James Appenzeller · EEG · quantum information

A quantum-information geometry for multichannel EEG

The trace-normalised covariance of an EEG window is a density matrix — by definition, not analogy. This site follows what happens when you take that literally: prepare the state on a register, ask it the questions quantum information asks, and derive its dynamics from the excitatory/inhibitory unit that produces it.

The arc, in five moves

  1. A circuit motivated by a neuron model. Paper 1 built the A-Gate: a quantum circuit whose rotations are set by EEG-derived features, motivated by the excitatory/inhibitory (PN) neuron. It underperformed a classical baseline (0.637 vs 0.742 AUC). Paper 1 · published
  2. The circuit exposed a sign. Some patients came out with AUC far below chance on a circuit with no trained parameters. Paper 2 showed why: the circuit measures in a fixed basis, and a fixed basis cannot absorb a sign flip. Paper 2 · preprint
  3. The circuit was found to be classically replaceable. Every gate after the encoding is a fixed unitary, so anything it measures is a transformed observable on three classical numbers per channel. The robust part of polarity is the sign of the ictal power change; the Tier-3 inverted set was axis-dependent, exactly as Paper 2’s mechanism predicts, and moves with the reference. Notes · 2026-09
  4. The object was replaced. The trace-normalised covariance is a density matrix — by definition, not analogy. So the framework becomes: the trace travels as a classical scalar; the state carries the shape; measure the state. Notebook 01
  5. The neuron model was rebuilt as a two-level system. The minimal coupled E/I unit is a 2×2 non-Hermitian generator with two knobs: net gain (sets the trace) and coupling-vs-asymmetry (sets the purity). Its balanced oscillatory state is a complex amplitude — envelope and phase — which is what the Hilbert encoding read all along, and why the absolute phase is a gauge. Notebook 02

Where to go

Notebook — the spine of the current work: numbered, chronological, one claim per page.

Papers — Paper 1 and Paper 2, each with what changed since it was written.

Learn — the math curriculum, from eigenvalues to quantum states.