Center-Breaking Bridges and Commutant Collapse in Projection-Generated Quantum Markov Semigroups

The present work develops a classification of finite-dimensional irreducible invariant subspaces for quantum Markov semigroups (GKSL semigroups) whose generators are algebraically generated from two non-commuting orthogonal projections. The central mechanism is commutant collapse: a Hermitian rank-1 operator—the “bridge”—couples a two-dimensional Halmos fiber to a trivial direction, forces the commutant of the generated C^*-algebra to become trivial, and thereby saturates the algebra to a full matrix algebra by the double commutant theorem.

We identify i-axis symmetry—the requirement that the noise operators generate a *-closed algebra—as the structural condition that guarantees bridges appear in Hermitian form. A single bridge connecting one Halmos fiber to one trivial direction forces the algebra M_3(\mathbb{C}); the resulting semigroup is primitive on that subspace. For k\ge 2 trivial directions coupled to the same fiber, we introduce the concept of center-breaking bridges. Whether the algebra saturates to M_{2+k}(\mathbb{C}) or stabilises at an intermediate barrier subalgebra depends on whether each successive bridge breaks the centre of the intermediate algebra. The dimension of the minimal non-trivial irreducible block is 2+k when all k bridges act in a centre-breaking manner.

The Lie envelope of the three-dimensional sector is \mathfrak{su}(3) for generic parameters, with degeneracies at symmetric projection angles. We discuss dynamical closure rigidity—the property that a finitely generated operator system becomes irreducible under C^*-closure—and distinguish kinematic capacity from dynamical accessibility.