Update MPS manual in docs

docs/Project.toml

@@ -11,7 +11,7 @@ NetworkLayout = "46757867-2c16-5918-afeb-47bfcb05e46a"
 Tenet = "85d41934-b9cd-44e1-8730-56d86f15f3ec"
 
 [sources]
-Tenet = {path = "/Users/mofeing/Developer/Tenet.jl/docs/.."}
+Tenet = {path = ".."}
 
 [compat]
 Documenter = "1"

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 +19,76 @@ 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, maxdim=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
+
+An `MPS` representation is not unique: a single `MPS` can be represented in different canonical [`Form`](@ref). The choice of canonical form can affect the efficiency and stability of algorithms used to manipulate the `MPS`. You can check the canonical form of an `MPS` by calling the `form` function:
+
+```@example
+mps = MPS([rand(2, 2), rand(2, 2, 2), rand(2, 2)])
+
+form(mps)
+```
+
+Currently, `Tenet` supports the [`NonCanonical`](@ref), [`CanonicalForm`](@ref) and [`MixedCanonical`](@ref) forms.
+
+#### `NonCanonical` Form
+In the `NonCanonical` form, the tensors in the `MPS` do not satisfy any particular orthogonality conditions. This is the default `form` when an `MPS` is initialized without specifying a canonical form. It is useful for general purposes but may not be optimal for certain computations that benefit from orthogonality.
+
+#### `Canonical` Form
+Also known as Vidal's form, the `Canonical` form represents the `MPS` using a sequence of isometric tensors (`Γ`) and diagonal vectors (`λ`) containing the Schmidt coefficients. The `MPS` is expressed as:
+
+```math
+| \psi \rangle = \sum_{i_1, \dots, i_N} \Gamma_1^{i_1} \lambda_2 \Gamma_2^{i_2} \dots \lambda_{N-1} \Gamma_{N-1}^{i_{N-1}} \lambda_N \Gamma_N^{i_N} | i_1, \dots, i_N \rangle \, .
+```
+
+You can convert an `MPS` to the `Canonical` form by calling `canonize!`:
+
+```@example
+mps = MPS([rand(2, 2), rand(2, 2, 2), rand(2, 2)])
+canonize!(mps)
+
+form(mps)
+```
+
+#### `MixedCanonical` Form
+In the `MixedCanonical` form, tensors to the left of the orthogonality center are left-canonical, tensors to the right are right-canonical, and the tensors at the orthogonality center (which can be `Site` or `Vector{<:Site}`) contains the entanglement information between the left and right parts of the chain. The position of the orthogonality center is stored in the `orthog_center` field.
+
+You can convert an `MPS` to the `MixedCanonical` form and specify the orthogonality center using `mixed_canonize!`:
+
+```@example
+mps = MPS([rand(2, 2), rand(2, 2, 2), rand(2, 2)])
+mixed_canonize!(mps, Site(2))
+
+form(mps)
+```
+
+##### Additional Resources
+For more in-depth information on Matrix Product States and their canonical forms, you may refer to:
+- Schollwöck, U. (2011). The density-matrix renormalization group in the age of matrix product states. Annals of physics, 326(1), 96-192.
+
+
 ## 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(MPO, n=10, maxdim=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`).