Are superpositions contagious?

Question

Does quantum mechanics really predict that a particle prepared in a state of superposition of spin will result, after being measured by an appropriate instrument (Stern-Gerlach device), in a macroscopic superposition of the instrument? Is there some theorem that proves that when an object in an unknown quantum state (the quantum state of the instrument is not in fact calculated) interacts with a particle in a superposition state this object will be found in a superposition state itself? And not any superposition, but a superposition of its classical states, like pointer going to the right - for spin-up or to the left - for spin-down?

Such an amplification of superpositions from the quantum level to macroscopic level seems to be at odds with observation. The actual state of a macroscopic object is in principle available everywhere due to the gravitational field associated with that object. There is no way to block the gravitational field of an object. So, either QM makes a wrong prediction (very unlikely) or there is some unjustified assumption behind the preparation of macroscopic superpositions.

Answer 1

You seem to be trying to play decoherence theory against the measurement problem, but the two are actually orthogonal. QM does not make wrong predictions, but we need to be precise about what a "prediction of QM" actually is:

The standard formalism of quantum mechanics gives a prescription for how to calculate the probabilities of the results of measurements of a quantum state (the Born rule), and it gives a prescription for how to calculate the evolution of quantum systems undisturbed by measurement (the Schrödinger equation). It does not, as such, explain what a measurement is or how it works. If we want to be precise, quantum mechanics without an interpretation is not a complete model of "how the world works" (because the Born rule and the Schrödinger equation contradict each other - the evolution induced by the first is not unitary, while the evolution induced by the latter is), but merely a tool that correctly predicts the results of measurements in a laboratory (or anywhere else) by chaining these to different evolutions together in the appropriate manner for the measurement being performed.

You can model the interaction between the measurement apparatus and the measured system as a quantum interaction itself, resulting in a superposition of states of the form $\sum_\lambda c_\lambda \lvert \lambda_s\rangle \otimes \lvert \lambda_a \rangle$ for the eigenvalues $\lambda$ of the quantity being measured, where the $s$ subscript denotes an eigenstate of the system for that quantity and the $a$ subscript denotes the corresponding "pointer state" of the measurement apparatus. This line of thinking began with things like the von Neumann measurement scheme and finally resulted in decoherence theory, explaining how measurement apparatus and measured system end up in this rather peculiar form of superposition. Decoherence does not explain away the measurement problem: It has no explanation for why we see only a single pointer state of the macroscopic measurement apparatus, nor does it claim to. It "merely" explains how the apparatus' state becomes entangled with the measured system in such a way that the pointer state always matches the actual value of the measured quantity for the corresponding state of the measured system.

But whether you model the interaction between the apparatus and the system thusly is irrelevant for the prediction of the probabilities of the results of the measurement (unless your measurement apparatus is imperfect and the $c_\lambda$ are not equal to the coefficients $c'_\lambda$ in the original state $\sum_\lambda c'_\lambda\lvert \lambda_s\rangle$ of the measured system alone) - the Born rule still holds. Adding gravity into the mix is likewise irrelevant - whether we "measure" the extended system of measured system + apparatus by looking at the apparatus it or by somehow deducing the position of the apparatus' pointer from its gravitational field doesn't change anything.

There is no universally accepted answer to the measurement problem. The lack of such an answer is one of the main reasons for the multitude of quantum interpretations, with each interpretation usually offering a different explanation of why we only see a single pointer state.

It is important to note that "the quantum state" of an object is inaccessible to us. All we can observe are the individual pointer states of quasi-classical measurement apparati, we cannot observe an actual quantum state. So whether a system is "in a superposition or not" is not, strictly speaking, a prediction of quantum mechanics. The predictions of quantum mechanics are the probabilities for results of measurements, and nothing else.

Answer 2

The problem you are describing is the complicated problem of the emergence of classical reality from quantum theory. The most accepted line of reasoning for this porblem today is based on decoherence developed by W. Zurek (a beginner friendly review of this work can be found here)

The short version is that the quantum system and the measurement apparatus get entangled as a result of the measurement and are found in a state

$$\left| \psi \right>_{SA}=\sum_{x} \left|x \right>_S\left|p_x \right>_A$$

where $x$ labels the states of the system and $p_x$ labels the corresponding states of the pointer of the measurement apparatus.

Since the apparatus is a macroscopic system it interacts very strongly and uncontrollably with its environment and gets entangled with it, resulting in the state

$$\left| \psi \right>_{SAE}=\sum_{x} \left|x \right>_S\left|p_x \right>_A\left|e_x \right>_E$$

Since the environment is not accessible (it is the EM fields and the thermal baths all around), we trace it out from the state resulting in the mixed state

$$\rho_{SA}=\sum_{x} \left|x \right> \left|p_x \right> \left<x\right|\left<p_x\right|$$

The superpositions of the apparatus become a statistical distribution.

How do we end up seeing only one possible outcome from this? It's an open problem that is commonly known as the quantum measurement problem. Nevertheless it still shows that superpositions of macroscopic object are unobservable because they "leak" into the environment very rapidly and become unobservable.

There are many many nuances here that I did not touch on but there is only so much you can say in one post.

Adding the gravitational effects to this opens a whole new can of worms that I saw a few papers on but it is still highly debatable and poorly understood.