XQM – Roof theory of the quantum-monadic field

The roof theory provides the formal foundation of the model and unifies mathematical and metaphysical ideas into a coherent field. Quantum monads are information- and energy-bearing units in the Hilbert space whose entanglement is captured via operators and couplings (VQM); coherence is assessed with IEQ.

Reader’s guide: formulas on the left, intuition/links on the right: XDM (ethics), VQM (relation), IEQ (measurement) and glossary: Hilbert space, operator.

(1) State space $ \mathcal{H} $ and field state $ \rho $.
(2) Couplings $ C_{ij} $ and field Hamiltonian $ H $.
(3) Open dynamics via the Lindblad equation.
(4) Coherence/resonance functionals $ \mathcal{K} $, $ \mathcal{R} $.
(5) Examples: XY/XXZ, Bell generation, local phase tuning.

Hilbert space & states

The quantum-monadic field lives on a separable Hilbert space $\mathcal{H}$. Individual monadic efficacy is modelled as a normalised vector $ \lvert m_i\rangle \in \mathcal{H} $; the field as density operator $\rho$ (mixed) or pure state $ \lvert\Psi\rangle $.

$$ \rho \succeq 0,\quad \mathrm{Tr}(\rho)=1,\quad \rho = \lvert\Psi\rangle\langle\Psi\rvert \ \text{(pure)}. $$

Observables are positive operators $ O \succeq 0 $ with expectation $ \langle O \rangle = \mathrm{Tr}(\rho O) $.

Explanation – Hilbert space & states

Carriers perish – monads persist: states/coupling patterns remain effective as carriers change.
Field view over subject view: relevance follows field effects, not ontological status.
Measurability: operators yield testable quantities (bridge to empirical ethics).

Couplings & field Hamiltonian

Hermitian couplings $ C_{ij}=C_{ji}^\dagger $ with time-dependent strengths $ J_{ij}(t) $:

$$ H(t) \;=\; \sum_i H_i(t) \;+\; \sum_{i< j} J_{ij}(t)\, C_{ij}. $$

For two-carrier baselines, use the Pauli matrices $ \sigma_x=\begin{pmatrix}0&1\\[2pt]1&0\end{pmatrix} $, $ \sigma_y=\begin{pmatrix}0&-i\\[2pt]i&0\end{pmatrix} $, $ \sigma_z=\begin{pmatrix}1&0\\[2pt]0&-1\end{pmatrix} $.

Explanation – Couplings & field Hamiltonian

Couplings are the bridges between monads: they set how strongly two states respond.
Hamiltonian = blueprint of dynamics: which resonance patterns arise or decay.
Ethics transfer: coherent couplings promote resonance (stable), chaotic ones cause disintegration (unstable).

Open dynamics & channels

Lindblad master equation for the field state $\rho$:

$$ \frac{d\rho}{dt} \;=\; -\,i\,[H,\rho] \;+\; \sum_k \Big( L_k \rho L_k^\dagger \;-\; \tfrac{1}{2}\{L_k^\dagger L_k,\rho\} \Big). $$

Birth/death as CPTP channels (completely positive, trace-preserving):

$$ \Phi(\rho)=\sum_a K_a\,\rho\,K_a^\dagger, \qquad \sum_a K_a^\dagger K_a=I. $$

“Power on/off” = discrete change of the channel composition over time.

Explanation – Open dynamics & channels

Why open? Carriers are transient; the field interacts with the environment → dissipation/dephasing.
Lindblad terms model loss/noise realistically (CPTP), beyond ideal Schrödinger dynamics.
Birth/death as channels: channel switches = “power on/off” → carriers change, coupling patterns may persist.
XD ethics: good design = coupling such that coherence remains viable under noise (resilience).

Coherence & resonance (assessment)

Weighted off-diagonal norm as a coherence measure:

$$ \mathcal{K}(\rho)=\sum_{\alpha\neq\beta} w_{\alpha\beta}\,|\rho_{\alpha\beta}|, \qquad w_{\alpha\beta}\ge 0. $$

Resonance functional via a positive operator field $R$:

$$ \mathcal{R}(\rho)=\mathrm{Tr}(\rho R), \qquad R=\sum_{i< j}\lambda_{ij}\,C_{ij}^\dagger C_{ij},\ \lambda_{ij}\ge 0. $$

Decision rule (locally in time): “good” if $ \Delta\mathcal{K}\ge 0 $ or $ \Delta\mathcal{R}\ge 0 $.

Explanation – Coherence & resonance

Measurability: $ \mathcal{K} $ (off-diagonal norm) and $ \mathcal{R} $ (operator field) give scalar scores.
Interpretation: large off-diagonals = strong superposition/resonance; large $ \mathrm{Tr}(\rho R) $ = strong “good” couplings.
Rule: “good” when $ \Delta \mathcal{K} \ge 0 $ or $ \Delta \mathcal{R} \ge 0 $ — locally in time.
Practice: IEQ as application-specific functional (e.g., de-escalation, fairness) → multi-objective optimisation.

Example I: XY coupling …

Hamiltonian (XY):

$$ H = J\big(\sigma_x\!\otimes\!\sigma_x + \sigma_y\!\otimes\!\sigma_y\big),\quad J>0. $$

Start $ \lvert\psi(0)\rangle=\lvert 01\rangle $, short time step $ U(t)\approx I-iHt $:

$$ H\lvert 01\rangle = 2J\,\lvert 10\rangle \;\Rightarrow\; \lvert\psi(t)\rangle \approx \lvert 01\rangle - i(2Jt)\,\lvert 10\rangle. $$

Density $ \rho(t)=\lvert\psi(t)\rangle\langle\psi(t)\rvert $ with off-diagonal

$$ \rho_{01,10}(t)\approx i\,2Jt \;\Rightarrow\; \big|\rho_{01,10}(t)\big|\approx 2Jt \quad(\text{linear for small }t). $$

Dephasing via $ L_1=\sqrt{\gamma}\,\sigma_z\!\otimes\!I,\ L_2=\sqrt{\gamma}\,I\!\otimes\!\sigma_z $:

$$ \rho_{01,10}(t)\approx i\,2Jt\,e^{-2\gamma t}. $$

Explanation – XY coupling

Mechanism: XY couples $|01\rangle \leftrightarrow |10\rangle$ directly → immediate superposition (seed of entanglement).
Coherence build-up: linear for small $t$ ($\propto Jt$); dephasing damps exponentially ($e^{-2\gamma t}$).
Didactics: minimal model to display coupling vs noise.
XD ethics: “good” couplings build resonance robustly against perturbations.

Example II: XXZ anisotropy & Bell generation

Hamiltonian (XXZ):

$$ H = J\big(\sigma_x\!\otimes\!\sigma_x + \sigma_y\!\otimes\!\sigma_y + \Delta\,\sigma_z\!\otimes\!\sigma_z\big). $$

In the subspace $ \{\lvert 01\rangle,\lvert 10\rangle\} $ we obtain

$$ H_{\text{sub}} = -J\Delta\,I + 2J\,\sigma_x, $$

giving the exact evolution for start $ \lvert 01\rangle $:

$$ \lvert\psi(t)\rangle = e^{\,iJ\Delta t}\Big(\cos(2Jt)\,\lvert 01\rangle - i\,\sin(2Jt)\,\lvert 10\rangle\Big). $$

At $ t^\star=\pi/(8J) $:

$$ \lvert\psi(t^\star)\rangle = e^{\,iJ\Delta t^\star}\,\frac{1}{\sqrt{2}}\big(\lvert 01\rangle - i\,\lvert 10\rangle\big), $$

… i.e. (up to a global phase) a Bell-like state (Bell).

Explanation – XXZ & Bell

Anisotropy $\Delta$: in the $\{01,10\}$ subspace only a global phase → Bell-time stays $ \pi/(8J) $.
Outside that subspace: $\Delta$ shifts relative phases ($|00\rangle,|11\rangle$) → interference/entanglement geometry is tunable.
Detuning via local fields: $h_1,h_2$ change the transition frequency $\Omega=\sqrt{(2J)^2+\delta^2}$ → timing/gate design.
Takeaway: $\Delta$ for phase landscape, $J$ for Rabi frequency — two levers of different nature.

Example III: Phase tuning → Bell canonisation

From $ (\lvert 01\rangle - i\,\lvert 10\rangle)/\sqrt{2} $ to the canonical Bell state via local Z-rotation:

$$ R_z(\phi) = e^{-\,i\,\frac{\phi}{2}\sigma_z} = \begin{pmatrix} e^{-i\phi/2} & 0 \\[2pt] 0 & e^{+i\phi/2} \end{pmatrix}. $$ $$ (R_z(\tfrac{\pi}{2})\!\otimes\!I)\,\frac{\lvert 01\rangle - i\,\lvert 10\rangle}{\sqrt{2}} \;=\; \frac{\lvert 01\rangle + \lvert 10\rangle}{\sqrt{2}} \;=\; \lvert\Psi^{+}\rangle. $$

Analogous for $ \lvert\Psi^{-}\rangle $ with $ \phi=-\tfrac{\pi}{2} $ (local $S^\dagger$).

Explanation – Phase tuning

Local Z-rotations change only relative phases (no population shift) → clean fine-tuning.
S-gate ($\phi=\pi/2$): turns $(|01\rangle - i|10\rangle)/\sqrt{2}$ into canonical $|\Psi^{+}\rangle$.
Cookbook: with S/S† and simple X/Y rotations all four Bell states are reachable.
XD ethics: precise phase design maximises measurable coherence without “more energy”.

Conclusion & outlook

The seven building blocks form a coherent model of the quantum-monadic field. The Hilbert space defines the space of possibilities — all states in which monadic efficacies can exist. Coupling operators and the field Hamiltonian describe how these efficacies interact and which resonances or conflicts arise.

Because carriers are transient, closed Schrödinger dynamics do not suffice. Open channels step in: Lindblad equations and CPTP channels model birth, death, noise, and disturbance — translating real-world contingency into the language of the field while keeping structural coherence in view.

To distinguish “good” from “bad” we need yardsticks. Coherence and resonance functionals provide them: they measure whether couplings thicken or fray the field. Ethics becomes operationalisable — not only discourse but quantitative effect-measurement in the spirit of XDM.

The three examples show how the principles apply in practice:

XY model: minimal couplings already create superpositions — the seed of entanglement.
XXZ model: anisotropy and parameter choice structure the phase landscape and shape entanglement.
Phase tuning: local operations turn an uncanonical state into a canonical Bell state — an ideal resonance figure.

Together this yields a toolbox for the field: the abstract idea of coherence becomes testable mathematical structure. The central challenge is solved: to grasp ethics not only normatively but as a field effect — quantifiable, analysable, and designable.

Outlook: next steps include spectral decomposition of the Hamiltonian, optimisation under IEQ constraints, and numerical simulation of large-scale monad fields.

XQM – FAQ

How does XQM differ from VQM?

XQM is the roof logic; VQM specifies couplings in the field.

How do you measure coherence?

Via IEQ as a functional combined with projections in the Hilbert space.

How does this flow into ethics?

XDM evaluates coupling patterns normatively as an ethics of resonance.