Source
Zurek (2009), LIT-1640, section III, p. 5 of the arXiv preprint 0903.5082; read in NOTE-1229.
What was actually shown
The setting. Postulates (i) states are Hilbert-space vectors, (ii) evolution is unitary, (iii) “Immediate repetition of a measurement yields the same outcome” (p. 4). Suppose states |u⟩ and |v⟩ of S “produce an imprint in a subsystem of E (which plays a role of an apparatus), but remain unperturbed (so they can produce more imprints)”: |u⟩|e₀⟩ ⇒ |u⟩|e_u⟩, |v⟩|e₀⟩ ⇒ |v⟩|e_v⟩ (p. 5).
The argument. A unitary preserves scalar products, so with ⟨e₀|e₀⟩ = 1, ⟨u|v⟩ = ⟨u|v⟩⟨e_u|e_v⟩ (eq. 5). This “can be satisfied only when; (a) either ⟨e_u|e_v⟩ = 1 (which means that copying was completely unsuccessful), or; (b) ⟨u|v⟩ = 0”, in which case ⟨e_u|e_v⟩ is arbitrary and a perfect record, ⟨e_u|e_v⟩ = 0, is possible (p. 5). The argument holds for imperfect records too: any ⟨e_u|e_v⟩ ≠ 1 forces orthogonality.
The reading. “As a consequence of unitarity, only sets of orthogonal states … can be so copied, explaining selection of a set of outcomes – terminal points of quantum jumps” (p. 5). It “relies on just two values of the scalar product – 0 and 1 – and, thus, does not appeal to Born’s rule” (p. 5). It is the reason copying many records into disjoint fragments is possible at all without violating no-cloning: what is copied is an observable’s eigenstates, not an unknown state (p. 4).
What could have come out otherwise. A record that distinguished two non-orthogonal states and left both undisturbed would falsify it; eq. (5) rules that out exactly, not approximately. The paper also relaxes the assumptions: E need not start pure (ref. [23]), and repeatability can be transferred to an apparatus A, whose orthogonal pointer states then correlate with non-orthogonal states of S, giving POVMs (p. 6).
What this does not say
- Not which orthogonal set. The argument shows the recorded set must be orthogonal; that it is the einselected pointer basis comes from the interaction (“Interaction Hamiltonians do not perturb observables that commute with them”, p. 5), not from eq. (5).
- Not that one outcome occurs. Eq. (5) explains the first part of the collapse postulate, the set of outcomes. The single outcome per run is handled elsewhere in the paper by the “perception of ‘collapse’” as observers acquire a branch fragment (p. 6), an interpretation, not a derivation.
- Not probabilities. Those come in section IV from envariance, a separate and contested argument.
- Not that non-orthogonal states cannot be measured. They can be, by moving repeatability to an apparatus (p. 6). The constraint is on what is repeatably recorded while left undisturbed.
- Not new in this paper. It cites ref. [23] (Zurek, Phys. Rev. A 76, 2007, “Quantum origin of quantum jumps”) for the result and its relaxations; that paper is not held.