Quantum states associated to mixed graphs and their algebraic characterization. (English) Zbl 1524.81020
Summary: 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.
MSC:
| 81P45 | Quantum information, communication, networks (quantum-theoretic aspects) |
| 81P16 | Quantum state spaces, operational and probabilistic concepts |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 15A67 | Applications of Clifford algebras to physics, etc. |
| 05C20 | Directed graphs (digraphs), tournaments |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
Software:
vertex-minorsReferences:
| [1] | T. A. I. M. H. Brun Devetak Hsieh, Correcting quantum errors with entanglement, Science, 314, 436-439 (2006) · Zbl 1226.81043 · 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. · Zbl 1404.81073 |
| [3] | S. R. P. K. Dutta Sarkar Panigrahi, Permutation symmetric hypergraph states and multipartite quantum entanglement, Internat. J. Theor. Phys., 58, 3927-3944 (2019) · Zbl 1428.81021 · doi:10.1007/s10773-019-04259-5 |
| [4] | F. R. Gantmakher, The Theory of Matrices, 2, AMS Chelsea Publishing Company, 2000. |
| [5] | M. W. J. R. M. H.-J. Hein Dür Eisert Raussendorf Van den Nest Briegel, Entanglement in graph states and its applications (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. · Zbl 1308.05082 |
| [8] | D. Markham and B. C. Sanders, Graph states for quantum secret sharing, Phys. Review A, 78 (2008), 042309. · Zbl 1255.81125 |
| [9] | D. W. Moore, Quantum hypergraph states in continuous variables, Phys. Rev. A, 100 (2019), 062301. |
| [10] | M. I. Nielsen Chuang, Quantum Computation and Quantum Information (2011) |
| [11] | A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995. · Zbl 0867.00010 |
| [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, 11-18 (1984) · Zbl 0556.15006 · 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. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
