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aotell comments on Welcome to Less Wrong! (July 2012) - Less Wrong
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I think it will be helpful if I briefly describe what my approach to understanding quantum theory is, so that you can put my statements in the correct context. I assume a minimal set of postulates, namely that the universe has a quantum state and that this state evolves unitarity, generated by the strictly local interactions. The usual state space is assumed. Specifically, there is no measurement postulate or any other postulates about probability measures or anything like that. Then I go on to define an observer as a mechanism within the quantum universe that is realized locally and gathers information about the universe by interacting with it. With this setup I am able to show that an observer is unable to reconstruct the (objective) density operator of a subsystem that he is part of himself. Instead he is limited to finding the eigenvector belonging to the greatest eigenvalue of this density operator. It is then shown that the measurement postulate follows as the observer's description of the universe, specifically for certain processes that evolve the density operator in a way that changes the order of the eigensubspaces sorted by their corresponding eigenvalues. That is really all. There are no extra assumptions whatsoever. So if the derivation is correct then the measurement postulate is already contained in the unitary structure (and the light cone structure) of quantum theory.
As you would know, the arxiv sees several papers every month claiming to have finally explained quantum theory. I would have seen yours in the daily listings and not even read it, expecting that it is based on some sort of fallacy, or on a "smuggled premise" - I mean that the usual interpretation of QM will be implicitly reintroduced (smuggled into the argument) in how the author talks about the mathematical objects, even while claiming to be doing without the Born rule. For example, it is very easy for this to happen when talking about density matrices.
It is a tedious thing to go through a paper full of mathematics and locate the place where the author makes a conceptual mistake. It means you have to do their thinking for them. I have had another look at your paper, and seen a little more of how it works. Since you are here and wanting to promote your idea, I hope you will engage with me even if I am somewhat "lazy", in the sense that I haven't gone through the whole thing and understood it.
So first of all, a very simple issue that you could comment on, not just for my benefit but for the benefit of anyone who wants to know what you're saying. An "observer" is a physical being who is part of the universe. The universe is described by a quantum state vector. The evolution of the state vector is deterministic. How do you get nondeterministic evolution of the observer's state, which ought to be just a part of the overall state of the universe? How do you get nondeterminism of the part, from determinism of the whole?
We know how this works in the many-worlds interpretation: the observer splits into several copies that exist in parallel, and the "nondeterminism" is just an individual copy wondering why it sees one eigenvalue rather than another. The copy in the universe next door is thinking the same thing but with a different eigenvalue, and the determinism applies at the multiverse level, where both copies were deterministically produced at the same time. That's the many-worlds story.
But you have explicitly said that only one branch exists. So how do you reconcile nondeterminism in the part with determinism in the whole?
Second, a slightly more technical issue. I see you writing about the observer as confined to a finite local region of space, into which particles unpredictably enter and scatter. But shouldn't the overall state vector be a superposition of such events? That is, it will be a superposition of "branches" where different particles enter the region at different times, or not at all. Are you implicitly supposing that the state vector outside the small region of space is already "reduced" to some very classical-looking basis?
I see it exactly like you. I too see the overwhelming number of theories that usually make more or less well hidden mistakes. I too know the usual confusions regarding the meaning of density matrices, the fallacies of circular arguments and all the back doors for the Born rule. And it is exactly what drives me to deliver something that is better and does not have to rely on almost esoteric concepts to explain the results of quantum measurements.
So I guarantee you that this is very well thought out. I have worked on this very publication for 4 years. I flipped the methods and results over and over again, looked for loopholes or logical flaws, tried to improve the argumentation. And now I am finally confident enough to discuss it with other physicists.
Unfortunately, you are not the only physicist that has developed an understandable skepticism regarding claims like I make. This makes it very hard for me to find someone who does exactly what you describe as being hard work, thinking the whole thing through. I'm in desperate need of someone to really look into the details and follow my argument carefully, because that is required to understand what I am saying. All answers that I can give you will be entirely out of context and probably start to look silly at some point, but I will still try.
I do promise that if you take the time to read the blog (leave the paper for later) carefully, you will find that I'm not a smuggler and that I am very careful with deduction and logic.
To answer your questions, first of all it is important that the observer's real state and the state that he assumes to be in are two different things. The objective observer state is the usual state according to unitary quantum theory, described by a density operator, or as I prefer to call them, state operator. There is no statistical interpretation associated with that operator, it's just the best possible description of a subsystem state. The observer does not know this state however, if he is part of the system that this state belongs to. And that is the key result and carefully derived: The observer can only know the eigenstate of the density operator with the greatest eigenvalue. Note that I'm not talking about eigenstates of measurement operators. The other eigensubspaces of the density operator still exist objectively, the observer just doesn't know about them. You could say that the "dominant" eigenstate defines the reality for the observer. The others are just not observable, or reconstructable from the dynamic evolution.
Once you understand this limitation of the observer, it follows easily that an evolution that changes the eigenvalues of the density operator can change their order too. So the dominant eigenstate can suddenly switch from one to another, like a jump in the state description. This jump is determined by external interactions, i.e. interactions of the system the observer describes with inaccessible parts of the universe. An incoming photon could be such an event, and in fact I can show that the information contained in the polarization state of an incoming photon is the source of the random state collapse that generates the Born rule. The process that creates this outcome is fully deterministic though and can be formulated, which I do in my blog and the paper. The randomness just comes from the unknown state of the unobserved but interacting photon.
So as you can see this is fundamentally different from MWI, and it is also much more precise about the mechanism of the state reduction and the source of the randomness. And the born rule follows naturally. No decision theory and artificial assumptions about state robustness, preferred basis or anything like that. Just a natural process that delivers an event with a probability measurable by counting events.
Your last question about the environment being classical is a very good one. I do not model the environment to be classical, in fact there is no assumption about it other than that it belongs to a greater quantum system and that it is not part of the system that the observer wants to describe. There are also no restrictions about anything being in a superposition. That problem resolves itself because the state described by the observer turns out to be a pure state of the local system, always. So even if you assume some kind of superposition of these events, you will always get a single outcome. The scattering process in fact has the property of sending superpositions to different eigensubspaces of the state operator, so that it cleans up everything and makes it more classical, just like the measurement postulate would.
I know I am demanding a lot here, but I really think you will not regret spending time on this. Let me know what else I can explain.
Here's another question. Suppose that the evolving wavefunction psi1(t), according to your scheme, corresponds to a sequence of events a, b, c,... and that the evolving wavefunction psi2(t) corresponds to another sequence of events A, B, C... What about the wavefunction psi1(t)+psi2(t)?
You really come up with tricky questions, good :-). I think there are several ways to understand your questions and I am not sure which one was intended, so I'll make a few assumptions about what you mean.
First, an event is a nonlinear jump in the time evolution of the subjectively perceived state. The objective global evolution is still unitary and linear however. In between the perceived nonlinear evolution events you have ordinary unitary evolution, even subjectively. So I assume you mean the subjective states psi1(t) and psi2(t). The answer is then that in general superpositions are not valid subjective evolutions anymore. You can still use linearity piecewise between the events, but the events themselves don't mix. There are exceptions, when both events happen at the same time and the output is compatible, as in can be interpreted as having measured an subspace instead of a single state, which requires mutual orthogonality. So in other words: In general there is no global state that would locally produce a superposition if there are nonlinear local events.
However if you mean that psi1 and psi2 are the global states that produce a list of events a,b,c and A,B,C respectively and you add up those, then the locally reconstructed state evolution will get complicated. If you add with coefficients psi(t) = c1 psi1(t) + c2 psi2(t) then you will get the event sequence a,b,c for |c1|>>|c2| and the sequence A,B,C for |c2|>>|c1|. What happens in between depends on the actual states and how their reduced state eigenspaces interact. You may see an interleaved mix of events, some events may disappear or you may see a brand new event not there before. I hope this answers your questions.
I find your reference to "the subjectively perceived state" problematic, when the physical processes you describe don't contain a brain or even a measuring device. Freely employing the formal elements and the rhetoric of the usual quantum interpretation, when developing a new one supposedly free of special measurement axioms and so forth, is another way for the desired conclusion to enter the line of reasoning unnoticed.
In an earlier comment you talk about the "objective observer state", which you describe as the usual density operator minus the usual statistical interpretation. Then you talk about "reality for the observer" as "the eigenstate of the density operator with the greatest eigenvalue", and apparently time evolution "for the observer" consists of this dominant eigenstate remaining unchanged for a while (or perhaps evolving continuously if the spectrum of the operator is changing smoothly and without eigenvalue crossings?), and then changing discontinuously when there is a sharp change in the "objective state".
Now I want to know: are we really talking about states of observers, or just of states of entities that are being observed? As I said, you're not describing the physics of observers, you're not even describing the physics of the measurement apparatus; you're describing simple processes like scattering. So what happens if we abolish references to the observer in your vocabulary? We have physical systems; they have an objective state which is the usual density operator; and then we can formally define the dominant eigenstate as you have done. But when does the dominant eigenstate assume ontological significance? For which physical systems, under which circumstances, is the dominant eigenstate meaningful - brains of observers? measuring devices? physical systems coupled to measuring devices?
Your question is absolutely valid and also important. In fact, most of what I write in my paper and the blog is about answering precisely this.
My observer is well defined, as a mechanism that is part of a quantum system and who interacts with the quantum system to gather information about it. He is limited by the locality of interaction and the unitary nature of the evolution. I imagine the observer to be a physicist, who tries to describe the universe mathematically, based on what he sees. But that is only a trick in order to have a mathematical formulation of the subjective view. The observer is prototypical for any mechanism that tries to create a model of his surrounding. This approach is very different from modeling cognitive mechanisms, and it's also much more general. The information restriction is so fundamental that you can talk about his subjective reconstruction of what is going on as local subjective reality, as everyone has to share it.
The meaning of the dominant eigensubspace is then derived from this assumption. Specifically, I am able to identify a non-trivial transformation on the objective density operator of the observer's subsystem that he cannot gain any knowledge about. This transformation creates a class of equivalent representations that are all equally valid descriptions which the observer could use for making a model of his environment (and himself). The arbitrariness of the representation connected with this reconstruction however forces him to reduce his state description to something more elementary, something that all equivalent descriptions have in common. And that turns out to be the dominant eigensubspace as his best option. This point is very important, and the derivation I provide in the blog is rigorous and detailed. The result is that the subjective reality as reconstructed by any observer like this evolves unitarily if the greatest eigenvalue does not intersect with other eigenvalues (the observer himself cannot know the value of the eigenvalues either) or discontinuous as a formerly smaller eigenvalue intersects with the greatest one to become the new dominant eigenvalue. This requires an interaction with a part of the system that is not contained in the objective local state description, like an incoming photon.
This approach also has the advantage that you don't have to actually model the observer. You still know what information is available to him. That is why the observer does not even have to be part of the system that you want to "subjectify". You already know how he would describe it. Specifically, you don't have to consider any kind of entanglement between observer states and observed states. The dominant eigensubspace is a valid description of every system that the describing entity is part of and that contains everything the observer is directly interacting with. If you want to get quantum jumps you also need an external inaccessible environment.
Summarizing, there's no need to postulate the ontology or relevance of the dominant eigensubspace. I was very careful to only make assumptions that are transparent and to derive everything from there. Specifically I am not adopting any definition or terminology from interpretations of quantum theory.
I finally got as far as your main calculation (part IV in the paper). You have a two-state quantum system, a "qubit", and another two-state quantum system, a "photon". You make some assumptions about how the photon scatters from the qubit. Then you show that, given those assumptions, if the coefficients of the photon state are randomly distributed, then applying the Born rule to the eigenvalues of the old "objective state" (density operator) of the qubit, gives the probabilities for what the "dominant eigenstate" of the new objective state of the qubit will be (i.e. after the scattering).
My initial thoughts are 1) it's still not clear that this has anything to do with real physical processes 2) it's not surprising that an algebraic combination of quantum coefficients with random variables is capable of yielding new random variables with a Born-rule distribution 3) if you try to make this work in detail, you will end up with a new modification of quantum mechanics - perhaps a stochastic, piecewise-linear Bohmian mechanics, or just a new form of "objective collapse" theory - and not a derivation of the Born rule from within quantum mechanics.
Are you saying that actual physical systems contain populations of photons with randomly distributed coefficients such as you describe? edit Or perhaps just that this is a feature of electromagnetically mediated measurement interactions? It sounds like a thermal state, and I suppose it's plausible that localized thermal states are generically involved in measurement interactions, but these details have to be addressed if anyone is to understand how this is related to actual observation.
There must be something that you have fundamentally misunderstood. I will try to clear up some aspects that I think may cause this confusion.
First of all, the scattering processes presented in the paper are very generic to demonstrate the range of possible processes. The blog contains a specific realization which you may find closer to known physical processes.
Let me explain in detail again what this section is about, maybe this will help to overcome our misunderstanding. A photon scatters on a single qubit. The photon and the qubit each bring in a two dimensional state space and the scattering process is unitary and agrees with conservation laws. The state of the qubit before the interaction is known, the state of the photon is external to the observer's system and therefore entirely unknown, and it is independent of the state of the qubit.
The result of the scattering process is traced over the external outgoing photon states to get a local objective state operator. You then write I apply the Born rule, but that's really exactly what I don't do. I use the earlier derived fact that a local observer can only reconstruct the eigenstate with the greatest eigenvalue. This will result in getting either the qubit's |0> or |1> state.
In order to get the exact probability distribution of these outcomes you have to assume exactly nothing about the state of the photon, because it is entirely unknown. If you assume nothing then all polarizations are equally likely, and you get an SU(2) invariant distribution of the coefficients. That's all. There are no assumptions whatsoever about the generation of the photons, them being thermal or anything. Just that all polarizations are equally likely. This is a very natural assumption and hard to argue against. The result in then not only the Born rule but also an orthogonal basis which the outcomes belong to.
So if you accept the derivation that the dominant eigensubspace is the relevant state description for a local internal observer and you accept that the state of the incoming photons is not known, then the Born rule follows for certain scattering processes. If you use precisely the process described in my blog is up to you. It merely stands for a class of processes that all result in the Born rule.
You don't need any modification of quantum mechanics for that. Why do you think you would? Also, this is not just a random combination of algebraic conditions and random distributions. Th assumption about the state distribution of the photon is the only valid assumption if you don't want to single out a specific photon polarization basis. And all the results are consequences of local observation and unitary interactions.
Have you worked through my blog posts from the beginning in the meantime? I ask because I was hoping that they describe all this very clearly. Please let me know if you disagree with how the internal observer reconstructs the quantum state, because I think that's the problem here.
I understand that you have an algebraic derivation of Born probabilities, but what I'm saying is that I don't see how to make that derivation physically meaningful. I don't see how it applies to an actual experiment.
Consider a Stern-Gerlach experiment. A state is prepared, sent through the apparatus, and the electron is observed coming out one way or the other. Repeat the procedure with identical state preparation, and you can get a different outcome.
For Copenhagen, this is just a routine application of the Born rule.
Suppose we try to explain this outcome using decoherence. Well, now we are writing a wavefunction for the overall system, measuring device as well as measured object, and we can show that the joint wavefunction splits into two parts which are entirely decohered for all practical purposes, corresponding to the two different outcomes. But you still have to apply the Born rule to "obtain" a specific outcome.
Now how does your idea explain the facts? I really don't see it. At the level of wavefunctions, each run of the experiment is the same, whether you look at just the wavefunction of the individual electron, or at the joint wavefunction of electron plus apparatus. How do we get physically different outcomes? Apparently it requires these random scattering events, that do not feature at all in the usual analysis of the experiment.
Are you saying that the electron that has passed through the Stern-Gerlach apparatus is really in a superposition, but for some reason I only see it as being located in one place, because that's the "dominant eigenstate"? Does this apply to the whole apparatus as well - really in a superposition, but experienced as being in a definite state, not because of decoherence, but because of scattering + my epistemic limitations??