We present a model of Selinger and Valiron's quantum lambda calculus based on von Neumann algebras, and show that the model is adequate with respect to the operational semantics. Keywords and phrases quantum lambda calculus, von Neumann algebras In 1925, Heisenberg realised, pondering upon the problem of the spectral lines of the hydrogen atom, that a physical quantity such as the x-position of an electron orbiting a proton is best described not by a real number but by an infinite array of complex numbers [12] . One current line of research (with old roots [6, 7, 9, 19] ) is the study of von Neumann algebras from a categorical perspective (see e.g. [4, 5, 30]). One example relevant to this paper is Kornell's proof that the opposite of the category vNA MIU of von Neumann algebras with the obvious structure preserving maps (i.e. the unital normal * -homomorphisms) is monoidal closed when endowed with the spatial tensor product [18] . He argues that vNA op MIU should be thought of as the quantum version of Set. We would like to focus instead on the category of von Neumann algebras and completely positive normal subunital maps, vNA CPsU , as it seems more appropriate for modelling quantum computation: the full subcategory of vNA op CPsU consisting of finite dimensional von Neumann algebras is equivalent to Selinger's category Q [35], which is used to model first order quantum programming languages. [36, 37] proposed a typed 1 lambda calculus for quantum computation, and they studied it in a series of papers [38][39][40]. A striking feature of this quantum lambda calculus is that functions * The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n o 320571 1 An untyped quantum lambda calculus had already been proposed by Van Tonder [43].
A model based on W*-algebras for a quantum lambda calculus without recursion or inductive datatypes was described in a recent manuscript REF .
Von Neumann Algebras form a Model for the Quantum Lambda Calculus
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