The Loss of Coherence

[DERIVATION (from AP09 open-systems formalism + factorisation assumptions) + STRUCTURAL INTERPRETATION (Axiom R as irreversibility postulate, pending D6)]

Now the algebra. You have the picture — environment writes records, records are irreversible, coherence disperses. What follows is the mathematics that makes this precise.

3.1 — Coherence as a phase relationship

In the Hilbert space (AP09 3.2), a superposition of two states is:

|ψ⟩ = α|A⟩ + β|B⟩

The coherence is the off-diagonal terms of the density matrix:

ρ = |ψ⟩⟨ψ| = |α|²|A⟩⟨A| + αβ|A⟩⟨B| + αβ|B⟩⟨A| + |β|²|B⟩⟨B|

where α, β ∈ ℂ with |α|² + |β|² = 1. The off-diagonal terms αβ|A⟩⟨B| and αβ|B⟩⟨A| are the coherence — they encode the phase relationship between states A and B.

The complex conjugation is essential: it is what makes the density matrix Hermitian (ρ = ρ†) and ensures the phase information is correctly represented. If these terms are nonzero, interference is possible.

If they vanish, the system behaves as a classical mixture: either A with probability |α|² or B with probability |β|², but not both simultaneously. You get one or the other. That is what classical means.

3.2 — What environmental records do to the coherence

After one environmental interaction, the system is entangled with the environmental particle:

|Ψ⟩ = α|A⟩|e_A⟩ + β|B⟩|e_B⟩

where |e_A⟩ and |e_B⟩ are the environmental states correlated with the system being at A or B.

The system’s density matrix — obtained by tracing over the environmental degree of freedom — is:

ρ_system = |α|²|A⟩⟨A| + αβ⟨e_B|e_A⟩|A⟩⟨B| + αβ⟨e_A|e_B⟩|B⟩⟨A| + |β|²|B⟩⟨B|

The coherence terms are now multiplied by ⟨e_B|e_A⟩ — the overlap between the two environmental states. If the environmental states are orthogonal (⟨e_B|e_A⟩ = 0), the coherence vanishes completely after a single interaction.

If the overlap is partial (0 < |⟨e_B|e_A⟩| < 1), the coherence is reduced but not eliminated.

Watch what happens next. One interaction reduces the coherence. What do a billion interactions do?

3.3 — The monoid amplifies the loss

Assumptions (not derived from the axioms; physically motivated). The following derivation assumes: (i) the environmental Hilbert space decomposes as H_E = ⊗ᵢ H_Eᵢ into effectively independent subsystems; (ii) the initial environment state factorises across these subsystems; (iii) the interactions are sequential and approximately independent, so the total coherence factor factorises as a product.

These are standard assumptions in collisional decoherence models (Joos et al., 2003; Zurek, 2003). They are not derived from {S, B, R, C}.

The monoid structure (Axiom R) guarantees that records compose and cannot be undone; it does not by itself guarantee that the composition yields a factorised product of overlaps.

The product form is a physically motivated ansatz for typical environments. A debt owed — and you hold the kill switch (see 6, Debts Owed).

Under these assumptions, after N environmental interactions the coherence factor becomes:

⟨e_B|e_A⟩_total = ∏ᵢ₌₁ᴺ ⟨e_B⁽ⁱ⁾|e_A⁽ⁱ⁾⟩

Each factor has magnitude ≤ 1. For generic environmental interactions where the typical overlap magnitude is strictly less than 1 — that is, where each environmental record carries nonzero information about the system — the magnitude of the product decays exponentially in N.

Define the per-interaction decoherence weight λᵢ = −ln|⟨e_B⁽ⁱ⁾|e_A⁽ⁱ⁾⟩| ≥ 0 for each interaction. Then |⟨e_B|e_A⟩_total| = e^(−∑ᵢλᵢ). For identical interactions with per-interaction weight λ > 0, after N interactions the coherence magnitude is e⁻ᴺλ.

Even if each individual environmental record is a weak measurement (λ small), the sum of many such weights drives the coherence magnitude to zero exponentially fast.

The condition λ > 0 (equivalently, |⟨e_B|e_A⟩| < 1 for at least some interactions) is the physical content: the environment must actually distinguish the two states, at least partially, for decoherence to occur.

An environment that writes no distinguishing records (λ = 0 for all interactions) produces no decoherence.

The monoid doing what it does. Each record is composed with the previous records. The composition is the monoid operation. The monoid grows monotonically (Axiom R — no inverses).

The magnitude of the coherence factor, which is the product of all individual overlap magnitudes, decays monotonically.

What Axiom R contributes specifically is the irreversibility interpretation: in standard quantum mechanics, the combined system+environment evolution is unitary and in principle reversible, making decoherence practically irreversible (because reversing a macroscopic entangling history is infeasible) but not structurally forbidden.

In the 420 Code, Axiom R upgrades this: the monoid has no inverse, so within the architecture’s allowed dynamics, environmental records cannot be undone. The irreversibility is structural, not merely practical.

A formal no-go theorem specifying the allowed dynamical class within which this holds is outstanding (see D6 in 6, Debts Owed).

The interpretive upgrade — from practical to structural irreversibility, pending D6 — is this paper’s specific contribution beyond standard decoherence theory.

Decoherence is Axiom R applied to the coherence of a quantum state. Records accumulate. Each record reduces the coherence magnitude. Under the Axiom R reading, the reduction is irreversible (see D6 above).

The coherence magnitude decays exponentially in the number of environmental records. The superposition becomes a classical mixture — not because the pre-state was destroyed, but because the phase information was dispersed into the monoid.

3.4 — The decoherence timescale

The rate of decoherence depends on two things:

The coupling strength: How efficiently each environmental interaction writes a record about the system. Strong coupling (|⟨e_B|e_A⟩| ≈ 0 per interaction) means each record is nearly complete — coherence is lost in a few interactions.

Weak coupling (|⟨e_B|e_A⟩| ≈ 1 per interaction) means each record is nearly empty — coherence decays slowly.

The interaction rate: How many environmental degrees of freedom interact with the system per unit time. A system in a dense environment (air, thermal radiation) decoheres in a fraction of a second.

A system in near-vacuum (deep space, cryogenic isolation) can maintain coherence much longer.

The decoherence timescale τ_D is determined by the product of these two factors.

From 3.3: if interactions occur at rate Γ (number of environmental interactions per unit time), then after time t the system has undergone N = Γt interactions.

Using the per-interaction decoherence weight λ = −ln|⟨e_B|e_A⟩| from 3.3, the coherence magnitude is |⟨e_B|e_A⟩_total| = e⁻ᴺλ = e^(−Γλt). The decoherence timescale: τ_D⁻¹ = Γλ = −Γ ln|⟨e_B|e_A⟩|.

Connection to the standard literature: for weak coupling (|⟨e_B|e_A⟩| ≈ 1), λ ≈ 1 − |⟨e_B|e_A⟩| ≈ (1 − |⟨e_B|e_A⟩|²)/2 to leading order.

The standard result from open quantum systems theory (Zurek, 2003, §II.C; Joos et al., 2003, Chapter 3) typically expresses the decoherence rate as Γ(1 − |⟨e_B|e_A⟩|²), which corresponds to the decay rate of the squared coherence magnitude (visibility squared) and equals 2Γλ to leading order.

The factor-of-2 difference reflects whether one tracks |coherence| or |coherence|²; both conventions appear in the literature. The qualitative content is the same: exponential decay at a rate set by Γ × (per-interaction distinguishability).

The axioms do not predict specific coupling strengths or interaction rates — those depend on the microphysical details of the system and its environment (see D2 in 6).

The quantitative expression for τ_D is adopted from standard open quantum systems theory, not derived from {S, B, R, C}.

What the axioms contribute is the structural identification (environmental record-writing) and the irreversibility interpretation (Axiom R, pending formal no-go theorem D6).

The mechanism is universal: it applies to every quantum system that is not perfectly isolated from the accumulated record.

Cross-reference: AP09 3.2: Hilbert space, density matrix. AP09 6: Born rule (probabilities from |ψ|²). AP20 4: Axiom R (monoid, no inverses).