bibkey: czaplinskimichalekseynnaeve2023umps
authors: Adam Czapliński; Mateusz Michałek; Tim Seynnaeve
year: 2023
title: "Uniform matrix product states from an algebraic geometer's point of view"
doi: 10.1016/j.aam.2022.102417
url: https://arxiv.org/abs/1904.07563v1
claim: "The set uMPS(D,d,N) of uniform (translation-invariant) matrix product states with bond dimension D, physical dimension d and N sites is the image of the trace-word map; its closure has dimension at most min{(d−1)D²+1, dim Cyc^N(C^d)}, and Conjecture 2.14 states that equality holds for every (D,d,N)."
strata_touched:
  - D5/S3/Quantum/TensorNetworks/UniformMPSExpectedDimensionRefutation
license: citation-only
triage: anchor

Czapliński, Michałek and Seynnaeve, Uniform matrix product states from an algebraic geometer’s point of view

A. Czapliński, M. Michałek and T. Seynnaeve, Adv. Appl. Math. 142 (2023) 102417; arXiv:1904.07563v1. Quotations are from the v1 TeX source, with the source’s macro \IMPS (typeset uMPS) kept as written.

For $D,d,N\ge 1$ and matrices $M_0,\dots ,M_{d-1}\in {\mathbb{C}}^{D\times D}$, the tensor $T_N(M_0,\dots ,M_{d-1})=\sum \mathrm{tr}(M_{i_1}\cdots M_{i_N})e_{i_1}\otimes \cdots \otimes e_{i_N}$ is a uniform $D$-matrix product state, and $\mathrm{uMPS}(D,d,N)$ is the image of $T_N$. It lies in the space ${\mathrm{Cyc}}^N({\mathbb{C}}^d)$ of cyclically symmetric tensors. For $N=2$ it consists of the symmetric $d\times d$ matrices of rank at most $D^2$. The dimension of its closure is at most $\min \{(d-1)D^2+1,\dim {\mathrm{Cyc}}^N({\mathbb{C}}^d)\}$, and the set has expected dimension when equality holds. Section 2 states:

For every choice of $(D,d,N)$, $\text{IMPS}(D,d,N)$ has expected dimension.

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Source: https://arxiv.org/abs/1904.07563v1

• DOI: https://doi.org/10.1016/j.aam.2022.102417 (Advances in Applied Mathematics 142, article 102417; title, authors, journal, volume and article number from Crossref).
• URL: https://arxiv.org/abs/1904.07563v1. The e-print member uMPS-final.tex has 1034 lines and md5 8f2a97c8c4c32b15d5873cfca15595b2. Locations:
  • the definition of $T_N$ and uMPS, l. 180–193; ${\mathrm{Cyc}}^N$, l. 209;
  • the trivial cases including $N=2$, l. 243–253; the dimension bound, l. 272–275;
  • $\dim {\mathrm{Cyc}}^N$, l. 279–293; expected dimension, l. 299–305;
  • Conjecture 2.14, l. 311–313.