A qubit prepared in a superposition α|0⟩ + β|1⟩ interacts with N independent environment fragments through a dephasing (CNOT-like) coupling Hint ∝ σzS ⊗ σxEk. Because this interaction commutes with σz, the populations |α|², |β|² in the {|0⟩,|1⟩} pointer basis are untouched — they are the states that survive contact with the environment ("einselection"). Coherence between them is not so lucky:
|ρ₀₁(t)| = |αβ*| · Πk cos(φk(t)), φk(t) = min(π/2, gk·t)
Each factor comes from one fragment partially entangling with the system; the product collapses exponentially, which is why real macroscopic pointer states decohere almost instantly. Meanwhile every fragment that reaches φk → π/2 becomes a near-perfect classical record of the same collapse outcome — this redundant, repeated imprinting across many independent fragments is Zurek's quantum Darwinism: it is what lets many observers, measuring disjoint pieces of the environment, independently agree on one objective classical outcome without ever touching the qubit itself.
The fragment fidelity Fk = sin²(φk) is the probability that fragment k's record matches the true outcome. Combining f fragments by majority vote, the chance of correctly inferring the outcome is approximated (Gaussian approximation to the binomial majority-vote tail) as:
Pcorrect(f) ≈ Φ( (f·F − f/2) / √(f·F·(1−F)) )
Redundancy R0.99 is the smallest fragment count for which Pcorrect ≥ 99% — the plateau this reaches with only a small fraction of N is the signature "redundancy spectrum" measured in quantum-Darwinism experiments on photons and spins.
- θ₀ — Bloch angle of the initial state away from the pointer axis; sets Born-rule probabilities P(0)=cos²(θ₀/2), P(1)=sin²(θ₀/2), and which outcome gets recorded.
- N — number of environment fragments arranged on the outer shell.
- g — coupling rate; higher g reaches full einselection (and full redundancy) sooner.