Normalization Across Circuit Response and Optimization
Node role: Cross-cutting semantic reference.
Normalization is a declared map that makes a quantity comparable, interpretable, or dimensionless. Residue normalization and residual normalization are different operations: residue normalization scales a local response around a pole, while residual normalization scales an optimization error against its target. Neither operation changes the physical circuit or silently changes which subsystem owns a quantity.
Every normalization declaration must name four things:
- the object being normalized;
- the map or operation applied;
- the invariant or meaning preserved; and
- the output units.
Normalization cannot choose physical ownership or create a Gate. Ownership is declared by the physical coordinates, operator, observable, or contract; a Gate requires its own accepted authority.
Reader task: identify what is being normalized, which map is applied, and which meaning and units survive. Use the family table for lookup, then the scalar LC calculation to see why residue and canonical scaling can coincide locally without being identical operations.
Six normalization families
Engineering choice. These six names organize the project’s semantic vocabulary; the table does not establish a single universal normalization.
These names are deliberately not interchangeable. Each row names the object, the invariant, and the feature that becomes readable.
| Family | Object normalized | Invariant preserved | What becomes readable |
|---|---|---|---|
| Coordinate/unit rescaling | A declared coordinate vector and every operator/map that acts on it. | The represented physical state and response, when all maps transform consistently. | Convenient units or a named coordinate basis. |
| Canonical/symplectic impedance scaling | Conjugate flux/charge pairs \((\Phi,Q)\). | The canonical/symplectic structure and classical quadratic energy. | A local oscillator amplitude and impedance scale. |
| Local pole/residue normalization | One selected simple root and its local response operator. | The selected root, branch, and response map. | Unit local diagonal slope and frequency-like retained couplings. |
| Modal energy/biorthogonal normalization | Closed modes or left/right open eigenvectors. | The declared energy metric or left/right pairing. | Mode amplitude, participation, and a reproducible phase/scale convention. |
| Port power/photon-flux wave normalization | Traveling port waves at declared planes and reference impedances. | Physical power flow or photon flux at those planes. | A calibrated \(S\) matrix and input/output amplitudes. |
| Dimensionless optimization residual normalization | A signed metric error. | The metric definition and its target; it does not preserve an energy or a response residue. | Comparable dimensionless objective terms and explicit trade-offs. |
Scalar LC: one local amplitude normalization seen two ways
This-page derivation. Start from the passive conservative LC energy (Vool and Devoret 2017, sec. 3.1.4.1). Rescale its conjugate coordinates exactly, then expand its scalar response at the positive root. The comparison belongs to this scalar model, not arbitrary dispersive Schur operators.
For one passive lossless LC coordinate,
\[ \Omega=\frac{1}{\sqrt{LC}}, \qquad Z=\sqrt{\frac LC}, \qquad X=\frac{\Phi}{\sqrt Z}, \qquad P=\sqrt Z Q. \]
The canonical impedance scaling gives
\[ H_{\rm cl}=\frac{Q^2}{2C}+\frac{\Phi^2}{2L} =\frac\Omega2(X^2+P^2), \qquad A=\frac{X+iP}{\sqrt2}, \qquad H_{\rm cl}=\Omega|A|^2. \]
The response view starts from the scalar dynamic operator
\[ D(\omega)=\frac1L-\omega^2 C, \qquad N=-D'(\Omega)=2\Omega C=\frac2Z. \]
Near the positive root,
\[ \frac Z2D(\omega) =\frac{(\Omega-\omega)(\Omega+\omega)}{2\Omega} \approx\Omega-\omega. \]
Canonical impedance scaling is an exact phase-space transform. Constant pole-slope normalization shares its LC scale, but additionally localizes the response at the positive-frequency branch and projects its first-order near-root form. The two are therefore a bridge in this simple case, not the same operation globally.
Do not globalize this result. An open, lossy, frequency-dependent complete-complement Schur operator need not have one global canonical impedance transform equivalent to every local pole/residue normalization. Residue normalization remains local to its selected root, branch, and response map.
Local residue normalization
The exact open-EOM operand is the local response coupling \(J_{\rm res}(\omega_*)\), obtained from a selected root/branch of the frequency-dependent Schur operator. Its derivative normalization, finite-band caveat, and comparison with Hamiltonian/RWA coefficients are owned by Poles, Zeros, and Residues. Pole-residue normalization is a local response operation; it is not an optimization residual normalization.
Which coupling is this?
Internal definitions. The \(J\) labels below follow the stated fixed-basis Hamiltonian and local response constructions. No single cited source establishes their cross-representation equality for an arbitrary open circuit; the approximation conditions remain with the linked response derivation.
The exact fixed-basis coefficient inheritance is \(J_{\rm RWA}=J_{\rm ex}=J_{\rm H}\) after the explicitly stated RWA drops \(\Lambda\); it does not make the reduced model exact. The local open-EOM quantity \(J_{\rm res}(\omega_*)\approx J_{\rm RWA}\) only under the response-side approximation conditions stated in the linked pole/residue node. The observed \(J_{\rm split}\) is separately conditional: it is a hybridized pole half-splitting, not a coefficient definition.
The concise coupling definitions name the fixed-anchored-basis exchange coefficient and its explicitly declared RWA inheritance. Their interpretation depends on the stated canonical normalization, basis and approximation; the full quadratic exchange/pairing block derivation is not developed in this selected material. The observable distinction for \(J_{\rm split}\) is in Poles, Zeros, and Residues.
Residual normalization for optimization
Engineering choice. Residual scales and weights describe a selected objective, not a source-mandated tolerance or scientific acceptance rule.
For a metric \(m_i\) and target \(t_i\), residual normalization commonly declares a nonzero scale \(s_i\) and forms the signed dimensionless residual
\[ e_i=\frac{m_i-t_i}{s_i}. \]
Weights then declare how residuals trade in the objective; they do not become tolerances or acceptance Gates by themselves. A relative-error choice \(s_i=|t_i|\) requires a separately declared scale when \(t_i=0\). The scale must state whether it represents a tolerance, uncertainty, nondimensionalization, or numerical tuning choice. See the normalized weighted objective.
Physics-to-engineering handoff
Circuit models use the physical-coordinate/open-operator path and its complete-complement Schur reduction. Network and simulation work uses the declared port waves, reference planes, and \(S/Y/Z\) maps. Optimization uses the already-defined operand and only then applies dimensionless residual scaling; it does not redefine the upstream frequency, coupling, pole, or observable.
Connections
| Field | Value |
|---|---|
| Status | Seed |
| Semantic state | Human-accepted common-language normalization semantics; stabilization pending. |
| Open review question | Are residue and residual normalization kept distinct wherever a quantity is defined or optimized? |