Upper and lower bounds for Tsallis- q entanglement measure
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Upper and lower bounds for Tsallis-q entanglement measure Mahboobeh Moslehi1 · Hamid Reza Baghshahi1 · Sayyed Yahya Mirafzali1 Received: 7 May 2020 / Accepted: 30 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Quantification of entanglement is important but very difficult task because, in general, different measures of entanglement, which have been introduced to determine the degree of entanglement, cannot be calculated easily for arbitrary mixed states. Hence, in order to have an approximation of entanglement, besides the numerical methods, a series of upper and lower bounds, which can be calculated analytically, have been introduced. In this paper using upper and lower bounds of concurrence and tangle, which are two measures of entanglement, and considering the relation between these measures and Tsallis-q entanglement (an another entanglement measure), upper and lower bounds for Tsallis-q entanglement for 2⊗d bipartite mixed states are introduced. Then, by comparing these bounds with upper and lower bounds of concurrence and tangle, it is shown that, in wide range of parameters, the Tsallis-q entanglement bounds are tighter than the bounds of concurrence and tangle. Keywords Concurrence · Tangle · Tsallis-q entanglement
1 Introduction Quantum entanglement is one of the most remarkable features of quantum mechanics and is the key resource central to much of quantum information applications [1,2]. Therefore, the characterization and quantification of entanglement have become an important problem in quantum information theory [3]. So far, different measures for quantification of entanglement have been introduced which some of them, such as
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Sayyed Yahya Mirafzali [email protected] Mahboobeh Moslehi [email protected] Hamid Reza Baghshahi [email protected]
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Department of Physics, Faculty of Science, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran 0123456789().: V,-vol
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entanglement of formation [4], concurrence [5], tangle [6] and Tsallis-q entanglement [7], have been defined based on the entropy of subsystem of the bipartite quantum system [8]. For two-qubit mixed states, an analytical formula for concurrence has been derived [5]. Moreover, analytical formulas for entanglement of formation and Tsallisq entanglement of two-qubit mixed states have been found, which can be expressed as a function of the squared concurrence [8]. Nevertheless, in general, for arbitrary dimensional mixed states, analytical formulas for these measures of entanglement, as well as the other entanglement measures, have not been found. For this reason, a series of upper and lower bounds for different entanglement measures have been introduced which can give us, besides the numerical methods, an approximation of entanglement [9–12]. In particular, several upper and lower bounds for concurrence have been introduced, which some of them can be expressed in terms of the expectation values of some Hermitian operators and, hence, can be calcul
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