Finite interval entanglement entropy

[Chenqitrg]

Hi developers,

I am wondering whether there exists a function to compute entanglement entropy of a finite MPS of an interval from the rest sites? For example, suppose the total length of the chain is 300, and need to compute the entanglement entropy of interval 50-150 with the rest of the lattice [1,49] cup [151, 300]?

[Chenqitrg]

And whether there is a function to transform a finite MPS to a central canonical form at a given center site?

[lkdvos]

The entanglement entropy for intervals is not a feature we currently have, although if you can come up with a nice interface for that I'm very happy to look into adding this (although typically this is a bit annoying to deal with efficiently if you want to consider many different regions.

For transforming the FiniteMPS structure, this is actually baked into the object, if you ask for mps.AL[i], mps.AR[i] or mps.AC[i] it will give you the left, right and center-gauged tensor for a given site automatically, while keeping track of minimal recomputations in case you start moving it around.

[Chenqitrg]

Thank you! I finally know the convention by calculating

ψ = FiniteMPS(10, ℂ^2, ℂ^2)
T = ψ.AL[5] * ψ.C[5]
L, Q = right_orth(transpose(T, ((1,), (3, 2))); positive = true)
@assert norm(transpose(Q, ((1, 3), (2,))) - ψ.AR[5]) ≈ 0

[lkdvos]

Indeed, this is also in the docstring of FiniteMPS:

MPSKit.jl/src/states/finitemps.jl

Lines 20 to 23 in 4712074

	By convention, we have that:
	- `AL[i] * C[i]` = `AC[i]` = `C[i-1] * AR[i]`
	- `AL[i]' * AL[i] = 1`
	- `AR[i] * AR[i]' = 1`

[Chenqitrg]

I see. Thank you. I tried the other way does not work, and finally realized because C is lower triangular and is a result of sweeping from the right hand side.

Below is what I can write for computing entanglement entropy on an interval:

function entropy(mps::FiniteMPS, interval::UnitRange{Int}) # Calculate the entanglement entropy for a region of the MPS
    @assert 1 ≤ interval.start < interval.stop ≤ length(mps) "Invalid interval: start and stop must be within the bounds of the MPS length and start must be less than stop."
    𝔼 = mps.C[interval[1] - 1] ⊗  transpose(mps.C[interval[1] - 1]') # Initialize the transfer matrix

    for i in interval
        @tensor 𝔼_new[-1 -2; -3 -4] := 𝔼[-1 -2; -5 -6] * mps.AR[i][-5 -7; -3] * mps.AR[i]'[-4; -6 -7]
        𝔼 = 𝔼_new
    end

    Λ = eigh_vals(transpose(𝔼, ((2, 4), (1, 3)))) # Compute the eigenvalue spectrum
    Λ = Λ[Λ .> 1e-12] # Filter out small eigenvalues to avoid numerical issues
    return -sum(Λ .* log.(Λ)) # Calculate the entanglement entropy using the eigenvalue spectrum of the transfer matrix
end

I have no idea of the case when there are multiple disjoint intervals are computed.