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I could not find any references for a basis-independent formulation of tensor networks: All papers I have found use pretty much (explicitly or implicitly) the canonical computational basis by defining tensors as lists of complex numbers.

While this is certainly ok from the perspective of "I want to compute stuff", it is not when trying to look at the bigger picture, i.e. realizing tensor networks as certain string diagrams in the category of finite dimensional Hilbert spaces.

Does anyone know of a resource that addresses this?

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  • $\begingroup$ The closest I've seen is in the TQFT literature surrounding state-sums, which seems like it could be what you're after since you mention a category. Check this out for instance: arxiv.org/abs/1010.4840 $\endgroup$
    – Ryan Thorngren
    Commented May 7, 2018 at 10:40
  • $\begingroup$ @JoBe ping me here or in chat in two days (when this question becomes eligible for a bounty) if you haven't gotten a good answer. $\endgroup$
    – Emilio Pisanty
    Commented May 7, 2018 at 11:56
  • $\begingroup$ @EmilioPisanty I already raised a mod flag, but given that you did that: Why did you tag this as "resource-recommendation"? I don't see how this matches the description on meta. $\endgroup$
    – Norbert Schuch
    Commented May 21, 2018 at 2:32
  • $\begingroup$ For administrative reasons, all reference requests are treated in the same way, even if they don't match the description on meta. $\endgroup$
    – Qmechanic ♦
    Commented May 21, 2018 at 4:33
  • $\begingroup$ @Qmechanic (I didn't get pinged for that.) Thanks. But nevertheless, the statement above "Answers containing only a reference to a book or paper will be removed!" might keep people from posting answers. I was very unsure whether I should post the link to the article below. What's the policy here? (I mean, it's not that these links would be easy to find for this kind of questions, so they are valuable.) $\endgroup$
    – Norbert Schuch
    Commented May 23, 2018 at 20:54

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https://arxiv.org/abs/1210.7710 develops a framework of MPS modulo their gauge degree of freedom (which includes basis choice) as a fibre bundle.

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