Towards Efficient Counter Example to Additivity of Minimum Output Renyi $p$ Entropy

Research in quantum information theory suggests that if a quantum channel is used to transmit classical information then the capacity (also referred to as classical capacity) of a quantum channel is super additive. This has evolved as a result of counter examples to additivity conjecture of quantum information theoretic quantities and their relation with one other.

In this talk, we discuss the counter example to additivity conjecture of minimal output Renyi p entropy of a quantum channel, which was given by Hayden and Winter. But the technique used by them doesn't give an efficient implementation of the underlying quantum channel, in the sense that their analysis is based on selecting channel unitaries at random according to Haar measure, which cannot be constructed efficiently.

We discuss the same counter example but the channel unitaries are now chosen from a finite size approximate t design which have efficient implementations. This is based on a geometric functional analytic result called Dvoretzky's theorem, as the additivity conjecture is equivalent to the multiplicativity of super operator norm which has a functional analytic form.