Let $M\in\mathrm{Lin}(\mathcal Y\otimes\mathcal X)$ be some linear operator whose input and output spaces are both $\mathcal Y\otimes\mathcal X$, for some pair of finite-dimensional Hilbert spaces $\mathcal X,\mathcal Y$. Moreover, suppose $M$ is positive semidefinite: $M\ge0$. It being positive semidefinite implies it admits a decomposition of the form $M=\... 5 The need for quantum fields There are two seemingly unrelated conceptual steps between quantum mechanics (QM) and quantum field theory (QFT). One step reconciles QM with special relativity (SR) and the other replaces a fixed finite number of particles with fields that have infinite degrees of freedom. These two steps appear unrelated, but in order to arrive ... 5 Choi operator of a linear map$\mathcal{E}$is defined as $$J(\mathcal{E}) = \sum_{ij} \mathcal{E}(|i\rangle\langle j|)\otimes |i\rangle\langle j|.\tag1$$ Substituting$\mathcal{E}(\rho)=\sum_k E_k\rho E_k^\dagger$into$(1)$, we have$\begin{align} J(\mathcal{E}) &= \sum_{ijk} \left(E_k|i\rangle\langle j| E_k^\dagger\right)\otimes |i\rangle\langle j|... 8 Every quantum channel has many Kraus representations that may differ in the number of Kraus operators. For example, for any positive integern$and numbers$p_i$with$i=1,\dots,n$and$\sum_{i=1}^np_i=1$the matrices$E_i=\sqrt{p_i}I$form a valid, if impractical, Kraus representation of the identity channel with$n\$ Kraus operators. This example also ...