Question about measurement and unitary dynamics

  • Thread starter S.Daedalus
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  • #1
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This is a question that's been in the back of my mind since I first learned quantum mechanics.

There seems to me to be some tension between the postulates of unitary evolution and state reduction upon measurement: basically, any quantum system ought to evolve unitarily; so in principle, every observer is a quantum system, so it ought to be possible to take the composite of the observer and the system she observes, treat it as a quantum system, and have it evolve unitarily. But then, where does the non-unitary reduction (i.e. the 'wave-function collapse') entailed by measurement come from?

Is it just that, if you have the system [itex]\rho_{OS}[/itex], i.e. the system made from the combination of the observer O and the observed system S, this will in general be highly entangled, since observation necessitates interaction, so the observer 'sees' the reduced system [itex]\mathrm{Tr}_O(\rho_{OS})[/itex], with herself 'traced out', which typically will have some von Neumann entropy -- which will tend to grow over time, as the entanglement grows with further observation, and since only non-unitary dynamics lead to rising entropy, the observer will tend to see non-unitary dynamics, even though the system made of herself and the observed system evolves unitarily?

Ugh, this turned into a bit of a run-on sentence. To be more clear, the observer is part of the system made out of the observer herself, and the observed system, described by the density operator [itex]\rho_{OS}[/itex]. However, she 'sees', from the inside, only the reduced system [itex]\rho_S = \mathrm{Tr}_O(\rho_{OS})[/itex]. As the system [itex]\rho_{OS}[/itex] evolves unitarily, as all (closed) quantum systems do, the observer 'sees' a nonunitary evolution of [itex]\rho_{S}[/itex], since entanglement between herself and the system she observes tends to grow, and thus, so does the von Neumann entropy of [itex]\rho_{S}[/itex]. Is that where the nonunitarity of measurement comes from?
 

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  • #2
kith
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Yes, this can be used to explain the measurement process, which is done in the Many Worlds interpretation and the ensemble interpretation.

However, it depends crucially on what density operators ρ and kets |ψ> are supposed to mean. Let's consider a simple example along your line of reasoning. We have a two-niveau system which initially is in a superposition state |ψ>=|ψ1>+|ψ2>. This state evolves into a mixed state ρ due to interactions with the environment given by the measurement apparatus. So there is non-unitarian time evolution in all interpretations, but it doesn't necessarily explain the reduction of the state vector. You start in a probabilistic state and you end up in a probabilistic state, but in experiments you always see definite outcomes.

The Copenhagen interpretation thus postulates, that performing a measurement reduces the state to one possibility. In the Many Worlds interpretation for example, all outcomes are equally real, so the explanation above explains the whole measurement process.
 

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