# Question about measurement and unitary dynamics

1. Nov 8, 2011

### S.Daedalus

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 $\rho_{OS}$, 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 $\mathrm{Tr}_O(\rho_{OS})$, 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 $\rho_{OS}$. However, she 'sees', from the inside, only the reduced system $\rho_S = \mathrm{Tr}_O(\rho_{OS})$. As the system $\rho_{OS}$ evolves unitarily, as all (closed) quantum systems do, the observer 'sees' a nonunitary evolution of $\rho_{S}$, since entanglement between herself and the system she observes tends to grow, and thus, so does the von Neumann entropy of $\rho_{S}$. Is that where the nonunitarity of measurement comes from?

2. Nov 8, 2011

### kith

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.