Tensor network state

A tensor network state contracts auxiliary indices of local tensors to encode a many-body wavefunction.

A matrix product state represents one-dimensional amplitudes as products of finite matrices. Its bond dimension bounds the Schmidt rank across a cut.

Matrices $A^i$ define both the completely positive map $\mathcal{E}(X)={\sum }_i{A}^{i}X(A^i{)}^{†}$ and an MPS. Repeatedly retaining the Kraus label $i$ purifies the channel output into the physical MPS chain; tracing those labels recovers repeated application of $\mathcal{E}$.

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.

An MPS tensor is injective after blocking when products of its physical matrices span the full virtual matrix algebra.

Blocking $L$ neighboring sites replaces their physical labels by one composite label and their tensor by the products $A^{i_1}\cdots A^{i_L}$. An MPS is injective when such products span the full virtual matrix algebra for some finite $L$.

A uniform matrix product state uses the same local tensor $A^i$ at every site. For a periodic chain its amplitudes are $\mathrm{Tr}(A^{i_1}\cdots A^{i_N})$.

Two injective uniform MPS tensors generate the same states on every sufficiently long periodic chain exactly when they differ by a virtual similarity transform and an overall phase: $A^i=e^{i\theta }X{B}^i{X}^{-1}$.

The virtual similarity transformation $A^i\mapsto X{A}^i{X}^{-1}$ leaves every periodic MPS amplitude invariant by cyclicity of the matrix trace. The fundamental theorem of matrix product states says that, up to an overall phase, this is the only ambiguity for injective uniform tensors.

The Affleck--Kennedy--Lieb--Tasaki state is an injective spin-one MPS whose virtual degrees of freedom are spin one-half. Its ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry acts linearly on physical spins and projectively on the virtual spins.

In a Cartesian basis of the physical spin-one space, the AKLT MPS matrices are proportional to ${\sigma }_x,{\sigma }_y,{\sigma }_z$. Products of two such matrices span the full virtual matrix algebra, and the two-site parent term projects onto total spin two.

Because ${\sigma }_i{\sigma }_j={\delta }_{ij}I+i{\epsilon }_{ijk}{\sigma }_k$, the two-site image of the Pauli-matrix AKLT tensor is the direct sum of the scalar and antisymmetric subspaces, which have total spin zero and one. Its orthogonal complement is the symmetric traceless total-spin-two subspace.

The AKLT parent Hamiltonian is $H={\sum }_j{P}_{j,j+1}^{(2)}$, where $P^{(2)}$ projects two neighboring spin-one particles onto total spin two. It is frustration free, and the periodic AKLT MPS is its unique ground state on a sufficiently long chain.

After suitable blocking, the AKLT and one-dimensional cluster states realize the same nontrivial ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry-protected topological phase. Their virtual symmetries carry the same nontrivial projective class and produce protected edge degrees of freedom.

The entanglement spectrum is the spectrum of $-\log {\rho }_A$, equivalently the negative logarithms of squared Schmidt coefficients. A nontrivial projective virtual symmetry can enforce degeneracies throughout this spectrum.

An MPS parent Hamiltonian is a sum of local projectors onto the orthogonal complement of the tensor's local support. The MPS is a frustration-free ground state.

A matrix product operator contracts a one-dimensional chain of tensors with physical input and output indices.

A projected entangled pair state generalizes an MPS to higher-dimensional lattices. Cutting a region intersects one virtual bond per boundary edge.

A projected entangled pair operator is a higher-dimensional tensor-network operator with physical input and output legs.