Activation and superactivation of single-mode Gaussian quantum channels

Activation of quantum capacity is a surprising phenomenon according to which the quantum capacity of a certain channel may increase by combining it with another channel with zero quantum capacity. Superactivation describes an even more particular occurrence, in which both channels have zero quantum capacity, but their composition has a nonvanishing one. We investigate these effects for all single-mode phase-insensitive Gaussian channels, which include thermal attenuators and amplifiers, assisted by a two-mode positive-partial-transpose channel. Our result shows that activation phenomena are special but not uncommon. We can reveal superactivation in a broad range of thermal attenuator channels, even when the transmissivity is quite low, or the thermal noise is high. This means that we can transmit quantum information reliably through very noisy Gaussian channels with the help of another Gaussian channel, whose quantum capacity is also zero. We further show that no superactivation is possible for entanglement-breaking Gaussian channels in physically relevant circumstances by proving the nonactivation property of the coherent information of bosonic entanglement-breaking channels with finite input energy.

Figure 1

Classification of phase-insensitive single-mode Gaussian channels. Axes defined by $\tau =detX,y=\sqrt{detY}$. Physical channels (CPTP) should satisfy the relation $y\ge |\tau -1|$. EB channels are on the (orange online) region of $y\ge \left|\tau \right|+1$. All the quantities plotted are dimensionless.

Figure 2

(a) Various regions having zero capacity (gray), zero maximum coherent information (white), and positive maximum coherent information (purple online, darker shading). EB region (orange online) and NP (nonphysical) region (green online) are also specified [34]. (b) Difference between the coherent information of the combined channel and the upper bound of quantum capacity. (c), (d) Comparing the difference between the coherent information of the combined channel and the upper bound of quantum capacity or maximum coherent information in the region $\tau >0.5$. See text for further details. All the quantities plotted are dimensionless.