  Extended versions of the Lambek Calculus currently used in computational
linguistics rely on unary modalities to allow for the controlled application of
structural rules affecting word order and phrase structure. These controlled
structural operations give rise to derivational ambiguities that are missed by
the original Lambek Calculus or its pregroup simplification. Proposals for
compositional interpretation of extended Lambek Calculus in the compact closed
category of FVect and linear maps have been made, but in these proposals the
syntax-semantics mapping ignores the control modalities, effectively
restricting their role to the syntax. Our aim is to turn the modalities into
first-class citizens of the vectorial interpretation. Building on the
directional density matrix semantics, we extend the interpretation of the type
system with an extra spin density matrix space. The interpretation of proofs
then results in ambiguous derivations being tensored with orthogonal spin
states. Our method introduces a way of simultaneously representing co-existing
interpretations of ambiguous utterances, and provides a uniform framework for
the integration of lexical and derivational ambiguity.
