  Consider a bipartite quantum system, where Alice and Bob jointly possess a
pure state $|\psi\rangle$. Using local quantum operations on their respective
subsystems, and unlimited classical communication, Alice and Bob may be able to
transform $|\psi\rangle$ into another state $|\phi\rangle$. Famously, Nielsen's
theorem [Phys. Rev. Lett., 1999] provides a necessary and sufficient algebraic
criterion for such a transformation to be possible (namely, the local spectrum
of $|\phi\rangle$ should majorise the local spectrum of $|\psi\rangle$).
  In the paper where Nielsen proved this theorem, he conjectured that in the
limit of large dimensionality, for almost all pairs of states $|\psi\rangle,
|\phi\rangle$ (according to the natural unitary invariant measure) such a
transformation is not possible. That is to say, typical pairs of quantum states
$|\psi\rangle, |\phi\rangle$ are entangled in fundamentally different ways,
that cannot be converted to each other via local operations and classical
communication.
  Via Nielsen's theorem, this conjecture can be equivalently stated as a
conjecture about majorisation of spectra of random matrices from the so-called
trace-normalised complex Wishart-Laguerre ensemble. Concretely, let $X$ and $Y$
be independent $n \times m$ random matrices whose entries are i.i.d. standard
complex Gaussians; then Nielsen's conjecture says that the probability that the
spectrum of $X X^\dagger / \operatorname{tr}(X X^\dagger)$ majorises the
spectrum of $Y Y^\dagger / \operatorname{tr}(Y Y^\dagger)$ tends to zero as
both $n$ and $m$ grow large. We prove this conjecture, and we also confirm some
related predictions of Cunden, Facchi, Florio and Gramegna [J. Phys. A., 2020;
Phys. Rev. A., 2021].
