Geometry on the manifold of Gaussian quantum channels

In the space of quantum channels, we establish the geometry that allows us to make statistical predictions about relative volumes of entanglement breaking channels among all the Gaussian quantum channels. The underlying metric is constructed using the Choi-Jamiołkowski isomorphism between the continuous-variable Gaussian states and channels. This construction involves the Hilbert-Schmidt distance in quantum state space. The volume element of the one-mode Gaussian channels can be expressed in terms of local symplectic invariants. We analytically compute the relative volumes of the one-mode Gaussian entanglement breaking and incompatibility breaking channels. Finally, we show that, when given the purities of the Choi-Jamiołkowski state of the channel, one can determine whether or not such channel is incompatibility breaking.

Figure 1

The range of $detM$ and $detN$ for which the complete positivity (gray), entanglement breaking (double-hatched), or incompatibility breaking (single-hatched) conditions are satisfied.

Figure 2

The relative volume of the entanglement breaking (dashed line) and incompatibility breaking (solid line) one-mode Gaussian channels as a function of the marginal purity of the CJ state.

Figure 3

The separability (double-hatched), coexistence (single-hatched), and entanglement (unhatched) regions for the two-mode Gaussian CJ states for different marginal purities: ${\mu }_{\sigma }=0.2$ (top), ${\mu }_{\sigma }=0.5$ (center), and ${\mu }_{\sigma }=0.8$ (bottom). The red line separates the steerable and nonsteerability states. The shading in the coexistence region is associated with the conditional relative volume of entangled CJ states. In terms of the one-mode channels, the regions correspond to the incompatibility breaking channels (nonsteerable CJ states), entanglement breaking channels (double-hatched region), and the coexistence of entanglement breaking and not-entanglement-breaking channels (single-hatched region).