On Hastings' Counterexamples to the Minimum Output Entropy Additivity Conjecture

  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.},
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… 
Entanglement of random subspaces via the Hastings bound
This paper uses Hastings’ method to derive new bounds for the entanglement of random subspaces of bipartite systems and uses these bounds to prove the existence of nonunital channels, which violate additivity of minimal output entropy.
Revisiting Additivity Violation of Quantum Channels
We prove additivity violation of minimum output entropy of quantum channels by straightforward application of $${\epsilon}$$ϵ-net argument and Lévy’s lemma. The additivity conjecture was disproved
Superadditivity of the regularized Minimum Output Entropy
A quantum channel for which the regularized minimum output entropy is super-additive is exhibited, and the strategy of proof relies on developing a variant of the Haagerup inequality optimized for a product of free groups.
Estimates for compression norms and additivity violation in quantum information
The free contraction norm (or the (t)-norm) was introduced by Belinschi, Collins and Nechita as a tool to compute the typical location of the collection of singular values associated to a random
Additivity violation of the regularized Minimum Output Entropy.
The problem of additivity of the Minimum Output Entropy is of fundamental importance in Quantum Information Theory (QIT). It was solved by Hastings in the one-shot case, by exhibiting a pair of
Almost One Bit Violation for the Additivity of the Minimum Output Entropy
The maximum of $$ell^p}$$ℓp norms on the set Kk,t is identified and it is proved that the maximum is attained on a vector of shape where a > b and the precise limit value of the minimum output entropy of a single random quantum channel is computed.
Asymptotically Well-Behaved Input States Do Not Violate Additivity for Conjugate Pairs of Random Quantum Channels
The limit output states for any sequence of well-behaved inputs, which consist of a large class of input states having a nice set of parameters, are computed and it is shown that among these input states tensor products of Bell states give asymptotically the least output entropy.
The minimum Rényi entropy output of a quantum channel is locally additive
The results demonstrate that the counterexamples to the Rényi additivity conjectures exhibit purely global effects of quantum channels, and the approach presented here cannot be extended to Rényu entropies with parameter $$\alpha <1$$α<1.
Additivity violation of quantum channels via strong convergence to semi-circular and circular elements
Using semi-circular systems and circular systems of free probability, the multiplicativity violation of maximum output norms in the asymptotic regimes is shown and the additivity violation via Haagerup inequality is proved for a new class of random quantum channels constructed by rectifying the above completely positive maps based on strong convergence.


Counterexamples to the Maximal p-Norm Multiplicativity Conjecture for all p > 1
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
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
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
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
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
Additivity for unital qubit channels
Additivity of the Holevo capacity is proved for product channels, under the condition that one of the channels is a unital qubit channel, with the other completely arbitrary, and provides an explicit formula for this classical capacity.
On Strong Superadditivity of the Entanglement of Formation
We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation EF and the conjugate function E* of the entanglement function E(ρ)=S(TrAρ). We then