The optical equivalence theorem relating c-number and q-number formulations of quantum optics is rigorously extended to cover various unbounded operators, and in particular those operators that directly yield counting rates.
Title:Homological Closure of the Planar Surface Code: A Fully Rigorous Relative Chain Complex Formulation Description: This paper presents a fully closed and homologically rigorous formulation of the planar open-boundary surface code. Unlike standard expositions that rely on heuristic parity arguments or generator counting, we derive every structural property directly from the relative chain complex C2→∂2C1→∂1C0C_2 \xrightarrow{\partial_2} C_1 \xrightarrow{\partial_1} C_0C2∂2C1∂1C0 over F2\mathbb{F}_2F2. The core results are: Abelian stabilizer structure derived solely from the exactness condition ∂1∘∂2=0\partial_1 \circ \partial_2 = 0∂1∘∂2=0. Logical qubit count identified with the relative homology group H1(Σ,∂rough)H_1(\Sigma,\partial_{\rm rough})H1(Σ,∂rough), computed via the long exact sequence. Anticommuting logical operators guaranteed by Poincaré–Lefschetz duality and nondegenerate pairing. Code distance d=Ld=Ld=L rigorously established using the min-cut/max-flow theorem (Menger’s theorem), eliminating geometric intuition gaps. All arguments are presented in quotient-space form, reducing the surface code to a purely homological stabilization mechanism. Structural Extension In the Appendix, we establish a structural isomorphism between: Homological stabilizer projection in quantum error correction, and Viability-preserving projection in Yang Jihoon Dynamics. Both systems are shown to be quotient constructions under constraint-preserving equivalence relations. This yields a unified structural perspective: Fault-tolerant homological protection≡Viability-preserving dynamical projection Significance This work provides: A mathematically sealed derivation of the planar surface code A homological closure free of informal parity reasoning A structural bridge between quantum error correction and viability dynamics A foundation for interpreting fault tolerance as constrained topological persistence The result is not an analogy, but a quotient-level structural correspondence.
These lecture notes originate from a one-semester course on the mathematical foundations of second quantization, first given at the Department of Physics of the Friedrich-Alexander-Universität Erlangen-Nürnberg in the Winter Semester 2025/26 to a mixed audience of physics and mathematics students. Their guiding thread is a class of physical models: spin-boson models, describing a two-level system coupled to a boson field. After summarizing the necessary background on unbounded operators, up to the Kato-Rellich and KLMN theorems, and developing direct sums and tensor products of Hilbert spaces and of unbounded operators, we treat single-mode boson fields, constructing the ladder and number operators and characterizing their domains, before moving to Fock spaces, second quantization, bosonic and fermionic annihilation and creation operators, and the canonical (anti)commutation relations. Throughout, spin-boson models provide both the motivation and the test bench for the formalism: we rigorously establish their self-adjointness, solve the pure dephasing and Jaynes-Cummings models explicitly, deriving their spectra and the qubit channels they induce, show how their multimode counterparts display irreversible decoherence and decay, and, in an appendix, prove the rotating-wave approximation for the quantum Rabi model with an explicit error bound. Finally, we use the KLMN theorem to construct spin-boson models with non-normalizable form factors, and briefly discuss stronger ultraviolet divergences and the need for renormalization. Exercises with full solutions are included.