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In our previous work we described quantized computation using Horn clauses and based the semantics, dubbed as entanglement semantics as a generalization of denotational and distribution semantics, and founded it on quantum probability by exploiting the key insight classical random variables have quantum decompositions. Towards this end we built a Hilbert space of H-interpretations and a corresponding non commutative von Neumann algebra of bounded linear operators. In this work we extend the formalism using second-quantized Horn clauses that describe processes such as Heisenberg evolutions in optical circuits, quantum walks, and quantum filters in a formally verifiable way. Our goal is to build a model of computation based on logic via Currry-Howard correspondence. Towards this end we can think of completely positive *-unital maps of Horn clauses as function types representing modus ponens (equation (19)). Recursions that result from inductive reasoning has a quantum analogue in terms of sequence of *-homomorphisms induced by completely positive *-unital maps (equation (18)). We base our system on a measure theoretic approach to handle infinite dimensional systems and demonstrate the expressive power of the formalism by casting an algebra used to describe interconnected quantum systems (QNET) in this language. The variables of a Horn clause bounded by universal or existential quantifiers can be used to describe parameters of optical components such as beam splitter scattering paths, cavity detuning from resonance, strength of a laser beam, or input and output ports of these components. Prominent clauses in this non commutative framework are Weyl predicates, that are operators on a Boson Fock space in the language of quantum stochastic calculus, martingales and conjugate Brownian motions compactly representing statistics of quantum field fluctuations. We formulate theorem proving as a quantum stochastic process in Heisenberg picture of quantum mechanics, a sequence of goals to be proved, using backward chaining.
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