Estimates for compression norms and additivity violation in quantum information

Abstract

ABSTRACT. The free contraction norm (or the (t)-norm) was introduced in [BCN12] as a tool to compute the typical location of the collection of singular values associated to a random subspace of the tensor product of two Hilbert spaces. In turn, it was used in [BCN13] in order to obtain sharp bounds for the violation of the additivity of the minimum output entropy for random quantum channels with Bell states. This free contraction norm, however, is difficult to compute explicitly. The purpose of this note is to give a good estimate for this norm. Our technique is based on results of super convergence in the context of free probability theory. As an application, we give a new, simple and conceptual proof of the violation of the additivity of the minimum output entropy.

Additivity violation of quantum channels via strong convergence to semi-circular and circular elements

“…In this section, we show that the MOpN of the random CP maps in (1.6) converge almost surely to some value formulated by the function in (2.21). Similar free-probability methods had been adopted in [6,12,13] to show the additivity/multiplicativity violation. Below, most of claims on Φ n and Φ c n are interchangeable because of Proposition A.1, and this fact will be repeatedly used without mentioning.…”

“…Remark 4.6. The method with the trace-less condition, also used in [13,11], requires trace-preserving property. This is why, a similar method would not work for the CP maps in (1.6).…”

On the Separability of Unitarily Invariant Random Quantum States: The Unbalanced Regime

“…Although there exist implicit algebraic characterizations of the support of the measures ⊞ in terms of the support of and , it is only in very simple circumstances that one can write down explicit formulas for the support. We recall below an approximation result obtained in [25 …”

Haagerup's inequality and additivity violation of the Minimum Output Entropy

“…By a standard compactness and continuity argument, the inequality (2) holds uniformly on any bounded choice of A. We refer for example to Proposition 7.3 of [CFZ16] for details.…”

“…We refer to Equation (18) of [CFZ16] for further detail. Dividing both sides of the above inequality by ||Φ n,k (X)− Ĩ|| 2 implies, that with probability one, lim sup…”