doi: 10.3934/amc.2021015
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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. BrunI. 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. 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.

[4]

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

[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. 
[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. BrunI. 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. 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.

[4]

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

[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. 
[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.

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