Evaluation of entanglement measures by a single observable

We present observable lower bounds for several bipartite entanglement measures including entanglement of formation, geometric measure of entanglement, concurrence, convex-roof extended negativity, and $G$ 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.

Figure 1

$H\left(\vec{\mu }\right)$ vs $\lambda$. $50000$ red dots represent randomly generated states with $m=4$. The lower boundary is a smooth blue curve, which corresponds to $R\left(\lambda \right)$.

Figure 2

$G\left(\vec{\mu }\right)$ vs $\lambda$. $50000$ red dots represent randomly generated states with $m=4$. The lower boundary is a smooth blue curve, which corresponds to both $Q\left(\lambda \right)$ and $\mathrm{co}\left[Q\right(\lambda \left)\right]$, since $\mathrm{co}\left[Q\right(\lambda \left)\right]=Q\left(\lambda \right)$.

Figure 3

$L\left(\vec{\mu }\right)$ vs $\lambda$. $50000$ red dots represent randomly generated states with $m=4$. The lower boundary is a smooth blue curve which corresponds to $P\left(\lambda \right)$, and the dashed black line corresponds to $\mathrm{co}\left[P\right(\lambda \left)\right]$.

Figure 4

$S\left(\vec{\mu }\right)$ vs $\lambda$. $50000$ red dots represent randomly generated states with $m=4$. The lower boundary is a smooth blue curve which corresponds to $K\left(\lambda \right)$, and the dashed black line corresponds to $\mathrm{co}\left[K\right(\lambda \left)\right]$.