I think that you are saying that there are 'real' values of position, momentum, spin, etc., but that quantum mechanics only describes our knowledge about them.

I am saying that the wave function (to be specific) describes one's knowledge about position, momentum, spin, etc., but I make no claim that these have any ‘real' values.

In the absence of a real post, here are some links:

By the way, you seem to have got this, but I'll say it anyway for the benefit of any other readers, since it's short and sums up the idea: The wave function exists in the map, not in the territory.

Comment author:endoself
09 July 2011 05:30:53AM
4 points
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I have not read the latter link yet, though I intend to.

I am saying that the wave function (to be specific) describes one's knowledge about position, momentum, spin, etc., but I make no claim that these have any ‘real' values.

What do you have knowledge of then? Or is there some concept that could be described as having knowledge of something without that thing having an actual value?

From Baez:

Probability theory is the special case of quantum mechanics in which ones algebra of observables is commutative.

This is horribly misleading. Bayesian probability can be applied perfectly well in a universe that obeys MWI while being kept completely separate mathematically from the quantum mechanical uncertainty.

Comment author:TobyBartels
10 July 2011 01:01:14AM
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3 points
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Probability theory is the special case of quantum mechanics in which ones algebra of observables is commutative.

This is horribly misleading. Bayesian probability can be applied perfectly well in a universe that obeys MWI while being kept completely separate mathematically from the quantum mechanical uncertainty.

As a mathematical statement, what Baez says is certainly correct (at least for some reasonable mathematical formalisations of ‘probability theory’ and ‘quantum mechanics’). Note that Baez is specifically discussing quantum statistical mechanics (which I don't think he makes clear); non-statistical quantum mechanics is a different special case which (barring trivialities) is completely disjoint from probability theory.

Of course, the statement can still be misleading; as you note, it's perfectly possible to interpret quantum statistical physics by tacking Bayesian probability on top of a many-worlds interpretation of non-statistical quantum mechanics. That is, it's possible but (I argue) unwise; because if you do this, then your beliefs do not pay rent!

The classic example is a spin-1/2 particle that you believe to be spin-up with 50% probability and spin-down with 50% probability. (I mean probability here, not a superposition.) An alternative map is that you believe that the particle is spin-right with 50% probability and spin-left with 50% probability. (Now superposition does play a part, as spin-right and spin-left are both equally weighted superpositions of spin-up and spin-down, but with opposite relative phases.) From the Bayesian-probability-tacked-onto-MWI point of view, these are two very different maps that describe incompatible territories. Yet no possible observation can ever distinguish these! Specifically, if you measure the spin of the particle along any axis, both maps predict that you will measure the spin to be in one direction with 50% probability and in the other direction with 50% probability. (The wavefunctions give Born probabilities for the observations, which are then weighted according to your Bayesian probabilities for the wavefunctions, giving the result of 50% every time.)

In statistical mechanics as it is practised, no distinction is made between these two maps. (And since the distinction pays no rent in terms of predictions, I argue that no distinction should be made.) They are both described by the same ‘density matrix’; this is a generalisation of the notion of quantum state as a wave vector. (Specifically, the unit vectors up to phase in the Hilbert space describe the pure states of the system, which are only a degenerate case of the mixed states described by the density matrices.) A lot of the language of statistical mechanics is frequentist-influenced talk about ‘ensembles’, but if you just reinterpret all of this consistently in a Bayesian way, then the practice of statistical mechanics gives you the Bayesian interpretation.

I am saying that the wave function (to be specific) describes one's knowledge about position, momentum, spin, etc., but I make no claim that these have any ‘real' values.

What do you have knowledge of then? Or is there some concept that could be described as having knowledge of something without that thing having an actual value?

This is the weak point in the Bayesian interpretation of quantum mechanics. I find it very analogous to the problem of interpreting the Born probabilities in MWI. Eliezer cannot yet clearly answer these questions that he poses:

What are the Born probabilities, probabilities of? Here's the map - where's the territory?

And neither can I (at least, not in a way that would satisfy him). In the all-Bayesian interpretation, the Born probabilities are simply Bayesian probabilities, so there's no special problems about them; but as you point out, it's still hard to say what the territory is like.

My best answer is simply what you suggest, that our maps of the universe assign probabilities to various possible values of things that do not (necessarily) have any actual values. This may seem like a counterintuitive thing to do, but it works, and we have no other way of making a map.

By the way, I've thought of a couple more references:

Baez (1993) is where I really learnt quantum statistical mechanics (despite having earlier taken a course in it), and my first (subtle) introduction to the Bayesian interpretation (not made explicit here). Note the talk about the ‘post-Everett school’, and recall that Everett is credited with founding the many-worlds interpretation (although he avoided the term ‘MWI’). The Bayesian interpretation could have been understood in the 1930s (and I have heard it argued, albeit unconvincingly, that it is what Bohr really meant all along), but it's really best understood in light of the modern understanding of decoherence that Everett started. We all-Bayesians are united with the many-worlders (and the Bohmians) in decrying the mystical separation of the universe into ‘quantum’ and ‘classical’ worlds and the reality of the ‘collapse of the wavefunction’. (That is, we do believe in the collapse of the wavefunction, but not in the territory; for us, it is simply the process of updating the map on the basis of new information, that is the application of a suitably generalised Bayes's Theorem.) We just think that the many-worlders have some unnecessary ontological baggage (like the Bohmians, but to a lesser degree).

Bartels (1998) is my first attempt to explain the Bayesian interpretation (on Usenet), albeit not a very good one. It's overly mathematical (and poorly so, since W*-algebras make a better mathematical foundation than C*-algebras). But it does include things that I haven't said here, (including mathematical details that you might happen to want). Still (even for the mathematics), if you read only one, read Baez.

Edit: I edited to use the word ‘world’ only in the technical sense of an interpretation.

Comment author:endoself
12 July 2011 05:21:10AM
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1 point
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As a mathematical statement, what Baez says is certainly correct.

I definitely don't disagree with that.

Specifically, if you measure the spin of the particle along any axis, both maps predict that you will measure the spin to be in one direction with 50% probability and in the other direction with 50% probability.

A final possibility is that there never was a pure state; the universe started off in a mixed state. In this example, whether this should be regarded as an ontologically fundamental mixed state or just a lack of knowledge on my part depends on which hypothesis is simpler. This would be too hard to judge definitively given our current understanding.

What are the Born probabilities, probabilities of? Here's the map - where's the territory?

In MWI, the Born probabilities aren't probabilities, at least not is the Bayesian sense. There is no subjective uncertainty; I know with very high probability that the cat is both alive and dead. Of course, that doesn't tell us what they are, just what they are not.

We all-Bayesians are united with the many-worlders (and the Bohmians) in decrying the mystical separation of the world into ‘quantum’ and ‘classical’ and the reality of the ‘collapse of the wavefunction’.

I think a large majority of physicists would agree that the collapse of the wavefunction isn't an actual process.

How would you analyze the Wigner's friend thought experiment? In order for Wigner's observations to follow the laws of QM, both versions of his friend must be calculated, since they have a chance to interfere with each other. Wouldn't both streams of conscious experience occur?

I don't understand what you're saying in these paragraphs. You're not describing how the two situations lead to different predictions; you're describing the opposite: how different set-ups might lead to the two states.

Possibly you mean something like this: In situation A, my friend intended to prepare one spin-down particle, but I predict with 50% chance that they hooked up the apparatus backward and produced a spin-up particle instead. In situation B, they intended to prepare a spin-right particle, with the same chance of accidental reversal. These are different situations, but the difference lies in the apparatus, my friend's mind, the lab book, etc, not in the particle. It would be much the same if I knew that the machine always produced a spin-up particle and the up/down/right/left dial did nothing: the situations are different, but not because of the particle produced. (However, in this case, the particle is not even entangled with the dial reading.)

A final possibility is that there never was a pure state; the universe started of in a mixed state.

I especially don't know what you mean by this. The states that most people talk about when discussing quantum physics (including Eliezer in the Sequence) are pure states, and mixed states are probabilistic mixtures of these. If you're a Bayesian when it comes to classical probability (even if you believe in the wave function when it comes to purely quantum indeterminacy), then you should never believe that the real wave function is mixed; you just don't know which pure state it is. Unless you distinguish between the map where the particle is spin-up or -down with equal odds from the map where the particle is definitely in the fullymixed state in the territory? Then you have an even greater plethora of distinctions between maps that pay no rent!

How would you analyze the Wigner's friend thought experiment?

For Schrödinger's Cat or Wigner's Friend, in any realistic situation, the cat or friend would quickly decohere and become entangled in my observations, leaving it in a mixed state: the common-sense situation where it's alive/happy/etc with 50% chance and dead/sad/etc with 50% chance. (Quantum physics should reproduce common sense in situations where that applies, and killing a cat with radioactive decay or a pseudorandom coin flip doesn't matter to the cat --ordinarily.) However, if we imagine that we keep the cat or friend isolated (where common sense doesn't apply), then it is in a superposition of these instead of a mixture --from my point of view. My friend's state of knowledge is different, of course; from that point of view, the state is completely determined (with or without decoherence). And how is it determined? I don't know, but I'll find out when I open the door and ask.

Comment author:endoself
19 July 2011 11:38:09PM
0 points
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I don't understand what you're saying in these paragraphs. You're not describing how the two situations lead to different predictions; you're describing the opposite: how different set-ups might lead to the two states.

I did not explain this very well. My point was that when we don't know the particle's spin, it is still a part of the simplest description that we have of reality. It should not be any more surprising that a belief about a quantum mechanical state does not have any observable consequences than that a belief about other parts of the universe that cannot be seen due to inflation does not have any observable consequences.

Unless you distinguish between the map where the particle is spin-up or -down with equal odds from the map where the particle is definitely in the fullymixed state in the territory? Then you have an even greater plethora of distinctions between maps that pay no rent!

I included this just in case a theory that implies such a thing ever turn out to be simpler than alternatives. I thought this was relevant because I mistakenly thought that you had mentioned this distinction.

And how is it determined? I don't know, but I'll find out when I open the door and ask.

What if your friend and the cat are implemented on a reversible quantum computer? The amplitudes for your friend's two possible states may both affect your observations, so both would need to be computed.

Comment author:TobyBartels
20 July 2011 04:02:36PM
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2 points
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My point was that when we don't know the particle's spin, it is still a part of the simplest description that we have of reality.

Sure, the spin of the particle is a feature of the simplest description that we have. Nevertheless, no specific value of the particle's spin is a feature of the simplest description that we have; this is true in both the Bayesian interpretation and in MWI.

To be fair, if reality consists only of a single particle with spin 1/2 and no other properties (or more generally if there is a spin-1/2 particle in reality whose spin is not entangled with anything else), then according to MWI, reality consists (at least in part) of a specific direction in 3-space giving the axis and orientation of the particle's spin. (If the spin is greater than 1/2, then we need something a little more complicated than a single direction, but that's probably not important.) However, if the particle is entangled with something else, or even if its spin is entangled with some another property of the particle (such as its position or momentum), then the best that you can say is that you can divide reality mathematically into various worlds, in each of which the particle has a spin in a specific direction around a specific axis.

(In the Bohmian interpretation, it is true that the particle has a specific value of spin, or rather it has a specific value about any axis. But presumably this is not what you mean.)

As for which is the simplest description of reality, the Bayesian interpretation really is simpler. To fully describe reality as best I can with the knowledge that I have, in other words to write out my map completely, I need to specify less information in the fully Bayesian interpretation (FBI) than in MWI with Bayesian classical probability on top (MWI+BCP). This is because (as in the toy example of the spin-1/2 particle) different MWI+BCP maps correspond to the same FBI map; some additional information must be necessary to distinguish which MWI+BCP map to use.

If you're an objective Bayesian in the sense that you believe that the correct prior to use is determined entirely by what information one has, then I can't even tell how one would ever distinguish between the various MWI+BCP maps that correspond to a given FBI map. (A similar problem occurs if you define probability in terms of propensity to wager, since there is no way to settle the wagers.) Even if I ask my friend who prepared the state, my friend's choice to describe it one way rather than another way only gives me information about other things (the apparatus, my friend's mind, their lab book, etc). It may be possible to always choose a most uniform MWI+BCP map (in the toy example, a uniform probability distribution over the sphere); I'll have to think about this.

For the record, I do believe in the implied invisible, if it really is implied by the simplest description of reality. In this case, it's not.

I mistakenly thought that you had mentioned this distinction.

I certainly didn't mean to; from my point of view, that makes the MWI only more ridiculous, and I don't want to attack a straw man.

What if your friend and the cat are implemented on a reversible quantum computer? The amplitudes for your friend's two possible states may both affect your observations, so both would need to be computed.

So compute both. There are theoretical problems with implementing an observer on a reversible computer (quantum or otherwise), because Bayesian updating is not reversible; but from my perspective, I'll compute my state and believe whatever that comes out to.

Probably I don't understand what your question here really is. Is there a standard description of the problem of Wigner's friend on a quantum computer, preferably together with the WMI resolution of it, that you can link to or write down? (I can't find one online with a simple search.)

Comment author:TobyBartels
23 July 2011 09:22:50AM
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1 point
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I wrote:

The classic example is a spin-1/2 particle that you believe to be spin-up with 50% probability and spin-down with 50% probability.

I've begun to think that this is probably not a good example.

It's mathematically simple, so it is good for working out an example explicitly to see how the formalism works. (You may also want to consider a system with two spin-1/2 particles; but that's about as complicated as you need to get.) However, it's not good philosophically, essentially since the universe consists of more than just one particle!

Mathematically, it is a fact that, if a spin-1/2 particle is entangled with anything else in the universe, then the state of the particle is mixed, even if the state of the entire universe is pure. So a mixed state for a single particle suggests nothing philosphically, since we can still believe that the universe is in a pure state, which causes no problems for MWI. Indeed, endoself immediately looks at situations where the particle is so entangled! I should have taken this as a sign that my example was not doing its job.

I still stand by my responses to endoself, as far as they go. One of the minor attractions of the Bayesian interpretation for me is that it treats the entire universe and single particles in the same way; you don't have to constantly remind yourself that the system of interest is entangled with other systems that you'd prefer to ignore, in order to correctly interpret statements about the system. But it doesn't get at the real point.

The real point is that the entire universe is in a mixed state; I need to establish this. In the Bayesian interpretation, this is certainly true (since I don't have maximal information about the universe). According to MWI, the universe is in a pure state, but we don't know which. (I assume that you, the reader, don't know which; if you do, then please tell me!) So let's suppose that |psi> and |phi> are two states that the universe might conceivably be in (and assume that they're orthogonal to keep the math simple). Then if you believe that the real state of the universe is |psi> with 50% chance and |phi> with 50% chance, then this is a very different belief than the belief that it's (|psi> + |phi>)/sqrt(2) with 50% chance and (|psi> - |phi>)/sqrt(2) with 50% chance. Yet these two different beliefs lead to identical predictions, so you're drawing a map with extra irrelevant detail. In contrast, in the fully Bayesian interpretation, these are just two different ways of describing the same map, which is completely specified upon giving the density matrix (|psi><psi| + |phi><phi|)/2.

Edit: I changed uses of ‘world’ to ‘universe’; the former should be reserved for its technical sense in the MWI.

Comment author:nshepperd
23 May 2013 06:22:02AM
0 points
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Specifically, if you measure the spin of the particle along any axis, both maps predict that you will measure the spin to be in one direction with 50% probability and in the other direction with 50% probability.

On the other hand, if the particle is spin up, the probability of observing "up" in an up-down measurement is 1, while the probability is 0 if the particle is down. So in the case of an up-down prior, observing "up" changes your probabilities, while in the case of a left-right prior, it does not.

That's a good point. It seems to me another problem with the MWI (or specifically, with Bayesian classical probability on top of quantum MWI) that making an observation could leave your map entirely unchanged.

However, in practice, followers of MWI have another piece of information: which world we are in. If your prior is 50% left and 50% right, then either way you believe that the universe is a superposition of an up world and a down world. Measuring up tells you that we are in the up world. For purposes of future predictions, you remember this fact, and so effectively you believe in 100% up now, the same as the person with the 50% up and 50% down prior. Those two half-Bayesians disagree about how many worlds there are, but not about what the up world —the world that we're in— is like.

Comment author:nshepperd
27 May 2013 04:53:56AM
0 points
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To be precise, if your prior is 50% left and 50% right, then you generally believe that the world you are in is either a left world or a right world, and you don't know which. A left or right world itself factorises into a tensor product of (rest of the world) × (superposition of up particle and down particle). Measuring the particle along the up/down axis causes the rest of the world to be become entangled with the particle along that axis, splitting it into two worlds, of which you observe yourself to be in the 'up' one.

Of course, observing the particle along the up/down axis tells you nothing about whether its original spin was left or right, and leaves you incapable of finding out, since the two new worlds are very far apart, and it's the phase difference between those two worlds that stores that information.

Comment author:Wei_Dai
10 July 2011 02:44:11PM
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5 points
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The wave function exists in the map, not in the territory.

Please explain how you know this?

ETA: Also, whatever does exist in the territory, it has to generate subjective experiences, right? It seems possible that a wave function could do that, so saying that "the wave function exists in the territory" is potentially a step towards explaining our subjective experiences, which seems like should be the ultimate goal of any "interpretation". If, under the all-Bayesian interpretation, it's hard to say what exists in the territory besides that the wave function doesn't exist in the territory, then I'm having trouble seeing how it constitutes progress towards that ultimate goal.

Comment author:TobyBartels
13 July 2011 11:04:53AM
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0 points
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Please explain how you know this?

I wouldn't want to pretend that I know this, just that this is the Bayesian interpretation of quantum mechanics. One might as well ask how we Bayesians know that probability is in the map and not the territory. (We are all Bayesians when it comes to classical probability, right?) Ultimately, I don't think that it makes sense to know such things, since we make the same physical predictions regardless of our interpretation, and only these can be tested.

Nevertheless, we take a Bayesian attitude toward probability because it is fruitful; it allows us to make sense of natural questions that other philosophies can't and to keep things mathematically precise without extra complications. And we can extend this into the quantum realm as well (which is good since the universe is really quantum). In both realms, I'm a Bayesian for the same reasons.

A half-Bayesian approach adds extra complications, like the two very different maps that lead to same predictions. (See this comment's cousin in reply to endoself.)

ETA: As for knowing what exists in the territory as an aid to explaining subjective experience, we can still say that the territory appears to consist ultimately of quark fields, lepton fields, etc, interacting according to certain laws, and that (built out of these) we appear to have rocks, people, computers, etc, acting in certain ways. We can even say that each particular rock appears to have a specific value of position and momentum, up to a certain level of precision (which fails to be infinitely precise first because the definition of any particular rock isn't infinitely precise, long before the level of quantum indeterminacy). We just can't say that each particular quark has a specific value of position and momentum beyond a certain level of precision, despite being (as far as we know) fundamental, and this is true regardless of whether we're all-Bayesian or many-worlder. (Bohmians believe that such values do exist in the territory, but these are unobservable even in principle, so this is a pointless belief).

Edit: I used ‘world’ consistently in a technical sense.

Comment author:Juno_Watt
21 May 2013 05:34:27PM
0 points
[-]

Nevertheless, we take a Bayesian attitude toward probability because it is fruitful; it allows us to make sense of natural questions that other philosophies can't and to keep things mathematically precise without extra complications. And we can extend this into the quantum realm as well

Where "extending" seems to mean "assuming". I find it more fruitful to come up with tests of (in)determinsm, such as Bell's Inequalitites.

I'm not sure what you mean by ‘assuming’. Perhaps you mean that we see what happens if we assume that the Bayesian interpretation continues to be meaningful? Then we find that it works, in the sense that we have mutually consistent degrees of belief about physically observable quantities. So the interpretation has been extended.

Yes, Bell's inequalities, along with Aspect's experiment to test them, really tell us something. Even before the experiment, the inequalities told us something theoretical: that there can be no local, single-world objective interpretation of the standard predictions of quantum mechanics (for a certain sense of ‘objective’); then the experiment told us something empirical: that (to a high degree of tolerance) those predictions were correct where they mattered.

Like Bell's inequalities, the Bayesian interpretation of quantum mechanics tells us something theoretical: that there can be a local, single-world interpretation of the standard predictions of quantum mechanics (although it can't be objective in the sense ruled out by Bell's inequalities). So now we want the analogue of Aspect's experiment, to confirm these predictions where it matters and tell us something empirical.

Bell's inequalities are basically a no-go theorem: an interpretation with desired features (local, single-world, objective true value of all potentially observable quantities) does not exist. There's a specific reason why it cannot exist, and Aspect's experiment tests that this reason applies in the real world. But Fuchs et al's development of the Bayesian interpretation is a go theorem: an interpretation with some desired features (local, single-world) does exist. So there's no point of failure to probe with an experiment.

We still learn something about the universe, specifically about the possible forms of maps of it. But it's a purely theoretical result. I agree that Bell's inequalities and Aspect's experiment are a more interesting result, since we get something empirical. But it wasn't a surprising result (which might be hindsight bias on my part). There seem to be a lot of people here (although that might be my bad impression) who think that there is no local, single-world interpretation of the standard predictions of quantum mechanics (or even no single-world interpretation at all, but I'm not here to push Bohmianism), so the existence of the Bayesian interpretation may be the more surprising result; it may actually tell us more. (At any rate, it was surprising once upon a time for me.)

Comment author:Peterdjones
10 July 2011 04:31:48PM
-1 points
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The wave function exists in the map, not in the territory.

That is not an uncontroversial fact. For instance, Roger Penrose, from the Emperor's New Mind

OBJECTIVITY AND MEASURABILITY OF QUANTUM STATES
Despite the fact that we are normally only provided with probabilities for the outcome of an experiment, there seems to
be something objective about a quantum-mechanical state. It is often asserted that the state-vector is merely a
convenient description of 'our knowledge' concerning a physical system or, perhaps, that the state-vector does not really
describe a single system but merely provides probability information about an 'ensemble' of a large number of similarly
prepared systems. Such sentiments strike me as unreasonably timid concerning what quantum mechanics has to tell us
about the actuality of the physical world.
Some of this caution, or doubt, concerning the 'physical reality' of state-vectors appears to spring from the fact that
what is physically measurable is strictly limited, according to theory. Let us consider an electron's state of spin, as
described above. Suppose that the spin-state happens to be |a), but we do not know this; that is, we do not know the
direction a in which the electron is supposed to be spinning. Can we determine this direction by measurement? No, we
cannot. The best that we can do is extract 'one bit' of information that is, the answer to a single yes no question. We
may select some direction P in space and measure the electron's spin in that direction. We get either the answer YES or
NO, but thereafter, we have lost the information about the original direction of spin. With a YES answer we know that
the state is now proportional to |p), and with a NO answer we know that the state is now in the direction opposite to p.
In neither case does this tell us the direction a of the state before measurement, but merely gives some probability
information about a.
On the other hand, there would seem to be something completely objective about the direction a itself, in which the
electron 'happened to be spinning' before the measurement was made For we might have chosen to measure the
electron's spin in the direction a -and the electron has to be prepared to give the answer YES, with certainty, if we
happened to have guessed right in this way! Somehow, the 'information' that the electron must actually give this
answer is stored in the electron's state of spin.
It seems to me that we must make a distinction between what is 'objective' and what is 'measurable' in discussing the
question of physical reality, according to quantum mechanics. The state- vector of a system is, indeed, not measurable,
in the sense that one cannot ascertain, by experiments performed on the system, This objectivity is a feature of our
taking the standard quantum-mechanical formalism seriously. In a non-standard viewpoint, the system might actually
'know', ahead of time, the result that it would give to any measurement. This could leave us with a different, apparently
objective, picture of physical reality.
precisely (up to proportionality) what that state is; but the state vector does seem to be (again up to proportionality) a
completely objective property of the system, being completely characterized by the results that it must give to
experiments that one might perform.
In the case of a single spin-one-half panicle, such as an electron, this objectivity is not unreasonable because it merely
asserts that there is some direction in which the electron's spin is precisely defined, even though we may not know what
that direction is.
(However, we shall be seeing later that this 'objective' picture is much stranger with more complicated systems- even
for a system which consists merely of a pair of spin-one-half particles. ) But need the electron's spin have any
physically defined state at all before it is measured? In many cases it will not have, because it cannot be considered as
a quantum system on its own; instead, the quantum state roust generally be taken as describing an electron inextricably
entangled with a large number of other particles.
In particular circumstances, however, the electron (at least as regards its spin) can be considered on its own. In such
circumstances, such as when its spin has actually previously been measured in some (perhaps unknown) direction and
then the electron has remained undisturbed for a while, the electron does have a perfectly objectively defined direction
of spin, according to standard quantum theory.
resolved. The possible relevance of quantum effects to brain
function will be considered in the final two chapters.

## Comments (190)

OldI

amsaying that the wave function (to be specific) describes one's knowledge about position, momentum, spin, etc., but I makenoclaim that these have any ‘real' values.In the absence of a real post, here are some links:

By the way, you seem to have got this, but I'll say it anyway for the benefit of any other readers, since it's short and sums up the idea:

The wave function exists in the map, not in the territory.I have not read the latter link yet, though I intend to.

What do you have knowledge of then? Or is there some concept that could be described as having knowledge of something without that thing having an actual value?

From Baez:

This is horribly misleading. Bayesian probability can be applied perfectly well in a universe that obeys MWI while being kept completely separate mathematically from the quantum mechanical uncertainty.

*3 points [-]As a mathematical statement, what Baez says is certainly correct (at least for some reasonable mathematical formalisations of ‘probability theory’ and ‘quantum mechanics’). Note that Baez is specifically discussing quantum

statisticalmechanics (which I don't think he makes clear); non-statistical quantum mechanics is a different special case which (barring trivialities) is completely disjoint from probability theory.Of course, the statement can still be misleading; as you note, it's perfectly possible to interpret quantum statistical physics by tacking Bayesian probability on top of a many-worlds interpretation of non-statistical quantum mechanics. That is, it's possible but (I argue) unwise; because if you do this, then your beliefs do not pay rent!

The classic example is a spin-1/2 particle that you believe to be spin-up with 50% probability and spin-down with 50% probability. (I mean probability here,

nota superposition.) An alternative map is that you believe that the particle is spin-right with 50% probability and spin-left with 50% probability. (Now superpositiondoesplay a part, as spin-right and spin-left are both equally weighted superpositions of spin-up and spin-down, but with opposite relative phases.) From the Bayesian-probability-tacked-onto-MWI point of view, these are two very different maps that describe incompatible territories. Yet no possible observation can ever distinguish these! Specifically, if you measure the spin of the particle alonganyaxis, both maps predict that you will measure the spin to be in one direction with 50% probability and in the other direction with 50% probability. (The wavefunctions give Born probabilities for the observations, which are then weighted according to your Bayesian probabilities for the wavefunctions, giving the result of 50% every time.)In statistical mechanics as it is practised, no distinction is made between these two maps. (And since the distinction pays no rent in terms of predictions, I argue that no distinction

shouldbe made.) They are both described by the same ‘density matrix’; this is a generalisation of the notion of quantum state as a wave vector. (Specifically, the unit vectors up to phase in the Hilbert space describe thepurestates of the system, which are only a degenerate case of themixedstates described by the density matrices.) A lot of the language of statistical mechanics is frequentist-influenced talk about ‘ensembles’, but if you just reinterpret all of this consistently in a Bayesian way, then the practice of statistical mechanics gives you the Bayesian interpretation.This is the weak point in the Bayesian interpretation of quantum mechanics. I find it very analogous to the problem of interpreting the Born probabilities in MWI. Eliezer cannot yet clearly answer these questions that he poses:

And neither can I (at least, not in a way that would satisfy him). In the all-Bayesian interpretation, the Born probabilities are simply Bayesian probabilities, so there's no special problems about them; but as you point out, it's still hard to say what the territory is like.

My best answer is simply what you suggest, that our maps of the universe assign probabilities to various possible values of things that do not (necessarily) have any actual values. This may seem like a counterintuitive thing to do, but it works, and we have no other way of making a map.

By the way, I've thought of a couple more references:

Baez (1993) is where I really learnt quantum statistical mechanics (despite having earlier taken a course in it), and my first (subtle) introduction to the Bayesian interpretation (not made explicit here). Note the talk about the ‘post-Everett school’, and recall that Everett is credited with founding the many-worlds interpretation (although he avoided the term ‘MWI’). The Bayesian interpretation could have been understood in the 1930s (and I have heard it argued, albeit unconvincingly, that it is what Bohr really meant all along), but it's really best understood in light of the modern understanding of decoherence that Everett started. We all-Bayesians are united with the many-worlders (and the Bohmians) in decrying the mystical separation of the universe into ‘quantum’ and ‘classical’ worlds and the reality of the ‘collapse of the wavefunction’. (That is, we

dobelieve in the collapse of the wavefunction, but not in the territory; for us, it is simply the process of updating the map on the basis of new information, that is the application of a suitably generalised Bayes's Theorem.) We just think that the many-worlders have some unnecessary ontological baggage (like the Bohmians, but to a lesser degree).Bartels (1998) is my first attempt to explain the Bayesian interpretation (on Usenet), albeit not a very good one. It's overly mathematical (and poorly so, since W*-algebras make a better mathematical foundation than C*-algebras). But it does include things that I haven't said here, (including mathematical details that you might happen to want). Still (even for the mathematics), if you read only one, read Baez.

Edit: I edited to use the word ‘world’ only in the technical sense of an interpretation.

*1 point [-]I definitely don't disagree with that.

They can give different predictions. Maybe I can ask my friend who prepared they quantum state and ey can tell me which it really is. I might even be able to use that knowledge to predict the current state of the apparatus ey used to prepare the particle. Of course, it's also possible that my friend would refuse to tell me or that I got the particle already in this state without knowing how it got there. That would just be belief in the implied invisible. "On August 1st 2008 at midnight Greenwich time, a one-foot sphere of chocolate cake spontaneously formed in the center of the Sun; and then, in the natural course of events, this Boltzmann Cake almost instantly dissolved." I would say that this hypothesis is

meaningfulandalmost certainly false. Not that it is "meaningless". Even though I cannot think of any possible experimental test that would discriminate between its being true, and its being false.A final possibility is that there never was a pure state; the universe started off in a mixed state. In this example, whether this should be regarded as an ontologically fundamental mixed state or just a lack of knowledge on my part depends on which hypothesis is simpler. This would be too hard to judge definitively given our current understanding.

In MWI, the Born probabilities aren't probabilities, at least not is the Bayesian sense. There is no subjective uncertainty; I know with very high probability that the cat is both alive and dead. Of course, that doesn't tell us what they are, just what they are not.

I think a large majority of physicists would agree that the collapse of the wavefunction isn't an actual process.

How would you analyze the Wigner's friend thought experiment? In order for Wigner's observations to follow the laws of QM, both versions of his friend must be calculated, since they have a chance to interfere with each other. Wouldn't both streams of conscious experience occur?

I don't understand what you're saying in these paragraphs. You're not describing how the two situations lead to different predictions; you're describing the opposite: how different set-ups might lead to the two states.

Possibly you mean something like this: In situation A, my friend intended to prepare one spin-down particle, but I predict with 50% chance that they hooked up the apparatus backward and produced a spin-up particle instead. In situation B, they intended to prepare a spin-right particle, with the same chance of accidental reversal. These are different situations, but the difference lies in the apparatus, my friend's mind, the lab book, etc, not in the particle. It would be much the same if I knew that the machine always produced a spin-up particle and the up/down/right/left dial did nothing: the situations are different, but not because of the particle produced. (However, in this case, the particle is not even entangled with the dial reading.)

I especially don't know what you mean by this. The states that most people talk about when discussing quantum physics (including Eliezer in the Sequence) are pure states, and mixed states are probabilistic mixtures of these. If you're a Bayesian when it comes to classical probability (even if you believe in the wave function when it comes to purely quantum indeterminacy), then you should never believe that the real wave function is mixed; you just don't know which pure state it is. Unless you distinguish between the map where the particle is spin-up or -down with equal odds from the map where the particle is definitely in the fullymixed state in the territory? Then you have an even greater plethora of distinctions between maps that pay no rent!

For Schrödinger's Cat or Wigner's Friend, in any realistic situation, the cat or friend would quickly decohere and become entangled in my observations, leaving it in a mixed state: the common-sense situation where it's alive/happy/etc with 50% chance and dead/sad/etc with 50% chance. (Quantum physics should reproduce common sense in situations where that applies, and killing a cat with radioactive decay or a pseudorandom coin flip doesn't matter to the cat --ordinarily.) However, if we imagine that we keep the cat or friend isolated (where common sense doesn't apply), then it is in a superposition of these instead of a mixture --from

mypoint of view. My friend's state of knowledge is different, of course; from that point of view, the state is completely determined (with or without decoherence). Andhowis it determined? I don't know, but I'll find out when I open the door and ask.I did not explain this very well. My point was that when we don't know the particle's spin, it is still a part of the simplest description that we have of reality. It should not be any more surprising that a belief about a quantum mechanical state does not have any observable consequences than that a belief about other parts of the universe that cannot be seen due to inflation does not have any observable consequences.

I included this just in case a theory that implies such a thing ever turn out to be simpler than alternatives. I thought this was relevant because I mistakenly thought that you had mentioned this distinction.

What if your friend and the cat are implemented on a reversible quantum computer? The amplitudes for your friend's two possible states may both affect your observations, so both would need to be computed.

*2 points [-]Sure, the spin of the particle is a feature of the simplest description that we have. Nevertheless, no specific value of the particle's spin is a feature of the simplest description that we have; this is true in both the Bayesian interpretation and in MWI.

To be fair, if reality consists only of a single particle with spin 1/2 and no other properties (or more generally if there is a spin-1/2 particle in reality whose spin is not entangled with anything else), then according to MWI, reality consists (at least in part) of a specific direction in 3-space giving the axis and orientation of the particle's spin. (If the spin is greater than 1/2, then we need something a little more complicated than a single direction, but that's probably not important.) However, if the particle is entangled with something else, or even if its spin is entangled with some another property of the particle (such as its position or momentum), then the best that you can say is that you can divide reality mathematically into various worlds, in each of which the particle has a spin in a specific direction around a specific axis.

(In the Bohmian interpretation, it

istrue that the particle has a specific value of spin, or rather it has a specific value about any axis. But presumably this is not what you mean.)As for which is the simplest description of reality, the Bayesian interpretation really is simpler. To fully describe reality as best I can with the knowledge that I have, in other words to write out my map completely, I need to specify less information in the fully Bayesian interpretation (FBI) than in MWI with Bayesian classical probability on top (MWI+BCP). This is because (as in the toy example of the spin-1/2 particle) different MWI+BCP maps correspond to the same FBI map; some additional information must be necessary to distinguish which MWI+BCP map to use.

If you're an objective Bayesian in the sense that you believe that the correct prior to use is determined entirely by what information one has, then I can't even tell how one would ever distinguish between the various MWI+BCP maps that correspond to a given FBI map. (A similar problem occurs if you define probability in terms of propensity to wager, since there is no way to settle the wagers.) Even if I ask my friend who prepared the state, my friend's choice to describe it one way rather than another way only gives me information about other things (the apparatus, my friend's mind, their lab book, etc). It may be possible to always choose a most uniform MWI+BCP map (in the toy example, a uniform probability distribution over the sphere); I'll have to think about this.

For the record, I do believe in the implied invisible, if it really is implied by the simplest description of reality. In this case, it's not.

I certainly didn't mean to; from my point of view, that makes the MWI only more ridiculous, and I don't want to attack a straw man.

So compute both. There are theoretical problems with implementing an observer on a reversible computer (quantum or otherwise), because Bayesian updating is not reversible; but from my perspective, I'll compute my state and believe whatever that comes out to.

Probably I don't understand what your question here really is. Is there a standard description of the problem of Wigner's friend on a quantum computer, preferably together with the WMI resolution of it, that you can link to or write down? (I can't find one online with a simple search.)

*1 point [-]I wrote:

I've begun to think that this is probably not a good example.

It's mathematically simple, so it

isgood for working out an example explicitly to see how the formalism works. (You may also want to consider a system withtwospin-1/2 particles; but that's about as complicated as you need to get.) However, it's not good philosophically, essentially since the universe consists of more than just one particle!Mathematically, it is a fact that, if a spin-1/2 particle is entangled with anything else in the universe, then the state of the particle is mixed, even if the state of the entire universe is pure. So a mixed state for a single particle suggests nothing philosphically, since we can still believe that the universe is in a pure state, which causes no problems for MWI. Indeed, endoself immediately looks at situations where the particle is so entangled! I should have taken this as a sign that my example was not doing its job.

I still stand by my responses to endoself, as far as they go. One of the minor attractions of the Bayesian interpretation for me is that it treats the entire universe and single particles in the same way; you don't have to constantly remind yourself that the system of interest is entangled with other systems that you'd prefer to ignore, in order to correctly interpret statements about the system. But it doesn't get at the real point.

The real point is that the

entire universeis in a mixed state; I need to establish this. In the Bayesian interpretation, this is certainly true (since I don't have maximal information about the universe). According to MWI, the universe is in a pure state, but we don't know which. (I assume that you, the reader, don't know which; if you do, then please tell me!) So let's suppose that |psi> and |phi> are two states that the universe might conceivably be in (and assume that they're orthogonal to keep the math simple). Then if you believe that the real state of the universe is |psi> with 50% chance and |phi> with 50% chance, then this is a very different belief than the belief that it's (|psi> + |phi>)/sqrt(2) with 50% chance and (|psi> - |phi>)/sqrt(2) with 50% chance. Yet these two different beliefs lead to identical predictions, so you're drawing a map with extra irrelevant detail. In contrast, in the fully Bayesian interpretation, these are just two different ways of describing the same map, which is completely specified upon giving the density matrix (|psi><psi| + |phi><phi|)/2.Edit: I changed uses of ‘world’ to ‘universe’; the former should be reserved for its technical sense in the MWI.

On the other hand, if the particle is spin up, the probability of observing "up" in an up-down measurement is 1, while the probability is 0 if the particle is down. So in the case of an up-down prior, observing "up" changes your probabilities, while in the case of a left-right prior, it does not.

That's a good point. It seems to me another problem with the MWI (or specifically, with Bayesian classical probability on top of quantum MWI) that making an observation could leave your map entirely unchanged.

However, in practice, followers of MWI have another piece of information: which world we are in. If your prior is 50% left and 50% right, then either way you believe that the universe is a superposition of an up world and a down world. Measuring up tells you that we are in the up world. For purposes of future predictions, you remember this fact, and so effectively you believe in 100% up now, the same as the person with the 50% up and 50% down prior. Those two half-Bayesians disagree about how many worlds there are, but not about what the up world —the world that we're in— is like.

To be precise, if your prior is 50% left and 50% right, then you generally believe that the world you are in is either a left world or a right world, and you don't know which. A left or right world itself factorises into a tensor product of (rest of the world) × (superposition of up particle and down particle). Measuring the particle along the up/down axis causes the rest of the world to be become entangled with the particle along that axis, splitting it into two worlds, of which you observe yourself to be in the 'up' one.

Of course, observing the particle along the up/down axis tells you nothing about whether its original spin was left or right, and leaves you incapable of finding out, since the two new worlds are very far apart, and it's the phase difference between those two worlds that stores that information.

*5 points [-]Please explain how you know this?

ETA: Also, whatever does exist in the territory, it has to generate subjective experiences, right? It seems possible that a wave function could do that, so saying that "the wave function exists in the territory" is potentially a step towards explaining our subjective experiences, which seems like should be the ultimate goal of any "interpretation". If, under the all-Bayesian interpretation, it's hard to say what exists in the territory besides that the wave function

doesn'texist in the territory, then I'm having trouble seeing how it constitutes progress towards that ultimate goal.*0 points [-]I wouldn't want to pretend that I

knowthis, just that this is the Bayesian interpretation of quantum mechanics. One might as well ask how we Bayesians know that probability is in the map and not the territory. (We are all Bayesians when it comes to classical probability, right?) Ultimately, I don't think that it makes sense to know such things, since we make the same physical predictions regardless of our interpretation, and only these can be tested.Nevertheless, we take a Bayesian attitude toward probability because it is fruitful; it allows us to make sense of natural questions that other philosophies can't and to keep things mathematically precise without extra complications. And we can extend this into the quantum realm as well (which is good since the universe is really quantum). In both realms, I'm a Bayesian for the same reasons.

A half-Bayesian approach adds extra complications, like the two very different maps that lead to same predictions. (See this comment's cousin in reply to endoself.)

ETA: As for knowing what exists in the territory as an aid to explaining subjective experience, we can still say that the territory appears to consist ultimately of quark fields, lepton fields, etc, interacting according to certain laws, and that (built out of these) we appear to have rocks, people, computers, etc, acting in certain ways. We can even say that each particular rock appears to have a specific value of position and momentum, up to a certain level of precision (which fails to be infinitely precise first because the definition of any particular rock isn't infinitely precise, long before the level of quantum indeterminacy). We just can't say that each particular quark has a specific value of position and momentum beyond a certain level of precision, despite being (as far as we know) fundamental, and this is true regardless of whether we're all-Bayesian or many-worlder. (Bohmians believe that such values do exist in the territory, but these are unobservable even in principle, so this is a pointless belief).

Edit: I used ‘world’ consistently in a technical sense.

Where "extending" seems to mean "assuming". I find it more fruitful to come up with tests of (in)determinsm, such as Bell's Inequalitites.

I'm not sure what you mean by ‘assuming’. Perhaps you mean that we see what happens if we assume that the Bayesian interpretation continues to be meaningful? Then we find that it works, in the sense that we have mutually consistent degrees of belief about physically observable quantities. So the interpretation has been extended.

*2 points [-]If the universe contains no objective probabilities, it will still contain subjective, ignorance based probabilities.

If the universe contains objective probabilities, it will also still contain subjective, ignorance based probabilities.

So the fact subjective probabilities "work" doesn't tell you anything about the universe. It isn't a test.

Aspect's experiment to test Bells theorem is a test. It tells you there isn't (local, single-universe) objective probability.

OK, I think that I understand you now.

Yes, Bell's inequalities, along with Aspect's experiment to test them, really tell us something. Even before the experiment, the inequalities told us something theoretical: that there can be

nolocal, single-world objective interpretation of the standard predictions of quantum mechanics (for a certain sense of ‘objective’); then the experiment told us something empirical: that (to a high degree of tolerance) those predictions were correct where they mattered.Like Bell's inequalities, the Bayesian interpretation of quantum mechanics tells us something theoretical: that there

canbe a local, single-world interpretation of the standard predictions of quantum mechanics (although it can't be objective in the sense ruled out by Bell's inequalities). So now we want the analogue of Aspect's experiment, to confirm these predictions where it matters and tell us something empirical.Bell's inequalities are basically a no-go theorem: an interpretation with desired features (local, single-world, objective true value of all potentially observable quantities) does not exist. There's a specific reason why it cannot exist, and Aspect's experiment tests that this reason applies in the real world. But Fuchs et al's development of the Bayesian interpretation is a

gotheorem: an interpretation with some desired features (local, single-world) does exist. So there's no point of failure to probe with an experiment.We still learn something about the universe, specifically about the possible forms of maps of it. But it's a purely theoretical result. I agree that Bell's inequalities and Aspect's experiment are a more interesting result, since we get something empirical. But it wasn't a surprising result (which might be hindsight bias on my part). There seem to be a lot of people here (although that might be my bad impression) who think that there is no local, single-world interpretation of the standard predictions of quantum mechanics (or even no single-world interpretation at all, but I'm not here to push Bohmianism), so the existence of the Bayesian interpretation may be the more surprising result; it may actually tell us more. (At any rate, it was surprising once upon a time for me.)

That is not an uncontroversial fact. For instance, Roger Penrose, from the Emperor's New Mind

OBJECTIVITY AND MEASURABILITY OF QUANTUM STATES Despite the fact that we are normally only provided with probabilities for the outcome of an experiment, there seems to be something objective about a quantum-mechanical state. It is often asserted that the state-vector is merely a convenient description of 'our knowledge' concerning a physical system or, perhaps, that the state-vector does not really describe a single system but merely provides probability information about an 'ensemble' of a large number of similarly prepared systems. Such sentiments strike me as unreasonably timid concerning what quantum mechanics has to tell us about the actuality of the physical world. Some of this caution, or doubt, concerning the 'physical reality' of state-vectors appears to spring from the fact that what is physically measurable is strictly limited, according to theory. Let us consider an electron's state of spin, as described above. Suppose that the spin-state happens to be |a), but we do not know this; that is, we do not know the direction a in which the electron is supposed to be spinning. Can we determine this direction by measurement? No, we cannot. The best that we can do is extract 'one bit' of information that is, the answer to a single yes no question. We may select some direction P in space and measure the electron's spin in that direction. We get either the answer YES or NO, but thereafter, we have lost the information about the original direction of spin. With a YES answer we know that the state is now proportional to |p), and with a NO answer we know that the state is now in the direction opposite to p. In neither case does this tell us the direction a of the state before measurement, but merely gives some probability information about a. On the other hand, there would seem to be something completely objective about the direction a itself, in which the electron 'happened to be spinning' before the measurement was made For we might have chosen to measure the electron's spin in the direction a -and the electron has to be prepared to give the answer YES, with certainty, if we happened to have guessed right in this way! Somehow, the 'information' that the electron must actually give this answer is stored in the electron's state of spin. It seems to me that we must make a distinction between what is 'objective' and what is 'measurable' in discussing the question of physical reality, according to quantum mechanics. The state- vector of a system is, indeed, not measurable, in the sense that one cannot ascertain, by experiments performed on the system, This objectivity is a feature of our taking the standard quantum-mechanical formalism seriously. In a non-standard viewpoint, the system might actually 'know', ahead of time, the result that it would give to any measurement. This could leave us with a different, apparently objective, picture of physical reality. precisely (up to proportionality) what that state is; but the state vector does seem to be (again up to proportionality) a completely objective property of the system, being completely characterized by the results that it must give to experiments that one might perform. In the case of a single spin-one-half panicle, such as an electron, this objectivity is not unreasonable because it merely asserts that there is some direction in which the electron's spin is precisely defined, even though we may not know what that direction is. (However, we shall be seeing later that this 'objective' picture is much stranger with more complicated systems- even for a system which consists merely of a pair of spin-one-half particles. ) But need the electron's spin have any physically defined state at all before it is measured? In many cases it will not have, because it cannot be considered as a quantum system on its own; instead, the quantum state roust generally be taken as describing an electron inextricably entangled with a large number of other particles. In particular circumstances, however, the electron (at least as regards its spin) can be considered on its own. In such circumstances, such as when its spin has actually previously been measured in some (perhaps unknown) direction and then the electron has remained undisturbed for a while, the electron does have a perfectly objectively defined direction of spin, according to standard quantum theory. resolved. The possible relevance of quantum effects to brain function will be considered in the final two chapters.