bibkey: critchmorton2014mps
authors: Critch, Andrew; Morton, Jason
year: 2014
title: "Algebraic geometry of matrix product states"
doi: 10.3842/SIGMA.2014.095
url: "https://arxiv.org/abs/1210.2812v4"
claim: "The paper studies the varieties of translation-invariant matrix product states with periodic and open boundary conditions, gives defining equations in small cases (among them the quartic hypersurface of open-boundary binary bond-dimension-two MPS on three sites), and states identifiability conjectures, including that each improper marginalization of open-boundary 2x2 binary MPS to three consecutive sites is a birational equivalence (Conjecture 8)."
strata_touched:
  - D5/S3/Quantum/TensorNetworks/OpenBoundaryMPSMarginalBirationality
license: citation-only
triage: anchor

Critch and Morton, Algebraic geometry of matrix product states

A. Critch and J. Morton, SIGMA 10 (2014), 095, DOI 10.3842/SIGMA.2014.095; arXiv:1210.2812v4 (10 September 2014). Quotations are from the arXiv v4 TeX source sigma14-095.tex; the author list is from the arXiv metadata.

With open boundary conditions the state is $\Psi ={\sum }_{i_1,\dots ,i_N}{b}_0^T{A}_{i_1}\cdots A_{i_N}{b}_1|i_1\dots i_N⟩$ for boundary vectors $b_0,b_1\in {\mathbb{C}}^D$ and $D\times D$ matrices $A_0,\dots ,A_{d-1}$; $\mathrm{OB}(D,d,N)$ is the set of such states and $\overline{\mathrm{OB}}(D,d,N)$ its Zariski closure in ${\mathbb{C}}^{d^N}$ (Section 3, l. 344–364). Theorem 7 (l. 378–403) shows that $\overline{\mathrm{OB}}(2,2,3)$ is a quartic hypersurface in ${\mathbb{C}}^8$. The improper marginalization maps sum all indices outside three consecutive sites (l. 420–428), and the paper conjectures (l. 430–434):

For $N\ge 4$, a~generic~$N$-qubit state can be recovered from its improper margina-li-zation to any three consecutive states. That is, each improper marginalization map $\overline{\text{OB}}(2,2,N)\to \overline{\text{OB}}(2,2,3)$ is a~birational equivalence of varieties.

Verified locator

• DOI: https://doi.org/10.3842/SIGMA.2014.095
• Source: https://arxiv.org/abs/1210.2812v4