  In this work, we prove a lower bound on the difference between the first and
second singular values of quantum channels induced by random isometries, that
is tight in the scaling of the number of Kraus operators. This allows us to
give an upper bound on the difference between the first and second largest (in
modulus) eigenvalues of random channels with same large input and output
dimensions for finite number of Kraus operators $k\geq 169$. Moreover, we show
that these random quantum channels are quantum expanders, answering a question
posed by Hastings. As an application, we show that ground states of infinite 1D
spin chains, which are well-approximated by matrix product states, fulfill a
principle of maximum entropy.
