We introduce a proof language for Intuitionistic Multiplicative Additive Linear Logic (IMALL), extended with a modality \(\mathcal B\) to capture mixed-state quantum computation. The language supports algebraic constructs such as linear combinations, and embeds pure quantum computations within a mixed-state framework via \(\mathcal B\) , interpreted categorically as a functor from a category of Hilbert Spaces to a category of finite-dimensional C*-algebras. Measurement arises as a definable term, not as a constant, and the system avoids the use of quantum configurations, which are part of the theory of the quantum lambda calculus. Cut-elimination is defined via a composite reduction relation, and shown to be sound with respect to the denotational interpretation. We prove that any linear map on \(\mathbb C^{2^n}\) can be represented within the system, and illustrate this expressiveness with examples such as quantum teleportation and the quantum switch.