Algebraic geometry of matrix product states

Andrew Critch, Jason Morton

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


We quantify the representational power of matrix product states (MPS) for entangled qubit systems by giving polynomial expressions in a pure quantum state's amplitudes which hold if and only if the state is a translation invariant matrix product state or a limit of such states. For systems with few qubits, we give these equations explicitly, considering both periodic and open boundary conditions. Using the classical theory of trace varieties and trace algebras, we explain the relationship between MPS and hidden Markov models and exploit this relationship to derive useful parameterizations of MPS. We make four conjectures on the identifiability of MPS parameters.

Original languageEnglish (US)
Article number095
JournalSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)
StatePublished - 2014

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematical Physics
  • Geometry and Topology


Dive into the research topics of 'Algebraic geometry of matrix product states'. Together they form a unique fingerprint.

Cite this