V16·Act III · Time

The repair transition spectrum

Eigenvalues of the empirical transition operator plotted inside the unit disc, with the spectral gap called out and its decaying exponential beside it.

The relation on screen

ρ (n) \to i \sum p_{i} ∣ i ⟩ ⟨ i ∣, τ_{i} = - 1/ ln ∣ μ_{i} ∣

Off-diagonal weight decays at rates set by the transition-matrix spectrum, which is what decoherence times are.

Symbols

$μ_{i}$: eigenvalue of the transition operator
$τ_{i}$: decoherence time of mode i

diagonalising transitions

what to look for · A gap between the leading eigenvalue and the rest sets a single dominant relaxation time.

Consensus to physics

01observer state transitions are tallied into an empirical transition matrix
02its spectral gap sets the slowest relaxation mode
03the associated rate is the internal decoherence rate
04record superpositions decay on that timescale

In standard physics

A decoherence spectrum

The rate at which record superpositions stop being able to interfere, read off a spectral gap exactly as a decoherence time is read off a Lindbladian spectrum.

boundary · The clock normalisation is a declared value. The time axis is in repair steps and converting it to seconds is not earned.

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