# On Hastings' Counterexamples to the Minimum Output Entropy Additivity Conjecture

@article{Brando2010OnHC,
title={On Hastings' Counterexamples to the Minimum Output Entropy Additivity Conjecture},
author={Fernando G. S. L. Brand{\~a}o and Michal Horodecki},
journal={Open Syst. Inf. Dyn.},
year={2010},
volume={17},
pages={31-52}
}
• Published 19 July 2009
• Mathematics
• Open Syst. Inf. Dyn.
Hastings recently reported a randomized construction of channels violating the minimum output entropy additivity conjecture. Here we revisit his argument, presenting a simplified proof. In particular, we do not resort to the exact probability distribution of the Schmidt coefficients of a random bipartite pure state, as in the original proof, but rather derive the necessary large deviation bounds by a concentration of measure argument. Furthermore, we prove non-additivity for the overwhelming…
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## References

SHOWING 1-10 OF 40 REFERENCES
Counterexamples to the Maximal p-Norm Multiplicativity Conjecture for all p > 1
• Mathematics, Computer Science
ArXiv
• 2008
For all p > 1, it is demonstrated the existence of quantum channels with non-multiplicative maximal output p-norms, and a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2.
Simplifying additivity problems using direct sum constructions
• Mathematics
• 2007
We study the additivity problems for the classical capacity of quantum channels, the minimal output entropy, and its convex closure. We show for each of them that additivity for arbitrary pairs of
The maximal p-norm multiplicativity conjecture is false
If the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of maximal p-norm multiplicativity, because the minimum output Renyi entropy of order p of a quantum channel is not additive for all 1 < p < 2.
On Some Additivity Problems in Quantum Information Theory
• Mathematics
• 2000
A class of problems in quantum information theory, having an elementary formulation but still resisting solution, concerns the additivity properties of various quantities characterizing quantum
Superadditivity of communication capacity using entangled inputs
The results show that the most basic question of classical capacity of a quantum channel remains open, with further work needed to determine in which other situations entanglement can boost capacity.
Classical information capacity of a class of quantum channels
• Computer Science
• 2005
The additivity of the minimal output Renyi entropies with entropic parameters α [0, 2], generalizing an argument by Alicki and Fannes, is proved and a weak form of covariance of a channel is introduced in order to relate these results to the classical information capacity.
Maximal output purity and capacity for asymmetric unital qudit channels
• Mathematics
• 2005
We consider generalizations of depolarizing channels to maps of the form with Vk being unitary and ∑kak = a < 1. We show that one can construct unital channels of this type for which the input which