Can the collapse of the wave function be modelled as a quantum system on its own?

Question

Imagine I have an observer $\mathcal O$, a quantum system $\mathcal S$ with Hilbert space $V_{\mathcal S}$, a Hamiltonian $H$, a self-adjoint operator $A$ acting on $V_{\mathcal S}$. The system is in the (normalized) state $|\psi_0\rangle$ at time $t=0$, and the observer measures the system at a later time $t>0$ with the operator $A$. We know that the state collapses onto an eigenfunction $|a_j\rangle$ of the operator $A$ with probability $$ |\langle{a_j,e^{-itH}\psi_0}\rangle|^2. $$

So far so good. Now, since the collapse of the wave function is ugly, I would like to understand how this phenomenon can be viewed and modelled as a "meta-system" that includes both the observer $\mathcal O$ and the original system $\mathcal S$. Since the act of measurement is essentially an interaction between the observer and the system, I would expect that there is some sort of hamiltonian $\mathbf H$, possibly time-dependent, that governs the process in a Hilbert space $V_{\mathcal S}\otimes V_{\mathcal O}$. Of course, $\mathbf H$ would depend on $H$ and $A$.

My question: is it possible to model the act of quantum measurement with a quantum system? In particular, how can the external observer be modelled? Is it a standard thing to do? Any references you know that discuss this and related topics?

My problem is that I have no idea how an external observer should look like as a quantum system, and as a consequence I don't know how the "meta-Hamiltonian" $\mathbf H$ would look like. Please don't be shy in giving math details if you feel like it :)

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For context, I am a math student and I am trying very hard to understand QM (I struggle like everyone, I guess). Some things I am currently trying to understand are (just some vague questions):

1. How does an internal observer (i.e., that is part of the universe) "feel" the state of (the rest of) the universe?

2. Would two different observers always agree on the wave function that models (the rest of) the universe? If not, what are the conditions to ensure they do?

3. What is the true state of the universe? If the universe has its own wave function, but there is no external observer that makes it collapse, does it really make sense to use the wave function to model the state of the universe? Or can something more precise be used, that takes into account the experiences of the internal observers? (Edit: here by “universe” I am not necessarily talking about the real universe, I mean “a closed quantum system that contains observers”; the question is about a quantum system as a model (with internal observers) and it is vague because I cannot make it more rigorous).

Answer 1

To model the act of measurement itself as an interaction of the measurement apparatus and the measured system as quantum systems is a perfectly standard thing to do, though you might get disagreements over how "real" this is depending on people's chosen quantum interpretation.

The main buzzword here is decoherence, where we have the system $H_S$ and the environment $H_E$ and then we stipulate that the environment has "pointer states" $\lvert i\rangle_E$ - imagine a classical measurement device with a large pointer on a number range and these states corresponding to the state of that apparatus pointing at the number $i$ - such that time evolution will lead to the system being in decohered states of the form $$ \sum_i \lvert s_i\rangle_S \otimes \lvert i\rangle_E.$$ We then say that the apparatus modelled by this setup measures the observable whose eigenstates the $\lvert s_i\rangle_S$ are in $H_S$. This is also called a "von Neumann measurement scheme" for this observable, and people using this model of measurement do not necessarily need to the larger "philosophical" underpinnings of decoherence theory. Decoherence theory is concerned, among other things, with how these pointer states arise and why time evolution takes this particular form.

To what extent this actually "resolves" the measurement problem is a matter of interpretational debate, but one line of thinking is that this now has done away with the necessity of wave function collapse: We're in the environment, and each of the $\lvert i\rangle_E$ "sees" only its corresponding $\lvert s_i\rangle_S$ and can no longer interfere with the others, so for every $\lvert i\rangle_E$, the wavefunction of $S$ has "effectively collapsed" although the full superposition is still there in the universal wavefunction and only unitary time evolution has happened. Branches of the universal wavefunction we can no longer interfere with are, for all practical purposes, non-existent, explaining the validity of the standard Born rule that the post-measurement state for a measurement that measured $i$ is the corresponding eigenstate $\lvert s_i\rangle_S$.

The reason we usually don't discuss this a lot in introductory QM is that this is pretty irrelevant for all the basic results and observations QM was designed to explain. For this model you need to know specifics about the measurement apparatus and its interaction with the system, while we often just want to say generalities about the results of idealized measurements if we're not doing a specific experiment. Nevertheless, there is a sizable body of literature about the details of how to set up such measurement schemes and results about the resulting uncertainties and errors, see e.g. Ozawa's "Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement" for a start with plenty of further references.

Answer 2

When a quantum system interacts with another system while it is undergoing interference information is copied out of the system that suppresses interference. This effect is called decoherence. Since most of the objects you see around interact with the environment on timescales a lot slower than their evolution, they don't undergo interference:

https://arxiv.org/abs/quant-ph/0306072

Decoherence doesn't cause collapse. Rather, it just causes prevents different measurement results from interacting so they act a bit like a collection of parallel universes:

https://arxiv.org/abs/1111.2189

This is consistent although some people claim this theory doesn't make sense:

https://arxiv.org/abs/0905.0624

For criticisms of this argument see

https://arxiv.org/abs/1508.02048

Answer 3

Decoherence and the Quantum to Classical Transition by Schlosshauer is my favorite book on the measurement problem and interpretations of quantum mechanics. Note that, as pointed out in this book, decoherence DOES NOT SOLVE THE MEASUREMENT PROBLEM. Studying decoherence is related to studying the measurement problem but it doesn't solve it.

My question: is it possible to model the act of quantum measurement with a quantum system? In particular, how can the external observer be modelled? Is it a standard thing to do? Any references you know that discuss this and related topics?

Mathematically the observer can be modeled as a Hilbert space with some number of states. This observer Hilbert space gets tensor multiplied with the system Hilbert space as you've written. Physically an observer is just a bunch of particles or fields depending on the granularity of your model. So the state of the observer is the state of the particles/fields that make up the observer.

Yes, this is a standard thing to do. See the Schlosshauer reference above.

How does an internal observer (i.e., that is part of the universe) "feel" the state of (the rest of) the universe? Would two different observers always agree on the wave function that models (the rest of) the universe? If not, what are the conditions to ensure they do? What is the true state of the universe? If the universe has its own wave function, but there is no external observer that makes it collapse, does it really make sense to use the wave function to model the state of the universe? Or can something more precise be used, that takes into account the experiences of the internal observers? (Edit: here by “universe” I am not necessarily talking about the real universe, I mean “a closed quantum system that contains observers”; the question is about a quantum system as a model (with internal observers) and it is vague because I cannot make it more rigorous).

1. How does an internal observer (i.e., that is part of the universe) "feel" the state of (the rest of) the universe?

This is a fraught question.

What can be said, mathematically, in brief, is that, under pure unitary dynamics (think Everettian interpretation if it helps), the observer is described by a Hilbert space, and the rest of the universe is described by another Hilbert space. The state of the total system is described by the tensor product of these two spaces. The state of the joint system is a vector in this tensor product space. In general the joint state is not separable into a simple product of a vector from one space tensor producted with a vector from the other. Instead, typical states are linear sums of such simple products. Non-separable states are entangled states. So what can be said, in general, is that the observer is entangled with the rest of the universe, which, is to say, there are certain statistical correlations between physical properties of the observer and physical properties of the universe.

Aside on interpretations of quantum mechanics

More about why this question is fraught as you've phrased it:

If by "feel" you are anthropomorphizing an unconscious object and asking something about its state or dynamics, then I require clarification about what you mean by this question (preferable with no anthropomorphization). Though perhaps my paragraphs above already addressed this meaning of the question.

However, I'm tempted to take the word "feel" at face value here to mean the sensations felt by a conscious subject within a physical system (such as a human performing an experiment). In this case you are asking a question that is outside the scope of physics alone and pivots on metaphysics/philosophy of the mind. However, this question is indeed, in my opinion, at the heart of the measurement problem. Naively we assume that mental states (what we subjectively "feel") are 1:1 with physical states. In the classical world this is fine because there is a well defined physical state. But in an Everettian quantum world, it is not so clear. In the Everettian world an observer (a human, or a human's brain) can be in a superposition state of, for example, having seen a screen light of blue and having seen a screen light up red. What mental state do we ascribe to such a physical state? This last sentence is, in a nutshell, the entire measurement problem. Or at least it is the measurement problem if you take unitary evolution seriously (e.g. Everettian interpration).

Approaches which permit non-linear evolution (Copenhagen and objective collapse theories) do not suffer from this problem as badly because things like brains which have physical states that may be correlated with mental states are, indeed, in well-defined states that could be mapped 1:1 with mental states. However, these approaches suffer from different problems such as specifying in physically and quantitatively rigorous terms precisely when and how non-linear evolution occurs.

2. Would two different observers always agree on the wave function that models (the rest of) the universe? If not, what are the conditions to ensure they do?

Again, different physicists may give different answers here depending on their philosophical inclinations. Many interpretations of quantum mechanics are consistent with there being a single universal wavefunction that observers can agree on. However, there is an issue that we don't directly measure wavefunctions of physical systems. We measure observables and, in some cases, can use those to infer what the wavefunction was before the measurement. But this requires repeated preparation and measurement of similar physical systems. All of this is to say that, if there IS a universal wavefunction then we wouldn't be able to, even in principle, measure it. While this point is important to appreciate it doesn't kill the possibility of a theory which uses a model of the universe described by a universal wavefunction to make predictions.

However, some interpretations of quantum mechanics explicitly reject the idea of a universal wavefunction and instead claim that the wavefunction reflects the state of knowledge of a specific observer instead. You can look up quantum Bayesianism (QBism, pronounce cube-ism) or debates about whether the wavefunction is a real thing (psi-ontic) or just something about our knowledge (psi-epistemic).

3. What is the true state of the universe? If the universe has its own wave function, but there is no external observer that makes it collapse, does it really make sense to use the wave function to model the state of the universe? Or can something more precise be used, that takes into account the experiences of the internal observers? (Edit: here by “universe” I am not necessarily talking about the real universe, I mean “a closed quantum system that contains observers”; the question is about a quantum system as a model (with internal observers) and it is vague because I cannot make it more rigorous).

Honestly this question doesn't make sense so I can't answer it. It's not vague, it's unclear. "If the universe has its own wave function, but there is no external observer that makes it collapse, does it really make sense to use the wave function to model the state of the universe?" Why does no external observer call into question using a wavefunction as a model?

Answer 4

A big-picture conceptual answer, which I think is helpful before digging too deeply into the math:

Any quantum that is isolated (in a "box") from the outside world will remain in a superposition state and will never ``collapse." Even if there is a measurement device inside the box exists or the system "decoheres," to an outside observer, outside the "box" will never invoke the Born rule. What happens inside the box must be just deterministic propagation of the wavefunction (until it is open).

Put a quantum system and heat it to a billion degrees. Add as much decoherence as you'd like. Add little measurement devices in it. No matter what. If it is isolated from you (in a perfect "box"), it will not collapse. The Born rule will not occur.

So how does that resolve with the fact that an observer inside the box does see collapse? This is the "Wigner's Friend problem." The "solution" is that a person outside the box see's the observer inside the box as being in a quantum superposition of having measured both outcomes. Measurement only collapses the wavefunction in your perspective. It does not collapse in others perspectives if they do not make the measurement.

Decoherence is simply the continuous version of such a situation. Sometimes a quantum system can be "measured" by many other things (which we call "interaction with the enviornment"), to someone "outside the box", you expect to see a superposition of all the possible outcomes entangled with each individual outcome that was measured. And by opening the box you collapse the state and you check which outcome has occurred.

If you check a quantum system that hasn't interacted with anything, you can sometimes see weird interference fringes as the probability amplitudes interfere. As you allow for many different things to make measurements of this origionally isolated quantum system (all inside a box that you don't look at yet), then you will see that as you ramp up the interaction, when you open the box you will see less and less interference fringes.

There was a point where this decoherence process was thought by some to "solve the measurement problem" as it shows that interaction with the enviornment makes the interference fringes appear to disappear. But it is well understood now that essentially what's happening is that you end up with a messy mush of entangled states (that are in an even more complicted superposition state) and this mess is why is seems like the "quantum weirdness" through interference has disappeared.

User @acuriousmind goes through some of the technical outline of how one might show how this entanglement forms when a quantum system interacts with the enviornment through "pointer states," which are essentially little things that are making their own measurements of the state of the origionally isolated system through interaction.

My problem is that I have no idea how an external observer should look like as a quantum system, and as a consequence I don't know how the "meta-Hamiltonian" H would look like.

Because of these reasons there's a big difference between an external observer (outside the box) and an internal observer (inside the box). An external observer, when making measurements will collapse the state, so you need more than the schrodinger equation. You would have to pair it with the Born rule and do some stocastic simulations. (This isn't so common)

Decoherence considers internal observers that are inside of the box, that are interacting with the origionally isolated system. This is simply unitary evolution of a larger Hamiltonian and an "observation" is literally just when the "measuring object" would have some information about the state on its own. Entanglement like $|0\rangle |H\rangle + |1\rangle |T\rangle$ shows that in the perspective of the second object, being in state H means it knows that the first object is 0. Since collapse never happens you simply just propagate the unitary evolution and you get your internal "observers."

1. How does an internal observer (i.e., that is part of the universe) "feel" the state of (the rest of) the universe?

An internal observer "feels" it by becoming a superpostion of measuring different outcomes.

An external observer "feels" it by experiencing itself in one of the possibilities of those outcomes. (This is why the many-worlds intepretation came out of a resolution to the Wigner's Friend problem. )

2. Would two different observers always agree on the wave function that models (the rest of) the universe? If not, what are the conditions to ensure they do?

But maybe a simpler example to get what I think you are looking for: Imagine my friend performs a measurement, and then tells me the result. Then this means that we are both correlated and you have what you want: two observers that always agree on the outcome that has occurred. Now you might ask, how do I know that what I consciously experience is the same outcome as the outcome that you consciuosly experience, as it's possible that we experience wavefunction collapse different. And the answer, I believe, is that particularly question is unanswerable unless you test it yourself.

3. What is the true state of the universe? If the universe has its own wave function, but there is no external observer that makes it collapse, does it really make sense to use the wave function to model the state of the universe? Or can something more precise be used, that takes into account the experiences of the internal observers? (Edit: here by “universe” I am not necessarily talking about the real universe, I mean “a closed quantum system that contains observers”; the question is about a quantum system as a model (with internal observers) and it is vague because I cannot make it more rigorous).

"but there is no external observer that makes it collapse, does it really make sense to use the wave function to model the state of the universe?"

To an external observer outside the universe that has not opened the box of the universe since the beginning of time: the universe is in a superposition of all possible outcomes and wavefunction collapse has never ever happened. It is in a superposition of all possible outcomes that could ever probabilisitically happen, and if you are to interpret these different outcomes as real then you believe in the many-worlds-interpretation of QM.

Or can something more precise be used, that takes into account the experiences of the internal observers?

This is maybe a bit more technical than what you are looking for but:

Something precise about the experience of others cannot be used.

I think you are flip-floping external and internal, but if you want to know what can happen in the perspective or experience of an observer, you cannot know what others will experience. In fact, I have written about how there can be other rules for what other observers experience and you would not be able to measure them (see here).

But you can say something loose like "it seems like other observers, in my perspective, will tell me that they observe the Born rule," which may or may not be satisfying to you.

Answer 5

Assume that the state of the compound system (observed system and observer) is $\phi \otimes \psi$, where $\phi = \alpha \phi_0 + \beta \phi_1$ and where $\phi_0$ and $\phi_1$ are the eigenstates of the measurement.

Assume that a measurement is in fact an interaction, that is, an evolution of the compound system, and assume moreover (this is a technical assumption) that the unitary operator describing this evolution is of the form $\vert \phi_0 \rangle \langle \phi_0 \vert \otimes U_0 + \vert \phi_1 \rangle \langle \phi_1\vert \otimes U_1$. You can imagine that if $\phi_0 = dead\ cat$, then $U_0 = get\ sad$, and if $\phi_1 = alive\ cat$, then $U_1 = get\ happy$.

Then the state of the compound system, after the interaction, is $\alpha \phi_0 \otimes U_0\psi + \beta \phi_1 \otimes U_1\psi$.

Now, what decoherence states is that $U_0 \psi$ and $U_1\psi$ are most likely almost orthogonal. Therefore, if the observer evolves on its own, the two parts of it, $U_0\psi$ and $U_1\psi$ never interfere again, and both of these parts "think" that the wavefunction of the state has collapsed: $U_0\psi$ "thinks" that $\phi$ has collapsed to $\phi_0$ and $U_1\psi$ "thinks" that $\phi$ has collapsed to $\phi_1$.