Algebraic Geometry of Matrix Product States

作者：Critch Andrew^*; Morton Jason

来源：Symmetry Integrability and Geometry-Methods and Applications, 2014, 10: 095.

DOI：10.3842/SIGMA.2014.095

• matrix product states
• trace varieties
• trace algebras
• quantum tomography

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.

• 出版日期2014