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arXiv:0704.0757 · Physical Review A

Bounds on negativity of superpositions

Yong-Cheng Ou, Heng Fan

Published Aug 22, 2007• 42 citations
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Abstract

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.

DOI
10.1103/physreva.76.022320

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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.
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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)].
We derive the lower and upper bounds on the entanglement of a given multipartite superposition state in terms of the entanglement of the states being superposed. The first entanglement measure we use is the geometric measure and the second is the qqq-squashed entanglement. These bounds allow us to estimate the amount of the multipartite entanglement of superpositions. We also show that two states of high fidelity to one another do not necessarily have nearly the same qqq-squashed entanglement.
Given a bipartite quantum state (in arbitrary dimension) and a decomposition of it as a superposition of two others, we find bounds on the entanglement of the superposition state in terms of the entanglement of the states being superposed. In the case that the two states being superposed are biorthogonal, the answer is simple, and, for example, the entanglement of the superposition cannot be more than one ebit more than the average of the entanglement of the two states being superposed. However, for more general states, the situation is very different.
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.