bibkey: sanzwolfperezgarciacirac2009mps
authors: M. Sanz; M. M. Wolf; D. Pérez-García; J. I. Cirac
year: 2009
title: "Matrix product states: Symmetries and two-body Hamiltonians"
doi: 10.1103/PhysRevA.79.042308
url: https://arxiv.org/abs/0901.2223v1
claim: "A translationally invariant MPS with SU(2) symmetry (irreducible representation) and even physical dimension cannot be injective; for U(1) symmetries the obstruction holds when the generator has eigenvalues −m/2,…,m/2 but fails for general generators (an injective example with physical dimension D²−D), physical dimension 2 with U(1) symmetry excludes injectivity, and the case of spin 3/2 is stated as open."
strata_touched:
  - D5/S3/Quantum/SpinChains/FourLevelU1SymmetricInjectiveMPS
license: citation-only
triage: anchor

Sanz, Wolf, Pérez-García and Cirac, Matrix product states: Symmetries and two-body Hamiltonians

M. Sanz, M. M. Wolf, D. Pérez-García and J. I. Cirac, Phys. Rev. A 79, 042308 (2009); arXiv:0901.2223v1. Quotations are from the v1 TeX source.

A translationally invariant MPS is $|\Phi ⟩={\sum }_{i_1,\dots ,i_N}\mathrm{tr}(A_{i_1}\cdots A_{i_N})|i_1\cdots i_N⟩$ with Kraus operators $A_i\in {\mathcal{M}}_D$. It is invariant under a group $G$ if $u_g^{\otimes N}|\Psi ⟩=e^{i{\theta }_g}|\Psi ⟩$ for a unitary representation $u_g$. It is injective if for some $n$ the map ${\Gamma }_n(X)={\sum }_{i_1,\dots ,i_n}\mathrm{tr}(X{A}_{i_1}\cdots A_{i_n})|i_1\cdots i_n⟩$ is injective. Section IV proves that an $SU(2)$-symmetric MPS with even physical dimension cannot be injective. It gives an injective MPS of physical dimension $D^2-D$ invariant under a $U(1)$ symmetry whose generator has only non-zero eigenvalues, and proves that physical dimension $2$ with $U(1)$ symmetry excludes injectivity. Between these:

Let us remark that this counter-example is applicable to spin $\ge \frac{5}{2}$. Indeed, one can prove Theorem \ref{} for $U(1)$ and spin $\frac{1}{2}$, which is the content of the following proposition. The case of spin $3/2$ remains an open question.

Verified locator

Source: https://arxiv.org/abs/0901.2223v1

• DOI: https://doi.org/10.1103/PhysRevA.79.042308 (Physical Review A 79(4), article 042308; title, authors, journal, volume, issue and article number from Crossref).
• URL: https://arxiv.org/abs/0901.2223v1. The e-print member Symmetries191208.tex has 1434 lines and md5 54c705c1d4b5323bbd6f9276ee90e022. Locations:
  • symmetry, Eq. (1), l. 134–140; the MPS, l. 222–237; injectivity, l. 267–276;
  • the $SU(2)$ theorem, l. 688–694, and its $U(1)$ remark, l. 719–721;
  • the $D^2-D$ example, l. 726–746; the open question, l. 748–751;
  • the physical-dimension-2 proposition, l. 752–755.