Method of developing analytical multipartite delocalization measures for mixed $W$-like states

We present a method of developing analytical measures of $k\text{−}\mathrm{partite}$ delocalization in arbitrary $n\text{−}\mathrm{body}$ $W$-like states, otherwise known as mixed states in the single-excitation subspace. These measures calculate the distance of a state to its closest reference state with $k-1$ entanglement. We find that the reference state is determined by the purity of the state undergoing measurement. Measures with up to 6-body delocalization for a 6-body system are derived in full, while an algorithm for general $k\text{−}\mathrm{partite}$ measures is given.

Figure 1

Distribution of ${\tau }_2$ vs purity for a set of 1 million randomly generated 3-body states (blue [gray], upper right corner) and 1 million randomly generated reference (separable) states (black diagonal line). The closest reference states are those with the same level of purity. Overlayed is a small circle indicating an example state, with a dashed line leading to its nearest separable state.

Figure 2

Distribution of ${\tau }_3$ vs purity for a set of 1 million randomly generated 3-body states (red [gray], upper right corner) and 1 million randomly generated reference (biseparable) states (blue [gray], center). The closest reference states lie along the border between these two regions (denoted by a black curve), running from $\text{purity}=1$ down to $\text{purity}=5/9$. Overlayed is a small circle indicating an example state, with a dashed line leading to its nearest biseparable state.

Figure 3

Early dephased evolution of multipartite delocalization in a 6-body system over the first 50 time steps. The inset shows the long-term evolution of multipartite delocalization. All measures have been normalized.