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 [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. 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 et al. [12].