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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.

Cutting an open-boundary matrix product state across one virtual bond of dimension $\chi$ gives Schmidt rank at most $\chi$, so its entanglement entropy obeysA periodic interval cuts two bonds and obeys $S(A)\le 2\log \chi$. Since the boundary of a one-dimensional interval has a constant number of points, this is an area law.

For a projected entangled pair state, a bipartition crossing $n_{\partial }$ virtual bonds has Schmidt rank at most ${\chi }^{n_{\partial }}$. The tensor-network area-law bound is thereforeBecause $n_{\partial }$ is proportional to the lattice boundary area, every fixed-bond-dimension PEPS satisfies an area law.

The Stinespring dilation isometry isApplying it successively to fresh environment systems givesContracting the final system with a boundary vector turns every amplitude into a boundary contraction of the matrices $K_{\alpha }$. This is a matrix product state with bond dimension at most the system dimension.

For an injective MPS, the fundamental gauge freedom istogether with the inverse transformation of boundary vectors; an overall phase is also immaterial. Its channel converges to a unique fixed state exactly when eigenvalue one is simple and there are no other peripheral eigenvalues, equivalently when it is a primitive quantum channel. A unique fixed state without convergence only requires the eigenvalue-one eigenspace itself to be one-dimensional.

Cutting an open-boundary matrix product state crosses one virtual bond of dimension $\chi$. Its Schmidt rank is at most $\chi$, soA periodic interval crosses two bonds and obeys $S(A)\le 2\log \chi$. The one-dimensional boundary has a constant number of points, so this is an area law.

For a projected entangled pair state, cut every virtual bond crossing the boundary of a region $A$. If $n_{\partial }$ bonds are cut, the Schmidt rank is at most ${\chi }^{n_{\partial }}$, and thereforeSince $n_{\partial }$ is proportional to the lattice boundary area, every fixed-bond-dimension PEPS satisfies an area law.