% The tensor network literature: foundational, modern, and formalization references.
\section{References}\label{ch:references}

\providecommand{\doi}[1]{\href{https://doi.org/#1}{\nolinkurl{doi:#1}}}
\providecommand{\arxiv}[1]{\href{https://arxiv.org/abs/#1}{\nolinkurl{arXiv:#1}}}

The \pkg{} package was developed for \textsc{TNLean}, a Lean~4 formalization of
the mathematical theory of tensor networks, and its diagram grammar reflects
the structural insights established across more than three decades of
quantum many-body research.  In \pkg{}, a diagram is not merely placed ink;
it is a typed mathematical record whose ports, contractions, and boundary
signatures respect the algebraic reality of tensor networks.

The references below gather the foundational and modern tensor-network literature
that forms the theoretical backbone of the language, with particular emphasis on
the seminal works of J.~I.~Cirac and collaborators on Matrix Product States
(MPS), Matrix Product Density Operators (MPDO), Matrix Product Unitaries (MPU),
and Projected Entangled Pair States (PEPS), alongside standard reviews,
diagrammatic lecture courses, and machine-checked formalizations.

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\begin{thebibliography}{99}

\bibitem{CiracRMP2021}
J.~I.~Cirac, D.~P{\'e}rez-Garc{\'i}a, N.~Schuch, and F.~Verstraete.
\newblock Matrix product states and projected entangled pair states: Concepts,
  symmetries, theorems.
\newblock \emph{Reviews of Modern Physics}, 93(4):045003, 2021.
\newblock \doi{10.1103/RevModPhys.93.045003}. \arxiv{2011.12127}.

\bibitem{PerezGarcia2007MPS}
D.~P{\'e}rez-Garc{\'i}a, F.~Verstraete, M.~M.~Wolf, and J.~I.~Cirac.
\newblock Matrix product state representations.
\newblock \emph{Quantum Information \& Computation}, 7(5):401--430, 2007.
\newblock \arxiv{quant-ph/0608197}.

\bibitem{VerstraeteCirac2004PEPS}
F.~Verstraete and J.~I.~Cirac.
\newblock Renormalization algorithms for quantum-many body systems in two and
  higher dimensions.
\newblock 2004.
\newblock \arxiv{cond-mat/0407066}.

\bibitem{Cirac2017MPDO}
J.~I.~Cirac, D.~P{\'e}rez-Garc{\'i}a, N.~Schuch, and F.~Verstraete.
\newblock Matrix product density operators: Renormalization fixed points and
  boundary theories.
\newblock \emph{Annals of Physics}, 378:100--149, 2017.
\newblock \doi{10.1016/j.aop.2016.12.030}. \arxiv{1606.00608}.

\bibitem{CiracMPU2017}
J.~I.~Cirac, D.~P{\'e}rez-Garc{\'i}a, N.~Schuch, and F.~Verstraete.
\newblock Matrix product unitaries: Structure, symmetries, and topological
  invariants.
\newblock \emph{Journal of Statistical Mechanics: Theory and Experiment},
  2017(8):083105, 2017.
\newblock \doi{10.1088/1742-5468/aa7e3b}. \arxiv{1703.09188}.

\bibitem{Molnar2018Normal}
A.~Moln{\'a}r, J.~Garre-Rubio, D.~P{\'e}rez-Garc{\'i}a, N.~Schuch, and
  J.~I.~Cirac.
\newblock Normal projected entangled pair states generating the same state.
\newblock \emph{New Journal of Physics}, 20(11):113017, 2018.
\newblock \doi{10.1088/1367-2630/aae94f}. \arxiv{1804.04964}.

\bibitem{Sanz2010Wielandt}
M.~Sanz, D.~P{\'e}rez-Garc{\'i}a, M.~M.~Wolf, and J.~I.~Cirac.
\newblock A quantum version of {Wielandt}'s inequality.
\newblock \emph{IEEE Transactions on Information Theory}, 56(9):4668--4673,
  2010.
\newblock \doi{10.1109/TIT.2010.2054562}. \arxiv{0909.5347}.

\bibitem{PerezGarcia2008StringOrder}
D.~P{\'e}rez-Garc{\'i}a, M.~M.~Wolf, M.~Sanz, F.~Verstraete, and J.~I.~Cirac.
\newblock String order and symmetries in quantum spin lattices.
\newblock \emph{Physical Review Letters}, 100(16):167202, 2008.
\newblock \doi{10.1103/PhysRevLett.100.167202}. \arxiv{0802.0447}.

\bibitem{Schuch2010PEPS}
N.~Schuch, I.~Cirac, and D.~P{\'e}rez-Garc{\'i}a.
\newblock {PEPS} as ground states: Degeneracy and topology.
\newblock \emph{Annals of Physics}, 325(10):2153--2192, 2010.
\newblock \doi{10.1016/j.aop.2010.05.008}. \arxiv{1001.3807}.

\bibitem{DeLasCuevas2017Irreducible}
G.~De las Cuevas, J.~I.~Cirac, N.~Schuch, and D.~P{\'e}rez-Garc{\'i}a.
\newblock Irreducible forms of matrix product states: Theory and
  applications.
\newblock \emph{Journal of Mathematical Physics}, 58(12):121901, 2017.
\newblock \doi{10.1063/1.4998639}. \arxiv{1708.00029}.

\bibitem{CiracGarreRubio2019OpenProblems}
J.~I.~Cirac, J.~Garre-Rubio, and D.~P{\'e}rez-Garc{\'i}a.
\newblock Mathematical open problems in projected entangled pair states.
\newblock \emph{Revista Matem{\'a}tica Complutense}, 32(3):579--599, 2019.
\newblock \doi{10.1007/s13163-019-00305-6}. \arxiv{1903.09439}.

\bibitem{Lu2026Autoformalization}
S.~Lu, E.~Tjoa, and J.~I.~Cirac.
\newblock Multi-agent autoformalization of tensor network theory.
\newblock 2026.
\newblock \arxiv{2607.07857}.

\bibitem{Fannes1992FCS}
M.~Fannes, B.~Nachtergaele, and R.~F.~Werner.
\newblock Finitely correlated states on quantum spin chains.
\newblock \emph{Communications in Mathematical Physics}, 144(3):443--490, 1992.
\newblock \doi{10.1007/BF02099178}.

\bibitem{Schollwock2011DMRG}
U.~Schollw{\"o}ck.
\newblock The density-matrix renormalization group in the age of matrix
  product states.
\newblock \emph{Annals of Physics}, 326(1):96--192, 2011.
\newblock \doi{10.1016/j.aop.2010.09.012}. \arxiv{1008.3477}.

\bibitem{Orus2014Practical}
R.~Or{\'u}s.
\newblock A practical introduction to tensor networks: Matrix product states
  and projected entangled pair states.
\newblock \emph{Annals of Physics}, 349:117--158, 2014.
\newblock \doi{10.1016/j.aop.2014.06.013}. \arxiv{1306.2164}.

\bibitem{BridgemanChubb2017Handwaving}
J.~C.~Bridgeman and C.~T.~Chubb.
\newblock Hand-waving and interpretive dance: An introductory course on tensor
  networks.
\newblock \emph{Journal of Physics A: Mathematical and Theoretical},
  50(22):223001, 2017.
\newblock \doi{10.1088/1751-8121/aa6dc3}. \arxiv{1603.03039}.

\bibitem{Wolf2012Notes}
M.~M.~Wolf.
\newblock Quantum channels \& operations: Guided tour.
\newblock Lecture notes, Technical University of Munich, 2012.
\newblock \url{https://mediatum.ub.tum.de/doc/1099654/1099654.pdf}.

\bibitem{TNLean}
The TNLean Project Contributors.
\newblock \textsc{TNLean}: A Lean~4 formalization of the mathematical theory of
  tensor networks.
\newblock 2026.
\newblock \url{https://github.com/LionSR/TNLean}.

\bibitem{tenkz}
S.~Lu.
\newblock \pkg{}: Declarative tensor-network diagrams in {\LaTeX}.
\newblock {\LaTeX} package, version 0.8.0, 2026.
\newblock \url{https://github.com/LionSR/tenkz}.

\end{thebibliography}