Pure matrix states on block Toeplitz matrices

Abstract

Let ${\mathcal{T}}_{n,m}={\mathcal{T}}_n(M_m(\mathbb{C}))$ denote the operator system of all block Toeplitz matrices $T=((T_{i-j}){)}_{i,j=1}^n$ with entries $T_k\in M_m(\mathbb{C})$ % T_{k} = \left[ t^{(k)}_{p-q} \right]_{p,q=1}^m. \[ T = \begin{pmatrix} T_0 & T_{-1} & \cdots & T_{-(n-1)} T_1 & T_0 & \cdots & T_{-(n-2)} \vdots & \vdots & \ddots & \vdots T_{n-1} & T_{n-2}& \cdots & T_0 \end{pmatrix} \in M_{mn}(\mathbb{C}). \] We characterize all pure unital completely positive ({\it{ucp}}) maps from ${\mathcal{T}}_{n,m}$ to $M_m(\mathbb{C})$. Working through the Stinespring isometry $V=(V_1,\dots ,V_n{)}^t:{\mathbb{C}}^m\to {\mathbb{C}}^{mn}$ and the matrix-valued polynomial $Q_V(z)={\sum }_{i=1}^n{z}^{n-i}{V}_i$, we prove that $\phi$ is pure if and only if it admits a unique pure \ucp extension to $M_{mn}(\mathbb{C})$ if and only if $Q_V$ has degree $n-1$ with all its roots on the unit circle $\T$. Every such pure $\phi$ induces a \ucp map ${\Phi }_{Q_V}$ on $C(\mathbb{T},M_m(\mathbb{C}))$ given by \[ Φ_{Q_V}(f) = \int_{\mathbb{T}} Q_V(z)^* f(z) Q_V(z)\, dz. \] Let ${\mathcal{Y}}_m$ be the compact convex set of all \ucp maps from $C(\mathbb{T},M_m(\mathbb{C}))$ to $M_m(\mathbb{C})$. Endowing ${\mathcal{Y}}_m$ with the matricial Monge-Kantorovich metric $\rho$, via an analysis of the extreme points of ${\mathcal{Y}}_m$ together with a point-splitting lemma, we show that the induced maps as above are $\rho$-dense in ${\mathcal{Y}}_m$. Consequently, if $\Bmn$ denotes the set of normalized ${\Phi }_{Q_V}$ where $Q_V$ is as above, then the Hausdorff distance $d_H(\Bmn, \mathcal{Y}_m) \to 0$ as $n\to \infty$, extending known results of approximation of positive regular Borel measures on the unit circle to the setting of matrix-valued completely positive maps.

20 pages