published on January 23, 2025
Limit formulas for the trace of the functional calculus of quantum channels for SU(2)
By Robin van Haastrecht
A generic quantum Wielandt's inequality for matrix and Lie algebras
By Angela Capel
Appears in collection : Jean Morlet Chair - Research school: Random quantum channels: entanglement and entropies / Chaire Jean Morlet - Ecole: Canaux quantiques aléatoires: Intrication et entropies
Here, we 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. This length generically is of order $\Theta(\log n)$, as opposed to the general case, in which the best bound is $O(n^2)$. This has implications for the primitivity index of random quantum channels, matrix product states and projected entangled pair states. Some results can be extended to Lie algebras. Joint work with Yifan Jia.
