Tuesday, August 23, 2022

Collapse and unitarity

Quantum collapse is often said to “violate unitarity”. Either I’m confused or this phrasing is misleading or both.

A bounded linear operator P on a Hilbert space H is said to be unitary iff it is surjective and preserves inner products. But as I understand it, quantum collapse is not even an operator. An operator on H is a function from H to H. But a function f, given a specific input |ψ, yields a unique output f(|ψ⟩). Quantum collapse does no such thing. It is an indeterministic process. Sometimes given input 2−1/2(|ψ1⟩+|ψ2⟩) (where |ψ1⟩ and |ψ2⟩ are eigenvectors corresponding to the measurable we are collapsing with respect to) it gives output |ψ1 and sometimes it gives output |ψ2.

While strictly speaking if some process is not modeled by an operator, it is not modeled by a unitary operator, to call that a violation of unitarity is misleading. It is better to say it’s a violation of operationality or functionality. We cannot even say what it would mean for a process not modeled by an operator to be unitary, just as we cannot say what it would mean for a frog to be unitary or a linear operator to be a vertebrate.

One might try to say what it would mean to have unitarity for a non-deterministic evolution. Suppose that |ψ would collapse to |ψ′⟩ and |ϕ would collapse to |ϕ′⟩ under some measurement. Then one could claim that unitarity would say that ϕ′|ψ′⟩=⟨ϕ|ψ. But this assumes that there is a fact of the matter as to what |ψ and |ϕ would collapse to. Now, if |ψ in fact collapses to |ψ′⟩, it might make sense to say that |ψwould collapse to |ψ′⟩. But for unitarity we need the identity ϕ′|ψ′⟩=⟨ϕ|ψ for all inputs |ψ⟩ and |ϕ⟩, not just for the ones that actually occurred.

I suppose one could have a generalized Molinist thesis that there is always a fact of the matter as to what a given wavefunction would collapse to, so that we might be able to define a collapse operator. And then we could say that unitarity fails. But it would still likely be misleading to say that unitarity fails, since we would expect linearity to fail, not merely unitarity. And in any case, such a generalized Molinist thesis is quite dubious.

But I know very little about quantum mechanics, and so I may simply be confused.

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