Canonical form

The sequential Schmidt decomposition creates a particular type of matrix-product state, called a “canonical form”. In the example cited above, the final MPS is in canonical form with respect to the first site. This means that we can write

\[|\psi\rangle = A^{(1)}_{1i_1\alpha_1}|i_1\rangle|\alpha_1\rangle\]

Here, \(i_1\) labels the physical state of the first subsystem and \(\alpha_1\) labels the possible states of the collective of remaining (N-1) quantum subsystems:

\[|\alpha_1\rangle = \sum_{\alpha,i} A^{(2)}_{\alpha_2i_2\alpha_3}\cdot A^{(N)}_{\alpha_Ni_N1}|\alpha_1,\ldots\alpha_N\rangle\]

The use of the Schmidt decomposition guarantees that the collective states of the remaining systems—the so called “environment”—forms an orthonormal basis.

\[\langle \alpha_N|\alpha_N'\rangle = \delta_{\alpha_N\alpha_N'}\]

This is extremely convenient for many types of calculations. For instance, to compute the expected value of an observable \(O\) acting on the first site, we can ignore the environment and use

\[\langle O\rangle = \sum_{i,j,\alpha_1} O_{ij} A^{(1) *}_{1i\alpha} A^{(1) *}_{1j\alpha}\]

Interestingly, the canonical form can be constructed with respect to any site, not just the first or the last one. In SeeMPS, the CanonicalMPS class is responsible for creating and maintaining these canonical forms, as explained in this manual. It is also possible to move the center of a canonical form using the recenter() method.

Consider the following example. We initially create a random matrix-product state with arbitrary tensors and transform it into canonical form with respect to the first site. This state is then destructively transformed into canonical form with respect to the last site. In both cases we can use a single tensor from the state to compute the norm, because the other tensors create an orthogonormal environment basis:

>>> import numpy as np
>>> from seemps import random_uniform_mps, CanonicalMPS
>>> state = CanonicalMPS(random_uniform_mps(2, 10), center=0)
>>> print(f"State norm: {np.linalg.norm(state[0])}")
>>> state.recenter(-1)
>>> print(f"State norm: {np.linalg.norm(state[-1])}")