Phase Transition in the Quantum Capacity of Quantum Channels

Despite years of extensive research into quantum channels, determining their capacities remains an extraordinarily challenging problem, even though it is of fundamental importance. Establishing precise lower and upper bounds for these capacities has proven to be difficult, making it essential to at least identify conditions under which the quantum capacity of a channel vanishes. In this paper, we prove that the quantum capacity of any quantum channel Λ, when contaminated by white noise, as described by Λₓ (ρ):=(1-x)Λ(ρ)+x/dId becomes zero when x≥ 1/2. To prove this, we first show that the depolarizing channel with x≥ 1/2 is anti-degradable, and we provide the exact form of the channel that connects it to its complement. The zero-capacity region we establish for this channel is significantly larger than the previously known entanglement-binding threshold, which depends on dimension and is given by x≥ d/d+1.