On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution

Let M denote the space of Borel probability measures on the real line. For every nonnegative t we consider the transformation $\mathbb B_t : M \to M$ defined for any given element in M by taking succesively the the (1+t) power with respect to free additive convolution and then the 1/(1+t) power with respect to Boolean convolution of the given element. We show that the family of maps {\mathbb B_t|t\geq 0} is a semigroup with respect to the operation of composition and that, quite surprisingly, every $\mathbb B_t$ is a homomorphism for the operation of free multiplicative convolution. We prove that for t=1 the transformation $\mathbb B_1$ coincides with the canonical bijection $\mathbb B : M \to M_{inf-div}$ discovered by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. Here M_{inf-div} stands for the set of probability distributions in M which are infinitely divisible with respect to free additive convolution. As a consequence, we have that $\mathbb B_t(\mu)$ is infinitely divisible with respect to free additive convolution for any for every $\mu$ in M and every t greater than or equal to one. On the other hand we put into evidence a relation between the transformations $\mathbb B_t$ and the free Brownian motion; indeed, Theorem 4 of the paper gives an interpretation of the transformations $\mathbb B_t$ as a way of re-casting the free Brownian motion, where the resulting process becomes multiplicative with respect to free multiplicative convolution, and always reaches infinite divisibility with respect to free additive convolution by the time t=1.Comment: 30 pages, minor changes; to appear in Indiana University Mathematics Journa

Convolution powers in the operator-valued framework

We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued distribution \mu one can define convolution powers with respect to free additive convolution and with respect to Boolean convolution, where the exponent considered in the power is a suitably chosen linear map \eta from B to B, instead of being a non-negative real number. More precisely, the Boolean convolution power is defined whenever \eta is completely positive, while the free additive convolution power is defined whenever \eta - 1 is completely positive (where 1 stands for the identity map on B). In connection to these convolution powers we define an evolution semigroup related to the Boolean Bercovici-Pata bijection. We prove several properties of this semigroup, including its connection to the B-valued free Brownian motion. We also obtain two results on the operator-valued analytic function theory related to the free additive convolution powers with exponent \eta. One of the results concerns analytic subordination for B-valued Cauchy-Stieltjes transforms. The other gives a B-valued version of the inviscid Burgers equation, which is satisfied by the Cauchy-Stieltjes transform of a B-valued free Brownian motion.Comment: 33 pages, no figure

Optimal control of a qubit coupled to a non-Markovian environment

A central challenge for implementing quantum computing in the solid state is decoupling the qubits from the intrinsic noise of the material. We investigate the implementation of quantum gates for a paradigmatic, non-Markovian model: A single qubit coupled to a two-level system that is exposed to a heat bath. We systematically search for optimal pulses using a generalization of the novel open systems Gradient Ascent Pulse Engineering (GRAPE) algorithm. We show and explain that next to the known optimal bias point of this model, there are optimal shapes which refocus unwanted terms in the Hamiltonian. We study the limitations of controls set by the decoherence properties. This can lead to a significant improvement of quantum operations in hostile environments.Comment: 5 pages, 3 figures, improved pulse shape

Free Infinite Divisibility for Ultrasphericals

We prove that the integral powers of the semicircular distribution are freely infinitely divisible. As a byproduct we get another proof of the free infnite divisibility of the classical Gaussian distribution.Comment: 10 page

BMO Estimates for the H∞(Bn)H^{\infty}(\mathbb{B}_n) Corona Problem

We study the $H^{\infty}(\mathbb{B}_{n})$ Corona problem $\sum_{j=1}^{N}f_{j}g_{j}=h$ and show it is always possible to find solutions $f$ that belong to $BMOA(\mathbb{B}_{n})$ for any $n>1$, including infinitely many generators $N$. This theorem improves upon both a 2000 result of Andersson and Carlsson and the classical 1977 result of Varopoulos. The former result obtains solutions for strictly pseudoconvex domains in the larger space $H^{\infty}\cdot BMOA$ with $N=\infty$, while the latter result obtains $BMOA(\mathbb{B}_{n})$ solutions for just N=2 generators with $h=1$. Our method of proof is to solve $\overline{\partial}$-problems and to exploit the connection between $BMO$ functions and Carleson measures for $H^{2}(\mathbb{B}_{n})$. Key to this is the exact structure of the kernels that solve the $\overline{\partial}$ equation for $(0,q)$ forms, as well as new estimates for iterates of these operators. A generalization to multiplier algebras of Besov-Sobolev spaces is also given.Comment: v1: 20 pg