Collapse and superposition vs mixture

Born's rule answers one question: the probability of outcome a is p(a) = ⟨ψ|Pₐ|ψ⟩, or Tr(Pₐ ρ) in general. It says nothing about the state afterwards, and the Schrödinger equation cannot supply that either, because unitary evolution never turns a superposition into one of its terms. The Lüders rule is the extra postulate: apply the projector belonging to the eigenvalue you actually got and renormalise, |ψ⟩ → Pₐ|ψ⟩/√⟨ψ|Pₐ|ψ⟩, or ρ → Pₐ ρ Pₐ / Tr(Pₐ ρ). Two details deserve naming. Pₐ projects onto the eigenspace, so for a non-degenerate eigenvalue it is |a⟩⟨a| and the result is |a⟩ whatever you started from, while for a degenerate one it is Σₖ |a, k⟩⟨a, k| and the result keeps whatever structure |ψ⟩ had inside that subspace. And the denominator is exactly √p(a), which is what makes the rule closed. Take |ψ⟩ = 0.6|0⟩ + 0.8|1⟩ and measure σ_z: outcome +1 with probability 0.36, and the state afterwards is 0.6|0⟩ divided by 0.6, which is |0⟩.

Ask what a second, immediate measurement of the same A returns. The state is ρₐ = Pₐ ρ Pₐ / Tr(Pₐ ρ), so the probability of a again is Tr(Pₐ ρₐ), and idempotence gives Pₐ Pₐ ρ Pₐ Pₐ = Pₐ ρ Pₐ, so that probability is 1. Nothing else was assumed: repeatability is Pₐ² = Pₐ wearing physical clothes. Three sharp edges. Immediate means no evolution in between — let H act for a time t with [H, A] ≠ 0 and the certainty decays. Not every measurement is of this kind: a photon absorbed at a detector has no post-measurement state at all, and such processes need the general instrument formalism, not a projector. And a continuous spectrum has no projector onto a point. Position offers only the projector for an interval [a, b], which multiplies ψ by that interval's indicator function; repeating it returns the same interval with probability 1, while the fabled collapse to δ(x − x₀) is not a vector in L², and a state squeezed into a narrow interval carries an energy growing as the inverse square of its width.

Take a three-level space where A has eigenvalue a twofold degenerate on |a₁⟩ and |a₂⟩ and eigenvalue b on |b⟩, in the state (|a₁⟩ + |a₂⟩ + |b⟩)/√3. Outcome a has probability 2/3. Lüders projects with the full Pₐ = |a₁⟩⟨a₁| + |a₂⟩⟨a₂| and leaves the pure state (|a₁⟩ + |a₂⟩)/√2. Von Neumann's original prescription instead summed over a chosen orthonormal basis inside the eigenspace, leaving the mixture ½|a₁⟩⟨a₁| + ½|a₂⟩⟨a₂|. These are different states and an experiment separates them: measure the observable whose eigenvector is (|a₁⟩ + |a₂⟩)/√2 and the Lüders state gives that outcome with probability 1, the von Neumann state with probability ½. Two reasons to prefer Lüders. It is basis-independent inside the eigenspace, where the older rule depends on a choice A itself never made. And it is minimally disturbing: read nothing and it leaves ⟨B⟩ unchanged for every B commuting with A, and it is the only update that does. A degenerate measurement asks a coarse question and should not take more than it asked for.

Write both qubit states as matrices. The superposition |+⟩ = (|0⟩ + |1⟩)/√2 gives ρₛ with all four entries ½. The half-and-half mixture gives ρₘ = ½I: the same diagonal, zeros off it. Every σ_z prediction is therefore identical, P(+1) = ρ₀₀ = ½ for both, and no quantity of data in that basis will separate them. Rotate the basis and they part at once. P(σₓ = +1) = ⟨+|ρ|+⟩ = ½(ρ₀₀ + ρ₁₁ + 2 Re ρ₀₁), which is 1 for ρₛ and ½ for ρₘ; equivalently ⟨σₓ⟩ = 2 Re ρ₀₁ = 1 against 0. One basis-free number says the same: Tr ρ² is 1 for the superposition and ½ for the mixture. Be careful with the word superposition — it is relative to a basis, since ρₛ is diagonal in the x basis and is an eigenstate of σₓ. What is not relative is purity. A pure state is an eigenstate of some observable, while the maximally mixed state is diagonal in every basis and an eigenstate of none, so no change of basis can carry one into the other.

Leave the outcome unread and the update is non-selective: ρ → Σₐ Pₐ ρ Pₐ, summed over the distinct eigenvalues. This is an ordinary linear, trace-preserving channel. It leaves every diagonal entry alone, deletes every coherence between different eigenvalues, and keeps the coherences inside a degenerate eigenspace. Coupling to an environment performs the same arithmetic with no measurement made at all. Let (|0⟩ + |1⟩)/√2 drive a marker into |E₀⟩ or |E₁⟩: the joint state is (|0⟩|E₀⟩ + |1⟩|E₁⟩)/√2, and tracing out the marker leaves ρ₀₁ = ½⟨E₁|E₀⟩. Fringe visibility is |⟨E₁|E₀⟩|, so at overlap 0.6 the contrast is 0.6 and the purity is ½(1 + 0.36) = 0.68; orthogonal pointer states drive both to the mixture's values. Now what the argument does not deliver. The joint state is a unit vector throughout, so its purity is 1 at every step, and the map is linear and deterministic. Neither feature can produce one outcome out of two.

Collapse is a computational rule with an odd status: discontinuous where the Schrödinger equation is continuous, stochastic where it is deterministic, and carrying no time of its own. Nobody has watched the transition; what is observed is that the statistics after a measurement match the projected state. Interpretations divide here, and it pays to know which differences are empirical. Everett keeps unitary evolution and reads the update as conditioning on a branch; de Broglie–Bohm gets effective collapse once the empty packets stop overlapping; textbook quantum mechanics simply postulates the projection. None of the three predicts a different number, so no experiment separates them. Dynamical-collapse models such as GRW and CSL do differ, adding a stochastic nonlinear term with a rate and a length scale that predicts spontaneous heating and X-ray emission, and experiments keep narrowing the allowed parameters. One check binds them all: tracing Alice's side out of Σₐ (Pₐ ⊗ I) ρ (Pₐ ⊗ I) returns ρ_B unchanged, which is why no collapse rule of this kind can signal.