Strong Converse Bound on the Two-Way Assisted Quantum Capacity

We show that the max-Rains information of a quantum channel is an efficiently computable, single-letter strong converse upper bound for transmitting quantum information over quantum channels when assisted by positive-partial-transpose (PPT) preserving channels between every use of the channel. This includes in particular the quantum capacity with local operations and classical communication (LOCC) assistance. For our proof we make use of the amortized entanglement of quantum channels, which is defined as the largest net amount of entanglement that can be generated if the sender and receiver are allowed to share an arbitrary state before using the channel. Our main technical result is that amortization does not enhance the entanglement of quantum channels when entanglement is quantified by the max-Rains relative entropy. We prove this statement by employing semi-definite programming (SDP) duality and SDP formulations for the max-Rains relative entropy and the channel's max-Rains information, found recently in [Wang et al., arXiv:1709.00200].