A conjecture on the absolute separability of eight by eight positive semi-definite matrices
DOI:
https://doi.org/10.62051/tdrp2f37Keywords:
Separability, Positive Semidefinite Matrix, Kronecker Product.Abstract
The separability problem, a long-standing open challenge in mathematics and quantum information theory. In this work, we propose and investigate a conjecture as a sub-question of the separability problem. Specifically, we investigate whether the form is separable for any eight-dimensional unit vector . This is actually the so-called absolute separability problem. For studying the conjecture, we classify the vectors in a finitely-dimensional bipartite Hilbert space, and construct their equivalence classes in terms of the so-called Schmidt rank, under both product general linear and product linear groups. Then we construct seven equivalence classes of , using which we partially prove the conjecture. Our results highlight the the complexity of the difference between bipartite and tripartite spaces.
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[1] P. Horodecki., “Separability criterion and inseparable mixed states with positive partial transposition.” Physics Letters A, vol. 232, no. 5, pp. 333–339, 1997. DOI: https://doi.org/10.1016/S0375-9601(97)00416-7
[2] R. Lockhart, “Optimal ensemble length of mixed separable states,” Journal of Mathematical Physics, vol. 41, no. 10, pp. 6766–6771, 2000. DOI: https://doi.org/10.1063/1.1290055
[3] S. Halder, S. Mal, A. Sen, et al., “Characterizing the boundary of the set of absolutely separable states and their generation via noisy environments,” Physical Review A, vol. 103, no. 5, p. 052431, 2021. DOI: https://doi.org/10.1103/PhysRevA.103.052431
[4] E. Størmer., “Separable states and positive maps.” Journal of Functional Analysis, vol. 254, no. 8, pp. 2303–2312, 2008. DOI: https://doi.org/10.1016/j.jfa.2007.12.017
[5] S. N. Filippov, K. Y. Magadov, and M. A. Jivulescu, “Absolutely separating quantum maps and channels,” New Journal of Physics, vol. 19, no. 8, p. 083010, 2017. DOI: https://doi.org/10.1088/1367-2630/aa7e06
[6] K.-C. Ha and S.-H. Kye, “Separable states with unique decompositions,” ArXiv e-prints, Oct. 2012, Available: https://arxiv.org/abs/1210.1088
[7] L. Gurvits and H. Barnum, “Largest separable balls around the maximally mixed bipartite quantum state,” Physical Review A, vol. 66, no. 6, p. 062311, 2002. DOI: https://doi.org/10.1103/PhysRevA.66.062311
[8] N. Johnston, “Separability from spectrum for qubit-qudit states,” Physical Review A, vol. 88, no. 6, p. 062330, 2013. DOI: https://doi.org/10.1103/PhysRevA.88.062330
[9] R. Hildebrand, “Positive partial transpose from spectra,” Physical Review A, vol. 76, no. 5, p. 052325, 2007. DOI: https://doi.org/10.1103/PhysRevA.76.052325
[10] S. Arunachalam, N. Johnston, and V. Russo, “Is absolute separability determined by the partial transpose?” arXiv preprint arXiv:1405.5853, 2014.
[11] M. A. Jivulescu, N. Lupa, I. Nechita, and D. Reeb, “Positive reduction from spectra,” Linear Algebra and its Applications, vol. 469, pp. 276–304, 2015. DOI: https://doi.org/10.1016/j.laa.2014.11.031
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