In what sense does MWI fail to predict the Born Rule?

[A. Neumaier]

What I gave is the modern version of the Born rule I think formulated by Von-Neumann[ - it's also in Ballentine.

Thus what you call the modern version of Born's rule is actually Ballentine's postulate. He doesn't call it Born's rule.

But I found that it is called so already by van Vraasen in 1950:

B.C. van Fraassen,
The Representation of Nature in Physics: A Reflection On Adolf
Grünbaum's Early Writings,
Mechanics, 17 (1950), 26--34.

So it should probably be called van Vraasen's postulate, or van Vraasen's version of Born's rule.

 

[A. Neumaier]

Does the Bayesian interpretation of probability require an observer? It is the degree of plausibility according to the cox axioms:

Plausibility to the observer!

 

[Derek P]

Such statements are meaningful in terms of decision theory. In MWI the world a hypothetical observer would find themselves in is considered like a bet or wager and that of course is exactly what decision theory is designed to sort out - what is the probability of the outcome of that wager.

I would have said such statements are meaningful just as they stand.

 

[bhobba]

The generalized form seems to have first been stated by Dirac in his Lectures on Quantum Mechanics, but I don't have a copy of the first edition, hence cannot check.

I have the fourth edition. Page 46 states it, without name, as the average of an observable ξ is for a system in state |x> is <x|ξ|x>. Its not the trace version, but of course easily derived from it. Nor does he mention mixed states - at least I can't find it - which the trace version also includes. In fact that's the very simple reason you need the trace version to even understand decoherence. The resulting state is a mixed state ∑pi |xi><xi|. That the pi are the probabilities of the pure states |xi><xi| if you observe it to see if its in the state |xi><xi| can only be deduced from that trace formula - at least as far as I know - someone may enlighten me to some other way.

I have always called the trace formula the Born rule - what would you call it?

Thanks
Bill

 

[bhobba]

Plausibility to the observer!

Of course - but you are faced with exactly the same issues as what is an observer in QM - I am sure you know all the arguments about that one. And to others its another thread - (mentors hat on) please don't do it here as it will simply be a side track that will make the this tread harder to follow.

Why not simply take plausibility like probability in the Kolmogorov axioms as an undefined term defined by the axioms themselves? Again something for another thread.

I think we are finally getting somewhere in this thread - but have to go off to dinner. Will see what has transpired when I return.

Thanks
Bill

 

[Derek P]

Since tracing is required to demonstrate decoherence, decoherence requires the Born Rule.
I suspect I am missing something.

The word "decoherence" is used to denote the physical process of entanglement with a large system. You are using it to denote diagonalisation. Both are legitimate uses of the word and the latter follows from the former. So deriving the trace heuristic from the dynamics is not circular.

 

[Derek P]

1. Does the Bayesian interpretation of probability require an observer?

You're asking me? I'd say probably not.

2. Can you expand on exactly what you mean by counting in this context. Of course numbers are required from the very definition of probability, but counting I do not quite get.

People have been using the word in several different contexts here. I can't attach much meaning to world-counting. https://www.physicsforums.com/threa...dict-the-born-rule.946467/page-6#post-5992983

 

[akvadrako]

The word "decoherence" is used to denote the physical process of entanglement with a large system. You are using it to denote diagonalisation. Both are legitimate uses of the word and the latter follows from the former. So deriving the trace heuristic from the dynamics is not circular.

The physical process of decoherence is fully unitary, I think we all can agree. After it's occurred, you can pick subsystems that will be evolving mostly independently. That doesn't seem to require the Born rule to observe in a simulation/calculation, does it? But I guess it requires some kind of measure which doesn't amplify the shared terms that have vanished.

 

[bhobba]

The physical process of decoherence is fully unitary, I think we all can agree. After it's occurred, you can pick subsystems that will be evolving mostly independently. That doesn't seem to require the Born rule to observe in a simulation/calculation, does it? But I guess it requires some kind of measure which doesn't amplify the shared terms that have vanished.

Decoherence transforms a superposition into a mixed state. Interpreting that mixed state requires the generalized Born Rule (it's what I will call the trace formula) as I explained. It is used elsewhere, but that is a very very basic place it is used. You can read about evarience etc that supposedly overcomes that, but again, mentors hat on, new thread please.

Thanks
Bill

 

[stevendaryl]

Decoherence transforms a superposition into a mixed state.

I would say that decoherence plus tracing out environmental degrees of freedom transforms a superposition into a mixed state, not decoherence alone.

 

[Derek P]

I would say that decoherence plus tracing out environmental degrees of freedom transforms a superposition into a mixed state, not decoherence alone.

Exactly! Until this thread I had always assumed that when people talk about decoherence they mean entanglement with those environmental degrees of freedom. I don't see how it can be avoided unless they think of decoherence as a sort of phase noise introduced by a decidedly non-unitary operator - which then leaves the model with no "handle" on probabilities except a dogmatic assertion of the Born Rule a priori. Counting states replaces the requirement for the Born Rule with a much weaker requirement for the Principle of Indifference to be applicable to those environmental degrees of freedom. Would you agree?

I didn't answer your post with the ∏ operator as I wanted time to digest it but this thread is racing too fast to keep up! It seems to have branched more than an Everettian universe and, appropriately enough, different subthreads seem to be unaware of each other I'll get back to ∏ when I can.

 

[Derek P]

The physical process of decoherence is fully unitary, I think we all can agree. After it's occurred, you can pick subsystems that will be evolving mostly independently. That doesn't seem to require the Born rule to observe in a simulation/calculation, does it? But I guess it requires some kind of measure which doesn't amplify the shared terms that have vanished.

See post #150 and my comment following it.

 

[DarMM]

The word "decoherence" is used to denote the physical process of entanglement with a large system. You are using it to denote diagonalisation. Both are legitimate uses of the word and the latter follows from the former. So deriving the trace heuristic from the dynamics is not circular.

Okay, entanglement with the measuring device and environment does not require the Born rule. If that is what you mean by Decoherence, then yes it is not circulur, it can be seen to be a dynamic property.

However, it is not enough for Many-Worlds, where the branch structure only emerges within subsystems. For that you do need the trace.

So more accurately the emergence of the branch structure requires tracing.

 

[A. Neumaier]

I have always called the trace formula the Born rule - what would you call it?

It is just the definition of an expectation value, without any commitment to its interpretation in terms of measurements. This is done routinely in the (purely mathematical) theory of $C^*$-algebras, and hence part of the theoretical (shut-up-and-calculate) fraction of quantum mechanics. See my post #140.

The interpretation only starts when one is postulating that ''Upon measuring $A$ an ensemble of systems independently prepared in a fixed state, the expected mean of the measurement results equals the formal expectation value $\langle A \rangle$ corresponding to this state.'' This is van Fraasen's postulate (p.9 of the article mentioned in post #141).

 

[A. Neumaier]

I have always called the trace formula the Born rule - what would you call it?

The trace formula says less than the standard form of Born's rule, as it is silent about the possible values of a measurement. One can prove from the trace rule only that a value from the spectrum occurs with probability 1, but this allows finitely many measurements with other results, as probabilities are insensitive to a finite number of results.

On a purist note, we can only take finitely many measurements on a system. But the expected mean is also insensitive to a finite number of results. Thus, strictly speaking, the trace rule says nothing at all about measurement.

 

[stevendaryl]

Okay, entanglement with the measuring device and environment does not require the Born rule. If that is what you mean by Decoherence, then yes it is not circulur, it can be seen to be a dynamic property.

However, it is not enough for Many-Worlds, where the branch structure only emerges within subsystems. For that you do need the trace.

So more accurately the emergence of the branch structure requires tracing.

I don't see that as a necessity. As I sketched in another thread, it seems to me that you can just define a "possible world" as a coarse-graining of the state to define macrostates. These coarse-grained states are subspaces of the Hilbert space that can be specified by projection operators. The evolution of the projection operators (in the Heisenberg representation) follows from ordinary quantum mechanics, and the probability for being in one of the subspaces is just the Born rule applied to the projection operators (which are Hermitian observables, after all). With this approach, it's not clear where tracing or decoherence comes into play.

My feeling is that the decoherence approach is approximately equivalent, in some sense, but it seems very different.

 

[DarMM]

Okay so here are Wallace's axioms on the set of acts $\mathcal{U}_{E}$ available at an event $E$ and his axioms on the preference order on those acts $>^{\psi}$
Once again events $E$ are either classical "worlds" or superpositions there of. The set of worlds, excluding their superpositions, is denoted $\mathcal{M}$, elements $M$.

There is also the set of awards $\mathcal{R}$ which is a coarsening of worlds, as you might have the same reward in separate worlds. Elements are $r$.

If I perform an act $U$ at event $E$, $\mathcal{O}_{U}$ is used to denote the smallest event resulting from that act, e.g. $E$ might be me ready to perform a spin measurement, $U$ the act of performing the measurement and $\mathcal{O}_{U}$ the branched world afterward. I'll just call $\mathcal{O}_{U}$ the outcome of $U$. For worlds the notation $U(M)$ is common for their outcome.

I will list the axioms informally, they can be found more accurately in Mandolesi's first paper on p.11-17

Axioms on $\mathcal{U}_{E}$:

1. Acts can be restricted to subevents, e.g. $F \subset E$ means for any $U \in \mathcal{U}_{E}$ there is $U |_{F} \in \mathcal{U}_{F}$. So an example might be if I perform a spin measurement and split the world in three. A position measurement being available upon the superposition of worlds, would mean it is available in any branch.
   
2. Branches do not later interfere, this is equivalent to existence of an identity act.
3. Composed acts are available. By composed acts I mean if I perform $U \in \mathcal{U}_{E}$, then if there is an act $V \in \mathcal{O}_{U}$, then $VU$ is a possible act.
4. There is an act available that results in a given award. If you are in a world $M$, then there is an act with outcome $U(M)$ such that $U(M) \subset r$ with $r$ some reward subspace.
5. There are acts that result in branching, but don't change the reward. i.e. acts that cause a world to evolve into a superposition of worlds, but remain in the same $r$
6. There are erasure acts. Essentially if two superpositions of worlds (including the special case of them being each a single world) are in the same award space, there exists an act on each that evolves them into the same state, i.e. an act where we "throw away" any differences between them.
7. Act continuity. If $U$ is available, acts "nearby" in the operator norm topology are available.

Axioms on $>^{\psi}$, these are given for $\mathcal{U}_{\mathcal{M}}$, that is they are the axioms on the preference order within one world:

1. $>^{\psi}$ is a total order on $\mathcal{U}_{\mathcal{M}}$
2. Acts that result purely in branching, without changing the reward space, are ignored, that is preferred as the identity act. If $M \subset r$ and $U(M) \subset r$ then $U \sim^{\psi} \mathbb{I}_{M}$. Nobody cares about pure branching with no reward change
3. Preferences depend only on final states, not acts or initial states (this is called State Supervenience in Wallace). There is a fully accurate form in Mandolesi, but let us just say if $U\psi = V\psi$ then $U \sim^{\psi} V$. This axiom basically encodes a weaker form of noncontexuality, or something related to noncontextuality (I'm not sure which yet). However it does not mean Wallace assumes a form of noncontextuality as strong as Gleason's theorem, hence criticism that it is a basically Gleason's theorem are unfounded I think. Of course he later uses the axioms to prove something like Gleason's noncontextuality, from which point the proof is like Gleason's.
4. The preference order is continuous. i.e. if $U >^{\psi} V$ then nearby $U', V'$ have $U' >^{\psi} V'$. Small changes don't alter act preferences.
5. Preferences only depend on branches where acts differ. If an act is preferred in all branches, then it is preferred overall. If two acts are only considered non-equivalent in only one branch, they are considered non-equivalent overall.
6. The situation under consideration is not degenerate, i.e. there are at least two acts with $U >^{\psi} V$.

So that's 13 axioms, with which Mandolesi finds 17 problems. I'll discuss this in the next post.

 

[DarMM]

I don't see that as a necessity. As I sketched in another thread, it seems to me that you can just define a "possible world" as a coarse-graining of the state to define macrostates. These coarse-grained states are subspaces of the Hilbert space that can be specified by projection operators. The evolution of the projection operators (in the Heisenberg representation) follows from ordinary quantum mechanics, and the probability for being in one of the subspaces is just the Born rule applied to the projection operators (which are Hermitian observables, after all). With this approach, it's not clear where tracing or decoherence comes into play.

My feeling is that the decoherence approach is approximately equivalent, in some sense, but it seems very different.

Wallace uses this form of coarse-graining as well. Tracing is used in the usual decoherence based approach to the emergence of worlds. I'll be discussing such coarse grainings as they are used by Wallace in his proof later.

You would in essence view the Born rule as arising from the "volume" of a coarse-graining, an idea like branch counting, but of course not exactly the same. Wallace says similar in his talks.

 

[akvadrako]

I don't see that as a necessity. As I sketched in another thread, it seems to me that you can just define a "possible world" as a coarse-graining of the state to define macrostates.

But in this scheme, wouldn't any course-graining be equally permissible? So you could have "possible worlds" that don't behave approximately classically. With decoherence and darwinian evolution, you end up with something recognisable as branching classical worlds.

 

[Derek P]

Okay, entanglement with the measuring device and environment does not require the Born rule. If that is what you mean by Decoherence, then yes it is not circulur, it can be seen to be a dynamic property.

Yay! Agreement!

However, it is not enough for Many-Worlds, where the branch structure only emerges within subsystems. For that you do need the trace.So more accurately the emergence of the branch structure requires tracing.

Oh dear! Why did you have to spoil it? Yes of course you have to perform a trace. But you do not need the Born Rule to calculate the matrix or to justify tracing.

 

[Derek P]

I don't see that as a necessity. As I sketched in another thread, it seems to me that you can just define a "possible world" as a coarse-graining of the state to define macrostates. These coarse-grained states are subspaces of the Hilbert space that can be specified by projection operators. The evolution of the projection operators (in the Heisenberg representation) follows from ordinary quantum mechanics, and the probability for being in one of the subspaces is just the Born rule applied to the projection operators (which are Hermitian observables, after all). With this approach, it's not clear where tracing or decoherence comes into play.

It comes from mapping the fine grain states to the coarse grained ones. Phenomenally the fine grain states add linearly. In the entanglement expression they add as vectors. Ergo Born's Rule. End of subject? No, probably not!

 

[DarMM]

Yay! Agreement!

Oh dear! Why did you have to spoil it? Yes of course you have to perform a trace. But you do not need the Born Rule to calculate the matrix or to justify tracing.

What's the justification for tracing if not to preserve Born weights?

 

[Derek P]

But in this scheme, wouldn't any course-graining be equally permissible?

Fine-grain states are tagged by the original interaction - a qubit or whatever - and also by the observer experience - which you can quantify by an operator ∏ as Steven does. Does that address what you're getting at?

 

[Derek P]

What's the justification for tracing if not to preserve Born weights?

? That doesn't make sense.

 

[Derek P]

But in this scheme, wouldn't any course-graining be equally permissible? So you could have "possible worlds" that don't behave approximately classically.

"that haven't behaved approximately classically to date". Non-classicality doesn't become baked into that branch for ever more. We are talking about a statistical ensemble based on frequencies in a history. There are plenty of rogue worlds.

 

[DarMM]

? That doesn't make sense.

How not, that's the justification for tracing given in most textbooks. If you have a justification of tracing that ignores Born Weights, what is it?

Also you still haven't said what proof of Born's rule you consider to close the issue.

 

[akvadrako]

Fine-grain states are tagged by the original interaction - a qubit or whatever - and also by the observer experience - which you can quantify by an operator ∏ as Steven does. Does that address what you're getting at?

I admit I didn't read his post carefully the first time. Now it's clear what you mean by micro-states and I imagine each one corresponds to the idea of a fixed volume of state space from Wallace's explanation. Maybe that works (I can't tell) and it would seem to be an alternative to Gleason's theorem — in the end what you have is a metric for Hilbert space. Is that accurate?

Oh and by non-classical, I didn't mean maverick worlds, I meant something like like |alive> + |dead>. But I guess you are assuming Alice's mind states are classical.

 

[bhobba]

What's the justification for tracing if not to preserve Born weights?

It will help I think to see exactly what partial tracing is. I generally do not like to give links to stuff by Lubos but to save me time will post the details from him on exactly what it is:
https://physics.stackexchange.com/q...ake-the-partial-trace-to-describe-a-subsystem

The the expectation value of a system composed of a number of likely entangled parts uses the Generalized Born Rule (it is what I will call it in view of the previous discussion - for simplicity now just called the Born Rule). Chug through the math and the trace breaks into two parts - the trace about what's being observed and the trace of the rest. Since we are only interested in what's being observed you can do the trace and give the formula as the trace of something else. This something else is of the form of the Born Rule on a mixed state. It is said the other stuff we are not interested in has been traced out and leads to, as all the stuff is likely entangled, interpreting the mixed state from this math as the state of the system. Mathematically the superposition has been converted to a mixed state by this manipulation which is the essence of decoherence. We say what we are not interested in has been traced out. The Born rule is central to it. Even interpreting the resulting mixed state needs the Born Rule. As I have I think said before given a general mixed state ∑pi |xi><xi| you need the Born Rule to be able to say pi is the probability of getting a yes if you observe it to see if it is in the state |xi><xi| (assuming the other states in the mixture are orthogonal).

The whole thing basically revolves around the Born Rule.

Thanks
Bill

 

[DarMM]

I'll have a separate post for each problem. This is not Mandolesi's list, as if the same problem affects two axioms, he lists them separately. I'm just stating the root problem. I'll call axioms on $\mathcal{U}_{E}$ U1, etc and those on $>^{\psi}$ O1, etc

Okay, so one of the major problems with Wallace's proof as far as Madolesi and others are concerned is related to posts we have seen here, the validity of assuming the existence, or rather the dominance, of robust quasi-classical branches. That is, a working solution of the preferred basis problem.

Wallace's axioms make use of this assumption both in the plain existence of a set $M$ of macroworlds and in axioms U2, U4, U5 and O3.

How so?

Axioms on $\mathcal{U}_{E}$:

U2. You need robust quasi-classical branches to have no interference
U4. To have an act, which is mathematically a unitary operator, result in a given fixed reward, there has to be no "leakage" under the Schrodinger evolution into separate reward spaces.
U5. Again to ensure branching does not cross reward subspace boundaries there can be no leakage. Physically I'd have to be guaranteed the existence of some experiment whose branches could never result in interference or overlap with Macroworlds with separate rewards.

Axioms on $>^{\psi}$:
O3. This is not as easy to see. If 2. is violated, it might still be the case that the proof can be recovered provided this axioms holds approximately. There might be a small amount of interference, but if this axiom is extended from:
$U\psi = V\psi \rightarrow U \sim^{\psi} V$
to
$U\psi = V\psi + \phi \rightarrow U \sim^{\psi} V$
Where $\phi$ is as "small" as interference terms between quasi-classical branches are. This would allow this axiom to generate an effective version of U2. There might be a small amount of interference, but it doesn't effect our ordering.

The big problem? How do you ensure the non-quasi classical histories are small or negligible without the Born rule?

As Schrodinger evolution does not preserve compact support of wavefunctions, the state will always "leak" into having a small overlap with another macroscopic world. Now such tails have small Born Weight, but how do you ignore them without the Born rule. Purely by counting they will be "most" of your branches.

Wallace claims one can use the Hilbert Space metric, small under the metric means little physical effect. However as has been pointed out by others, states can be very similar physically and be "far" under the metric, or very different physically and be "close", to quote Mandolesi:
"Note that, no matter how far apart two wave packets get, their states remain at an almost constant distance in Hilbert space. Which is smaller than the distance between the physically equivalent states $\psi$ and $-\psi$. If this metric is such a lousy measure of how different two states are, why, without a Born-like rule, should we expect it to be a good measure of similarity?"

It has also been claimed that one can use the volume of the macrostates in Hilbert space, ones with small volumes being negligible, but it is hard to see how this affects probabilities.

Say the state has become as follows after a measurement:
$\psi = \alpha_{M}\psi_{M} + \Sigma_{i} \alpha_{A_i}\psi_{A_i}$

where $\psi_{M}$ is a microstate which at the macro level looks like the classical world $M$ and the $\psi_{A_i}$ are correspond to non-classical bizarro macroworlds $A_i$. Yes the world volume of states with appearance $M$ might be larger, but the $A_i$ are still part of your state, simply states like them are rarer. In terms of resulting branches from your act, bizarro worlds are more common, they just have lower "world entropy" in the sense of being less generic.

(Also note the above even assumes you can carve out these world volumes consistently in Hilbert space, for which there is no clear demonstration).

I don't know of any reason that these tail evolutions can be discarded without the Born rule. They are equally real parts of the wavefunction, the only difference being they have small coefficients. Since no meaning has been given to the coefficients that would allow them to be discarded, they can't be, especially seeing as how they dominate world count following a measurement.

Mandolesi has an idea, that I will return to at the end. It's not a bad one in my opinion, but it has a common feature with many attempts to save the many worlds, the transformation of the theory into a many-minds theory. Zurek's envariance proof has also gone to a Many-Minds form to prevent certain problems, which is interesting as it follows a completely separate line of reasoning.

EDIT: I think i should add that this is not to say one cannot define negligible worlds or prove the existence of a branching structure without Born's Rule, just that this has currently not been done and is more a hope rather than a rigorous result of some research program.

I think many will have expected this problem, so I will move onto more subtle ones.

 

[A. Neumaier]

Which is smaller than the distance between the physically equivalent states ψ and −ψ

Shouldn't one use the standard metric on the projective space?

volume of the macrostates in Hilbert space

The Hilbert spaces are infinite-dimensional. So volume is not really defined!

 

[DarMM]

Shouldn't one use the standard metric on the projective space?

Yes, probably. However would this show the other terms can be discarded?

The Hilbert spaces are infinite-dimensional. So volume is not really defined!

Yes, of course. I think the hope is that the quasi-classical macroworld subspaces are larger in some appropriate manner than odd macroworld subspaces, for which volume is a vague shorthand. However since the actual definition of these subspaces is quite loose, just assumed to exist, at least in Wallace, I'm not sure how this is realized or justified.

For the purpose of honesty let me just say my preferred interpretation of QM is "I'm really confused and the more I learn about QM the less I understand", that is the "I wish I could shut up and calculate" interpretation.

 

[A. Neumaier]

my preferred interpretation of QM is "I'm really confused and the more I learn about QM the less I understand", that is the "I wish I could shut up and calculate" interpretation.

Maybe you'd like my thermal interpretation! It is the result of my gradual coping with the confusion I found when I studied the available options many years ago.

 

[Derek P]

I admit I didn't read his post carefully the first time. Now it's clear what you mean by micro-states and I imagine each one corresponds to the idea of a fixed volume of state space from Wallace's explanation. Maybe that works (I can't tell) and it would seem to be an alternative to Gleason's theorem — in the end what you have is a metric for Hilbert space. Is that accurate?

I don't know Wallace's explanation so I can't really say. And I'm not sure what "a metric for Hilbert space" means. It sounds right. You should ask @stevendaryl. "My" version would be something more simplistic like the one I posted by Price, which was instantly torn to shreds. <shrug> I'm not arguing one way or the other, I'm just asking why the more-or-less obvious approach is said to fail as well as all others. I didn't make the claim. If you read my OP you'll see I'm rusty as hell after nearly half a century.

Oh and by non-classical, I didn't mean maverick worlds, I meant something like like |alive> + |dead>. But I guess you are assuming Alice's mind states are classical.

Yes it's quite possible that there is an assumption of a preferred basis.

 

[stevendaryl]

Oh and by non-classical, I didn't mean maverick worlds, I meant something like like |alive> + |dead>. But I guess you are assuming Alice's mind states are classical.

I'm not at all suggesting a principled reason for choosing one coarse-graining over another. But I was postulating a very specific coarse-graining, which is that the coarse-grained state determines, for every volume of space down to some minimal volume, the particle content, total energy, total momentum, average electric field, average magnetic field, etc. I'm assuming that a single coarse-grained state would not be compatible with a cat being both alive and dead. As far as what's going on in Alice's mind, I don't actually know about that. That could require very fine-grained information, or maybe not.

The idea of the coarse-grained state is that if you care about gross quantities, then such things are approximately commuting---they can have simultaneous values.

 

[DarMM]

Next points.

Contradiction between U2 and U6:
The axiom U2 supposes that branches can never re-interfere once branching has occurred. In essence there is thermodynamic irreversibility post measurement. However the erasure axiom, U6, requires there to be a way to undo differences between acts if they don't affect rewards, including branchings that don't affect rewards, hence some branching acts need to be reversible.

The overt nonphysical power of U6:
Is the capacity to reverse the branchings, as is required by erasure in some cases, really physically possible?
Consider a superposition of two microstates resulting in separate macroworlds (which are orthogonal), i.e. $\psi_{1} \in M_{1}$ and $\psi_{2} \in M_{2}$.

Now assume both Macroworlds are in the same reward subspace. Hence we can erase differences between them by some acts $U$ and $V$:

$$U\psi_{1} = V\psi_{2}$$

However in the superposition

$$\psi = \psi_{1} + \psi_{2}$$

One could perform both acts of erasure, one in each branch and hence $U$ and $V$ are just restrictions of some $W$ and so:

$$W\psi_{1} = V\psi_{2}$$

in violation of unitarity

Branching indifference not justified:
O2 assumes the agent does not care about branching, if the branching remains within the same reward subspace, that is the agent does not care about branching alone. Many (Kent, Maudlin, Mallah) have criticised this for basically assuming the agent reasons in a one world manner. If $\psi_{1} \in M_{1}$ has me receive a reward and $\psi_{2} \in M_{2}$ has me receive the same reward, yes I would be indifferent between them, but why would I also consider them as valuable as:

$$\psi = \psi_{1} + \psi_{2}$$

as there are now two of me enjoying the reward?

This even links back into the definition of the reward sets as vector subspaces. Is a superposition of two worlds with the same reward, again itself the same reward? If not, the view of the reward space as a vector subspace fails, affecting the entire proof.

This one has quite a bit of back and forth within the literature, see Mandolesi's second paper, p.23

Branch indifference not justified:
Axiom O5 declares that act preferences only depends on branches where they differ. I won't go into Mandolesi's specific example, but this axiom is equivaklenbt to saying your decisions shouldn't take other branches into account, only your own and its descendants.

Again it can be argued that this sneaks one-world style reasoning into your notion of rationality. Other worlds don't exist from you perspective and hence don't matter.

I'm not sure how I feel about this one.

Do preferences only depend on final states:
This is claimed by axiom O3. However this is ultimately the very strong claim that I have no reason to care about the path taking me to the final state, i.e. the intermediate states I pass through. Wallace acknowledges this as a weakness of his taking fixed unitary operators (i.e. ones that simply map intial state to final state, without the path), but defends it by saying one can consider the unitary operators to act on time scales to small to notice.