25,724 research outputs found
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
A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs
The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier
analysis of real-valued functions on the Boolean cube. In this paper we present
a version of this inequality for matrix-valued functions on the Boolean cube.
Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also
present a number of applications. First, we analyze maps that encode
classical bits into qubits, in such a way that each set of bits can be
recovered with some probability by an appropriate measurement on the quantum
encoding; we show that if , then the success probability is
exponentially small in . This result may be viewed as a direct product
version of Nayak's quantum random access code bound. It in turn implies strong
direct product theorems for the one-way quantum communication complexity of
Disjointness and other problems. Second, we prove that error-correcting codes
that are locally decodable with 2 queries require length exponential in the
length of the encoded string. This gives what is arguably the first
``non-quantum'' proof of a result originally derived by Kerenidis and de Wolf
using quantum information theory, and answers a question by Trevisan.Comment: This is the full version of a paper that will appear in the
proceedings of the IEEE FOCS 08 conferenc
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