From 5d81735fc88ff4e6d619c8f1b53692bd12f1d87b Mon Sep 17 00:00:00 2001
From: jofrevalles <jofrevalles99@gmail.com>
Date: Mon, 25 Nov 2024 12:51:24 +0100
Subject: [PATCH] Update MPS manual

---
 docs/src/manual/ansatz/mps.md | 67 ++++++++++++++++++++++++++---------
 1 file changed, 50 insertions(+), 17 deletions(-)

diff --git a/docs/src/manual/ansatz/mps.md b/docs/src/manual/ansatz/mps.md
index 91190139..a936dd38 100644
--- a/docs/src/manual/ansatz/mps.md
+++ b/docs/src/manual/ansatz/mps.md
@@ -1,8 +1,9 @@
 # Matrix Product States (MPS)
 
-Matrix Product States (MPS) are a Quantum Tensor Network ansatz whose tensors are laid out in a 1D chain.
-Due to this, these networks are also known as _Tensor Trains_ in other mathematical fields.
-Depending on the boundary conditions, the chains can be open or closed (i.e. periodic boundary conditions).
+Matrix Product States ([`MPS`](@ref)) are a Quantum Tensor Network ansatz whose tensors are laid out in a 1D chain.
+Due to this, these networks are also known as _Tensor Trains_ in other scientific fields.
+Depending on the boundary conditions, the chains can be open or closed (i.e. periodic boundary conditions), currently
+only `Open` boundary conditions are supported in `Tenet`.
 
 ```@setup viz
 using Makie
@@ -18,36 +19,68 @@ using NetworkLayout
 ```@example viz
 fig = Figure() # hide
 
-tn_open = rand(MatrixProduct{State,Open}, n=10, χ=4) # hide
-tn_periodic = rand(MatrixProduct{State,Periodic}, n=10, χ=4) # hide
+open_mps = rand(MPS, n=10, χ=4) # hide
 
-plot!(fig[1,1], tn_open, layout=Spring(iterations=1000, C=0.5, seed=100)) # hide
-plot!(fig[1,2], tn_periodic, layout=Spring(iterations=1000, C=0.5, seed=100)) # hide
+plot!(fig[1,1], open_mps, layout=Spring(iterations=1000, C=0.5, seed=100)) # hide
 
 Label(fig[1,1, Bottom()], "Open") # hide
-Label(fig[1,2, Bottom()], "Periodic") # hide
 
 fig # hide
 ```
 
+### Canonical Forms
+
+A Matrix Product State ([`MPS`](@ref)) representation is not unique. Instead, a single `MPS` can be represented in different canonical forms. We can check the canonical form of an `MPS` by calling the [`form`](@ref) function.
+
+```@example
+mps = MPS([rand(2, 2), rand(2, 2, 2), rand(2, 2)])
+
+form(mps)
+```
+
+Each canonical form can be useful in different situations, and the choice of the canonical form can affect the efficiency of the algorithms used to manipulate the `MPS`. Currently, `Tenet` supports the [`NonCanonical`](@ref), [`CanonicalForm`](@ref) and [`MixedCanonical`](@ref) forms.
+
+#### `NonCanonical` Form
+The default form of an `MPS` when we do not specify a canonical form.
+
+#### `CanonicalForm`
+Also known as Vidal's form. This form stores each `Tensor` of the `MPS` as a sequence of $\Gamma$ unitary tensors and $\lambda$ vectors:
+
+```math
+| \psi \rangle = \sum_{i_1, \dots, i_N} \Gamma_1^{i_1} \lambda_2^{i_2} \Gamma_2^{i_2} \dots \lambda_{N-1}^{i_{N-1}} \Gamma_{N-1}^{i_{N-1}} \lambda_N^{i_N} \Gamma_N^{i_N} | i_1, \dots, i_N \rangle \, .
+```
+This form can be obtained by calling [`canonize!`](@ref) on an `MPS`:
+
+```@example
+mps = MPS([rand(2, 2), rand(2, 2, 2), rand(2, 2)])
+canonize!(mps)
+
+form(mps)
+```
+
+#### `MixedCanonical` Form
+This form stores the `Tensor`s in an `MPS` as left or right canonical wether the `Tensor` is on the left or right of the ortogonality center, which is stored in the field `orthog_center` of the `MixedCanonical` form.
+
+```@example
+mps = MPS([rand(2, 2), rand(2, 2, 2), rand(2, 2)])
+mixed_canonize!(mps, Site(2))
+
+form(mps)
+```
+
 ## Matrix Product Operators (MPO)
 
-Matrix Product Operators (MPO) are the operator version of [Matrix Product State (MPS)](#matrix-product-states-mps).
-The major difference between them is that MPOs have 2 indices per site (1 input and 1 output) while MPSs only have 1 index per site (i.e. an output).
+Matrix Product Operators ([`MPO`](@ref)) are the operator version of [Matrix Product State (MPS)](#matrix-product-states-mps).
+The major difference between them is that MPOs have 2 indices per site (1 input and 1 output) while MPSs only have 1 index per site (i.e. an output). Currently, only `Open` boundary conditions are supported in `Tenet`.
 
 ```@example viz
 fig = Figure() # hide
 
-tn_open = rand(MatrixProduct{Operator,Open}, n=10, χ=4) # hide
-tn_periodic = rand(MatrixProduct{Operator,Periodic}, n=10, χ=4) # hide
+open_mpo = rand(MatrixProduct{Operator,Open}, n=10, χ=4) # hide
 
-plot!(fig[1,1], tn_open, layout=Spring(iterations=1000, C=0.5, seed=100)) # hide
-plot!(fig[1,2], tn_periodic, layout=Spring(iterations=1000, C=0.5, seed=100)) # hide
+plot!(fig[1,1], open_mpo, layout=Spring(iterations=1000, C=0.5, seed=100)) # hide
 
 Label(fig[1,1, Bottom()], "Open") # hide
-Label(fig[1,2, Bottom()], "Periodic") # hide
 
 fig # hide
 ```
-
-In `Tenet`, the generic `MatrixProduct` ansatz implements this topology. Type variables are used to address their functionality (`State` or `Operator`) and their boundary conditions (`Open` or `Periodic`).