@article{22813,
  abstract     = {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 [28] provides a necessary and sufficient
algebraic criterion for such a transformation to be possible (namely, the entanglement
spectrum of |φ should majorise the entanglement 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.ViaNielsen’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 XX†/ tr(XX†) majorises the spectrum of YY†/ tr(YY†) tends to zero as both n and
m grow large. We prove this conjecture, and we also confirm some related predictions
of Cunden et al. [12].},
  author       = {Jain, Vishesh and Kwan, Matthew Alan and Michelen, Marcus},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  number       = {10},
  publisher    = {Springer},
  title        = {{Entangled states are typically incomparable}},
  doi          = {10.1007/s00220-026-05711-4},
  volume       = {407},
  year         = {2026},
}