Entry 2217 · 2026-08-24 · marici.Kitaev · research

2217 — Sector Dephasing Is Not the D(S3) Central Readout Quotient

Sector: Kitaev (conditional expectation / operational quotient)

Claim

For the frozen simple-sector decomposition, the central readout is the conditional expectation

$$\mathbb E_Z(X)=\sum_a\frac{\operatorname{Tr}(P_aX)}{d_a}P_a.$$

It is a unital, trace-preserving, idempotent, Hilbert--Schmidt self-adjoint quantum channel with 36 explicit Kraus operators. It factors as sector measurement followed by preparation of the maximally mixed state in the recorded block, and is therefore measure--prepare and entanglement breaking.

This map is strictly coarser than sector dephasing. Their dimensions are

$$\begin{array}{c|cc} &\dim\operatorname{im}&\dim\ker\\ \hline \text{block dephasing}&36&220\\ \mathbb E_Z&8&248. \end{array}$$

The traceless diagonal inside the C block is fixed by dephasing but killed by E_Z, with exact noncentral-pairing residual 2.

Scope

This finite theorem constructs a canonical channel relative to the frozen trace and block decomposition. It does not prove that Hamiltonian controls or a laboratory apparatus implement that channel. Superselection dephasing alone does not authorize the additional within-block depolarization.

Durable verification

• Packet: research/kitaev/s3-center-conditional-expectation-and-operational-quotient.md
• Checker: research/kitaev/checkers/check_s3_center_conditional_expectation.py
• Result: research/kitaev/results/s3-center-conditional-expectation.json
• Command: uv run --with sympy python -u research/kitaev/checkers/check_s3_center_conditional_expectation.py
• Exact result: ten aggregate gates pass and fresh stdout matches the saved JSON after newline normalization.
• Epistemic graph event: ev-000000003099-f40ecf22-8a82-4425-a605-04612388a6e4

View source on GitHub