i)

$\mathcal{E}_{+}=\mathcal{E}=$ UEB$(\mathcal{S},B(H))$.

  2. (ii)

If $\Gamma\in\mathcal{E}$ is given by
$\Gamma(X)=\sum_{i=1}^{m}T_{i}^{*}(g_{i}(X)I_{H})T_{i}\in\mathcal{E}$, then
$\Gamma\in\mathcal{E}_{+}$ and there exists $B_{i}\in B(H)^{+}$ such that
$\sum_{i=1}^{m}B_{i}^{2}=I_{H}$ and

$\Gamma(X)=\sum_{i=1}^{m}B_{i}(g_{i}(X)I_{H})B_{i},$

for all $X\in\mathcal{S}$.

###### Proof.

Evidently, $\mathcal{E}_{+}\subset\mathcal{E}\subset$ UEB$(\mathcal{S},B(H))$.
To complete the proof of part $(i)$, it suffices to show that
UEB$(\mathcal{S},B(H))$ $\subset\mathcal{E}_{+}$. To this end, let $\Phi\in$
UEB$(\mathcal{S},B(H)).$ By Lemma 4.6, it follows that for each
$X\in\mathcal{S}$, $\Phi(X)=\sum_{j=1}^{k}\phi_{j}(X)S_{j}$, where $\phi_{j}$
are distinct linear extremal states on $\mathcal{S}$ and $S_{j}$ are positive
operators in $B(H)$ satisfying $\sum_{j=1}^{k}S_{j}=I_{H}$. Rewrite

$\Phi(X)=\sum_{j=1}^{k}\phi_{j}(X)S_{j}=\sum_{j=1}^{k}\sqrt{S_{j}}(\phi_{j}(X)I_{H})\sqrt{S_{j}}.$

Thus $\Phi\in\mathcal{E}_{+}$. To prove part (ii), write $T_{i}$ in its polar
decomposition, i.e., $T_{i}=U_{i}B_{i}$, where $U_{i}$ is a unitary operator
and $B_{i}=\sqrt{T_{i}^{*}T_{i}}\in B(H)^{+}$. Note that
$\sum_{i=1}^{m}B_{i}^{2}=\sum_{i=1}^{m}T_{i}^{*}T_{i}=I_{H}$. By part $(i)$,
it follows that $\Gamma\in\mathcal{E}_{+}$. Also, for each $X\in\mathcal{S}$,

$\displaystyle\Gamma(X)$
$\displaystyle=\sum_{i=1}^{m}T_{i}^{*}(g_{i}(X)I_{H})T_{i}=\sum_{i=1}^{m}B_{i}U_{i}^{*}(g_{i}(X)I_{H})U_{i}B_{i}$
$\displaystyle=\sum_{i=1}^{m}B_{i}(g_{i}(X)I_{H})B_{i}.\qed$

###### Proof of Theorem 1.12.

$(i)\Rightarrow(ii):$ Using the alternate definition of a $C^{*}$-extreme UEB
map given in Lemma 5.1, let
$\Phi(X)=T_{1}^{*}\Phi_{1}(X)T_{1}+T_{2}^{*}\Phi_{2}(X)T_{2}$ for all
$X\in\mathcal{S}$, where $\Phi_{1},\Phi_{2}:\mathcal{S}\to B(H)$ are UEB maps
and $T_{1},T_{2}\in B(H)$ are invertible operators such that
$\sum_{i=1}^{2}T_{i}^{*}T_{i}=I_{H}$. It suffices to show that $\Phi_{1}$ and
$\Phi_{2}$ are unitarily equivalent to $\Phi$.

Define an isometry $V:H\to H\oplus H$ via $V(x)=(T_{1}(x),T_{2}(x))$ and a
linear map $\Psi:\mathcal{S}\to B(H\oplus H)$ by

$\Psi(X)=\begin{bmatrix}\Phi_{1}(X)&0\\\ 0&\Phi_{2}(X)\end{bmatrix},$

for all $X\in\mathcal{S}$. Observe that $\Psi$ being a direct sum of UEB maps,
is a UEB map. In fact, it is a UEB dilation of $\Phi$ since
$V^{*}\Psi(X)V=\Phi(X)$ for all $X\in\mathcal{S}$. By hypothesis, it follows
that $\Psi$ is a trivial UEB dilation of $\Phi$, that is, $\Psi(X)V=V\Phi(X)$
for all $X\in\mathcal{S}$. Equivalently,

$\begin{bmatrix}\Phi_{1}(X)&0\\\
0&\Phi_{2}(X)\end{bmatrix}\begin{bmatrix}T_{1}\\\
T_{2}\end{bmatrix}=\begin{bmatrix}T_{1}\\\ T_{2}\end{bmatrix}\Phi(X)$

for all $X\in\mathcal{S}$. Consequently, we have

$\Phi_{1}(X)=T_{1}\Phi(X)T_{1}^{-1}\,\text{ and
}\,\Phi_{2}(X)=T_{2}\Phi(X)T_{2}^{-1}$

for all $X\in\mathcal{S}$.

Using the polar decomposition of the invertible operator $T_{i}^{*}$, $i=1,2$,
there exists a unitary operator $W_{i}\in B(H)$ and a positive invertible
operator $P_{i}\in B(H)$ given by $\sqrt{T_{i}T_{i}^{*}}$ such that
$T_{i}=P_{i}W_{i}$, $i=1,2$. Hence

$\Phi(X)=W_{i}^{*}P_{i}^{-1}\Phi_{i}(X)P_{i}W_{i},$ (5.1)

for $i=1,2.$ Since $\Psi$ is a trivial UEB dilation of $\Phi$, it follows that

$\Psi(X)VV^{*}=VV^{*}\Psi(X),$ (5.2)

for all $X\in\mathcal{S}$, (See Remark 1.9). Applying the definition of $\Psi$
and $VV^{*}$ in Equation (5.2), one obtains

$\displaystyle\Phi_{i}(X)T_{i}T_{i}^{*}=$ $\displaystyle
T_{i}T_{i}^{*}\Phi_{i}(X),$

for $i=1,2$. Note that $T_{i}T_{i}^{*}$ is a positive operator and
$\Phi_{i}(X)$ commutes with $T_{i}T_{i}^{*}$. Hence

$\Phi_{i}(X)P_{i}=P_{i}\Phi_{i}(X),$ (5.3)

for each $X\in\mathcal{S}$ and $i=1,2$.

It follows from equations (5.1) and (5.3) that

$\displaystyle\Phi(X)=W_{i}^{*}P_{i}^{-1}P_{i}\Phi_{i}(X)W_{i}=W_{i}^{*}\Phi_{i}(X)W_{i},$

for each $X\in\mathcal{S}$ and $i=1,2$. Since $W_{1}$ and $W_{2}$ are
unitaries, the proof is complete.

The following proof of the implication $(ii)\Rightarrow(i)$ is adapted from
[FM93, Theorem 4.1]. Recall the notations $\mathcal{K}$, $\mathcal{E}$ and
$\mathcal{E}_{+}$ from Lemma 5.2. Let $\Phi:\mathcal{S}\rightarrow B(H)$ be a
$C^{*}$-extreme UEB map. Using Lemma 5.2, write
$\Phi(X)=\sum_{i=1}^{m}T_{i}^{*}\Psi_{i}(X)T_{i}$, where
$\Psi_{i}(X)=g_{i}(X)I_{H}$ for all $X\in\mathcal{S}$, the $g_{i}$’s are
linear extremal states on $\mathcal{S}$, and
$\sum_{i=1}^{m}T_{i}^{*}T_{i}=I_{H}$. By part (ii) of Lemma 5.2, there is no
loss of generality in assuming that the $T_{i}$’s are positive. Let $m$ be the
least number of coefficients required to represent $\Phi$ as a positive
$C^{*}$-convex combination, i.e., as a $C^{*}$-convex combination of elements
of $\mathcal{K}$ with positive coefficients (or equivalently to represent
$\Phi$ as an element of $\mathcal{E}_{+}$.) If $m=1$, then there is nothing to
prove, due to Theorem 1.10. So assume $m\geq 2$. Note that there must exist an
$i$ such that $\|T_{i}\|=1$. Otherwise $\|T_{i}\|<1$ for all $i$ and, by
modifying Technique-A in [FM93] to our current setting, one can rewrite $\Phi$
as a proper $C^{*}$-convex combination of some $\Gamma_{j}\in\mathcal{E}$ with
the additional property that each $\Gamma_{j}$ is a $C^{*}$-convex combination
of less than $m$ of the $\Psi_{i}$’s. Since $\Phi$ is $C^{*}$\- extreme, each
$\Gamma_{j}$ is unitarily equivalent to $\Phi$. An application of part (ii) of
Lemma 5.2 implies that $\Phi$ is a positive $C^{*}$-convex combination of
fewer than $m$ of the $\Psi_{i}$’s, which contradicts the minimality of $m$.
Thus at least one of the $T_{i}$’s has unit norm. Without loss of generality,
assume that $\|T_{1}\|=1$. Due to the unitality of $\Phi$, note also that
$\sum_{i=1}^{m}T_{i}^{2}=I_{H}$. Since $T_{1}\geq 0$ and $\|T_{1}\|=1$, it
follows that there exists a unitary $U\in B(H)$ such that

$U^{*}T_{1}U=\begin{pmatrix}I&0\\\ 0&Y_{1}\end{pmatrix}$

where $Y_{1}$ is a diagonal matrix satisfying $Y_{1}\geq 0$, $\|Y_{1}\|<1$ and
$I$ is the identity operator of suitable size. Since $T_{i}\geq 0$ for all
$i\geq 2$ and $\sum_{i=1}^{m}T_{i}^{2}=I_{H}$,

$U^{*}T_{i}U=\begin{pmatrix}0&\\\ &Y_{i}\end{pmatrix},\text{ with }Y_{i}\geq
0\text{ for all }i\geq 2.$

Let $W_{i}=U^{*}T_{i}U$ for each $i\in\\{1,2,\dots,m\\}$. Observe that
$W_{i}\geq 0$ and

$U^{*}\Phi(X)U=\sum_{i=1}^{m}W_{i}(U^{*}\Psi_{i}(X)U)W_{i}=\sum_{i=1}^{m}W_{i}\Psi_{i}(X)W_{i}.$
(5.4)

Since $\sum_{i=1}^{m}W_{i}^{2}=I_{H}$, it follows that
$\sum_{i=1}^{m}Y_{i}^{2}=I$. Since $\|Y_{1}\|<1$, $I-Y_{1}^{2}$ invertible.
Also since

$\sum_{i\geq 2}\begin{pmatrix}0&0\\\ 0&Y_{i}^{2}\end{pmatrix}=\sum_{i\geq
2}W_{i}^{2}=I_{H}-W_{1}^{2}=\begin{pmatrix}0&0\\\ 0&I-Y_{1}^{2}\end{pmatrix},$

it follows that $\sum_{i\geq 2}Y_{i}^{2}$ is invertible. Adapting Technique-C
from [FM93] to our setting and applying it here allows us to write
$\sum_{i\geq 2}W_{i}\Psi_{i}(X)W_{i}$ as a single term
$T_{0}^{*}\Gamma_{0}(X)T_{0}$ for some $\Gamma_{0}\in\mathcal{E}$. Note that
$T_{0}$ need not be positive. By using the polar decomposition
$T_{0}=U_{0}W_{0}$ where $U_{0}$ is unitary and $W_{0}$ is a positive
operator, observe that $T_{0}^{*}\Gamma_{0}(X)T_{0}=W_{0}\Psi_{0}(X)W_{0},$
for some $\Psi_{0}\in\mathcal{E}$. Thus

$\sum_{i\geq 2}W_{i}\Psi_{i}(X)W_{i}=W_{0}\Psi_{0}(X)W_{0}$ (5.5)

for some positive operator $W_{0}$ and $\Psi_{0}\in\mathcal{E}$.

Indeed $W_{0}^{2}+W_{1}^{2}=I_{H}$ and $W_{0}=\begin{pmatrix}0&0\\\
0&Y_{0}\end{pmatrix}$ for some positive operator $Y_{0}$. Recall the positive
matrix $Y_{1}$ and rewrite it as $Y_{1}=\begin{pmatrix}Z_{1}&0\\\
0&0\end{pmatrix}$ where $Z_{1}$ is a positive invertible matrix with
$\|Z_{1}\|<1.$ Since $Y_{0}^{2}+Y_{1}^{2}=I$, $Y_{0}=\begin{pmatrix}Z_{0}&0\\\
0&I\end{pmatrix}$ for some matrix $Z_{0}$. Thus

$W_{1}=\begin{pmatrix}I&0&0\\\ 0&Z_{1}&0\\\
0&0&0\end{pmatrix}\quad\text{and}\quad W_{0}=\begin{pmatrix}0&0&0\\\
0&Z_{0}&0\\\ 0&0&I\end{pmatrix},$

where $Z_{1},Z_{0}$ are positive matrices. Since $W_{0}^{2}+W_{1}^{2}=I_{H}$,
it must be the case that $Z_{0}^{2}+Z_{1}^{2}=I$, which in turn implies that
$Z_{0}$ is invertible, since $\|Z_{1}\|<1$. Recall that $Z_{1}$ is also
invertible. It follows from equations (5.4) and (5.5) and the fact that
$\Psi_{1}\in\mathcal{K}$ that

$\displaystyle U^{*}\Phi(X)U=W_{0}\Psi_{0}(X)W_{0}+W_{1}\Psi_{1}(X)W_{1}$
$\displaystyle=\begin{pmatrix}I&0&0\\\ 0&Z_{0}&0\\\ 0&0&I\\\
\end{pmatrix}\Theta(X)\begin{pmatrix}I&0&0\\\ 0&Z_{0}&0\\\ 0&0&I\\\
\end{pmatrix}+\begin{pmatrix}0&0&0\\\ 0&Z_{1}&0\\\
0&0&0\end{pmatrix}\Psi_{1}(X)\begin{pmatrix}0&0&0\\\ 0&Z_{1}&0\\\
0&0&0\end{pmatrix},$ (5.6)

where

$\displaystyle\Theta(X)$ $\displaystyle=\begin{pmatrix}I&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix}\Psi_{1}(X)\begin{pmatrix}I&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix}+\begin{pmatrix}0&0&0\\\ 0&I&0\\\
0&0&I\end{pmatrix}\Psi_{0}(X)\begin{pmatrix}0&0&0\\\ 0&I&0\\\
0&0&I\end{pmatrix}$ (5.7) $\displaystyle\in\mathcal{E}.$

By the invertibility of $Z_{0}$, it is immediate that
$Y_{0}=\begin{pmatrix}Z_{0}&0\\\ 0&I\end{pmatrix}$ is also invertible.
Applying Technique-B from [FM93] allows for writing $U^{*}\Phi U$ as a proper
$C^{*}$-convex combination of $\Theta$ and some other $\Gamma\in\mathcal{E}$.
By hypothesis, $\Phi$ is a $C^{*}$-extreme UEB map. Hence so is $U^{*}\Phi U$.
It follows that $\Phi$ is unitarily equivalent to $\Theta$. Let $Q_{1}$ and
$Q_{1}^{\perp}$ denote the projections $\begin{pmatrix}I&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix}$ and $\begin{pmatrix}0&0&0\\\ 0&I&0\\\ 0&0&I\end{pmatrix}$
respectively. Indeed, with respect to the decomposition
$H=\operatorname{Range}(Q_{1})\oplus\operatorname{Range}(Q_{1}^{\perp})$,

$\Theta(X)=Q_{1}\Psi_{1}(X)|_{\operatorname{Range}(Q_{1})}\oplus
Q_{1}^{\perp}\Psi_{0}(X)|_{\operatorname{Range}(Q_{1}^{\perp})},$ (5.8)

for all $X\in\mathcal{S}$. Since
$Q_{1}^{\perp}\Psi_{0}(X)|_{\operatorname{Range}(Q_{1}^{\perp})}\in$
UEB($\mathcal{S},B(\operatorname{Range}(Q_{1}^{\perp})))$, one can write

$Q_{1}^{\perp}\Psi_{0}(X)|_{\operatorname{Range}(Q_{1}^{\perp})}:=\sum_{i=1}^{k}S_{i}^{*}\Gamma_{i}(X)S_{i},$

where $\Gamma_{i}\in$
UEB($\mathcal{S},B(\operatorname{Range}(Q_{1}^{\perp})))$ are given by
$\Gamma_{i}(X)=\gamma_{i}(X)I_{\operatorname{Range}(Q_{1}^{\perp})}$,
$\gamma_{i}$ are linear extremal states defined on $\mathcal{S}$, $S_{i}\in
B(\operatorname{Range}(Q_{1}^{\perp}))$ satisfy
$\sum_{i=1}^{k}S_{i}^{*}S_{i}=I_{\operatorname{Range}(Q_{1}^{\perp})}.$ Since
$\Theta\in$ UEB($\mathcal{S},B(H))$ is a $C^{*}$-extreme map, it follows from
an argument similar to the one on page 770 in [FM93], that each direct summand
of $\Theta$ in equation (5.8) is also a $C^{*}$-extreme UEB map. In
particular, the UEB map
$Q_{1}^{\perp}\Psi_{0}(X)|_{\operatorname{Range}(Q_{1}^{\perp})}$ is a
$C^{*}$-extreme point of
UEB($\mathcal{S},B(\operatorname{Range}(Q_{1}^{\perp})))$. As before, there is
no loss of generality in assuming that $\|S_{1}\|=1$. We can now repeat the
arguments from before to the current set up and conclude (by taking
appropriate direct sums with the zero operator) that there exists a projection
$Q_{2}$ in $B(H)$ such that

$\Theta(X)=Q_{1}\Psi_{1}(X)Q_{1}+Q_{2}\Delta_{1}(X)Q_{2}+\text{an EB map},$

where $Q_{1}$ and $Q_{2}$ are mutually orthogonal projections in $B(H)$,
$\Delta_{1}(X)=\gamma_{1}(X)I_{H}$ and $\gamma_{1}$ is a linear extremal state
on $\mathcal{S}$. This process has to end after a finite number of steps, due
to our finite dimensionality assumptions. This along with the fact that
$\Theta$ is unitarily equivalent to $\Phi$ implies that
$\Phi(X)=\sum_{i=1}^{\ell}\phi_{i}(X)P_{i}$, for all $X\in\mathcal{S}$, where
the $\phi_{i}$’s are linear extremal states on $\mathcal{S}$ and the $P_{i}$’s
are mutually orthogonal projections in $B(H)$ satisfying
$\sum_{i=1}^{\ell}P_{i}=I_{H}$. An application of Theorem 1.10 implies that
$\Phi$ is a maximal UEB map and the proof is complete.

To prove the implication (ii) $\implies$ (iii), observe that since $\Phi$ is a
$C^{*}$-extreme UEB map, by the (proved) equivalence of statements (i) and
(ii) and by an application of Lemma 4.2, $\Phi$ has commutative range. That
$\Phi$ is a linear extreme UEB map follows from an easy and direct adaptation
of Theorem 2.2.2 in [Z] to UEB maps.

Finally, to prove the implication (iii) $\implies$ (i), observe that since
$\Phi$ is a UEB map with commutative range, Lemma 4.1 implies that

$\Phi(X)=\sum_{i=1}^{k}\phi_{i}(X)P_{i},$

for some $k\in{\mathbb{N}}$, where the $\phi_{i}$’s are distinct states on
$\mathcal{S}$ and the $P_{i}$’s are mutually orthogonal projections in $B(H)$
satisfying $\sum_{i=1}^{k}P_{i}=I_{H}$. By appealing to Theorem 1.10, it
suffices to show that $\phi_{i}$ is a linear extremal state for each $i$. Fix
$i\in\\{1,\dots,k\\}$. Let $\sigma$ and $\tau$ be states defined on
$\mathcal{S}$ such that

$\phi_{i}=t\sigma+(1-t)\tau,$

for some $t\in(0,1)$. It is enough to show that $\sigma=\tau$. Observe that

$\displaystyle\Phi(X)=t\left(\sigma(X)P_{i}+\sum_{j\neq
i}\phi_{j}(X)P_{j}\right)+(1-t)\left(\tau(X)P_{i}+\sum_{j\neq
i}\phi_{j}(X)P_{j}\right),$

for all $X\in\mathcal{S}$. Define the linear maps
$\Psi,\Gamma:\mathcal{S}\rightarrow B(H)$ by

$\Psi(X)=\sigma(X)P_{i}+\sum_{j\neq i}\phi_{j}(X)P_{j}\text{ and
}\Gamma(X)=\tau(X)P_{i}+\sum_{j\neq i}\phi_{j}(X)P_{j}.$

Observe that $\Psi,\Gamma\in$ UEB$(\mathcal{S},B(H))$ and
$\Phi(X)=t\Psi(X)+(1-t)\Gamma(X)$, for all $X\in\mathcal{S}$. From the
assumption that $\Phi$ is linear extreme, it follows that $\Psi=\Gamma$. This
in turn implies that $\sigma=\tau$ and the proof is complete. ∎

######  Remark 5.3.

The following remarks concern the above proof of Theorem 1.12.

  * (i)

The proof of the implication $(i)\Rightarrow(ii)$ in the above theorem works
equally well for UCP maps or even just unital positive maps.

  * (ii)

In the proof of the implication $(ii)\implies(i)$, it is assumed that $T_{1}$
is not invertible and hence the appearance of the zero block in the definition
of $Y_{1}$. If the operator $T_{1}$ is invertible, even then the proof works
just fine by letting $Y_{1}=Z_{1}$, $Y_{0}=Z_{0}$ and by deleting the last row
and column in the coefficient matrices occuring in equations (5.6) and (5.7).

###### Proof of Corollary 1.13.

Since $\Phi$ is a $C^{*}$-extreme point in UEB$(\mathcal{S},B(H))$, it follows
from Theorems 1.12 and 1.10 that $\Phi(X)=\sum_{i=1}^{\ell}\phi_{i}(X)P_{i},$
where the $\phi_{i}$’s are linear extremal states on $\mathcal{S}$ and the
$P_{i}$’s are orthogonal projections in $B(H)$ such that
$\sum_{i=1}^{\ell}P_{i}=I_{H}$. By [Z, Proposition 1.2.4] there exist linear
extremal states $\psi_{i}:B(E)\to{\mathbb{C}}$ such that
$\psi_{i}\big{|}_{\mathcal{S}}=\phi_{i}$, for each $i$. Define $\Psi:B(E)\to
B(H)$ by $\Psi(X)=\sum_{i=1}^{\ell}\psi_{i}(X)P_{i}.$ Observe that
$\Psi\big{|}_{\mathcal{S}}=\Phi$. It follows from Theorems 1.10 and 1.12 that
$\Psi$ is a $C^{*}$-extreme UEB extension of $\Phi$. ∎

The following example shows that the converse of Corollary 1.13 is not true in
general, i.e., the restriction of a $C^{*}$-extreme UEB map on an operator
system need not always be $C^{*}$-extreme. Recall that for states defined on
operator systems, $C^{*}$-extremality coincides with linear extremality. Let
$\mathcal{S}\subset M_{2}({\mathbb{C}})$ denote the operator system

$\mathcal{S}:=\left\\{\begin{pmatrix}a&b\\\ c&a\end{pmatrix}\in
M_{2}({\mathbb{C}}):a,b,c\in{\mathbb{C}}\right\\}.$

Define $\phi:\mathcal{S}\to{\mathbb{C}}$ by $\phi(X)=\operatorname{trace}(XE)$
for all $X\in\mathcal{S}$, where $E=\begin{pmatrix}1/2&0\\\
0&1/2\end{pmatrix}$. Then $\phi$ is a state on $\mathcal{S}$. In particular,
$\phi$ is a UEB map. Define the (distinct) states
$\phi_{1},\phi_{2}:\mathcal{S}\to{\mathbb{C}}$ by
$\phi_{i}(X)=\operatorname{trace}(XF_{i})$, for all $X\in\mathcal{S}$, i = 1,
2, where $F_{1}=\begin{pmatrix}1/2&1/2\\\ 1/2&1/2\end{pmatrix}$ and
$F_{2}=\begin{pmatrix}1/2&-1/2\\\ -1/2&1/2\end{pmatrix}$. It follows that

$\displaystyle\phi(X)$
$\displaystyle=\operatorname{trace}\left(X\begin{pmatrix}1/2&0\\\
0&1/2\end{pmatrix}\right)=\frac{1}{2}\operatorname{trace}(X(F_{1}+F_{2}))$
$\displaystyle=\frac{1}{2}\left(\phi_{1}(X)+\phi_{2}(X)\right).$

This in turn implies that $\phi$ is not a linear extremal state and hence not
a $C^{*}$-extreme state on $\mathcal{S}$. Let $\psi:M_{2}\to{\mathbb{C}}$ be
the state defined by $\psi(Y)=\operatorname{trace}(YE_{11})$, where
$E_{11}=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}$. Since $E_{11}$ is a
projection of rank one, it is well-known that $\psi$ has to be a linear
extremal state (See [St] and [Wa]) or equivalently a $C^{*}$-extremal state on
$M_{2}$. Finally, observe that $\psi|_{\mathcal{S}}=\phi$.

We end this section with a proof of Corollary 1.14, which is a Krein-Milman
type theorem for UEB maps.

###### Proof of Corollary 1.14.

Recall the notations $\mathcal{E}$ and $\mathcal{K}$ from Lemma 5.2. It
follows from Lemma 5.2 that $\mathcal{E}=$ UEB($\mathcal{S},B(H))$. By
Theorems 1.10 and 1.12, it follows that each element of $\mathcal{K}$ is a
$C^{*}$-extreme UEB map in UEB($\mathcal{S},B(H))$. This completes the proof.
∎

##  6\. $C^{*}$-extreme UEB maps between Matrix algebras

This section contains an improved version of [BDMS, Theorem 5.3], which
includes various characterizations of a $C^{*}$-extreme UEB map between matrix
algebras (See Theorem 6.2). Recall the convention 1.1 and the definitions of
the Choi-rank, EB rank, Schmidt rank and Schmidt number from Section 1. For a
positive matrix $X=\sum_{i=1}^{\ell}A_{i}\otimes B_{i}\in B(E\otimes H)$, the
partial trace of $X$ with respect to the first coordinate, is denoted by
$\operatorname{trace}_{1}(X)$ and is defined as

$\operatorname{trace}_{1}(X)=\displaystyle\sum_{i=1}^{\ell}\operatorname{trace}(A_{i})B_{i}\in
B(H).\\\ $

The following lemma is a variant of [HSR, Lemma 8].

######  Lemma 6.1.

Let $\Psi:M_{d}\rightarrow M_{n}$ be an EB map and
$C_{\Psi}=\sum_{j=1}^{m}\xi_{j}\xi_{j}^{*}$ with $SR(\xi_{j})=1$ for each
$j\in\\{1,2,\dots,n\\}$. If the Choi-rank of $\Psi=n$ and
$\operatorname{trace}_{1}(C_{\Psi})=\Psi(I_{d})\in M_{n}$ is invertible, then
$m\geq n$ and there exists
$\gamma_{1},\dots,\gamma_{n}\in{\mathbb{C}}^{d}\otimes{\mathbb{C}}^{n}$ with
$SR(\gamma_{k})=1$ such that
$C_{\Psi}=\sum_{k=1}^{n}\gamma_{k}\gamma_{k}^{*}$.

###### Proof.

Since the Choi-rank of $\Psi$ equals the rank of the Choi matrix $C_{\Psi}$
(See [Wa] and [St]), it is clear that $m\geq n$. Suppose that $m>n$ and
$C_{\Psi}=\sum_{j=1}^{m}\xi_{j}\xi_{j}^{*}$ where $SR(\xi_{j})=1$ for all
$1\leq j\leq m$. For each $1\leq j\leq m$, let $\xi_{j}=x_{j}\otimes y_{j}$
with $\|x_{j}\|=1$. By hypothesis,
$\Psi(I_{d})=\operatorname{trace}_{1}(C_{\Psi})=\sum_{j=1}^{m}\operatorname{trace}(x_{j}x_{j}^{*})y_{j}y_{j}^{*}=\sum_{j=1}^{m}y_{j}y_{j}^{*}$
is invertible. This implies that
Span$\\{y_{1},\dots,y_{m}\\}={\mathbb{C}}^{n}$. Without loss of generality,
assume that $\\{y_{1},\dots,y_{n}\\}$ is a basis for
${\mathbb{C}}^{n}=\operatorname{Range}(\operatorname{trace}_{1}(C_{\Psi}))$.
Note that $B=\\{\xi_{j}\,:\,1\leq j\leq n\\}$ is linearly independent. By the
Douglas range inclusion property ([D, Theorem 1]) and due to the hypothesis
that the rank of $C_{\Psi}=n$, it follows that $\\{\xi_{j}\,:\,1\leq j\leq
m\\}\subset\operatorname{Range}(C_{\Psi})$ and that $B$ is in fact a basis for
$\operatorname{Range}(C_{\Psi})$. Consider the sum
$\sum_{j=1}^{n+1}\xi_{j}\xi_{j}^{*}$. There exists $\alpha_{j}$ such that
$\xi_{n+1}=\sum_{j=1}^{n}\alpha_{j}\xi_{j}$. It follows that
$x_{j}=\lambda_{j}x_{n+1}$, for some scalars $\lambda_{j}$, whenever
$\alpha_{j}\neq 0$, $1\leq j\leq n$. Thus $|\lambda_{j}|=1$ (since the
$x_{j}$’s are unit vectors) and

$x_{n+1}\otimes y_{n+1}=\xi_{n+1}=\sum_{j:\alpha_{j}\neq
0}\alpha_{j}\xi_{j}=\sum_{j:\alpha_{j}\neq
0}x_{n+1}\otimes(\lambda_{j}\alpha_{j})y_{j}.$

Consequently, $y_{n+1}=\displaystyle\sum_{j:\alpha_{j}\neq
0}(\lambda_{j}\alpha_{j})y_{j},$ and

$\displaystyle\sum_{j=1}^{n+1}\xi_{j}\xi_{j}^{*}$
$\displaystyle=x_{n+1}x_{n+1}^{*}\otimes\left(\left(\sum_{j:\alpha_{j}\neq
0}y_{j}y_{j}^{*}\right)+y_{n+1}y_{n+1}^{*}\right)$
$\displaystyle+\sum_{j:\alpha_{j}=0}x_{j}x_{j}^{*}\otimes y_{j}y_{j}^{*}.$
(6.1)

Let $r$ denote the cardinality of the set $\\{j:\alpha_{j}=0\\}$. Since
$\\{y_{j}:\alpha_{j}\neq 0\\}$ is linearly independent and
$y_{n+1}\in\operatorname{span}\\{y_{j}:\alpha_{j}\neq 0\\}$,

$\sum_{j:\alpha_{j}\neq
0}y_{j}y_{j}^{*}+y_{n+1}y_{n+1}^{*}=\sum_{k=1}^{n-r}w_{k}w_{k}^{*},$

for some non-zero vectors $w_{k}\in\operatorname{span}\\{y_{j}:\alpha_{j}\neq
0\\}\subseteq{\mathbb{C}}^{n}.$ Thus,

$\sum_{j=1}^{n+1}\xi_{j}\xi_{j}^{*}=\sum_{k=1}^{n-r}\eta_{k}\eta_{k}^{*}+\sum_{j:\alpha_{j}=0}\xi_{j}\xi_{j}^{*},$

where $\eta_{k}=x_{n+1}\otimes w_{k}$. Note that the right hand side is a sum
consisting of exactly $n$ summands that are vectors in
${\mathbb{C}}^{d}\otimes{\mathbb{C}}^{n}$ with Schmidt rank equal to one.
Repeat the above process, by adding the $(n+2)^{th}$ term of $C_{\Psi}$ to
$\sum_{j=1}^{n+1}\xi_{j}\xi_{j}^{*}$. Continuing this way, after a finite
number of iterations, one can conclude that $C_{\Psi}$ is of the desired form,
i.e., $C_{\Psi}=\sum_{k=1}^{n}\gamma_{k}\gamma_{k}^{*}$ with
$\gamma_{k}\in{\mathbb{C}}^{d}\otimes{\mathbb{C}}^{n}$ and $SR(\gamma_{k})=1$.
This completes the proof. ∎

######  Theorem 6.2.

Let $\Phi:M_{d}\to M_{n}$ be a UEB map. The following statements are
equivalent.

  * (i)

$\Phi$ is a $C^{*}$-extreme UEB map.

  * (ii)

$\Phi$ is a linear extreme UEB map with commutative range.

  * (iii)

$\Phi$ is a maximal UEB map.

  * (iv)

The map $\Phi$ has the form $\Phi(X)=\sum_{i=1}^{\ell}\phi_{i}(X)P_{i},$ where
the $\phi_{i}$’s are distinct linear extremal states defined on $M_{d}$ and
the $P_{i}$’s are mutually orthogonal projections in $M_{n}$ satisfying
$\sum_{i=1}^{\ell}P_{i}=I_{n}$.

  * (v)

The Choi-rank of $\Phi$ is $n$.

  * (vi)

If $C_{\Phi}=\sum_{j=1}^{m}\xi_{j}\xi_{j}^{*}$ with $SR(\xi_{j})=1$ for each
$j\in\\{1,2,\dots,m\\}$, then $m\geq n$ and there exists
$\gamma_{1},\dots,\gamma_{n}\in{\mathbb{C}}^{d}\otimes{\mathbb{C}}^{n}$ with
$SR(\gamma_{j})=1$ such that
$C_{\Phi}=\sum_{j=1}^{n}\gamma_{j}\gamma_{j}^{*}$.

  * (vii)

If $\Phi(X)=\sum_{i=1}^{k}V_{i}^{*}XV_{i}$ and rank of $V_{i}$ is one for each
$i\in\\{1,2,\dots,k\\}$, then $k\geq n$ and there exists matrices
$W_{1},\dots,W_{n}$ of rank one such that
$\Phi(X)=\sum_{i=1}^{n}W_{i}^{*}XW_{i}$.

  * (viii)

The EB-rank of $\Phi$ is $n$.

###### Proof.

The equivalence of $(i),(ii)$ and $(iii)$ follows by letting
$\mathcal{S}=B(E)$ in Theorem 1.12

The equivalence $(iii)\iff(iv)$ follows by letting $\mathcal{S}=B(E)$ in
Theorem 1.10.

To prove $(iv)\implies(v),$ rewrite
$\Phi(X)=\sum_{k=1}^{n}\psi_{k}(X)w_{k}w_{k}^{*}$, where
$\psi_{k}\in\\{\phi_{i}\,:\,1\leq i\leq\ell\\}$ and $\\{w_{k}\,:\,1\leq k\leq
n\\}$ is an orthonormal set of vectors satisfying
$\sum_{k=1}^{n}w_{k}w_{k}^{*}=I_{n}$. For each $k\in\\{1,2,\dots,n\\}$,
Observe that $\psi_{k}(X)=\operatorname{trace}(Xu_{k}u_{k}^{*})=\langle
Xu_{k},u_{k}\rangle$ for some unit vector $u_{k}$. Recall the Choi matrix
$C_{\Phi}\in M_{d}\otimes M_{n}$. It follows that

$\displaystyle C_{\Phi}$
$\displaystyle=\sum_{i,j=1}^{d}e_{i}e_{j}^{*}\otimes\Phi(e_{i}e_{j}^{*})=\sum_{k=1}^{n}\sum_{i,j=1}^{d}e_{i}e_{j}^{*}\otimes(u_{k}^{*}e_{i}e_{j}^{*}u_{k})w_{k}w_{k}^{*}$
$\displaystyle=\sum_{k=1}^{n}\left(\sum_{i=1}^{d}(u_{k}^{*}e_{i})e_{i}\otimes
w_{k}\right)\left(\sum_{i=1}^{d}(e_{i}^{*}u_{k})e_{i}^{*}\otimes
w_{k}^{*}\right)=\sum_{k=1}^{n}z_{k}z_{k}^{*},$

where $z_{k}=\sum_{i=1}^{d}(u_{k}^{*}e_{i})e_{i}\otimes w_{k}$ . Since
$C_{\Phi}$ is a sum of rank-one operators, the rank of $C_{\Phi}$, i.e., the
Choi-rank of $\Phi$ is at most $n$. Also since $C_{\Phi}$ is a sum of the
positive rank-one operators $z_{k}z_{k}^{*}$, the Douglas range inclusion
property ([D, Theorem 1]) implies that $\\{z_{k}\,:\,1\leq k\leq n\\}$ is
contained in $\operatorname{Range}(C_{\Phi})$. By the orthonormality of
$\\{z_{k}\,:\,1\leq k\leq n\\}\subset\operatorname{Range}(C_{\Phi})$ it
follows that Choi-rank$(\phi)\geq n$. Thus in fact, the Choi-rank of $\Phi$
equals $n$.

Since $\Phi$ is unital, the implication $(v)\implies(vi)$ follows from the
observation that $\operatorname{trace}_{1}(C_{\Phi})=I_{n}$ and a direct
application of Lemma 6.1.

The implication $(vi)\implies(vii)$, is a direct consequence of the well-known
fact (See [St, Proposition 4.1.6] and [Bh, Theorem 3.1.1] for instance), that
every spectral decomposition of $C_{\Phi}$ as a sum of $\ell$ rank one
positive matrices $\xi\xi^{*}$ with $SR(\xi)=1$, yields a Choi-Kraus
decomposition of $\Phi$ with exactly $\ell$ Choi-Kraus coefficients, each
having rank one (and vice-versa).

The implication $(vii)\implies(viii)$ is immediate from the definition of the
EB-rank of $\Phi$.

The below proof of the implication $(viii)\implies(iii)$ is essentially the
same as that given in [BDMS]. We include it here for the sake of completeness.
Let $\Phi(X)=\sum_{j=1}^{n}(v_{j}u_{j}^{*})X(u_{j}v_{j}^{*})$, where
$\\{u_{1},\dots,u_{n}\\}\subset{\mathbb{C}}^{d}$ are unit vectors and
$\\{v_{1},\dots,v_{n}\\}\subset{\mathbb{C}}^{n}$. It suffices to show that
$\\{v_{1},\dots,v_{n}\\}$ forms an orthonormal basis for ${\mathbb{C}}^{n}$ or
equivalently, the matrix $W:=\left(v_{1}\,|\,v_{2}\,|\cdots|\,v_{n}\right)\in
M_{n}$ is unitary. This easily follows from the fact that
$I_{n}=\Phi(I_{d})=\sum_{j=1}^{n}v_{j}v_{j}^{*}=WW^{*}$. Let
$\\{u_{j_{k}}:1\leq k\leq\ell\\}$ be the distinct unit vectors among
$u_{1},\dots,u_{n}$. For $1\leq k\leq\ell$, define the states $\phi_{k}$ on
$M_{d}$ by $\phi_{k}(X)=\operatorname{trace}(Xu_{j_{k}})$. Rewrite $\Phi$ such
that

$\Phi(X)=\sum_{j=1}^{n}(v_{j}u_{j}^{*})X(u_{j}v_{j}^{*})=\sum_{k=1}^{\ell}\phi_{k}(X)P_{k}.$

It is easily seen that the $P_{k}$’s are mutually orthogonal projections
satisfying $\sum_{k=1}^{\ell}P_{k}=I_{n}$. This completes the proof. ∎

Acknowledgements: The authors thank Professor Scott McCullough (UF) for
several discussions and many helpful suggestions, Mr. Devendra Repana (IITM)
for a clarification/suggestion on Lemma 6.1 and also for pointing us to some
important references and Mr. Chinmay Ajay Tamhankar (IITM) for his
comments/suggestions on Theorem 1.10.

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