Comments on Hastings’ Additivity Counterexamples

Abstract

Hastings [12] recently provided a proof of the existence of channels which violate the additivity conjecture for minimal output entropy. In this paper we present an expanded version of Hastings' proof. In addition to a careful elucidation of the details of the proof, we also present bounds for the minimal dimensions needed to obtain a counterexample.where the supremum runs over ensembles of input states, and where S(ρ) denotes the von Neumann entropy of the state ρ: S(ρ) = −Trρ log ρ (1.2)

Towards a state minimizing the output entropy of a tensor product of random quantum channels

“…Background. One of the most important questions in quantum communication theory is whether a quantum channel has additive properties or not [1,4,9,10,11,12,13,14,18]. If a channel Φ is additive for the Holevo capacity χ(·) in the sense that ∃N ∈ N, ∀n N χ(Φ ⊗n ) = nχ(Φ) (1) then the classical capacity of the quantum channel equals the Holevo capacity, giving a one-shot (non-asymptotic) formula for the classical capacity C(·):…”

Estimates for compression norms and additivity violation in quantum information

“…Figure 2 indicates the minimum possible dimension for the violation based on our method; k = 31114 (with t = k 1.387 ), which gives g(k, t) = −6.71108 × 10 −12 . This value of k is smaller than 3.9 × 10 4 which was obtained in [FKM10] with t = k −1 in a similar setting, but larger than 183 obtained in [BCN13], which is known to be the best estimate in this context.…”

“…On the other hand, additivity violation for H min was proven by Hastings' [Has09] . Later, generalizations and improvements were made in [FKM10,BH10,FK10,ASW11,Fuk14,BCN13]. Since the resolution of the additivity conjecture, efforts have been made to understand more the additivity violation in terms of tensor-products of quantum channels in [CN10b, CN10a, CN11b, CN11a, CFN12, CFN13a, FN14] into understanding, extending and improving the deviations from additivity.…”

Revisiting Additivity Violation of Quantum Channels

“…The technical discussion on this issue is written in Section 3 after stating additivity violation in Section 2. One can see that our result is stronger than all the existing proofs [Has09,FKM10,BH10,FK10,ASW11] in the sense that we can prove the additivity violation asymptotically as long as the dimensions of input and output are proportional to each other and proportionally larger than or equal to square of the dimension of environment; there is no restriction on the ratios. We make some analysis on our method and compare it to Hastings' in Section 4.…”

“…For example, see [FKM10], where the important idea tubal neighborhood was reformulated as TUBE. However we believe that we arrive at the same goal, or at least get convinced, if we look at (4.2) in the Hilbert-Schmidt norm.…”