# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021015
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Quantum states associated to mixed graphs and their algebraic characterization

 1 Department of Computer Science, Electrical Engineering and Mathematical Sciences, Western Norway University of Applied Sciences, 5020 Bergen, Norway 2 Department of Informatics, University of Bergen, Postboks 7803, N-5020, Bergen, Norway 3 Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943-5212, U.S.A

* Corresponding author: Constanza Riera

Received  January 2021 Early access May 2021

Graph states are present in quantum information and found applications ranging from quantum network protocols (like secret sharing) to measurement based quantum computing. In this paper, we extend the notion of graph states, which can be regarded as pure quantum graph states, or as homogeneous quadratic Boolean functions associated to simple undirected graphs, to quantum states based on mixed graphs (graphs which allow both directed and undirected edges), obtaining mixed quantum states, which are defined by matrices associated to the measurement of homogeneous quadratic Boolean functions in some (ancillary) variables. In our main result, we describe the extended graph state as the sum of terms of a commutative subgroup of the stabilizer group of the corresponding mixed graph with the edges' directions reversed.

Citation: Constanza Riera, Matthew G. Parker, Pantelimon Stǎnicǎ. Quantum states associated to mixed graphs and their algebraic characterization. Advances in Mathematics of Communications, doi: 10.3934/amc.2021015
##### References:
 [1] T. A. Brun, I. Devetak and M. H. Hsieh, Correcting quantum errors with entanglement, Science, 314 (2006), 436-439.  doi: 10.1126/science.1131563. [2] A. Dahlberg and S. Wehner, Transforming graph states using single-qubit operations, Philosophical Trans. Royal Society A, 376 (2018), 20170325. doi: 10.1098/rsta.2017.0325. [3] S. Dutta, R. Sarkar and P. K. Panigrahi, Permutation symmetric hypergraph states and multipartite quantum entanglement, Internat. J. Theor. Phys., 58 (2019), 3927-3944.  doi: 10.1007/s10773-019-04259-5. [4] F. R. Gantmakher, The Theory of Matrices, 2, AMS Chelsea Publishing Company, 2000. [5] M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest and H.-J. Briegel, Entanglement in graph states and its applications, in Proc. Internat. School of Physics "Enrico Fermi", Quantum Computers, Algorithms and Chaos (Eds., G. Casati, D. L. Shepelyansky, P. Zoller, G. Benenti), 162, IOS Press, 2006. [6] N. Jacobson, Basic Algebra 1, 2$^{nd}$ edition, Dover Publications, 2009. [7] D. W. Lyons, D. J. Upchurch, S. N. Walck and C. D. Yetter, Local unitary symmetries of hypergraph states, J. Phys. A: Math. Theor., 48 (2015), 095301. doi: 10.1088/1751-8113/48/9/095301. [8] D. Markham and B. C. Sanders, Graph states for quantum secret sharing, Phys. Review A, 78 (2008), 042309. doi: 10.1103/PhysRevA.78.042309. [9] D. W. Moore, Quantum hypergraph states in continuous variables, Phys. Rev. A, 100 (2019), 062301. [10] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, 10$^th$ edition, Cambridge University Press, Cambridge, 2011. [11] A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995. [12] C. Riera, Spectral Properties of Boolean Functions, Graphs and Graph States, Doctoral Thesis, Universidad Complutense de Madrid, 2006. [13] D. Schlingemann and R. F. Werner, Quantum error-correcting codes associated with graphs, Phys. Review A, 65 (2001), 012308. [14] D. Shemesh, Common Eigenvectors of Two Matrices, Linear Alg. Applic., 62 (1984), 11-18.  doi: 10.1016/0024-3795(84)90085-5. [15] M. Van den Nest, Local Equivalences of Stabilizer States and Codes, Ph.D thesis, Faculty of Engineering, K.U. Leuven (Leuven, Belgium), May 2005.

show all references

##### References:
 [1] T. A. Brun, I. Devetak and M. H. Hsieh, Correcting quantum errors with entanglement, Science, 314 (2006), 436-439.  doi: 10.1126/science.1131563. [2] A. Dahlberg and S. Wehner, Transforming graph states using single-qubit operations, Philosophical Trans. Royal Society A, 376 (2018), 20170325. doi: 10.1098/rsta.2017.0325. [3] S. Dutta, R. Sarkar and P. K. Panigrahi, Permutation symmetric hypergraph states and multipartite quantum entanglement, Internat. J. Theor. Phys., 58 (2019), 3927-3944.  doi: 10.1007/s10773-019-04259-5. [4] F. R. Gantmakher, The Theory of Matrices, 2, AMS Chelsea Publishing Company, 2000. [5] M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest and H.-J. Briegel, Entanglement in graph states and its applications, in Proc. Internat. School of Physics "Enrico Fermi", Quantum Computers, Algorithms and Chaos (Eds., G. Casati, D. L. Shepelyansky, P. Zoller, G. Benenti), 162, IOS Press, 2006. [6] N. Jacobson, Basic Algebra 1, 2$^{nd}$ edition, Dover Publications, 2009. [7] D. W. Lyons, D. J. Upchurch, S. N. Walck and C. D. Yetter, Local unitary symmetries of hypergraph states, J. Phys. A: Math. Theor., 48 (2015), 095301. doi: 10.1088/1751-8113/48/9/095301. [8] D. Markham and B. C. Sanders, Graph states for quantum secret sharing, Phys. Review A, 78 (2008), 042309. doi: 10.1103/PhysRevA.78.042309. [9] D. W. Moore, Quantum hypergraph states in continuous variables, Phys. Rev. A, 100 (2019), 062301. [10] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, 10$^th$ edition, Cambridge University Press, Cambridge, 2011. [11] A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995. [12] C. Riera, Spectral Properties of Boolean Functions, Graphs and Graph States, Doctoral Thesis, Universidad Complutense de Madrid, 2006. [13] D. Schlingemann and R. F. Werner, Quantum error-correcting codes associated with graphs, Phys. Review A, 65 (2001), 012308. [14] D. Shemesh, Common Eigenvectors of Two Matrices, Linear Alg. Applic., 62 (1984), 11-18.  doi: 10.1016/0024-3795(84)90085-5. [15] M. Van den Nest, Local Equivalences of Stabilizer States and Codes, Ph.D thesis, Faculty of Engineering, K.U. Leuven (Leuven, Belgium), May 2005.
 [1] Cesare Tronci. Momentum maps for mixed states in quantum and classical mechanics. Journal of Geometric Mechanics, 2019, 11 (4) : 639-656. doi: 10.3934/jgm.2019032 [2] Jaime Angulo Pava, Nataliia Goloshchapova. On the orbital instability of excited states for the NLS equation with the δ-interaction on a star graph. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5039-5066. doi: 10.3934/dcds.2018221 [3] Gökhan Mutlu. On the quotient quantum graph with respect to the regular representation. Communications on Pure and Applied Analysis, 2021, 20 (2) : 885-902. doi: 10.3934/cpaa.2020295 [4] Barton E. Lee. Consensus and voting on large graphs: An application of graph limit theory. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1719-1744. doi: 10.3934/dcds.2018071 [5] Roberto de A. Capistrano–Filho, Márcio Cavalcante, Fernando A. Gallego. Controllability for Schrödinger type system with mixed dispersion on compact star graphs. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022019 [6] Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68. [7] Oded Schramm. Hyperfinite graph limits. Electronic Research Announcements, 2008, 15: 17-23. doi: 10.3934/era.2008.15.17 [8] J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413 [9] John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16. [10] Roberto De Leo, James A. Yorke. The graph of the logistic map is a tower. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5243-5269. doi: 10.3934/dcds.2021075 [11] Roy H. Goodman. NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2203-2232. doi: 10.3934/dcds.2019093 [12] Mario Roy, Mariusz Urbański. Random graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 261-298. doi: 10.3934/dcds.2011.30.261 [13] Rui Wang, Rundong Zhao, Emily Ribando-Gros, Jiahui Chen, Yiying Tong, Guo-Wei Wei. HERMES: Persistent spectral graph software. Foundations of Data Science, 2021, 3 (1) : 67-97. doi: 10.3934/fods.2021006 [14] Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems and Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036 [15] Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6029-6045. doi: 10.3934/dcds.2018260 [16] James B. Kennedy, Jonathan Rohleder. On the hot spots of quantum graphs. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3029-3063. doi: 10.3934/cpaa.2021095 [17] Martin Gugat, Falk M. Hante, Markus Hirsch-Dick, Günter Leugering. Stationary states in gas networks. Networks and Heterogeneous Media, 2015, 10 (2) : 295-320. doi: 10.3934/nhm.2015.10.295 [18] Tai-Ping Liu, Zhouping Xin, Tong Yang. Vacuum states for compressible flow. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 1-32. doi: 10.3934/dcds.1998.4.1 [19] Chun-Xiang Guo, Guo Qiang, Jin Mao-Zhu, Zhihan Lv. Dynamic systems based on preference graph and distance. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1139-1154. doi: 10.3934/dcdss.2015.8.1139 [20] Liu Hui, Lin Zhi, Waqas Ahmad. Network(graph) data research in the coordinate system. Mathematical Foundations of Computing, 2018, 1 (1) : 1-10. doi: 10.3934/mfc.2018001

2021 Impact Factor: 1.015