Hostname: page-component-7479d7b7d-qs9v7 Total loading time: 0 Render date: 2024-07-10T22:51:22.688Z Has data issue: false hasContentIssue false
  • English
  • Français

Combinatorial laplacians and positivity under partial transpose

Published online by Cambridge University Press:  01 February 2008

ROLAND HILDEBRAND ,
STEFANO MANCINI  and
SIMONE SEVERINI
ROLAND HILDEBRAND
Affiliation:
LJK Université Joseph Fourier, Tour IRMA, 51 rue des Mathématiques, 38400 St. Martin d'Héres, France
STEFANO MANCINI
Affiliation:
Dipartimento di Fisica, Universitá di Camerino, Via Madonna delle Carceri 9, 62032 Camerino, Italy
SIMONE SEVERINI
Affiliation:
Department of Computer Science, University of York, Heslington, Y010 5DD York, United Kingdom
Get access Rights & Permissions [Opens in a new window]

Abstract

The density matrices of graphs are combinatorial laplacians normalised to have trace one (Braunstein et al. 2006b). If the vertices of a graph are arranged as an array, its density matrix carries a block structure with respect to which properties such as separability can be considered. We prove that the so-called degree-criterion, which was conjectured to be necessary and sufficient for the separability of density matrices of graphs, is equivalent to the PPT-criterion. As such, it is not sufficient for testing the separability of density matrices of graphs (we provide an explicit example). Nonetheless, we prove the sufficiency when one of the array dimensions has length two (see Wu (2006) for an alternative proof). Finally, we derive a rational upper bound on the concurrence of density matrices of graphs and show that this bound is exact for graphs on four vertices.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 18 , Special Issue 1: in memory of Sauro Tulipani , February 2008 , pp. 205 - 219
Copyright
Copyright © Cambridge University Press2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alber, G. et al. (eds.) (2001) Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments. Springer-Verlag Tracts in Modern Physics 173.Google Scholar
Bennett, C. H., DiVincenzo, D. P., Smolin, J. A. and Wootters, W. K. (1996) Mixed-state entanglement and quantum error correction. Phys. Rev. A 54 (5)3824–3851. (quant-ph/9604024.)CrossRefGoogle ScholarPubMed
Braunstein, S. L., Ghosh, S. and Severini, S. (2007) The laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states. (To appear in Ann. Comb.) (quant-ph/0406165.)Google Scholar
Braunstein, S. L., Ghosh, S., Mansour, T.Severini, S. and Wilson, R. C. (2006) Some families of density matrices for which separability is easily tested. Phys. Rev. A 73 1012320. (quant-ph/0508020.)CrossRefGoogle Scholar
Bruß, D. (2002) Characterizing entanglement. Quantum information theory. J. Math. Phys. 43 94237–4251. (quant-ph/0110078.)CrossRefGoogle Scholar
Chen, K. and Wu, L.-A. (2003) A matrix realignment method for recognizing entanglement. Quantum Inf. and Comput. 3 3193–202. (quant-ph/0205017.)Google Scholar
Duan, L. M., Giedke, G., Cirac, J. I. and Zoller, P. (2000) Inseparability criterion for continuous variable systems. Phys. Rev. Lett. 84 2722–2725. (quant-ph/9908056.)CrossRefGoogle ScholarPubMed
Ghosh, S. (2006) Personal communication.Google Scholar
Godsil, C. and Royle, G. (2001) Algebraic graph theory. Springer-Verlag Graduate Texts in Mathematics 207.CrossRefGoogle Scholar
Grötschel, M., Lovász, L. and Schrijver, A. (1988) Geometric Algorithms and Combinatorial Optimization, Springer-Verlag.CrossRefGoogle Scholar
Gurvits, L. (2003) Classical deterministic complexity of Edmonds' Problem and quantum entanglement. Proceeding of the thirty-fifth ACM Symposium on Theory of Computing, ACM Press 10–19. (quant-ph/0303055.)Google Scholar
Hildebrand, R. (2006) Concurrence of Lorentz-positive maps. (quant-ph/0612064.)Google Scholar
Hill, S. and Wootters, W. K. (1997) Entanglement of a Pair of Quantum Bits. Phys. Rev. Lett. 78 5022–5025.CrossRefGoogle Scholar
Horodecki, M., Horodecki, P. and Horodecki, R. (1996) Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223 (1–2)1–8. (quant-ph/9605038.)CrossRefGoogle Scholar
Horodecki, P. and Lewenstein, M. (2000) Bound entanglement and continuous variables. Phys. Rev. Lett. 85 132657–2660. (quant-ph/0001035.)CrossRefGoogle ScholarPubMed
Ioannou, L. M. (2006) Deterministic Computational Complexity of the Quantum Separability Problem. (quant-ph/0603199.)Google Scholar
Li, Y.-Q. and Zhu, G.-Q. (2003) Concurrence Vectors for Entanglement of High-dimensional Systems. (quant-ph/0308139.)Google Scholar
Mintert, F., Kuś, M. and Buchleitner, A. (2005) Concurrence of mixed multipartite quantum states. Phys. Rev. Lett. 95 26260502. (quant-ph/0411127.)CrossRefGoogle ScholarPubMed
Mohar, B. (1988) The Laplacian spectrum of graphs. In: Graph theory, combinatorics, and applications, Vol. 2, Kalamazoo.Google Scholar
Nielsen, M. and Chuang, I. (2000) Quantum Computation and Quantum Information, Cambridge University Press.Google Scholar
Peres, A. (1996) Separability criterion for density matrices, Phys. Rev. Lett. 77 81413–1415. (quant-ph/9604005.)CrossRefGoogle ScholarPubMed
Plenio, M. B. and Virmani, S. (2005) An introduction to entanglement measures. (quant-ph/0504163.)CrossRefGoogle Scholar
Rudolph, O. (2005) Further results on the cross norm criterion for separability. Quantum Inf. Process. 4 3219–239. (quant-ph/0202121.)CrossRefGoogle Scholar
Rungta, P., Bužek, V., Caves, C. M.Hillery, M. and Milburn, G. J. (2001) Universal state inversion and concurrence in arbitrary dimensions. Phys. Rev. A 64 4042315. (quant-ph/0102040.)CrossRefGoogle Scholar
Simon, R. (2000) Peres-Horodecki Separability Criterion for Continuous Variable Systems. Phys. Rev. Lett. 84 2726–2729. (quant-ph/9909044.)CrossRefGoogle ScholarPubMed
Mancini, S. and Severini, S. (2007) The Quantum Separability Problem for Gaussian States, In: Proceedings of Logic, Models and Computer Science 2006. Electronic Notes in Theoretical Computer Science 169 121–127. (cs.CC/0603047.)CrossRefGoogle Scholar
Vidal, G. and Werner, R. (2002) Computable measure of entanglement. Phys. Rev. A 65 032314.CrossRefGoogle Scholar
Vollbrecht, K. G. H. and Werner, R. (2001) Entanglement measures under symmetry. Phys. Rev. A 64 062307. (quant-ph/0010095.)CrossRefGoogle Scholar
Wu, C.-W. (2006) Conditions for separability in generalized Laplacian matrices and diagonally dominant matrices as density matrices. Phys. Lett. A 351 (1-2)18–22. (quant-ph/0508163.)CrossRefGoogle Scholar
Wootters, W. K. (1998) Entanglement of Formation of an Arbitrary State of Two Qubits. Phys. Rev. Lett. 80 2245–2248.CrossRefGoogle Scholar