Abstract
We use the concept of entangled graphs with weighted edges to present a classification for four-qubit entanglement which is based neither on the LOCC nor the SLOCC. Entangled graphs, first introduced by Plesch et al. [Phys. Rev. A 67, 012322 (2003)], are structures such that each qubit of a multi-qubit system is represented as a vertex and an edge between two vertices denotes bipartite entanglement between the corresponding qubits. Our classification is based on the use of generalized Schmidt decomposition of pure states of multi-qubit systems. We show that for every possible entangled graph one can find a pure state such that the reduced entanglement of each pair, measured by concurrence, represents the weight of the corresponding edge in the graph. We also use the concept of tripartite and quadripartite concurrences as a proper measure of global entanglement of the states. In this case a circle including the graph indicates the presence of global entanglement.
Graphical abstract
Similar content being viewed by others
References
A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 (1935)
E. Schrödinger, Naturwissenschaften 23, 807 (1935)
S.M. Barnett, Quantum Information (Oxford University Press, Oxford, 2009)
E. Schmidt, Math. Ann. 63, 433 (1906)
A. Peres, Phys. Lett. A 202, 16 (1995)
A. Acín et al., Phys. Rev. Lett. 85, 1560 (2000)
A. Acín et al., J. Phys. A 34, 6725 (2001)
H.A. Carteret, A. Higuchi, A. Sudbery, J. Math. Phys. 41, 7932 (2000)
D.M. Greenberger, M. Horn, A. Zeilinger, in Bell’s Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos (Kluwer Academic Publishers, Dordrecht, Holland, 1989), pp. 69–72
W. Dür, G. Vidal, J.I. Cirac, Phys. Rev. A 62, 062314 (2000)
A. Acín et al., Phys. Rev. Lett. 87, 040401 (2001)
C. Sabín, G. García-Alcaine, Eur. Phys. J. D 48, 435 (2008)
J.I. de Vicente et al., Phys. Rev. Lett. 108, 060501 (2012)
X. Li, D. Li, Phys. Rev. A 88, 022306 (2013)
F. Verstraete et al., Phys. Rev. A 65, 052112 (2002)
L. Lamata et al., Phys. Rev. A 75, 022318 (2007)
Y. Cao, A.M. Wang, Eur. Phys. J. D 44, 159 (2007)
D. Li et al., Quant. Inf. Comput. 9, 0778 (2009)
L. Borsten et al., Phys. Rev. Lett. 105, 100507 (2010)
E. Jung, D.K. Park, Quant. Inf. Comput. 14, 0937 (2014)
D.K. Park, Phys. Rev. A 89, 052326 (2014)
W. Dür, Phys. Rev. A 63, 020303(R) (2001)
M. Plesch, V. Bužek, Phys. Rev. A 67, 012322 (2003)
M. Plesch et al., J. Phys. A 37, 1843 (2004)
W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)
P.J. Love et al., Quant. Inf. Process. 6, 187 (2007)
V. Coffman, J. Kundu, W.K. Wootters, Phys. Rev. A 61, 052306 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ghahi, M., Akhtarshenas, S. Entangled graphs: a classification of four-qubit entanglement. Eur. Phys. J. D 70, 54 (2016). https://doi.org/10.1140/epjd/e2016-60729-1
Received:
Published:
DOI: https://doi.org/10.1140/epjd/e2016-60729-1