Energy-constrained LOCC-assisted quantum capacity of bosonic dephasing channel

We study the LOCC-assisted quantum capacity of bosonic dephasing channel with energy constraint on input states. We start our analysis by focusing on the energy-constrained squashed entanglement of the channel, which is an upper bound for the energy-constrained LOCC-assisted quantum capacity. As computing energy-constrained squashed entanglement of the channel is challenging due to a double optimization (over the set of density matrices and the isometric extensions of a squashing channel), we first derive an upper bound for it, and then we discuss how tight that bound is for energy-constrained LOCC-assisted quantum capacity of bosonic dephasing channel. We prove that the optimal input state is diagonal in the Fock basis. Furthermore, we prove that for a generic channel, the optimal squashing channel belongs to the set of symmetric quantum Markov chain inducer (SQMCI) channels of the channel system-environment output, provided that such a set is non-empty. With supporting arguments, we conjecture that this is instead the case for the bosonic dephasing channel. Hence, for it we analyze two explicit examples of squashing channels which are not SQMCI, but are symmetric. Through them, we derive explicit upper and lower bounds for the energy-constrained LOCC-assisted quantum capacity of the bosonic dephasing channel in terms of its quantum capacity with different noise parameters. As the difference between upper and lower bounds is at most of the order $10^{-1}$, we conclude that the bounds are tight. Hence we provide a very good estimation of the LOCC-assisted quantum capacity of the bosonic dephasing channel.