One success of the probabilistic method in quantum information theory is Hastings’s example of quantum channels, for which the capacity to transmit information is non-additive. The exegesis of that example has generated lot of activity in the community.

Here we consider a related question: for , can we find two quantum channels and such that

This question is well-known to be much simpler than the one solved by Hastings (which corresponds to ). However, one can make it a bit harder by demanding some extra properties from the channels , for example of being *PPT-inducing *(meaning that applying the channel on one part on a bipartite system always outputs PPT states). Existence of such channels was proved by Moto Fukuda and Ion Nechita for by using tools from free probability. We noticed recently that one can push down to if we use Asymptotic Geometric Analysis instead.

One motivation for such a question is to understand how PPT-inducing channels are different from entanglement-breaking channels (i.e., channels which output separable states), as the is always multiplicative for them. One famous open problem in this direction is the “PPT squared problem”: must the composition of two PPT-inducing channels be entanglement-breaking ?

As often in such questions, the example is obtained most efficiently by taking , with induced by a random isometric embedding . It remains to find the right dimensions: if we set and , one can check (applying Dvoretzky’s theorem for the Schatten -norm) that the “non-multiplicativity” holds provided , while the resulting channel will be PPT-inducing provided (this is essentially the “threshold theorem” for the PPT property of random states, since the PPT-inducing property can be checked on the Choi matrix, which here is random). And there is room for satisfying both inequalities whenever . Some technical details have to enter the picture, but you don’t show these on blogs, and anyway here is a complete argument.

The apparent need for all the dimensions to go simultaneously to infinity makes the situation less feasible to analyze from the free probability point of view, while AGA handles that routinely.

And here is a challenge to our readers: extend the above observation to , the context of Hastings’s result. At the first sight, the constraints on needed for additivity violations are fairly tight and appear to be incompatible with the PPT-inducing property being generic. But perhaps there is a trick…