### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 095, 10 pages      arXiv:1210.2812      https://doi.org/10.3842/SIGMA.2014.095

### Algebraic Geometry of Matrix Product States

Andrew Critch a and Jason Morton b
a) Jane Street Capital, 1 New York Plaza New York, NY 10004, USA
b) Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

Received February 28, 2014, in final form August 22, 2014; Published online September 10, 2014

Abstract
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.

Key words: matrix product states; trace varieties; trace algebras; quantum tomography.

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