
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.
pdf (352 kb)
tex (29 kb)
References

Biamonte J., Bergholm V., Lanzagorta M., Tensor network methods for invariant theory, J. Phys. A: Math. Theor. 46 (2013), 475301, 19 pages, arXiv:1209.0631.

Bray N., Morton J., Equations defining hidden Markov models, in Algebraic Statistics for Computational Biology, Cambridge University Press, New York, 2005, 237249.

Cox D., Little J., O'Shea D., Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra, 2nd ed., Undergraduate Texts in Mathematics, SpringerVerlag, New York, 1997.

Critch A., Binary hidden Markov models and varieties, J. Algebr. Stat. 4 (2013), 130, arXiv:1206.0500.

Eisenbud D., Grayson D.R., Stillman M., Sturmfels B. (Editors), Computations in algebraic geometry with Macaulay 2, Algorithms and Computation in Mathematics, Vol. 8, SpringerVerlag, Berlin, 2002.

Erickson R.V., Functions of Markov chains, Ann. Math. Statist. 41 (1970), 843850.

Grayson D.R., Stillman M.E., Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.

Greuel G.M., Pfister G., A singular introduction to commutative algebra, SpringerVerlag, Berlin, 2008.

Leron U., Trace identities and polynomial identities of $n\times n$ matrices, J. Algebra 42 (1976), 369377.

Morton J., Tensor networks in algebraic geometry and statistics, Lecture at Networking Tensor Networks (Benasque, Spain, 2012), available at http://benasque.org/2012network/talks_contr/106_morton.pdf.

Pachter L., Sturmfels B., Tropical geometry of statistical models, Proc. Natl. Acad. Sci. USA 101 (2004), 1613216137, qbio/0311009.

PerezGarcia D., Verstraete F., Wolf M.M., Cirac J.I., Matrix product state representations, Quantum Inf. Comput. 7 (2007), 401430, quantph/0608197.

Procesi C., The invariant theory of $n\times n$ matrices, Adv. Math. 19 (1976), 306381.

Sibirskii K.S., Algebraic invariants of a system of matrices, Sib. Math. J. 9 (1968), 115124.

The online encyclopedia of integer sequences (OEIS), A000031 Number of $n$bead necklaces with 2 colors when turning over is not allowed, available at http://oeis.org.

Turner J., Morton J., The invariant ring of $m$ matrices under the adjoint action by a subgroup of a product of general linear groups, arXiv:1310.0370.

Verstraete F., Cirac J., Matrix product states represent ground states faithfully, Phys. Rev. B 73 (2006), 094423, 8 pages, condmat/0505140.

Vidyasagar M., The complete realization problem for hidden Markov models: a survey and some new results, Math. Control Signals Systems 23 (2011), 165.

