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 $V = (V_1, \ldots, V_n)^t \colon \mathbb{C}^m \to \mathbb{C}^{mn}$ and the matrix-valued polynomial $Q_V(z) = \sum_{i=1}^n z^{n-i} V_i$, we prove that $\varphi$ 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 $\varphi$ induces a \ucp map $\Phi_{Q_V}$ on $C(\mathbb{T}, M_m(\mathbb{C}))$ given by \[ \Phi_{Q_V}(f) = \int_{\mathbb{T}} Q_V(z)^* f(z) Q_V(z)\, dz. \] 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.