Operator entanglement of an MPO

ITensor Julia Questions
1

Hi,

I would like to compute the operator entanglement of an MPO, defined as S_{op} = - \sum_{\alpha} \lambda_\alpha^2 \log_2{\lambda_\alpha^2} where \lambda_\alpha are the Schmidt values across a cut of the system (see this paper). This is the same definition as the entanglement entropy for MPS states but extended to MPOs. Unlike the MPS case, the operator entanglement is not a genuine measure of entanglement. For example, the operator entanglement across bond b (with N_A on the left and N_B sites on the right) of the maximally mixed state \rho_{mm} = \frac{1}{2^N} \mathbb{I}^{\otimes N} is

S_{op}( \rho_{mm}, b ) = N_A \log_2(2)

since the reduced density matrix on \rho_{N_A} is

\rho_{N_A} = \frac{1}{2^N_A} \mathbb{I}^{\otimes N_A} = \sum_{\alpha_A} \frac{1}{2^{N_A}} | \alpha_A \rangle \langle \alpha_A | \\

where | \alpha_A \rangle is an orthonormal basis, making the schmidt values \lambda_\alpha ^2 = \frac{1}{2^{N_A}} the Schmidt values.

I thought I could compute the operator entanglement the same was as computing the von Neumann entanglement for the MPS, but I’m not getting the expected result for the maximally mixed state. Do you know what I’m doing wrong?

using ITensors, ITensorMPS

function operator_entanglement(state::Union{MPS, MPO}, bond::Int)
    state = orthogonalize(state, bond)

    U,S,V = svd(state[bond], (linkinds(state, bond-1)..., siteinds(state, bond)...))
    SvN = 0.0
    for n=1:dim(S, 1)
        p = S[n,n]^2
        SvN -= p * log2(p)
    end
    return SvN
end

let 
    N = 4
    s = siteinds("Qubit", N)

    # Operator entanglement of the maximally mixed state
    max_mixed = MPO(s, "Id")/2^N
    Na = 3
    Sop_mix = operator_entanglement(max_mixed, Na+1)
    @show maxlinkdim(max_mixed)
    println("Operator Entanglement = ", Sop_mix)
    println("Exact Operator Entanglement = ", Na*log2(2))
    println("Does it match the expected value?", Sop_mix == Na * log2(2) )

end

gives the following output:

maxlinkdim(max_mixed) = 1
Operator Entanglement = 0.25
Exact Operator Entanglement = 3.0
Does it match the expected value?false
1 Like
2

I would like to push this person’s question up as I have the same question. I am having a hard time calculating operator entanglement in ITensor. I am finding ITensor really hates when you contract a big part of an MPO. I am wondering if there is an ITensor function to do this or anyone has efficient ways to calculate operator entaglement of an MPO that is not the denity matrix.

3

Also, just to add. I have seen some chatter in papers about calculating the entaglement of an MPO by using the bra and ket indices and proceeding as if it is a matrix. Again, I would like to try this, but I do not see good ways to fuse indices of an MPO.

4

Sorry, last post. If someone could share how to do this process efficiently then that would be fantastic. I know the contractions are expensive, but this quantity could really help me answer some scientific questions. I know how to do this in other libraries like quinb, but ITensor doesn’t like me messing with my own tensors from its MPO object.

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