Self-Referential Dephasing

Fixed-Point Structure of the Nonlinear Map F(ρ) = Δ_ρ(Φ(ρ))

Jona Heidsick

This page summarizes the proven results of an ongoing research program. Claims are limited to what is formally established. Open problems are listed explicitly.

Core object: Given a quantum channel Φ, define the nonlinear map F(ρ) = Δ_ρ(Φ(ρ)), where Δ_ρ dephases in the spectral decomposition of the input state ρ. The nonlinearity is entirely from the state-dependent basis choice.

Proved Results

Machine-checked (Coq, general d + independent d=2, d=3)

Strict inclusion: Fix(Φ) ⊊ Fix(F)

For mixing channels Φ(ρ) = (1-p)ρ + pσ with non-diagonal σ and pairwise distinct diagonal entries, the self-referential loop creates additional fixed points for all d ≥ 2. The witness is ρ* = diag(σ). Zero physics axioms.

Machine-checked (Coq, general d + independent d=2)

Unital no-enlargement: Fix(F) = Fix(Φ)

When σ is diagonal, every F-fixed point is a Φ-fixed point. For σ = I/2, the unique F-fixed point is the maximally mixed state.

Proved

Primitive unital no-enlargement

For any primitive unital channel, Fix(F) = {I/d}. The key ingredient: primitivity plus unitality implies strict entropy production S(Φ(ρ)) > S(ρ) for ρ ≠ I/d. No additional hypothesis needed.

Proved

Partial entropy monotonicity

S(F(ρ)) ≥ (1-p)S(ρ) + pS(σ) for mixing channels. Entropy increases when S(ρ) ≤ S(σ). Global monotonicity is false (explicit counterexample). F stabilizes entropy near S(ρ*), pulling from both directions.

Proved

Local contraction

Near nondegenerate fixed points, if the Lipschitz constant of Φ plus the dephasing perturbation constant divided by the spectral gap is less than 1, the iteration Fⁿ converges geometrically.

Proved

Discontinuity at degenerate spectra

F does not extend continuously to states with degenerate eigenvalues. Different approach directions select different limiting eigenbases, making F naturally set-valued at degeneracies.

Construction

Start with a quantum channel Φ on finite-dimensional density matrices.
Define F(ρ) = Δ_ρ(Φ(ρ)), where Δ_ρ dephases in the eigenbasis of ρ.
The fixed-point equation F(ρ*) = ρ* means: the channel output, projected onto the state's own eigenbasis, returns the state.
For non-unital mixing channels, the state ρ* = diag(σ) satisfies this but is not a channel fixed point.
The unital/non-unital dichotomy determines whether extra fixed points exist.

Open Problems

Beyond mixing channels

Strict inclusion is proved for mixing channels in all d ≥ 2. Extension to general CPTP maps is expected but unverified.

Global Lyapunov function

Partial entropy monotonicity is proved. Von Neumann entropy is not globally monotone along F-trajectories. The natural candidate D(ρ || ρ*) fails because the data processing inequality yields the wrong reference point: the dephasing basis depends on ρ, not ρ*. A Lyapunov argument must account for this basis mismatch.

Degenerate fixed-point structure

The set-valued nature of F at degenerate points is established, but the degenerate fixed-point equation ("there exists a spectral decomposition such that F(ρ) = ρ") is unexplored.

Operational realization

The paper develops a structural connection to Zurek's einselection: F-fixed points are self-consistent pointer states whose eigenbasis survives one cycle of dynamics plus self-measurement. A rigorous embedding into measurement theory remains open.

Falsifiability

If the strict inclusion result does not extend beyond mixing channels to physically relevant CPTP maps, the construction is mathematically correct but physically inert.
If the unital/non-unital dichotomy does not survive in realistic models, the structural claim about non-unitality breaks down.
If standard decoherence theory already produces the same fixed-point structure without the self-referential step, F adds no new content.

Summary

Result: For non-unital mixing channels, self-referential dephasing strictly enlarges the fixed-point set for all d ≥ 2 and stabilizes states at a characteristic entropy level. For primitive unital channels, it does not. The strict inclusion and unital dichotomy are machine-checked in Coq for general d, with independent cross-checks at d=2 and d=3. The fixed points admit an operational reading as self-consistent pointer states in the sense of einselection.