  Given a random subspace $H_n$ chosen uniformly in a tensor product of Hilbert
spaces $V_n\otimes W$, we consider the collection $K_n$ of all singular values
of all norm one elements of $H_n$ with respect to the tensor structure. A law
of large numbers has been obtained for this random set in the context of $W$
fixed and the dimension of $H_n$ and $V_n$ tending to infinity at the same
speed in a paper of Belinschi, Collins and Nechita. In this paper, we provide
measure concentration estimates in this context. The probabilistic study of
$K_n$ was motivated by important questions in Quantum Information Theory, and
allowed to provide the smallest known dimension (184) for the dimension an an
ancilla space allowing Minimum Output Entropy (MOE) violation. With our
estimates, we are able, as an application, to provide actual bounds for the
dimension of spaces where violation of MOE occurs.
