Entangled states are typically incomparable

Consider a bipartite quantum system, where Alice and Bob jointly possess a pure state |ψ. Using local quantum operations on their respective subsystems, and unlimited classical communication, Alice and Bob may be able to transform |ψ into another state |φ. 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 |φ should majorise the local spectrum of |ψ). In the paper where Nielsen proved this theorem, he conjectured that in the limit of large dimensionality, for almost all pairs of states |ψ, |φ (according to the natural unitary invariant measure) such a transformation is not possible. That is to say, typical pairs of quantum states |ψ, |φ 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 × 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^ / tr(X X⁾ majorises the spectrum of Y Y^ / tr(Y Y⁾ 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].