Entangled States are Typically Incomparable

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