Energy-Constrained Two-Way Assisted Private and Quantum Capacities of Quantum Channels*

With the rapid growth of quantum technologies, knowing the fundamental characteristics of quantum systems and protocols is essential to their effective implementation. The maximum rates at which a quantum channel can reliably transfer private and quantum information are respectively called the private and quantum capacities. Our contribution begins by formalizing the notion of energy-constrained private and quantum communication with the assistance of local operations and classical communication (LOCC). We then define the energy-constrained squashed entanglement of a channel and prove that it upper bounds the energy-constrained LOCC-assisted private and quantum capacities of an arbitrary channel with an arbitrary Hamiltonian when the channel and Hamiltonian are subject to certain physically well motivated conditions. We shift our focus to actual bosonic channels as an example of our general theory and prove that the two-mode squeezed vacuum optimizes the squashed entanglement for any single-mode phase-insensitive Gaussian channel whenever the squashing channel is constrained to be phase-insensitive and Gaussian. Finally, we extend the theory to the broadcast setting with one sender and multiple receivers.