Channel capacities versus entanglement measures in multiparty quantum states

For quantum states of two subsystems, highly entangled states have a higher capacity of transmitting classical as well as quantum information, and vice versa. We show that this is no more the case in general: Quantum capacities of multiaccess channels, motivated by communication in quantum networks, do not have any relation with genuine multiparty entanglement measures. Importantly, the statement is demonstrated for arbitrary multipartite entanglement measures. Along with revealing the structural richness of multiaccess channels, this gives us a tool to classify multiparty quantum states from the perspective of its usefulness in quantum networks, which cannot be visualized by any genuine multiparty entanglement measure.

Figure 1

From RVB state to ferromagnet. The unassisted capacity ${\mathcal{C}}_{\mathit{ua}}$ (in ebits) is plotted on the vertical axis against a base consisting of the measurement parameter $x$ (for arbitrary projective measurements at two parties in the basis $\{|\underset{¯}{x}\rangle ,|\underset{¯}{x}\rangle {}^{\perp }\}$, with $|\underset{¯}{x}\rangle =\cos x|0\rangle +\exp (i\phi )\sin x|1\rangle$, and $|\underset{¯}{x}\rangle {}^{\perp }=\exp (-i\phi )\sin x|0\rangle -\cos x|1\rangle$) and the state parameter $\mu$, for the family of four-qubit quantum states $|{\varphi }_{\mathrm{RF}}^{\mu }\rangle$. The base variables $x$ and $\mu$ are dimensionless. The capacity for this strategy can be readily read off from the figure for any $\mu$. For example, for the RVB state, situated at $\mu =0$, the capacity is $1/3$ ebit, while for the ferromagnetic state at $\mu \to \infty$, the corresponding quantity is unit ebit.

Figure 2

The capacities and the measures. The assisted capacities (triangles), unassisted capacities (boxes), generalized geometric measures (hexagons), and geometric measures (stars) are plotted on the vertical axis, for a selection of four-party quantum states, that are important from a quantum networks perspective, on the horizontal axis. See text for the definitions of the states on the horizontal axis. While the capacities are measured in ebits, the measures are dimensionless.