Bond dimension witnesses and the structure of homogeneous matrix product states

Miguel Navascues1,2 and Tamas Vertesi3

1Department of Physics, Bilkent University, Ankara 06800, Turkey
2Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmangasse 3, 1090 Vienna, Austria
3Institute for Nuclear Research, Hungarian Academy of Sciences, H-4001 Debrecen, P.O. Box 51, Hungary

full text pdf

For the past twenty years, Matrix Product States (MPS) have been widely used in solid state physics to approximate the ground state of one-dimensional spin chains. In this paper, we study homogeneous MPS (hMPS), or MPS constructed via site-independent tensors and a boundary condition. Exploiting a connection with the theory of matrix algebras, we derive two structural properties shared by all hMPS, namely: a) there exist local operators which annihilate all hMPS of a given bond dimension; and b) there exist local operators which, when applied over any hMPS of a given bond dimension, decouple (cut) the particles where they act from the spin chain while at the same time join (glue) the two loose ends back again into a hMPS. Armed with these tools, we show how to systematically derive `bond dimension witnesses', or 2-local operators whose expectation value allows us to lower bound the bond dimension of the underlying hMPS. We extend some of these results to the ansatz of Projected Entangled Pairs States (PEPS). As a bonus, we use our insight on the structure of hMPS to: a) derive some theoretical limitations on the use of hMPS and hPEPS for ground state energy computations; b) show how to decrease the complexity and boost the speed of convergence of the semidefinite programming hierarchies described in [Phys. Rev. Lett. 115, 020501 (2015)] for the characterization of finite-dimensional quantum correlations.


► BibTeX data

► References

[1] Dorit Aharonov, Daniel Gottesman, Sandy Irani, and Julia Kempe. The power of quantum systems on a line. Communications in Mathematical Physics, 287 (1): 41-65, jan 2009. 10.1007/​s00220-008-0710-3. URL https:/​/​​10.1007.

[2] P. W. Anderson. Limits on the energy of the antiferromagnetic ground state. Physical Review, 83 (6): 1260-1260, sep 1951. 10.1103/​physrev.83.1260. URL https:/​/​​10.1103.

[3] MOSEK ApS. The MOSEK optimization toolbox for MATLAB manual. Version 7.1 (Revision 28)., 2015. URL http:/​/​​7.1/​toolbox/​index.html.

[4] A. C. Doherty, Pablo A. Parrilo, and Federico M. Spedalieri. Distinguishing separable and entangled states. Physical Review Letters, 88 (18), apr 2002. 10.1103/​physrevlett.88.187904. URL https:/​/​​10.1103.

[5] Glen Evenbly and Guifre Vidal. Quantum criticality with the multi-scale entanglement renormalization ansatz. In Springer Series in Solid-State Sciences, pages 99-130. Springer Berlin Heidelberg, 2013. 10.1007/​978-3-642-35106-8_4. URL https:/​/​​10.1007.

[6] M. Fannes, B. Nachtergaele, and R. F. Werner. Finitely correlated states on quantum spin chains. Communications in Mathematical Physics, 144 (3): 443-490, mar 1992. 10.1007/​bf02099178. URL https:/​/​​10.1007.

[7] Edward Formanek. The Polynomial Identities and Variants of $n \times n$ Matrices. American Mathematical Society, jan 1991. 10.1090/​cbms/​078. URL https:/​/​​10.1090.

[8] D. Gross, J. Eisert, N. Schuch, and D. Perez-Garcia. Measurement-based quantum computation beyond the one-way model. Physical Review A, 76 (5), nov 2007. 10.1103/​physreva.76.052315. URL https:/​/​​10.1103.

[9] Leonid Gurvits. Classical complexity and quantum entanglement. Journal of Computer and System Sciences, 69 (3): 448-484, nov 2004. 10.1016/​j.jcss.2004.06.003. URL https:/​/​​10.1016.

[10] M. Hein, J. Eisert, and H. J. Briegel. Multiparty entanglement in graph states. Physical Review A, 69 (6), jun 2004. 10.1103/​physreva.69.062311. URL https:/​/​​10.1103.

[11] Michael Karbach, Kun Hu, and Gerhard Muüller. Introduction to the bethe ansatz II. Computers in Physics, 12 (6): 565, 1998. 10.1063/​1.168740. URL https:/​/​​10.1063.

[12] Robert König and Renato Renner. A de finetti representation for finite symmetric quantum states. Journal of Mathematical Physics, 46 (12): 122108, dec 2005. 10.1063/​1.2146188. URL https:/​/​​10.1063.

[13] Michael Levin and Cody P. Nave. Tensor renormalization group approach to two-dimensional classical lattice models. Physical Review Letters, 99 (12), sep 2007. 10.1103/​physrevlett.99.120601. URL https:/​/​​10.1103.

[14] Chanchal K. Majumdar and Dipan K. Ghosh. On next-nearest-neighbor interaction in linear chain. i. Journal of Mathematical Physics, 10 (8): 1388-1398, aug 1969. 10.1063/​1.1664978. URL https:/​/​​10.1063.

[15] Miguel Navascués and Tamás Vértesi. Bounding the set of finite dimensional quantum correlations. Physical Review Letters, 115 (2), jul 2015. 10.1103/​physrevlett.115.020501. URL https:/​/​​10.1103.

[16] Miguel Navascués, Adrien Feix, Mateus Araújo, and Tamás Vértesi. Characterizing finite-dimensional quantum behavior. Physical Review A, 92 (4), oct 2015. 10.1103/​physreva.92.042117. URL https:/​/​​10.1103.

[17] Roberto Oliveira and Barbara M. Terhal. The complexity of quantum spin systems on a two-dimensional square lattice. Quant. Inf, Comp., 8, 2008.

[18] Román Orús. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of Physics, 349: 117-158, oct 2014. 10.1016/​j.aop.2014.06.013. URL https:/​/​​10.1016.

[19] Asher Peres. Separability criterion for density matrices. Physical Review Letters, 77 (8): 1413-1415, aug 1996. 10.1103/​physrevlett.77.1413. URL https:/​/​​10.1103.

[20] D. Perez-García, F. Verstraete, M. M. Wolf, and J.I. Cirac. Matrix product state representations. Quantum Inf. Comput., 7: 401, sep 2007.

[21] Ho N. Phien, Johann A. Bengua, Hoang D. Tuan, Philippe Corboz, and Román Orús. Infinite projected entangled pair states algorithm improved: Fast full update and gauge fixing. Physical Review B, 92 (3), jul 2015. 10.1103/​physrevb.92.035142. URL https:/​/​​10.1103.

[22] David Poulin and Matthew B. Hastings. Markov entropy decomposition: A variational dual for quantum belief propagation. Physical Review Letters, 106 (8), feb 2011. 10.1103/​physrevlett.106.080403. URL https:/​/​​10.1103.

[23] Norbert Schuch, Ignacio Cirac, and David Pérez-García. PEPS as ground states: Degeneracy and topology. Annals of Physics, 325 (10): 2153-2192, oct 2010. 10.1016/​j.aop.2010.05.008. URL https:/​/​​10.1016.

[24] Neil J. A. Sloane. The on-line encyclopedia of integer sequences. In Towards Mechanized Mathematical Assistants, pages 130-130. Springer Berlin Heidelberg. 10.1007/​978-3-540-73086-6_12. URL https:/​/​​10.1007.

[25] Stellan Östlund and Stefan Rommer. Thermodynamic limit of density matrix renormalization. Physical Review Letters, 75 (19): 3537-3540, nov 1995. 10.1103/​physrevlett.75.3537. URL https:/​/​​10.1103.

[26] Barbara M. Terhal. Bell inequalities and the separability criterion. Physics Letters A, 271 (5-6): 319-326, jul 2000. 10.1016/​s0375-9601(00)00401-1. URL https:/​/​​10.1016.

[27] Lieven Vandenberghe and Stephen Boyd. Semidefinite programming. SIAM Review, 38 (1): 49-95, mar 1996. 10.1137/​1038003. URL https:/​/​​10.1137.

[28] F. Verstraete, J. J. García-Ripoll, and J. I. Cirac. Matrix product density operators: Simulation of finite-temperature and dissipative systems. Physical Review Letters, 93 (20), nov 2004. 10.1103/​physrevlett.93.207204. URL https:/​/​​10.1103.

[29] F. Verstraete, V. Murg, and J.I. Cirac. Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems. Advances in Physics, 57 (2): 143-224, mar 2008. 10.1080/​14789940801912366. URL https:/​/​​10.1080.

[30] R.F. Werner. Finitely correlated states. In Encyclopedia of Mathematical Physics, pages 334-340. Elsevier, 2006. 10.1016/​b0-12-512666-2/​00379-5. URL https:/​/​​10.1016.

[31] Eric Ziegel, William Press, Brian Flannery, Saul Teukolsky, and William Vetterling. Numerical recipes: The art of scientific computing. Technometrics, 29 (4): 501, nov 1987. 10.2307/​1269484. URL https:/​/​​10.2307.

► Cited by (beta)

[1] Miguel Navascués, "Resetting Uncontrolled Quantum Systems", Physical Review X 8, 031008 (2018).

(The above data is from Crossref's cited-by service. Unfortunately not all publishers provide suitable and complete citation data so that some citing works or bibliographic details may be missing.)