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We establish a generalisation of the fundamental state convertibility theoremin quantum information to the context of bipartite quantum systems modelled bycommuting semi-finite von Neumann algebras. Namely, we establish ageneralisation to this setting of Nielsen's theorem on the convertibility ofquantum states under local operations and classical communication (LOCC)schemes. Along the way, we introduce an appropriate generalisation of LOCCoperations and connect the resulting notion of approximate convertibility tothe theory of singular numbers and majorisation in von Neumann algebras. As anapplication of our result in the setting of $II_1$-factors, we show that theentropy of the singular value distribution relative to the unique tracial stateis an entanglement monotone in the sense of Vidal, thus yielding a new way toquantify entanglement in that context. Building on previous work in theinfinite-dimensional setting, we show that trace vectors play the role ofmaximally entangled states for general $II_1$-factors. Examples are drawn frominfinite spin chains, quasi-free representations of the CAR, and discretisedversions of the CCR.

State convertibility in the von Neumann algebra framework