41.9 Completely Positive Maps into $M_{n}(\complex)$
Lemma 41.9.1.label Let $A$ be a unital $C^{*}$-algebra, $S \subset A$ be an operator system, and $\phi \in L(S; M_{n}(\complex))$. Define
then:
- (1)
Let $\xi = \sum_{j = 1}^{n} e_{j} \otimes e_{j} \in \complex^{n} \otimes \complex^{n}$, then
\[\dpn{a, \tilde \phi}{M_n(S)}= \dpn{\phi_n(a)\xi, \xi}{\complex^n \otimes \complex^n}\]for all $a \in M_{n}(S)$.
- (2)
The mapping $\phi \mapsto \tilde \phi$ is a linear isomorphism between $M_{n}(S)^{*}$ and $L(S; M_{n}(\complex))$.
Proof. (1): For any $a \in A$ and $1 \le k, l \le n$,
(2): Let $\phi_{0}: M_{n}(S) \to \complex$ be a linear map, and let
then for any $1 \le k, l \le n$ and $a \in A$,
Therefore the mapping $\phi \to \tilde \phi$ is an isomorphism.$\square$
Theorem 41.9.2.label Let $S$ be an operator system, $\phi \in L(S; M_{n}(\complex))$, and
then the following are equivalent:
- (1)
$\phi$ is completely positive.
- (2)
$\phi$ is $n$-positive.
- (3)
$\tilde \phi \in M_{n}(S)^{*}$ is a positive linear functional.
Proof. See Proposition 41.11.6.$\square$
Theorem 41.9.3 (Krein’s Extension Theorem).label Let $A$ be a unital $C^{*}$-algebra, $S \subset A$ be an operator system, and $\phi: S \to M_{n}(\complex)$ be completely positive, then there exists a completely positive map $\Phi: A \to M_{n}(\complex)$ such that $\Phi|_{S} = \phi$.
Proof. Let
then by Theorem 41.9.2, $\tilde \phi$ is a positive linear functional. As $S$ is an operator system, $\normn{\tilde \phi}_{S^*}= \tilde \phi(1_{A})$. By the Hahn-Banach Theorem and Lemma 41.9.1, there exists $\Phi \in L(A; M_{n}(\complex))$ such that $\tilde \Phi|_{M_n(S)}= \tilde \phi$ and $\Phi|_{S} = \phi$. By Theorem 39.9.2, $\tilde \Phi$ is positive, so $\Phi$ is completely positive by Theorem 41.9.2.$\square$
Definition 41.9.4 (Non-Entangled Cone).label Let $A$ be a unital $C^{*}$-algebra, $S \subset A$ be an operator system, then
is the non-entangled positive cone of $M_{n}(S)$.
Lemma 41.9.5.label Let $A$ be a unital $C^{*}$-algebra, $S \subset A$ be an operator system, $\phi \in L(S; M_{n}(\complex))$, and
then $\phi$ is positive if and only if $\tilde \phi|_{M_n(\complex)_+ \otimes S_+}\ge 0$.
Proof. ($\Rightarrow$): Let $a \in M_{n}(\complex)_{+}$ and $b \in S_{+}$. Using the Spectral Theorem, assume without loss of generality that $a$ is a rank one projection. That is, there exists $\alpha = (\alpha_{1}, \cdots, \alpha_{n}) \subset \complex^{n}$ such that $a = \sum_{i, j = 1}^{n} \ol \alpha_{i} \alpha_{j} E_{ij}$. As such,
($\Leftarrow$): Let $b \in S_{+}$, $\alpha = (\alpha_{1}, \cdots, \alpha_{n}) \in \complex^{n}$, and $a = \alpha\alpha^{*}$, then
$\square$
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