2
$\begingroup$

Suppose I've got a tensor network (TN) representing some bipartite quantum state, $|\Psi\rangle$. Using the Schmidt decomposition, I can write $|\Psi\rangle = \sum_{k=1}^r \sqrt{\lambda_k}|\chi_k\rangle_A \otimes |\phi_k\rangle_B$, for some upper bond dimension $r$. My question is: can the mutual information $I(A:B)$ be efficiently and accurately computed for this system in the case where the Hilbert space of $A$ is small but $B$ is large? (For example, consider a 1D chain, where $A$ corresponds to two spins in the centre, and $B$ is the rest of the chain).

Here's my best understanding of the problem: I know that $I(A:B) = S(A) + S(B) - S(AB)$. Because $|\Psi\rangle$ is a pure state, we know that $S(A) = S(B) = -\sum_{k=1}^r \lambda_k \log \lambda_k$. For a reasonable bond dimension, I feel like this calculation should be straightforward to implement. However, I have no good idea for how difficult it is to estimate $S(AB) \equiv S(\rho_{AB})$, where $\rho_{AB}$ is the density matrix corresponding to the whole system.

Also--does this problem become easier if we assume some structure in the TN? For example, if we assume that it's an MPS, then can we more easily calculate the mutual information?

Thanks for all the help!

Improve this question
$\endgroup$
6
  • 2
    $\begingroup$ If the system as a whole is in pures state $|\Psi\rangle$, then the entropy of the entire state $S(AB) = 0$ $\endgroup$
    – By Symmetry
    Commented Feb 5, 2022 at 12:03
  • 1
    $\begingroup$ ... and S(A) = S(B). $\endgroup$
    – Norbert Schuch
    Commented Feb 5, 2022 at 12:34
  • 1
    $\begingroup$ For a general TN, this is potentially hard. For an MPS, this is easy, and a standard task. But what do you know about TNs, and for which TNs do you want this question answered? Otherwise, you are asking for a full introduction to the field, and I would recommend a good review article. (Do you know about how to get the entropy in theory, and you want to know how to do it computationally?) $\endgroup$
    – Norbert Schuch
    Commented Feb 5, 2022 at 12:35
  • $\begingroup$ @NorbertSchuch I am intersted in TNs in general, but if the MPS example is particularly illuminating, I would love to hear it. The review article would also be very helpful -- I am just getting started out in the field. And I am primarily interested in a computational approach. $\endgroup$
    – redfive
    Commented Feb 5, 2022 at 18:28
  • 2
    $\begingroup$ For MPS this is a standard calculation. arxiv.org/abs/1008.3477 should be a good starting point - Sec. 4 should contain everything you need to know. Keep in mind that you are just looking for the entanglement entropy of one of the two parts. $\endgroup$
    – Norbert Schuch
    Commented Feb 5, 2022 at 22:46

0

Reset to default