Constrained bounds on measures of entanglement

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.

Figure 1

(Color online) A histogram of the difference ${\widehat{n}}_{\Phi }\text{−}{n}_{\Phi }$ for five million $4\times 4$ pure states picked randomly from the Haar measure. The bin size in the histogram is 0.001. The difference is always found to be positive. These results carry over to $4N$ states when $N>4$ ($N$ even), since the difference only depends on the Schmidt coefficients. We have tried to prove that the difference is non-negative without success and thus rely on this numerical demonstration instead.

Figure 2

On the top is the bound on the EOF based on a constrained ${\widehat{n}}_{\Phi }$, Eq. (4.8). The plot on the bottom is the bound on the EOF based on a constrained negativity, Eq. (4.12).

Figure 3

In Region 1, the singly constrained $n_{\Phi }$ bound is better than the singly constrained $n_T$ bound. In Region 2, the opposite is true.

Figure 4

The plot on the top shows the bounds on the tangle and the concurrence based on the $n_{\Phi }$ constraint. The solid line is the bound on the tangle and the dashed line is the bound on the concurrence. On the bottom is a plot of the analogous bounds based on the $n_T$ constraint.

Figure 5

Region 1 is where the $n_{\Phi }$ constraint is better than the $n_T$ constraint for bounding the tangle and concurrence. Region 2 is where the converse is true.

Figure 6

The pure-state region in the ${\widehat{n}}_{\Phi }\text{−}{n}_T$ plane for $4N$ systems.

Figure 7

The part of the two-constraint region in which a value for $\tilde{H}\left(\mathbf{n}\right)$ can be computed is shown for four values of ${\mu }_4=0$, 0.02, 0.1, and 0.25. The two lines are the boundaries of the pure-state region.

Figure 8

(Color online) Plots of $\tilde{H}\left(\mathbf{n}\right)$, the minimum of the entropy of formation, $H\left(\mathbf{\mu }\right)$, in the pure-state region. On the left side is a three-dimensional plot of $\tilde{H}\left(\mathbf{n}\right)$ and on the right is a contour plot of the same function.

Figure 9

(Color online) The doubly constrained bound $\mathcal{H}\left(\mathbf{n}\right)$ on the EOF of all $4N$ states. On the right side is a contour plot of the same function.

Figure 10

The thick black line is the doubly constrained bound on the EOF for isotropic states. The dashed white line, lying on top of the black line, is the singly constrained bound $\mathcal{H}\left(n_T\right)$ from Eq. (4.12).

Figure 11

(Color online) The doubly constrained bound $\mathcal{T}\left(\mathbf{n}\right)$ on the tangle of $4N$ states. On the right side is a contour plot of the same function.

Figure 12

(Color online) The doubly constrained bound $\mathcal{C}\left(\mathbf{n}\right)$ on the concurrence of $4N$ states. On the right side is a contour plot of the same function.