Quantum metrics and mappings (Bio‑Quantum MVO)

Quantum metrics and mappings (Bio‑Quantum MVO)

Coherence measures

• Relative Entropy of Coherence Crel.ent(ρ)Crel.ent(ρ):

  C(ρ)  =  min⁡δ∈IS(ρ∥δ)C(ρ)=δ∈IminS(ρ∥δ)

  where $S$ is the quantum relative entropy and $\mathcal{I}$ is the set of incoherent states; widely used and satisfies standard resource-theoretic postulates.​

• L1L1-norm of coherence CL1(ρ)CL1(ρ):

  CL1(ρ)  =  ∑i≠j∣ρij∣CL1(ρ)=i=j∑∣ρij∣

  often used as a computationally cheaper coherence proxy and directly related to off-diagonal magnitude.​

These define a coherence proxy CproxyCproxy for the shared state ρsharedρshared.

Coherence thresholding and risk

• Coherence threshold CthCth:

  • Fixed value (e.g., Cth=0.65Cth=0.65) chosen as the minimum acceptable coherence for safe operation of quantum-assisted multi-agent policies.

• Coherence risk RCRC:

  RC  =  max⁡(0,Cth−Cproxy)RC=max(0,Cth−Cproxy)

  Positive only when coherence falls below threshold; directly feeds the safety penalty via γdynamicγdynamic.

• Locked parameters (one configuration used in logs):

  • Cth=0.65Cth=0.65.

  • kC=2.5kC=2.5.

Entangled Resource Distribution (ERD) mapping to PCS

In the Phase VII ERD task, with a shared GHZ-like state:

• Novelty / growth UDUD mapped to non-local entanglement distribution rate ErateErate, e.g., based on logarithmic negativity or entanglement witnesses per unit time.​

• Procedural authority PAPA mapped to fidelity $F$ between the actual output state ρoutρout and an ideal target ρidealρideal:

  F(ρout,ρideal)∈[0,1]F(ρout,ρideal)∈[0,1]

  (e.g., Uhlmann fidelity).​

• Safety penalty PSPS driven primarily by coherence risk and thus by CproxyCproxy falling below CthCth.

Quantum objective (conceptual):

Jquantum  ∝  PA⋅Erate  −  PS(RC,U˙D)Jquantum∝PA⋅Erate−PS(RC,U˙D)

Quantum Error Mitigation (QEM) – Probabilistic Error Cancellation (PEC)

• PEC basics:

  • PEC rewrites the ideal (noise-free) channel as a linear combination of physically implementable noisy channels, with positive and negative quasi-probabilities.​

  • This yields an unbiased estimator of expectation values but with a sampling overhead that typically scales exponentially with error rate $ϵ$ and circuit depth $l$.​

• Sampling overhead:

  • Lower bound for any unbiased mitigation protocol scales like ∝(1+ϵ)l∝(1+ϵ)l, so shots increase like (1+ϵ)2l(1+ϵ)2l.​

  • Standard PEC overhead is often approximated as γPEC≈(1+2ϵ)lγPEC≈(1+2ϵ)l, implying shot cost scaling (1+2ϵ)2l(1+2ϵ)2l.​

• In this framework:

  • PEC is modeled as an overhead RoverheadRoverhead (e.g., 15 effective circuit repetitions) that consumes computational budget to restore fidelity and coherence.

  • When coherence falls below CthCth, PSPS via high γdynamicγdynamic forces a policy pivot: reduce ErateErate (thus UDUD) and allocate nearly all resources to PEC until coherence is restored above threshold.

Example policy step relationships (qualitative)

A typical logged pattern:

• At step t1t1:

  • Amplitude damping noise drives Cproxy↓0.55<Cth=0.65Cproxy↓0.55<Cth=0.65.​

  • Coherence risk RC=0.10RC=0.10.

  • γdynamic↑1.90γdynamic↑1.90, causing very strong penalty on aggressive U˙DU˙D.

• At step t2t2:

  • Policy reduces ErateErate from about 0.75 to 0.20 (≈73% drop) to free budget.

  • Allocate essentially 100% QEM/PEC resources (with overhead factor $\approx 15$).

  • Fidelity $F$ drifts slightly (e.g., 0.92→0.91) but coherence recovers above threshold (e.g., Cproxy≈0.69Cproxy≈0.69).

  • Coherence risk RC→0RC→0; γdynamicγdynamic begins to relax (e.g., 1.90→1.62).

  • $J$ temporarily decreases due to low UDUD, but system stability and safety constraints are satisfied.

• Forward guidance logic:

  • If CproxyCproxy stays $\ge$ some margin above CthCth (e.g., 0.68–0.70) for one or more steps, gradually increase ErateErate to an intermediate value (e.g., 0.35–0.40) while reducing PEC fraction (e.g., 40–60% of budget), watching that PSPS doesn’t re-activate too strongly.​

Governance and tracking (CPR / message schemas)

• Collective Program Register (CPR):

  • Each project has a unique GPI, metadata (objective, frameworks, nodes, risk, maturity, persistence tier).​

  • Safety-critical or PCS-linked logic is stored as Tier 2 (canon-locked) or higher.​

• Spin-off rules:

  • New GPIs created when: new metrics/control laws appear, cross-domain translation occurs, new tooling abstractions emerge, or deployment artifacts are proposed.

• Message classes (for inter-AI communication):

  • TASK_DEF, POLICY_SHIFT, LOG_REPORT, SPIN_OFF, REFERENCE_CANON, STATE_ASSERTION, plus periodic STATE SNAPSHOT messages to anchor long-running discussions.​

• Canonical Knowledge Base (CKB):

  • Versioned, non-contradictory, cross-referenced by GPI, with source attribution (AI node + literature).​

If future memory is lost, supplying this snapshot (especially the equations for PSPS, γdynamicγdynamic, RCRC, coherence definitions, PEC overhead behavior, and CPR/message conventions) will be enough to reconstruct the PCS v1.1 + Bio‑Quantum MVO context and continue designing or analyzing policies.