Entangled States are Typically Incomparable

Consider a bipartite quantum system, where Alice and Bob jointly possess a pure state $|\psi ⟩$ . Using local quantum operations on their respective subsystems, and unlimited classical communication, Alice and Bob may be able to transform $|\psi ⟩$ into another state $|\varphi ⟩$ . Famously, Nielsen's theorem [28] provides a necessary and sufficient algebraic criterion for such a transformation to be possible (namely, the entanglement spectrum of $|\varphi ⟩$ should majorise the entanglement spectrum of $|\psi ⟩$ ). In the paper where Nielsen proved this theorem, he conjectured that in the limit of large dimensionality, for almost all pairs of states $|\psi ⟩,|\varphi ⟩$ (according to the natural unitary invariant measure) such a transformation is not possible. That is to say, typical pairs of quantum states $|\psi ⟩,|\varphi ⟩$ 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 $X{X}^{†}/tr\left(X{X}^{†}\right)$ majorises the spectrum of $Y{Y}^{†}/tr\left(Y{Y}^{†}\right)$ 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].