Speaker
Description
In this talk, I will provide a generic version of quantum Wielandt’s inequality, which
gives an optimal upper bound on the minimal length such that products of that
length of n-dimensional matrices in a generating system span the whole matrix
algebra with probability one. I will show that this length generically is of order
Θ(log n), as opposed to the general case, in which the best bound to the date is
O(n2log n). We will discuss the implications of this result as a new bound on the
primitivity index of a random quantum channel, as well as to show that almost any
translation-invariant (with periodic boundary conditions) matrix product state with
length of order Ω(log n) is the unique ground state of a local Hamiltonian. Finally,
we will comment on the possibility of extending these results to Lie algebras. This
is based on joint work with Yifan Jia.
