  In a recent paper, a realizability technique has been used to give a
semantics of a quantum lambda calculus. Such a technique gives rise to an
infinite number of valid typing rules, without giving preference to any subset
of those. In this paper, we introduce a valid subset of typing rules, defining
an expressive enough quantum calculus. Then, we propose a categorical semantics
for it. Such a semantics consists of an adjunction between the category of
distributive-action spaces of value distributions (that is, linear combinations
of values in the lambda calculus), and the category of sets of value
distributions.
