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Applications of the Nielsen’s theorem. (English) Zbl 1187.81031

Introduction: Entanglement plays a fundamental role in quantum information theory, being the key resource in many processes, such as quantum cryptography, quantum teleportation, superdense coding, quantum computation [M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, Cambridge, U. K. (2000; Zbl 1049.81015)]. Due to the fact that the entanglement is a multipartite resource shared by many observers, the most important tasks are performed by applying local operations and classical communication (LOCC). One may address the question if it is possible to transform an arbitrary bipartite entangled state into another one by LOCC. M. A. Nielsen [Phys. Rev. Lett. 83, 436–439 (1999), M. A. Nielsen and G. Vidal, Quantum Inf. Comput. 1, 76–93 (2001; Zbl 1187.81053)]. has investigated this problem and found a connection between the theory of majorization and entanglement transformations.
Another interesting problem is the relation between the entanglement of two given states and the entanglement of their superposed state. N. Linden, S. Popescu, J. A. Smolin [Phys. Rev. Lett. 97, 100502 (2006)] have recently found an upper bound on the entanglement of superpositions. This result was generalized by G. Gour [Phys. Rev. A 76, 052320 (2007)], who has shown there are lower and upper bounds on the entanglement of superpositions of two bipartite states in terms of the entanglement of the two states constituting the superposition.
Our task in this paper is to investigate the possibility of conversion between superpositions of two states by LOCC. By applying the Nielsen’s theorem, we find the necessary and sufficient conditions for this transformation to be performed.

MSC:

81P40 Quantum coherence, entanglement, quantum correlations
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