Abstract
We supplement some foundational theorems which connect majorization inequalities with quantum mechanics. To begin, we give the sufficient condition for bi-partite pure states entanglement transformation from the control to the controlled states which are impossible to be transformed even by catalysts. Furthermore, a sufficient and necessary condition about the existence of an entangled assisted state is discussed. Also, the sufficient and necessary condition to give rise to the same given ensemble for a group of vectors is demonstrated. At last, owing to the unique distinguishability of orthogonal states, it is natural to establish local distinguishability of two bi-partite orthogonal quantum states by using ensembles of majorization in LOCC.
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References
Bennett, C.H., Brassard, G., Crepeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)
Duan, R., Feng, Y., Ying, M.: Entanglement is not necessary for perfect discrimination between unitary operations. Phys. Rev. Lett. 98(10), 100503 (2007)
Duan, R., Feng, Y., Ying, M.: Local distinguishability of multipartite unitary operations. Phys. Rev. Lett. 100(2), 020503 (2008)
Duan, R., Feng, Y., Ying, M.: Perfect distinguishability of quantum operations. Phys. Rev. Lett. 103(21), 210501 (2009)
Dou, Z., Xu, G., Chen, X.B., Yuan, K.: Rational non-hierarchical quantum state sharing protocol. Comput. Mater. Contin. 58(2), 335–347 (2019)
Feng, Y., Duan, R., Ying, M.: Catalyst-assisted probabilistic entanglement transformation. IEEE Trans. Inf. Theory 51(3), 1090–1101 (2005)
Henderson, L., Vedral, V.: Classical, quantum and total correlations. Phys. Rev. A 34(35), 6899–6905 (2001)
Jonathan, D., Plenio, M.B.: Entanglement-assisted local manipulation of pure quantum states. Phys. Rev. Lett. 83(17), 3566–3569 (2012)
Jozsa, R., Linden, N.: On the role of entanglement in quantum-computational speed-up. Proc. Roy. Soc. A-Math. Phys. 459(2036), 2011–2032 (2003)
Li, Y., Qiao, Y., Wang, X., Duan, R.: Tripartite-to-bipartite entanglement transformation by stochastic local operations and classical communication and the structure of matrix spaces. Commun. Math. Phys. 358(2), 791–814 (2018). https://doi.org/10.1007/s00220-017-3077-5
Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and Its Applications. Academic Press, Cambridge (1979)
Nielsen, M.A.: Conditions for a class of entanglement transformations. Phys. Rev. Lett. 83(2), 436–439 (1998)
Nielsen, M.A.: An introduction to majorization and its applications to quantum mechanics. Lecture Notes (2002)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, 10th Anniversary edn. Cambridge University Press, Cambridge (2011)
Partovi, M.H.: Majorization formulation of uncertainty in quantum mechanics. Phys. Rev. A 84(5), 13724–13731 (2011)
Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77(8), 1413 (1996)
Sun, X., Duan, R., Ying, M.: The existence of quantum entanglement catalysts. IEEE Trans. Inf. Theory 51(1), 75–80 (2005)
Xiao, H., Zhang, J., Huang, W.H., Zhou, M., Hu, W.C.: An efficient quantum key distribution protocol with dense coding on single photons. Comput. Mater. Contin. 61(2), 759–775 (2019)
Walgate, J., Short, A.J., Hardy, L., Vedral, V.V.: Local distinguishability of multipartite orthogonal quantum states. Phys. Rev. Lett. 85(23), 4972–4975 (2000)
Zhang, S.B., Chang, Y., Yan, L., Sheng, Z.W., Yang, F.: Quantum communication networks and trust management: a survey. Comput. Mater. Contin. 61(3), 1145–1174 (2019)
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Xu, M., Liu, Z., Chen, H., Zheng, S. (2020). The Mathematical Applications of Majorization Inequalities to Quantum Mechanics. In: Sun, X., Wang, J., Bertino, E. (eds) Artificial Intelligence and Security. ICAIS 2020. Lecture Notes in Computer Science(), vol 12239. Springer, Cham. https://doi.org/10.1007/978-3-030-57884-8_46
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DOI: https://doi.org/10.1007/978-3-030-57884-8_46
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