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

Quantum Control in the Unitary Sphere: Lambda-S1 and its Categorical Model