Variational characterizations of separability and entanglement of formation
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In this paper we develop a mathematical framework for the characterization of separability and entanglement of formation (EOF) of general bipartite states. These characterizations are variational in nature, meaning that separability and EOF are given in terms of a function that is to be minimized over the manifold of unitary matrices. A major benefit of such a characterization is that it directly leads to a numerical procedure for calculating EOF. We present an efficient minimization algorithm and apply it to the bound entangled $3\ifmmode\times\else\texttimes\fi{}3$ Horodecki states; we show that their EOF is very low and that their distance to the set of separable states is also very small. Within the same variational framework we rephrase the results by Wootters [W. Wootters, Phys. Rev. Lett. 80, 2245 (1998)] on EOF for $2\ifmmode\times\else\texttimes\fi{}2$ states and also present some progress in generalizing these results to higher-dimensional systems.
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For a suitably chosen set {zα}, the expression S1 - ∑j Sj can be significantly larger than the corresponding expression for a single matrix Tα.
“For a suitably chosen set { z α } , the expression S 1 -∑ j S j can be significantly larger than the corresponding expression for a single matrix T α [64].”
From Measures and dynamics of entangled states (2005) · cites this paper as [64]
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References 16 · Cited by 176
Research works printed but not linked: 7
- ͓3͔ M. Lewenstein, D. Bruß, J. Cirac, B. Kraus, M. Kus, J. Samsonowicz, A. Sanpera, and R. Tarrach, e-print quant-ph/0006064. · arXiv
- ͓4͔ B. Terhal, e-print quant-ph/0101032. · arXiv
- ͓8͔ B. Terhal and K. Vollbrecht, e-print quant-ph/0005062. · arXiv
- ͓12͔ A. Uhlmann, e-print quant-ph/9704017 ͑1997͒. · arXiv
- ͓17͔ J. Dehaene, Cheng Yi, and B. De Moor, IEEE Trans. Autom. Control 42, 1596 ͑1997͒.
- ͓19͔ J.-F. Cardoso, in Proceedings of ICASSP '91, ͑IEEE, Piscat- away, NJ, 1991͒, Vol. 5, pp. 3109-3112.
- ͓20͔ R. Zippel, in Proceedings of the International Symposium on Symbolic and Algebraic Manipulation (EUROSAM '79), Marseille, France, 1979, edited by E. Ng, Lecture Notes in Computer Science Vol. 72 ͑Springer, New York 1979͒, pp. 216 -226.
- Authors
- Audenaert, K., Verstraete, F., & De Moor, B.
- Affiliations
- KU Leuven (BE)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2001-10-10
- Volume
- 64
- Issue
- 5
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2048468143
- Cited by
- 176
- References
- 16
- Citations by year
- 2026: 1 · 2025: 4 · 2024: 7 · 2023: 35 · 2022: 5 · 2021: 7 · 2020: 6 · 2019: 5 · 2018: 5 · 2017: 1 · 2016: 3 · 2015: 6 · 2014: 7 · 2013: 4 · 2012: 3
- FWCI
- 5.21
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Physical Review A (publishedVersion)
- Ghent University Academic Bibliography (Ghent University) (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Ghent University Academic Bibliography (Ghent University) (submittedVersion, other-oa)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Royal Holloway Digital Repository (Royal Holloway University of London) (submittedVersion)
- arXiv (Cornell University)
- Funders
- European Commission
- Fonds Wetenschappelijk Onderzoek
- Vlaamse regering
- Eurostars
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- entanglement of formation, separability, bipartite quantum states, variational characterization, unitary matrices, minimization algorithms, bound entanglement, 2x2 systems
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