# Completely Positive Maps and their dual in Separable Hilbert Space

Consider a separable Hilbert space $\mathcal{H}$, we can first define $T(\mathcal{H})$ the trace class of $\mathcal{H}$, then $D(\mathcal{H})$ to denote the set of positive operator with trace less than 1.

The set of positive maps consists of all elements that maps positive operator to positive operator. The set of completely positive maps contains all elements $a$ that $id_k\otimes a$ is still positive for any integer $k$.

We are interested in $C$, a subset of the completely positive maps contains all $c$ such that for all $b\in D(\mathcal{H})$, $$\mathrm{tr}[c(b)]\leq \mathrm{tr}[b].$$

In finite dimensional case, we always have the so-called Kraus decomposition that any $c$ can be written as $$c(b)=\sum E_i bE_i^{*},$$ with $E_i^{*}E_i\leq I$.

Moreover, we have $c^*$, the dual map of $c$, satisfied $$c^*(b)=\sum E_i^{*} bE_i.$$

The dual map is defined as $$\langle d,c(b)\rangle = \langle c^*(d),b \rangle$$ for all $b,d$.

In the separable Hilbert space, do we have similar version of Kruas decomposition, maybe with infinite sum? Do we have similar version of dual map?
