 
Summary: Around Dvoretzky theorem in quantum information
theory
Guillaume AUBRUN
Universit´e Lyon 1, France
Guillaume AUBRUN (Lyon) Dvoretzky theorem and QIT MarneLaVall´ee, Dec. 2008 1 / 27
Dvoretzky theorem
If K Rn is a convex body, let x K = inf{t > 0 s.t. x tK}.
Theorem (V. Milman)
Let K Rn or be a convex body and X a random vector uniformly
distributed on Sn1. Let M = E X K and b = sup X K . Then with high
probability, a random kdimensional subpsace E Rn satisfies
x E Sn1
, (1  )M x K (1 + )M,
with k = c2n(M/b)2 .
Probability is given on the Grassman manifold is the Haar measure.
Also true for unit balls of complex normed spaces.
Combined with the fact that every convex body has an affine image
for which M/b
log n/
