Canonical form MPS (CanonicalMPS)

There is a particular form of matrix-product states where most of the tensors are isometries that, when contracted from the left or the right, they give the identity. In SeeMPS we call those states being in “canonical form” and have a special class CanonicalMPS for them.

Assume we have a state \(\psi\) in canonical form with respect to the n-th site. This state can be written as

\[|\psi\rangle = A_{\alpha_n,i_n,\alpha_{n+1}}\ket{\alpha_n}\ket{i_n}\ket{\alpha_{n+1}}\]

where the quantum states \(\ket{\alpha_{n}}\) and \(\ket{\alpha_{n+1}}\) enclose all subsystems to the left (j < i) or to the right (j > i). Because the state is in canonical form, these “environment” states satisfy orthogonalitiy relations

\[\braket{\alpha_{n}'|\alpha_{n}} = \delta_{\alpha_{n}'\alpha_{n}}, \mbox{ and } \braket{\alpha_{n+1}'|\alpha_{n+1}} = \delta_{\alpha_{n+1}'\alpha_{n+1}}\]

This simplifies many tasks, such as the computation of expectation values:

\[\braket{O_n} = \sum_{i,j,\alpha,\beta} (O_n)_{ji} A^*_{\alpha j \beta} A_{\alpha i \beta}\]

Creation

Many of the algorithms described in other parts of this documentation (e.g., the algorithms section) produce MPS in canonical form. Alternatively, we can create a CanonicalMPS object from a MPS by a process known as orthogonalization, specifying both the center for the canonical form, as well as the truncation errors and bond dimensions that we will allow.

CanonicalMPS(data[, center, normalize, ...])	Canonical MPS class.
from_vector(ψ, dimensions[, strategy, ...])	Create an MPS in canonical form starting from a state vector.