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For an MPS tensor $A^i$, the transfer map is the completely positive map $\mathcal{E}(X)={\sum }_i{A}^{i}X(A^i{)}^{†}$. Its fixed points control normalization and local expectation values, while its subleading eigenvalues control correlation lengths.

The map $\Lambda$ is a completely positive map when $\Lambda \otimes {\mathrm{id}}_m$ maps positive operators to positive operators for every $m$. In a basis $\{|i⟩\}$ define the unnormalized Choi matrixIf $\Lambda$ is completely positive, $J(\Lambda )=(\mathrm{id}\otimes \Lambda )(|\Omega ⟩⟨\Omega |)\ge 0$, where $|\Omega ⟩={\sum }_i|ii⟩$. Conversely, decompose $J={\sum }_k|a_k⟩⟨a_k|$ and reshape each vector into an operator $A_k$. The Choi reconstruction formula gives $\Lambda (X)={\sum }_k{A}_{k}X{A}_k^{†}$, which is completely positive. Thus

Let $\Phi :M_d(\mathbb{C})\to M_D(\mathbb{C})$ be a linear map. It is positive when $X\ge 0$ implies $\Phi (X)\ge 0$, and completely positive whenis positive for every ancillary dimension $r$. In finite dimensions it is enough to check $r=d$.

A finite family of Kraus operators $A^{\alpha }:{\mathbb{C}}^d\to {\mathbb{C}}^D$ defines the Kraus representationThis map is completely positive because, for every positive semidefinite matrix $Y$ on the enlarged space,

Conversely, use the unnormalized maximally entangled vector $|\Omega ⟩={\sum }_{j=1}^d|j⟩|j⟩$. Complete positivity makes the Choi matrixpositive semidefinite. By the spectral theorem for normal operators, $C_{\Phi }={\sum }_{\alpha =1}^R|v_{\alpha }⟩⟨v_{\alpha }|$, where $R=\mathrm{rank}{C}_{\Phi }\le dD$. Reshape each $v_{\alpha }\in {\mathbb{C}}^D\otimes {\mathbb{C}}^d$ into a matrix $A^{\alpha }$ by $|v_{\alpha }⟩={\sum }_{\mu j}{A}_{\mu j}^{\alpha }|\mu ⟩|j⟩$. The Choi matrix inversion formulathen gives $\Phi (X)={\sum }_{\alpha }{A}^{\alpha }X(A^{\alpha }{)}^{†}$. Thus finite Kraus representations characterize finite-dimensional completely positive maps.

The additional normalizationmakes $\Phi$ trace preserving and hence a quantum channel; ${\sum }_{\alpha }{A}^{\alpha }(A^{\alpha }{)}^{†}=I_D$ instead makes it unital.

An open-boundary matrix product state with physical dimension $q$ and bond dimension $\chi$ iswhere the $A^i$ are $\chi \times \chi$ matrices and $|\ell ⟩,|r⟩$ are boundary vectors. For periodic boundary conditions, replace the boundary contraction by the matrix trace $\mathrm{Tr}(A^{i_1}\cdots A^{i_N})$.

The same matrices define the matrix product state transfer mapWhen ${\sum }_i(A^i{)}^{†}{A}^i=I$, the Stinespring dilationis an isometry. Repeatedly applying $V$ stores each Kraus label in a fresh physical register:Contracting the remaining virtual system with $⟨\ell |$ gives an MPS, while tracing over all recorded labels gives repeated application of the completely positive map $\mathcal{E}$. The MPS is therefore a coherent unravelling of the channel. Equivalently, retaining the Kraus-label registers realizes a purification of its output. A non-normalized MPS tensor gives the same construction with a general completely positive map; an appropriate canonical gauge normalizes the transfer map on its support.

A positive linear map sends every positive semidefinite matrix to a positive semidefinite matrix. Positivity need not persist after tensoring with the identity map on an ancillary system; requiring that stronger property gives a completely positive map.