How to calculate tri-partite entanglement or mutual information

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1

Hi,

I am trying to calculate the tri-partite entanglement entropy or mutual information (I) in a one dimension (1D) spin-1/2 chain, defined as,
I (A, B, C) ≡ S(A) + S(B) + S(C) − S(A ∪ B) − S(A ∪ C) − S(B ∪ C) + S(A ∪ B ∪ C).

where, A, B, C are three regions (connected or disconnected) in the 1D chain.

c.f.: Phys. Rev. B 101, 060301(R) (2020), PRE 98 052205 (2018),

I am not sure how SVD can help in this case, especially for the S(A ∪ B ∪ C) term.
Can you please point me in the right direction of thinking?

2

One way could be to divide the system into “connected” A,B,C,D region as shown in the paper you referred Phys. Rev. B 101, 060301(R) (2020).

Except… make sure that length of (no. of spin sites in) A+B+C is less than D.

Then S(A ∪ B ∪ C) can be calculated by doing an SVD at site between the regions A+B+C and D.