Minimal unitary representations of time-invariant causal quantum channels
In this work, we study quantum channels that respect causality in single-input-single-output linear time-invariant systems. We show that positive semidefinite kernels satisfying certain analyticity constraints, which we call Schur-class kernels, provide a suitable description of such channels through multiplicative maps (Schur multipliers). Using the Moore-Aronszajn Theorem, we establish that these multipliers admit a minimal analytic Stinespring representation, which, in the unital case, is given by multiplication by a left inner function. We then relate the output rank of pure-state inputs to the dimension of the associated reproducing kernel Hilbert space. Using de Branges-Rovnyak realizations, we derive a stability criterion for the existence of two-sided inner completions of these vector-valued functions. We further show that the adjoint of the Julia embedding associated with the realization is minimal, and that its characteristic function essentially coincides with the canonical two-sided inner completion suggested by the Beurling-Lax-Halmos Theorem. Finally, we provide algorithms for explicitly computing minimal two-sided inner completions in the case of finite McMillan degree.