Parametrized entanglement monotone

Entanglement concurrence has been widely used for featuring entanglement in quantum experiments. As an entanglement monotone it is related to specific quantum Tsallis entropy. Our goal in this paper is to propose a parametrized bipartite entanglement monotone which is named as $q$-concurrence inspired by general Tsallis entropy. We derive an analytical lower bound for the $q$-concurrence of any bipartite quantum entanglement state by employing positive partial transposition criterion and realignment criterion, which shows an interesting relationship to the strong separability criteria. The parametrized entanglement monotone is used to characterize bipartite isotropic states. Finally, we provide a computational method to estimate the $q$-concurrence for any entanglement by superposing two bipartite pure states. It shows that the superposition operations can at most increase one ebit for the $q$-concurrence in the case that the two states being superposed are biorthogonal or one-sided orthogonal. These results reveal a series of phenomena about the entanglement, which may be interesting in quantum communication and quantum information processing.

Figure 1

Schematic six-body system with long-range correlations. The dotted line denotes a long-range correlation between two particles. Two gray regions represent a bipartite cut of $\{1,2,3\}$ and $\{4,5,6\}$.

Figure 2

Comparison of the $q$-concurrence and Tsallis-$q$ entanglement in terms of $q$. Here, we consider three evolution systems with 5, 6, and 7 qubits.

Figure 3

2-concurrence (green) of isotropic states ${\rho }_F$ and lower bounds (blue) in Example 1. (a) $3\le d\le 10$ and $1/d\le F\le 4(d-1)/d^2$. (b) $3\le d\le 10$ and $4(d-1)/d^2\le F\le 1$.

Figure 4

Increase $\mathrm{\Delta }{C}_q\left(\right|\mathrm{\Gamma }\rangle )$ of the $q$-concurrence for the superposition state $|\mathrm{\Gamma }\rangle$ in Example 2. Here, $\alpha =\beta =1/\sqrt{2}$ and $q=4$.

Figure 5

Increase of the $q$-concurrence $\mathrm{\Delta }{C}_q\left(\right|\mathrm{\Gamma }\rangle )$ for the superposition state $|\mathrm{\Gamma }\rangle$ in Example 3. Here, $\alpha =\beta =1/\sqrt{2}$ and $q=6$.

Figure 6

$q$-concurrence (blue) and upper bounds (green) of the superposition state $|\mathrm{\Gamma }\rangle$ in Example 4. Here, $2\le q\le 4$ and $\theta \in (0,\pi /2)$. The upper bound of the $q$-concurrence is restricted to be no larger than 1.

Figure 7

Schematic three types of channels. (a) Quantum channels ${\varepsilon }_1$ and ${\varepsilon }_2$. (b) The classical mixing channel ${\varepsilon }_3$. (c) Quantum superposition channel ${\varepsilon }_4$. Here, the input $a_i$ is sent through the channel ${\varepsilon }_i$ corresponding to the output $b_i$.