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arXiv:quant-ph/0612049 · Physical Review A

Constrained bounds on measures of entanglement

Animesh Datta, Steven T. Flammia, Anil Shaji, Carlton M. Caves

Published Jun 28, 2007• 11 citations
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Abstract

Entanglement measures constructed from two positive, but not completely positive, maps on density operators are used as constraints in placing bounds on the entanglement of formation, the tangle, and the concurrence of $4N$ mixed states. The maps are the partial transpose map and the phi map introduced by Breuer [H.-P. Breuer, Phys. Rev. Lett. 97, 080501 (2006)]. The norm-based entanglement measures constructed from these two maps, called negativity and phi negativity, respectively, lead to two sets of bounds on the entanglement of formation, the tangle, and the concurrence. We compare these bounds and identify the sets of $4N$ density operators for which the bounds from one constraint are better than the bounds from the other. In the process, we present a derivation of the already known bound on the concurrence based on the negativity. We compute bounds on the three measures of entanglement using both the constraints simultaneously. We demonstrate how such doubly constrained bounds can be constructed. We discuss extensions of our results to bipartite states of higher dimensions and with more than two constraints.

DOI
10.1103/physreva.75.062117

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Entanglement measures constructed from two positive, but not completely positive, maps on density operators are used as constraints in placing bounds on the entanglement of formation, the tangle, and the concurrence of 4N4N4N mixed states. The maps are the partial transpose map and the phi map introduced by Breuer [H.-P. Breuer, Phys. Rev. Lett. 97, 080501 (2006)]. The norm-based entanglement measures constructed from these two maps, called negativity and phi negativity, respectively, lead to two sets of bounds on the entanglement of formation, the tangle, and the concurrence. We compare these bounds and identify the sets of 4N4N4N density operators for which the bounds from one constraint are better than the bounds from the other. In the process, we present a derivation of the already known bound on the concurrence based on the negativity. We compute bounds on the three measures of entanglement using both the constraints simultaneously. We demonstrate how such doubly constrained bounds can be constructed. We discuss extensions of our results to bipartite states of higher dimensions and with more than two constraints.
Employing a recently proposed separability criterion we develop analytical lower bounds for the concurrence and for the entanglement of formation of bipartite quantum systems. The separability criterion is based on a nondecomposable positive map which operates on state spaces with even dimension N >= 4, and leads to a class of nondecomposable optimal entanglement witnesses. It is shown that the bounds derived here complement and improve the existing bounds obtained from the criterion of positive partial transposition and from the realignment criterion.
We discuss properties of entanglement measures called I-concurrence and tangle. For a bipartite pure state, I-concurrence and tangle are simply related to the purity of the marginal density operators. The I-concurrence (tangle) of a bipartite mixed state is the minimum average I-concurrence (tangle) of ensemble decompositions of pure states of the joint density operator. Terhal and Vollbrecht [Phys. Rev. Lett. 85, 2625 (2000)] have given an explicit formula for the entanglement of formation of isotropic states in arbitrary dimensions. We use their formalism to derive comparable expressions for the I-concurrence and tangle of isotropic states.
We study the entanglement of general (pure or mixed) two-mode Gaussian states of continuous-variable systems by comparing the two available classes of computable measures of entanglement: entropy-inspired Gaussian convex-roof measures and positive partial transposition--inspired measures (negativity and logarithmic negativity). We first review the formalism of Gaussian measures of entanglement, adopting the framework introduced in M. M. Wolf et al., Phys. Rev. A 69, 052320 (2004), where the Gaussian entanglement of formation was defined. We compute explicitly Gaussian measures of entanglement for two important families of nonsymmetric two-mode Gaussian state: namely, the states of extremal (maximal and minimal) negativities at fixed global and local purities, introduced in G. Adesso et al., Phys. Rev. Lett. 92, 087901 (2004). This analysis allows us to compare the different orderings induced on the set of entangled two-mode Gaussian states by the negativities and by the Gaussian measures of entanglement. We find that in a certain range of values of the global and local purities (characterizing the covariance matrix of the corresponding extremal states), states of minimum negativity can have more Gaussian entanglement of formation than states of maximum negativity. Consequently, Gaussian measures and negativities are definitely inequivalent measures of entanglement on nonsymmetric two-mode Gaussian states, even when restricted to a class of extremal states. On the other hand, the two families of entanglement measures are completely equivalent on symmetric states, for which the Gaussian entanglement of formation coincides with the true entanglement of formation. Finally, we show that the inequivalence between the two families of continuous-variable entanglement measures is somehow limited. Namely, we rigorously prove that, at fixed negativities, the Gaussian measures of entanglement are bounded from below. Moreover, we provide some strong evidence suggesting that they are as well bounded from above.
In this paper we investigate two different entanglement measures in the case of mixed states of two qubits. We prove that the negativity of a state can never exceed its concurrence and is always larger than √[(1 − C ) 2 + C 2 ] − (1 − C ), where C is the concurrence of the state. Furthermore, we derive an explicit expression for the states for which the upper or lower bound is satisfied. Finally we show that similar results hold if the relative entropy of entanglement and the entanglement of formation are compared.
It is well known that for two qubits the upper bounds of the relative entropy of entanglement (REE) for a given concurrence as well as the negativity for a given concurrence are reached by pure states. We show that, by contrast, there are two-qubit mixed states for which the REE for some range of a fixed negativity is higher than that for pure states. Moreover, we demonstrate that a mixture of a pure entangled state and pure separable state orthogonal to it is likely to give the maximal REE. By noting that the negativity is a measure of entanglement cost under operations preserving positivity of partial transpose, our results provide an explicit example of operations such that, even though the entanglement cost for an exact preparation is the same, the entanglement of distillation of a mixed state can exceed that of pure states. This means that the entanglement manipulation via a pure state can result in a larger entanglement loss than that via a mixed state.