doi: 10.3934/amc.2021015

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 Published  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. BrunI. Devetak and M. H. Hsieh, Correcting quantum errors with entanglement, Science, 314 (2006), 436-439.  doi: 10.1126/science.1131563.  Google Scholar

[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.  Google Scholar

[3]

S. DuttaR. 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.  Google Scholar

[4]

F. R. Gantmakher, The Theory of Matrices, 2, AMS Chelsea Publishing Company, 2000.  Google Scholar

[5] M. HeinW. DürJ. EisertR. RaussendorfM. 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.   Google Scholar
[6]

N. Jacobson, Basic Algebra 1, 2$^{nd}$ edition, Dover Publications, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. W. Moore, Quantum hypergraph states in continuous variables, Phys. Rev. A, 100 (2019), 062301. Google Scholar

[10] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, 10$^th$ edition, Cambridge University Press, Cambridge, 2011.   Google Scholar
[11]

A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.  Google Scholar

[12]

C. Riera, Spectral Properties of Boolean Functions, Graphs and Graph States, Doctoral Thesis, Universidad Complutense de Madrid, 2006. Google Scholar

[13]

D. Schlingemann and R. F. Werner, Quantum error-correcting codes associated with graphs, Phys. Review A, 65 (2001), 012308. Google Scholar

[14]

D. Shemesh, Common Eigenvectors of Two Matrices, Linear Alg. Applic., 62 (1984), 11-18.  doi: 10.1016/0024-3795(84)90085-5.  Google Scholar

[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. Google Scholar

show all references

References:
[1]

T. A. BrunI. Devetak and M. H. Hsieh, Correcting quantum errors with entanglement, Science, 314 (2006), 436-439.  doi: 10.1126/science.1131563.  Google Scholar

[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.  Google Scholar

[3]

S. DuttaR. 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.  Google Scholar

[4]

F. R. Gantmakher, The Theory of Matrices, 2, AMS Chelsea Publishing Company, 2000.  Google Scholar

[5] M. HeinW. DürJ. EisertR. RaussendorfM. 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.   Google Scholar
[6]

N. Jacobson, Basic Algebra 1, 2$^{nd}$ edition, Dover Publications, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. W. Moore, Quantum hypergraph states in continuous variables, Phys. Rev. A, 100 (2019), 062301. Google Scholar

[10] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, 10$^th$ edition, Cambridge University Press, Cambridge, 2011.   Google Scholar
[11]

A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.  Google Scholar

[12]

C. Riera, Spectral Properties of Boolean Functions, Graphs and Graph States, Doctoral Thesis, Universidad Complutense de Madrid, 2006. Google Scholar

[13]

D. Schlingemann and R. F. Werner, Quantum error-correcting codes associated with graphs, Phys. Review A, 65 (2001), 012308. Google Scholar

[14]

D. Shemesh, Common Eigenvectors of Two Matrices, Linear Alg. Applic., 62 (1984), 11-18.  doi: 10.1016/0024-3795(84)90085-5.  Google Scholar

[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. Google Scholar

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