Upon detection by an observer or measurement device, an isolated system changes (or collapses) from it's superposition of states to a single state. In decoherence, does an isolated system undergo the same kind of (or dare I say exact?) transition when decohered by it's environment?
2 Answers
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Short answer: No. Decoherence describes the interaction between a system and its environment. After this interaction, the system and environment will together be in a superposition of different states. This occurs by default because quantum mechanics is a linear theory. The collapse postulate is an additional rule that justifies why we only see a single outcome in experiments.
Long answer: Zurek has written extensive reviews of decoherence (2001, 2003) which address this question. I've sketched out the general ideas in an example below:
Consider a particle moving through space. The state of this particle may be described as a superposition of different position states. I'll denote two of them using $|x\rangle$ and $|y\rangle$.
Suppose there is a slight interaction between the particle and its immediate surroundings, which I'll call the environment. (This is generally the case$-$quantum systems are trickly to isolate for extended periods of time.) Like the particle, the environment can also be treated quantum-mechanically. I'll call the initial state of the environment $|\varepsilon\rangle$. During the interaction, the state of the environment will change in response to the system. I'll introduce the states $|\varepsilon_x\rangle$ and $|\varepsilon_y\rangle$ to represent the final state of the environment, for a particle sitting in the $|x\rangle$ and $|y\rangle$ states respectively. Then, because quantum mechanics is linear, an arbitrary superposition of these position states evolves to
$$ (a |x\rangle + b |y\rangle) |\varepsilon\rangle \rightarrow a |x\rangle |\varepsilon_x\rangle + b |y\rangle |\varepsilon_y\rangle $$
Often, the environment states will be orthogonal to each other, that is $\langle \varepsilon_x | \varepsilon_y\rangle = 0$. This entanglement with orthogonal environment states destroys the interference between the original position states; if you let the particle evolve and measured it later, you would no longer see interference effects.
(Feynman actually exemplifies this in volume III, part 3$-$2 of his lectures; the $x$ and $y$ states represent the particle at the locations of the slits, and the light source is the environment. While he doesn't introduce the term decoherence, I think this works as an example.)
So, decoherence can describe the loss of interference by entanglement with the environment, but it doesn't seem to explain why only one outcome occurs. Rather, this is handled by the collapse postulate. In our example, suppose we perform a measurement right after the decoherence has occurred. The collapse postulate then says that the state changes to $$ a |x\rangle |\varepsilon_x\rangle + b |y\rangle |\varepsilon_y\rangle \rightarrow \begin{cases} |x\rangle|\varepsilon_x\rangle & \text{w.p.}\,\,\,|a|^2 \\ |y\rangle|\varepsilon_y\rangle & \text{w.p.}\,\,\,|b|^2 \\ \end{cases} $$
which is what we observe experimentally. In particular, you could think of the measurement apparatus like the environment this way, since it's just another system that is sensitive to the quantum state.
That said, the question of if, when and how the collapse occurs constitutes the measurement problem, and is currently unsolved. (Or at least, there is no consensus on its resolution.) For instance, the Many-Worlds formulation of quantum mechanics argues that the collapse doesn't need to be postulated at all; observers would go into superpositions just like the environment does, and decoherence of the detector/observer explains why they only perceive one outcome.
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$\begingroup$ Thank you for engaging. I just left a comment, but I want to reword it. When the system and environment are together in that superposition of different states once they've interacted with each other, doesn't the part of the environment (we can say particle) that interacted with the initial system still need to be explained? I.e., since everything is a quantum system, doesn't something in the environment's environment enter into a superposition of different states with the environment before the environment interacts with our aforementioned initial system? $\endgroup$– Ant6491Commented Nov 10 at 13:57
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1$\begingroup$ Let's denote $|\varepsilon\rangle$ as the "environment", and $|\delta\rangle$ as the "environment's environment". I can certainly see how, before the interaction with the system, the environment's environment may become entangled with the environment, such that the combined state is like $|\varepsilon_1\rangle |\delta_1 \rangle + |\varepsilon_2 \rangle |\delta_2 \rangle$. If I understand correctly, your question roughly is "how are we justified in invoking the un-entangled environment state $|\varepsilon\rangle$ when the environment is constantly interacting with its surroundings?" $\endgroup$ Commented Nov 10 at 22:55
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1$\begingroup$ You could invoke a combined state $|\omega\rangle$, which is just whatever state the environment and environment's environment is in, immediately before the experiment. We can still say that $|\omega\rangle$ evolves into a superposition of $|\omega_x\rangle$ and $|\omega_y\rangle$ after interaction with the "initial system", and make the same argument about them being orthogonal, and interference being lost. In this sense, decoherence is quite agnostic to the exact state the environment is in. But I suppose you could then ask, "what about the environment's environment's environment..?" $\endgroup$ Commented Nov 10 at 22:56
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1$\begingroup$ Of course, it is rather unclear what "measurement" even means in this context, since we can't draw a clean line between system and observer. In particular, it might seem inconsistent to write the environment as a state which enters a quantum superposition, yet also claim that, at some undefined moment, it collapses back into a single state. (This is what formulations of QM such as Many-Worlds aim to clear up.) $\endgroup$ Commented Nov 23 at 0:45
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1$\begingroup$ But for now, my point is that measurement/collapse is what breaks entanglement, providing the origin of the factorised states. If we can accept that this occurs for a quantum particle and a detector, and if we think of measurement as a generic interaction between systems, then it seems plausible that a similar process occurs within the environment itself. One's attitude towards the measurement problem will specify how this process occurs. In conclusion, there doesn't need to be an "initial" unentangled state, since factorised states are constantly being broken by the process of measurement. $\endgroup$ Commented Nov 23 at 0:45
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The equations of motion of quantum theory describe the evolution of physical quantities in terms of Hermitian operators called observables whose eigenvalues are the possible results of measuring those observables. In many experiments the result can only be predicted by taking into account what happens to all of the possible values: quantum interference. See Section 2 of this paper for an example:
https://arxiv.org/abs/math/9911150
A measurement is a physical process that produces a record of some property of the measured system that can be copied, e.g. - you can look at the measurement result and then copy into a notebook, or you can copy a file containing a measurement result and send the copy to somebody else and so on. If we describe records that can be copied using quantum theory we find that in general copying suppresses interference: this is called decoherence
https://arxiv.org/abs/1911.06282
You stated that measurement causes collapse: it changes a system so that it only has one value. The decoherence description of measurement doesn't do that. To produce collapse you would have to modify the equations of motion of quantum theory, e.g. - spontaneous collapse theory
https://arxiv.org/abs/2310.14969
It should be noted that such theories can't currently reproduced the predictions of relativistic quantum theories, which comprise the vast bulk of experimentally tested predictions of quantum theory:
https://arxiv.org/abs/2205.00568
Quantum theory without such modifications describes reality in terms of multiple versions of each system and predicts that in everyday life those versions sort themselves into layers each of which looks approximately like the universe as described by classical physics
https://arxiv.org/abs/1111.2189
https://arxiv.org/abs/quant-ph/0104033
This is often called the many worlds interpretation and some physicists don't like it and prefer to modify quantum theory with collapse or other modifications such as pilot waves:
https://arxiv.org/abs/2408.05403
which all have the problems pointed out above. Others just refuse to describe the implications of quantum theory and call this an interpretation of quantum theory, e.g. - the Copenhagen and statistical interpretations.
