We obtain a formula for the density of the free convolution of an arbitrary probability measure on the unit circle of C with the free multiplicative analogues of the normal distribution on the unit circle. This description relies on a characterization of the image of the unit disc under the subordination function, which also allows us to prove some regularity properties of the measures obtained in this way. As an application, we give a new proof for Biane's classic result on the densities of the free multiplicative analogue of the normal distributions. We obtain analogue results for probability measures on R + . Finally, we describe the density of the free multiplicative analogue of the normal distributions as an example and prove unimodality and some symmetry properties of these measures.

We prove superconvergence results for all freely infinitely divisible distributions. Given a nondegenerate freely infinitely divisible distribution ν, let µn be a sequence of probability measures and let kn be a sequence of integers tending to infinity such that µ ⊞kn n converges weakly to ν. We show that the density dµ ⊞kn n /dx converges uniformly, as well as in all L p -norms for p > 1, to the density of ν except possibly in the neighborhood of one point. Applications include the global superconvergence to freely stable laws and that to free compound Poisson laws over the whole real line.

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Abstract. We consider a pair of probability measures µ, ν on the unit circle such that Σ λ (η ν (z)) = z/η µ (z). We prove that the same type of equation holds for any t ≥ 0 when we replace ν by ν ⊠λ t and µ by M t (µ), where λ t is the free multiplicative analogue of the normal distribution on the unit circle of C and M t is the map defined by Arizmendi and Hasebe. These equations are a multiplicative analogue of equations studied by Belinschi and Nica. In order to achieve this result, we study infinite divisibility of the measures associated with subordination functions in multiplicative free Brownian motion and multiplicative free convolution semigroups. We use the modified S-transform introduced by Raj Rao and Speicher to deal with the case that ν has mean zero. The same type of the result holds for convolutions on the positive real line. In the end, we give a new proof for some Biane's results on the densities of the free multiplicative analogue of the normal distributions.

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.

Let |ψ ψ| be a random pure state on C d 2 ⊗ C s , where ψ is a random unit vector uniformly distributed on the sphere in C d 2 ⊗ C s . Let ρ 1 be random induced states ρ 1 = Tr C s (|ψ ψ|) whose distribution is µ d 2 ,s ; and let ρ 2 be random induced states following the same distribution µ d 2 ,s independent from ρ 1 . Let ρ be a random state induced by the entanglement swapping of ρ 1 and ρ 2 . We show that the empirical spectrum of ρ − 1l/d 2 converges almost surely to the Marcenko-Pastur law with parameter c 2 as d → ∞ and s/d → c.As an application, we prove that the state ρ is separable generically if ρ 1 , ρ 2 are PPT entangled.

This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law. Superconvergence to the marginal law of free multiplicative Brownian motion at a specified time is also studied. In the unitary case, the superconvergence to free Brownian motion and that to the Haar measure are shown to be uniform over the entire unit circle, implying further a free entropic limit theorem and a universality result for unitary free Lévy processes. Finally, the method of proofs on the positive half-line gives rise to a new multiplicative Boolean to free Bercovici-Pata bijection.