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arXiv:2005.01881 · Modern Physics Letters A

Lower bounds on concurrence and negativity from a trace inequality

K. V. Antipin

Published Aug 11, 2020• 5 citations
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Abstract

For bipartite quantum states we obtain lower bounds on two important entanglement measures, concurrence and negativity, studying the inequalities for the expectation value of a projector on some subspace of the Hilbert space. Several applications, including analysis of stability of entanglement under various perturbations of a state, are discussed.

DOI
10.1142/s0217732320502545

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For bipartite quantum states we obtain lower bounds on two important entanglement measures, concurrence and negativity, studying the inequalities for the expectation value of a projector on some subspace of the Hilbert space. Several applications, including analysis of stability of entanglement under various perturbations of a state, are discussed.
We present observable lower bounds for several bipartite entanglement measures including entanglement of formation, geometric measure of entanglement, concurrence, convex-roof extended negativity, and GGG concurrence. The lower bounds facilitate estimates of these entanglement measures for arbitrary finite-dimensional bipartite states. Moreover, these lower bounds can be calculated analytically from the expectation value of a single observable. Based on our results, we use several real experimental measurement data to get lower bounds of entanglement measures for these experimentally realized states. In addition, we also study the relations between entanglement measures.
For pure bipartite superposed states, the entanglement quantified by negativity is studied. If the entanglement is quantified by concurrence, we show that two pure states with high fidelity to one another have nearly the same entanglement. We deduce an inequality in which the concurrence is known to be a continuous function in infinite dimensions. The main result of this paper is to give the bounds on the negativity of a bipartite state in terms of the entanglement of the states being superposed. These bounds may be used in estimating the entanglement of a given state.
Abstract For a multipartite quantum state, the maximal violation of all Bell inequalities constitutes a measure of its nonlocality (Loubenets 2012 J . Math . Phys . 53 , 022201). In the present article, for the maximal violation of Bell inequalities by a pure bipartite state, possibly infinite-dimensional, we derive a new upper bound expressed in terms of the Schmidt coefficients of this state . This new upper bound allows us also to specify general analytical relations between the maximal violation of Bell inequalities by a bipartite quantum state, pure or mixed, and such entanglement measures for this state as ‘negativity’ and ‘concurrence’. To our knowledge, no any general analytical relations between measures for bipartite nonlocality and entanglement have been reported in the literature though, for a general bipartite state, specifically such relations are important for the entanglement certification and quantification scenarios. As an example, we apply our new results to finding upper bounds on nonlocality of bipartite coherent states intensively discussed last years in the literature in view of their experimental implementations.
The entanglement content of superpositions of quantum states is investigated based on a measure called concurrence. Given a bipartite pure state in arbitrary dimension written as the quantum superposition of two other such states, we find simple inequalities relating the concurrence of the state to that of its components. We derive an exact expression for the concurrence when the component states are biorthogonal and provide elegant upper and lower bounds in all other cases. For quantum bits, our upper bound is tighter than the previously derived bound [N. Linden et al., Phys. Rev. Lett. 97, 100502 (2006)].
Concurrence is one of the most well-known entanglement measures. We show that, for arbitrary bipartite states, the lower bound of concurrence induced by symmetric measurements is always tighter than the one obtain from realignment. Besides, we derive an extended version of lower bound of concurrence induced by symmetric measurements, which can also be experimentally identified without state tomography, and further present a corresponding lower bound of genuine tripartite entanglement concurrence.