Dissociation and annihilation of multipartite entanglement structure in dissipative quantum dynamics

We study the dynamics of the entanglement structure of a multipartite system experiencing a dissipative evolution. We characterize the processes leading to a particular form of output-system entanglement and provide a recipe for their identification via concatenations of particular linear maps with entanglement-breaking operations. We illustrate the applicability of our approach by considering local and global depolarizing noises acting on general multiqubit states. A difference in the typical entanglement behavior of systems subjected to these noises is observed: the originally genuine entanglement dissociates by splitting off particles one by one in the case of local noise, whereas intermediate stages of entanglement clustering are present in the case of global noise. We also analyze the definitive phase of evolution when the annihilation of the entanglement compound finally takes place.

Figure 1

Schematic of sets ${\mathcal{S}}_{k\text{-sep}}$ (dashed) and ${\mathcal{S}}_{r\text{-ent}}$ (solid) for a four-body system.

Figure 2

Local channels: (a) general; (b) homo-geneous.

Figure 3

Elementary blocks of entanglement dissociation for the six-body system $ABCDEF$ constructed via concatenation of a linear Hermitian map $\Xi$ and measure-and-prepare (EB) operations. Semicircles and triangles depict projections onto $|{\psi }_n\rangle$ and preparations of $|{\phi }_n\rangle$ of Kraus operators $A_n\propto |{\phi }_n\rangle \langle {\psi }_n|$, respectively, and double lines depict the classical information transfer. Only one restriction is imposed: $\Xi \left[{\varrho }_{\mathrm{in}}\right]$ becomes positive semidefinite after performing the “measure” part of EB operations (red dotted compound). Partitions: (a) $A\left|B\right|C\left|D\right|E|F$, (b) $AB\left|CD\right|EF$, (c) $A\left|B\right|C|DEF$, (d) $ABC|DEF$, and (e) $A|BCDEF$.

Figure 4

Scaling of the entanglement degradation properties of an $N$-qubit local depolarizing channel ${\Phi }_q^{\otimes N}$ with increasing $N$: (a) entanglement annihilation; (b) dissociation of genuine entanglement.

Figure 5

Tracks of the typical entanglement structure dynamics subject to local depolarizing noise (green dotted line) and global depolarizing noise (purple dash-dotted line) for $N$-qubit systems: (a) $N=3$, (b) $N=4$, and (c) $N=6$. The state space $\mathcal{S}\left({\mathcal{H}}_2^{\otimes N}\right)$ is divided into areas of $k$-separable states (red dashed lines) and $r$-entangled states (blue solid lines), with representatives of the states being depicted. Stars on the tracks denote points detected in Sec. 6 and listed in Tables 1 and 2.