bibkey: delazzarimotwaniseynnaeve2022umpsspan
authors: Claudia De Lazzari; Harshit J. Motwani; Tim Seynnaeve
year: 2022
title: "The Linear Span of Uniform Matrix Product States"
doi: 10.3842/SIGMA.2022.099
url: https://arxiv.org/abs/2204.10363v3
claim: "uMPS(m,n,d) is the closure of the image of the trace parametrization (A_0,…,A_{n−1}) ↦ Σ Tr(A_{i_1}···A_{i_d}) e_{i_1}⊗···⊗e_{i_d}; the paper computes the linear span of uMPS(2,2,d) for d ≤ 20 and conjectures its dimension: (d⁴ − 4d² + 192d + 192)/192 for even d and (d⁴ − 10d² + 192d + 201)/192 for odd d."
strata_touched:
  - D5/S3/Quantum/TensorNetworks/UniformMPSTwoByTwoTraceWords
  - D5/S3/Quantum/TensorNetworks/UniformMPSTwoByTwoLinearSpan
license: citation-only
triage: anchor

De Lazzari, Motwani and Seynnaeve, The Linear Span of Uniform Matrix Product States

C. De Lazzari, H. J. Motwani and T. Seynnaeve, SIGMA 18 (2022), 099; arXiv:2204.10363v3. Quotations are from the v3 TeX source.

The uniform matrix product state parametrization sends $(A_0,\dots ,A_{n-1})\in ({\mathbb{C}}^{m\times m}{)}^n$ to ${\sum }_{i_1,\dots ,i_d}\mathrm{Tr}(A_{i_1}\cdots A_{i_d})e_{i_1}\otimes \cdots \otimes e_{i_d}$; $\mathrm{uMPS}(m,n,d)$ is the closure of its image, and the paper studies its linear span. For bond and physical dimension two it states:

Conjecture 3.8 (Corollary of Conjecture 3.7). $\dim ⟨\mathrm{uMPS}(2,2,d)⟩=\frac{1}{192}(d^4-4d^2+192d+192)$ for $d$ even, and $\frac{1}{192}(d^4-10d^2+192d+201)$ for $d$ odd.

Verified locator

Source: https://arxiv.org/abs/2204.10363v3

• DOI: https://doi.org/10.3842/SIGMA.2022.099 (Symmetry, Integrability and Geometry: Methods and Applications 18 (2022), article 099; title and authors from Crossref).
• URL: https://arxiv.org/abs/2204.10363v3. The e-print member sigma22-099.tex has 1122 lines and md5 0049029a527b4d77a0f6a19e9d2d901b. Locations:
  • the parametrization, l. 214–225; the linear-span question, l. 245–247;
  • Table 1, l. 498–524; Conjecture 3.7, l. 529–537; Conjecture 3.8, l. 540–547.