Why are measurements considered irreversible?

Question

In quantum mechanics, every interaction is described by a unitary Hamiltonian operator. We expect that a measurement is no different from any other interaction, yet in the standard way of treating QM the "state collapse" postulate is included as an extra non-reversible and instantanous evolution to describe measurements.

But why can't measurements be reversible? For example, in the double slit experiment where a detector is placed before the slits, the global state is: $|x1\rangle|D_{1}\rangle + |x2\rangle|D_{2}\rangle$, such an experiment does not exhibit interference, not because of a instantaneous state collapse, but because the interference term is proportional to $\langle D_{1}|D_{2}\rangle$ and this amplitude is really small.

In fact, I would expect that one could even modulate the factor $\langle D_{1}|D_{2}\rangle$ in an experiment to observe a continuous of interference patterns, clearly showing that there's nothing special about measurements where two sharp bands are observed, it's just a strongly suppressed interference pattern.

Is the measurement problem just a consequence of this misunderstanding? Or am I missing something deeper about it?

Answer 1

In the canonical interpretation of quantum mechanics (Copenhagen interpretation), the superposition of two eigenstates (e.g., spin, $(|+\rangle + |-\rangle)/\sqrt{2}$) means that the state, which is exhaustive in describing the system, itself has no definitive value for the spin of the particle. When you take the atom in this state and pass it through a Stern-Gerlach apparatus, you will measure $+\hbar/2$ or $-\hbar/2$. If you repeat the experiment with the same atom, you will always get the same result, so there is experimental evidence for the projection postulate. This is irreversible because after you measure, you get an eigenstate (or a state in the eigenspace of) some eigenvalue you have measured. For example, if you have measured $\hbar/2$, the original state can be $(|+\rangle + |-\rangle)/\sqrt{2}$, but it can also be $(|+\rangle + \sqrt{2}|-\rangle)/\sqrt{3}$. It can be any state $(a|+\rangle + b|-\rangle)/\sqrt{a^2+b^2}$, for $a\neq 0$. So, the function that associates the state before and after the experiment is not bijective, not invertible.

In other interpretations, such as the statistical one, there's no projection postulate because the state just represents an ensemble of similarly prepared experiments, but there is no superposition in the sense of a property of the particle itself. The particle is always well defined, so the experiment is theoretically invertible. But in these interpretations, the state does not represent an exhaustive description of reality, so the problem remains in every practical way. The interpretation of pilot-waves has similar properties. The problem of measurement, as well as those concerning uncertainty, needs to be interpreted as fundamental ones arising from the formalism of the theory, and not because the quantities are very small.

Answer 2

Empirical evidence tells us that we never directly measure or observe a superposition of states. In an interference pattern we average many separate observations over a period of time to infer the superposition state of the original photons - but we are not observing the superposition state directly. If we observe each individual photon we see that it hits the screen at a specific location, not in a superposition of locations.

Answer 3

Irreversibility of measurement is what we see in experiments. There really is no why: the theory is a story we tell about it. If you believe you have designed a method for reversible measurement, go try it: fame and fortune await you if you succeed.

Answer 4

In this answer I'm going to assume that the equations of motion of quantum theory are an accurate description of reality. This is not consistent with wave function collapse, which is incompatible with the vast bulk of measurement theory and quantum field theory in any case:

https://arxiv.org/abs/1604.05973

Now, let's suppose you have a system in the state $\tfrac{1}{\sqrt{2}}(|x_1\rangle+|x_2\rangle)$. If you want to measure this system in the $x$ basis then you need an interaction that couples the system to a detector and you get $$\tfrac{1}{\sqrt{2}}(|x_1\rangle|D_1\rangle+|x_2\rangle|D_2\rangle)$$

That detector then becomes coupled to other systems around it called the environment: $$\tfrac{1}{\sqrt{2}}(|x_1\rangle|D_1\rangle|E_1\rangle+|x_2\rangle|D_2\rangle|E_2\rangle)$$ This includes gas atoms in the atmosphere, photons from light bulbs and so on. These interactions copy the information about the measurement result with great redundancy and the only states that can be copied this way are orthogonal states, see Section II of:

https://arxiv.org/abs/0707.2832

These measurements suppress interference in a process called decoherence so that it is unobservable on the macroscopic scale:

https://arxiv.org/abs/1911.06282

But they don't eliminate the superposition so the measurement device, the environment and you are still in a superposition. As a result reality as described by quantum theory without collapse looks like a collection of parallel universes on the scale of everyday life to a good approximation:

https://arxiv.org/abs/1111.2189

Since you can't control all of the gas atoms and photons in the room you won't be able to undo this interaction and it is irreversible.

As for the measurement problem, there was a problem of why you only see one outcome of an experiment if you're in a superposition but in substance this has been solved. Some people don't like this for various reasons that are usually not clearly explained. For an unusually clear attempt at criticism see

https://arxiv.org/abs/0905.0624

For replies see

https://arxiv.org/abs/0906.2718

https://arxiv.org/abs/1508.02048