# Looking for a simple example of generating unequal probabilities in QM

I am trying to understand the problem of branch counting in Everettian interpretations of QM, so I thought I would try to analyze a simple example of starting with equal branch amplitudes that evolve into unequal amplitudes and show for myself that a naive uniform counting measure conflicts with the Born rule. Note that I don't want to just start with unequal amplitudes by fiat, since in such a case the meaning of unequal amplitudes must be defined rather than derived and the argument will be circular. So I'm looking for a simple example, such as a spin 1/2 particle and an arrangement of Stern-Gerlach experiments, that takes some initial state with equal amplitudes like $\frac{1}{\sqrt{2}}$(|+z>+|-z>) and generates unequal amplitudes from it. My problem is I haven't been able to figure out such an arrangement. Can someone give an example?

One attempt I had was to start with a Stern-Gerlach along +Z followed by Stern-Gerlachs along +X and then combining paths on only the +X side using another Stern-Gerlach along -Z, resulting in a total of 5 possible measurement outcomes { (+Z$\rightarrow$+X$\rightarrow$+Z), (+Z$\rightarrow$+X$\rightarrow$-Z or -Z$\rightarrow$+X$\rightarrow$+Z), (-Z$\rightarrow$+X$\rightarrow$-Z), (+Z$\rightarrow$-X), (-Z$\rightarrow$-X) }. However then it's not clear to me how to apply the Born rule to find the experimental expectation because different outcomes correspond to different observables being measured simultaneously.

Note that the answer by Punk_Physicist is not what I'm after, since it tacitly assumes the Born rule in defining the functional dependence of the amplitude on the angle $\theta$ to be sin and cosine.

EDIT:

After comments by both Timaeus and Rococo I realize that my question can be boiled down to a far simpler experimental setup. A single electron being deflected by a single Stern-Gerlach magnet. The wave function after deflection will be something like $\frac{1}{\sqrt{2}}|+>+\frac{1}{\sqrt{2}}|->$, and my question amounts to: where did the $\frac{1}{\sqrt{2}}$ factors come from? Do they come from pure Schrodinger evolution, or do you assume unitarity and therefore essentially apply the $\frac{1}{\sqrt{2}}$ by hand because you know the probabilities have to sum to 1? Ie can it be proven from pure Schrodinger evolution without any renormalization that the wave function evolves to $\frac{1}{\sqrt{2}}|+>+\frac{1}{\sqrt{2}}|->$ rather than the more intuitive $\frac{1}{2}|+>+\frac{1}{2}|->$ that is subsequently renormalized to enforce the expected unitarity?

• do you want an example of a state $\alpha|1\rangle + \alpha|2\rangle$ evolving in $\alpha|1\rangle + \beta|2\rangle$ with $\alpha \ne \beta$ ? – image Apr 6 '15 at 0:58
• @MarcelKöpke, yes, but only using something like SG at 90 degrees that split the wave function into halves for example so that branch counting can be followed directly. Punk_Physicist gave an example with SG rotated at $\theta$<90 degrees that violates the spirit of what I'm asking for. – user1247 Apr 6 '15 at 3:15
• you might be interested in Rabi-oscillations. If two states of different energy evolve over time, the probability of finding on system in a certain energy-state oscillates, such that for some time $|\alpha|^2 = |\beta|^2$ but later $|\alpha|^2 \ne |\beta|^2$ (thus $\alpha \ne \beta$). This is actually always the case for non-stationary states ((1) and (2) ) – image Apr 6 '15 at 11:02

Actually, I think that what you ask for is impossible, or at least not as physically meaningful as one might expect. This is because of the following fact: any situation in which you have unequal amplitudes between two states can be re-formulated as a bunch of states of equal amplitude. So, as a result, it is possible to describe any state as coming from a bunch of Everettian splits that were all individually equal.

The proof for this is due to Zurek, and comes out of his work on einvariance in which he tries to derive the Born rule. See, for example, here (if you have institutional access, there is also a popular account in Physics Today here). In essence, what he does is show, using symmetry arguments, that an entangled state with equal amplitudes for each of its outcomes must have these outcomes occur with equal probability. Then he goes on to show (in section IID of the first paper) that for unequal amplitudes, using an auxiliary entangled state, you can always decompose the wavefunction into a series of equal branches. So, for example, in a rotated Stern-Gerlach measurement it is always possible to describe the process as having $N$ branches that are equally weighted, in which case $N\cos^2{\theta}$ will be spin-up (or whatever) and $N\sin^2{\theta}$ will be spin-down.

Where does this leave the problem of branch counting? Well, in lieu of my opinion here is Scott Aaronson's take (from his blog):

In particular, if we agree to talk in this way about a probability distribution over various copies of yourself in the different Everett branches, then lots of well-known arguments (Gleason’s theorem, Zurek’s “envariance” argument, etc.) make a strong case that, given the framework of Hilbert spaces and unitary evolution, the distribution must be given by the standard Born rule and not some other rule. So the question “why the Born rule?” is not one that keeps me up at night; and at any rate, it certainly isn’t a special problem for MWI (one could just as well ask it in any interpretation).

Edit: Maybe an example will be helpful. Start with a Mach-Zender interfermeter (from wikipedia): This is already, I guess, what you want: you shine in light which is evenly split among two paths, but given a certain arrangement of path lengths light will only come out at detector 1. Furthermore, you can replace both the mirrors with two more beamsplitters, and then you will have three outputs: the two new outputs each have probability 25%, and detector 1 has probability 50%.

The point of the above, however, is that you can regard this as a wavefunction with four branches, equally weighted, in which two have the identical outcome of "detector 1." Here is the explicit way this mapping works:

The state coming out of this modified Mach-Zender interferometer is:

$\frac{1}{\sqrt{2}}|1>+\frac{1}{2}|2>+\frac{1}{2}|3>$,

where |1> is detector 1 and the other two kets are the other two outputs. When this is measured, it becomes entangled with the environment, which can be represented as:

$\frac{1}{\sqrt{2}}|1 1_E>+\frac{1}{2}|2 2_E>+\frac{1}{2}|3 3_E>$

Now, suppose you can rewrite the environment state* as:

$|1_E>=\frac{1}{\sqrt{2}}(|1_E'>+|1_E''>)$

This allows you to reframe the overall state as something with equally weighted coefficients:

$\frac{1}{2}(|1 1_E'>+|1 1_E''>+|2 2_E>+|3 3_E>)$

at which point the probability of a particular "you" experiencing any one of the results is found just by branch counting. But, and maybe this is the real point, it would make no sense to do this branch counting procedure at any time before the measurement, because, in this example, everything is completely coherent before that point and there is no sense in which the universes have split.

*Note for the careful reader: Zurek actually uses an additional ancillary state that is entangled with the environment to provide the splitting necessary to make all the states evenly weighted. This lets him define his branches uniquely by the Schmidt decomposition, which avoids the arbitrary nature of a change of basis in the way I've done it.

• What specifically are you saying is impossible? That there is no conceivable experiment that takes two equal amplitudes (say spin up and down) and uses combinations of 50-50 splitting (say various 90 degree SG) to produce unequal amplitudes? – user1247 Apr 5 '15 at 20:48
• Because in such a case it seems prima facie that there is a contradiction, regardless of whether what you say is true, since you explicitly evolve, say, two equal numbers of branches into, say, three possible detectors with 25% of the branches, 50% of the branches, and 25% of the branches, then the branch counting clearly leads to a contradiction with the Born rule. – user1247 Apr 5 '15 at 20:56
• Also the quote of Aaronson (someone whose opinion I greatly respect) is a good one, but I think is not a quote directly addressing the point, which isn't whether or not Born is necessary given the Hilbert Space framework (we know it is), but whether a uniform branch counting measure is consistent with indexical uncertainty giving rise to the Born rule, given that the Born rule seems to weight double-counted indices higher than others. – user1247 Apr 5 '15 at 21:01
• Hi user1247, please see my edit and let me know if this isn't getting more directly at what you are interested in. – Rococo Apr 6 '15 at 0:25
• So what I still don't understand is that in your example by branch counting I would derive that the experimental probabilities should be 50% |detector1 you1> and 25% for each of the others. But the Born rule would tell you than the probability for |detector1 you1> is actually 2$^2$/($\sqrt{2^2+1^2+1^2}$) = 66%, and 17% for each of the others. So somewhere there is a contradiction, no? – user1247 Apr 6 '15 at 0:41

Prepare your spin-1/2 particle to be aligned in the $+z$ direction, so you have as your state $\left|+z\right\rangle$. Then send your particle through a Stern-Gerlach oriented such that you are measuring between $z$ and $x$ (and at some angle $\theta$ relative to the $z$-axis). Call this new rotated axis $z'$.

Now if we label the eigenstates of your detector as $\left|\pm z'\right\rangle$, then in terms of these eigenstates your original state can be written as $$\left|+z\right\rangle = \cos(\theta)\left|+z'\right\rangle + \sin(\theta)\left|-z'\right\rangle,$$ and thus you now have a way of choosing continuously any amplitude between zero and one.

Alternatively you can rotate the particle (e.g. by putting the particle in a strong magnetic field and slowly/adiabatically rotating the particle). In this case $\left|+z\right\rangle\to\left|+z'\right\rangle$ and when you measure you will find that the state is $$\left|+z'\right\rangle = \cos(\theta)\left|+z\right\rangle - \sin(\theta)\left|-z\right\rangle.$$ If in addition you don't believe in the inner-product rules of standard quantum mechanics then simply replace $\cos(\theta)$ by $\langle z|z'\rangle$, which on physical grounds still needs to be a monotonically increasing function that continuously maps $\theta\in[0,\pi/2]\mapsto|\langle z|z'(\theta)\rangle|\in[0,1]$ (i.e. gives you a continuous range of values, not just $1/\sqrt{2}$

• Thanks, this does answer my question as-stated, but it shows that I didn't state my question clearly enough, which is my fault. The problem with your above solution (and why I was trying to focus on combining orthogonally oriented Stern-Gerlach magnets) is that the amplitudes you propose are tautologically the result of assuming the Born rule, since we know that sin$^2$($\theta$)+cos$^2$($\theta$)=1 – user1247 Mar 12 '15 at 17:19
• This post only assumes geometry and the linearity of QM, i.e. $\left|+z\right\rangle = \left\langle +|+z\right\rangle \left|+\right\rangle + \left\langle -|+z\right\rangle\left|-\right\rangle = \cos(\theta)\left|+\right\rangle + \sin(\theta)\left|-\right\rangle$. – Punk_Physicist Mar 12 '15 at 18:58
• How do you calculate <+|+z>=cos($\theta$) without assuming that <+|+z>$^2$+<-|+z>$^2$=1, ie that the Born probabilities sum to 1? This is a subtle issue related to the fact that as Gleason showed, the Born rule is inevitable given the Hilbert inner-product space. This is basically the point of my question. I want to find an example that allows me to count branches without making circular assumptions. One way would be to combine orthogonal Stern-Gerlach experiments in ways like I described in my post. – user1247 Mar 12 '15 at 19:42
• If you were to integrate the Schroedinger equation to obtain the time evolution of the |+z> state as it passes through the magnetic field in the Stern-Gerlach device (before "measurement" occurs) you'd find that you obtain two wave packets. Each packet has the spin orientation and amplitude from Punk_Physicist's answer. No Born rule required, just the Schroedinger equation. – adipy Apr 8 '15 at 0:38
• @adipy, yes I am equating the two, intentionally, in the sense that it seems like one implies the other. Maybe see my question here. That my thinking here is not obviously stupid is justified by Gleason's theorem. Normalization is related to Born in that it is saying that the squared amplitudes sum to probability = 100%. While I may ultimately be wrong, this is the heart of my question, and it cannot be so trivially dismissed without significantly more engagement and explanation. – user1247 Apr 12 '15 at 6:00

I don't understand your example, when you say you have 5 possibilities and the second clearly has two different final outcomes. It seems you want to find a flaw in MWI, and I want to warn you that you going to fail, because it is an interpretation, not a separate theory, so all you can hope for is to show that all of QM is internally inconsistent (unlikely) or that you misunderstood the interpretation or misused it. edit: You say in comments that an experimental signature is based on which particle detector flashes, but that is simply false, and thinking it can lead to problems. The first cited paper should explain that a SG oriented in the +z direction and one oriented in the -z direction deflect positive spin and negative spin states to the same detector, i.e. deflect the beams in the same manner. However when a general spin state goes through, the resulting deflected beams aren't just deflected, they become spin polarized, and the two devices polarize oppositely. To be an experiment, you have to look at the whole wavefunction, including the spin part, being deflected one way with one spin, and being deflected the same way but with a different spin is totally different. Different enough to be completely orthogonal. Presumably different detectors firing correspond to orthogonal states, but just because the same detector fires doesn't mean the states weren't orthgonal

There is an more detailed another answer on this issue at https://physics.stackexchange.com/a/170250/57780 but I will summarize: Firstly, the Born rule can in a sense be derived from the Schrödinger equation. And that derivation shows that the case when the Born rule can be applied is when the different possibilities have wavepackets that are mutually orthogonal both now and ever after. The Schrödinger equation also spells out exactly how large the different wavepackets are (see This article(arxiv version) for explicit examples for Stern-Gerlach machines). The born rule assigned relative probabilities to the outcomes which are equal to the relative squared norms of the wavepackets. That's what you need from the linked answer. edit: Isn't the Born rule built into the definition of a Hilbert space through definition of wave function normalization? The Born rule is technically superfluous (at best). Since the Schrödinger equation gives unitary evolution, the norm is indeed preserved, but the Born rule is simply a heuristic device to generate information from one wavefunction describing one system about the actual Schrödinger evolution for an ensemble of systems. It is built-in in the sense that the Schrödinger equation was already enough and that the Born rule merely (at best) attempts to describe something the Schrödinger equation already predicts. The Born rule talks about ensembles of systems, it is definitely not a mere aspect of the fact that the wavefunction for a single system is normalizable. Since the actual ensemble we measure in the lab is described by the Schrödinger equation, and since the relative frequency of the results within an ensemble can also be described by the Schrödinger equation, the Born rule is akin to a clever shortcut about relating a property of wavefunctions that describe ensembles to some properties of a single system.

Now, for how this relates to your hopes. The unitary evolution of the Schrödinger equation says that the squared norm of the wavefucntion at two times is the same, and the orthogonality of the different possibilities post measurement means the square of the sum is the sum of the squares (that's a generic or even defining characteristic of being orthogonal). So the probabilities of the born rule actual add up to one.

In short every time you feed it through a device, either the measurement finished happening and the outcomes are now orthogonal (and by your desire, the two orthogonal parts are equally sized) and must stay orthogonal, or it hasn't. And if it hasn't you can neither use branching nor the born rule.

Since they stay mutually orthogonal that is exactly when the born rule allows you to have frequencies that can be used like we normally use frequencies from probability theories (even though that isn't what happened). But that means you start with a wavefunction and then decompose it into two equal sized chucks and break those, etc. Nothing else will happen. They have to have the squared norms add up to one because they have to stay orthogonal (that's what it means to make a measurement). So now when you look at an experimental outcome, you notice that some of thsoe waves have that outcome, and some don't. But since they are all orthogonal, you can add the squared length of all the parts to get the probability, or you can add them up and then take the square, but you get the same answer. For your situation, those parts always have a squared length that is a negative power of two. And their sum is exactly what the born rule predicts.

We don't assign frequencies (or measurement outcomes) until after the simple adding of the frequencies gives us the squared norm of a collection of mutually orthogonal waves. And that's also when branching occurs.

All interpretations of QM do this. They all only provide frequencies for outcomes when the waves of those different outcomes are orthogonal and will evolve to things that are orthogonal. And that's because it is only internally consistent to do it then, and not earlier.

For vectors the sum of the squared lengths does not equal the squared length of the sum unless they are orthogonal. It wouldn't work to assign frequencies when the possibilities aren't orthogonal, and absolutely no interpretation of quantum mechanics does that.

Edit

All you have to do is learn how the Schrödinger equation evolves wavefunctions. See how to evolves ensembles. See how it evolves devices that sum the results of ensembles. Then you will notice that there are situations where you can get some of those sums by summing the results of smaller systems. You will also notice that sometimes you can get those frequencies by using regular run of the mill probability theory. You'll notice that this happens when the parts are orthogonal now and forevermore. You'll notice that at this point you (the person using the theory to make predictions) can pretend that the outcomes happen with a certain probability. That is literally when probability enters the theory. All evidence is the Schrödinger equation doesn't have some twin or supplement. You can choose to insert probability theory if you (the person using the theory to make predictions) wish, and you can get the same predictions about the sum of frequencies of large n systems of many identically prepared subsystems. All without having to model the actual system or the device that totals them up. And any interpretation you wish that helps you (the person using the theory to make predictions) choose when to start using probability theory is totally fine as long as it tells you to do it in a way that agrees with what the Schrödinger equation already predicts for an actual ensemble.

Absolutely every interpretation strives to do this. Taking any of them too seriously about the stories they tell about when why or how to ignore using the Schrödinger equation to model the actual setup is potentially dangerous.

For the MWI the "worlds", rightly, are the orthogonal parts of the wavefunction that evolve to forever more remain orthogonal. The frequencies predicted are the frequencies of ensembles, and a particular outcome of a particular ensemble (and outcome of an ensemble is the relative frequency of members of the ensemble) is itself a property of just ... one ... world. The talk about branching, is again (as always) just talk about how to use information about a single subsystem to describe the collective attributes of an ensemble of identically prepared subsystems. In this case when a single subsystem evolves into parts that are orthogonal and couple to other parts (the well known environment) to remain orthogonal forever and the single subsystem interacts with a single part (the sum device) that only cares about some parts (the sum device doesn't care about the whole universe, it is just sensitive to the thing we designed it to care about). Then we are free to use regular probability theory to describe how that sum device acts. The talk about probability is, as always (for every interpretation), just about when you can use it and what numbers to use.

So do you need anything fancy? No. You don't need to use anything at all except the Schrödinger equation. Do you need the born rule to assign probabilities to the branches? Since the branches (worlds) in the "end" are orthogonal, there is an observable that has them as eigenvectors. And you can evolve them backwards in time to get initial states that are orthogonal (since the Schrödinger evolution is unitary). Those states each evolve to a distinct world. But a normalized linear combination of the initial states above is also a valid wavefunction. We can take those linear combinations and call them subsystems and make a giant ensemble and then we know how the ensembles of those subsystems evolve and how a sum device evolves. And it evolves according to the Schrödinger equation. The born rule is a valid way to describe the likely (for large n) approximate (relative) number of subsystems with particular results.

Probabilities are simply not a fundamental part of quantum mechanics. Mathematicians do probability theory by fixing a sample space with a measure and then placing random variables on it and then proving theorems about the results. If you started out trying to assume there was a sample space and that observables are random variables on it, then you actually get mathematical contradictions because random variables on a sample space commute but operators on a Hilbert space do not.

But the Schrödinger equation doesn't give contradictions and handles non commuting operators just fine. It models the things we actually do when we run a series of identically prepared subsystems and look at the aggregation of the results. Probabilities come out of the Schrödinger equation (for every single interpretation, not just MWI) when you try to relate the aggregate property of an actual evolution of an actual series of subsystems to the predictions a probabilistic theory would have made, had we had a sample space (which depended on the actual evolution of the actual experiments). You are trying to say which probabilistic theory has results about relative frequencies that matches what we actually predicted with the actual Schrödinger equation.

It is not (not for any interpretation) a true Stochastic Process in the sense meant by mathematicians. We simply cannot (not for any interpretation) assign some measure to a sample space then get probabilities for various outcomes of a set a random variables corresponding to different possible experiments. You can say this is because operators don't commute and random variables do. But really it is because the things we call measurements don't reveal preexisting properties like people like to think they do.

For instance a SG device polarizes the spin even though it might not have been polarized before. What we call measurements are actually just evolutions with a certain consistency. For instance a SG device sends a beam of spin 1/2 particles that is polarised along an axis by a fixed deflection, and the consistency aspect is that it deflects all beams of spin 1/2 particles into two possible fixed deflections and it polarizes them in the direction that makes the reproduce that deflection upon repetition. We know exactly how it deflects and polarizes from looking at the Schrödinger equation evolution, I cited the paper earlier. So they evolve into a bunch of orthogonal states.

And they do it so that a certain volume (in the squared norm sense) gets deflected in each direction. The actual direction a piece of volume goes depends not just on which observable is measured, but also on details of exactly how you set it up. For instance you can make SG that deflects spin up to the right, or one that deflects spin up to the left, they both work fine. They have the same relative frequency for up and down. But different parts of the wave end up joining different end packets, so end up getting polarized in different ways. You aren't measuring a prexisting condition.

You are evolving. The evolutions depend on the actual setups. Certain aggregate properties of collections of end states for subsystems don't depend on the details. Regular probability theory can be used post-facto to describe those same aggregate properties of collections of end states. You do not get those probabilities (not in any interpretation) from a measure on some sample space. If you have a whole ensemble in mind, you can of course make a measure to get those probabilities and push it backwards, but it is just as artificial for MWI as it is for any other interpretation. It is not causal. It is not actual. It is not even detectable. It is just a way to make something that corresponds to what what actually happens for the ensemble of subsystems.

I think you might have a very simple question buried inside all the misconceptions Can you take equal splits to get unequal probabilities?

Now that we can be clear that the probabilities are about a different theory, we can describe both set ups. We can send say an electron through some SG machines. The subsystem will split a +z state into a +x and a -x state heading in different directions. Each of those beams can encounter a $\pm$z oriented SG splitting them into +z and -z states, and we can steer the beams with standard electron optics so that each aims for one of three quite large spheres. If the electron beams was original very high energy, and the spheres are quite large the slight charge build up can avoid distorting the dynamics very much. So the relative charge on the spheres indicates the relative number of times each was hit. By having electron optics that steer two of the four beams to the same sphere we can end up expecting about half the charge to end up on the middle sphere for thousands of electrons fired. That could be all you want.

We can send it so the one that goes left left is steered to the left sphere, the one that goes right right goes to the right sphere, and the rest goes t to the middle sphere. If you thought the Born rule gives something else other than 1/4, 1/2, 1/4 then you are mistaken. The final ensemble frequencies can be measured from the deflection of a low energy electron sent near the spheres after all the fast electrons go through the SG machines and are collected by the spheres.

You get 1/4 for the right sphere because the squared amplitude of the right-right electron is 1/4, and same for the left-left electron. Thus by unitary evolution the rest has 1/2 squared amplitude. Since all interpretations assign probabilities to match the actual predictions for a real ensemble, the born rule must assign 1/4,1/2,1/4 for a single subsystem doing one run.

If you think adjusting the exact up-down of the two changes something, then you are just making a mistake. For instance with a MZ device with photons you can have a beam splitter that where each spilt beam hit s a beam splitter and angle them so transmit-then-reflect and reflect-then-reflect head to the same location. Transmit-transmit and reflect-transmit can be sent to their own detectors. The transmit-then-reflect and reflect-then-reflect that head to the same location can meet another beam splitter. If the phase is adjusted just right, the transmit-then-reflect-then-transmit can interfere perfectly with the reflect-then-reflect-then-reflect to cancel out perfectly. Since each beam splitter sends have the squared length through (and reflected) the amplitude reflected (or transmitted) is a phase times 1/$\sqrt{2}$. So two beams that have 1/2 the amplitude escape (1/4 the probability) and the other two have 1/2 the amplitude coming in to the same location, each on it's own would have (1/2)/ $\sqrt{2}$ an amplitude, but when the reflect-and-transmit one option has them cancel completely (equal and opposite amplitudes) and the other they combine 2(1/2)/$\sqrt{2}$=1/$\sqrt{2}$, which has a probability of 1/2.

So 1/4, 1/2, 1/4, this time with photons. Each device sending 1/2 the probability through the beam splitter. I don't see what the issue is except that maybe you misunderstand the Born rule. So hopefully one way or another you know the answer to your question. Note that I still don't understand what your your initial setup was supposed to be.

Next edit

Consider a single electron being deflected by a single Stern-Gerlach magnet. Its covered in detail how the experiment goes according to solely Schrödinger equation evolution in the paper cited above, just ignore the evolution of the particle position since it affects nothing (it only reacts). But let's see it here. Note: I'll use the product notation to specify the spin state and the spatial state. The wavepacket before entering the SG device is $\left(\frac{1}{\sqrt{2}}|+>+\frac{1}{\sqrt{2}}|->\right)|entering>=\frac{1}{\sqrt{2}}|+>|entering>+\frac{1}{\sqrt{2}}|->|entering>$, and after it goes through the SG device it looks like $\frac{1}{\sqrt{2}}|+>|deflected-up>+\frac{1}{\sqrt{2}}|->|deflected-down>$. Where did the $1/\sqrt{2}$ come from? Well, the states $|+>$ and $|->$ are orthogonal, so if normalized, you need $1/\sqrt{2}$ to have the initial state be normalized. We know that $|+>|entering>$ evolves to $|+>|deflected-up>$ and that $|->|entering>$ evolves to $|->|deflected-down>$ because (1) the SG device is designed to be consistent in a way to produce reproducible results and (2) as described in the cited paper a Hamiltonian for a particle with a magnetic moment in a $\vec{B}$ field is $\beta \mu \vec{\sigma}\cdot \vec{B}$, and so the regular Schrödinger equation evolution for a particle with a magnet moment in an inhomogeneous $\vec{B}$ field produces exactly that evolution and we know electrons have a magnetic moment and that a SG device is basically just a concentrated inhomogeneous $\vec{B}$ field.

So $|+>|entering>$ evolves to $|+>|deflected-up>$ and $|->|entering>$ evolves to $|->|deflected-down>$. Thus, by linearity, since the Schrödinger equation evolution is linear we know that $\left(\frac{1}{\sqrt{2}}|+>+\frac{1}{\sqrt{2}}|->\right)|entering>$ evolves to $\frac{1}{\sqrt{2}}|+>|deflected-up>+\frac{1}{\sqrt{2}}|->|deflected-down>$.

The born rule relates the ratio of the squared amplitudes of the results to the squared amplitude of the input to the relative frequency of the outcome. The outcomes are always orthogonal. You are free to normalize the incoming state since the overall magnitude affects nothing. The evolution is unitary and it is linear. But this is again (like everything) a consequence of the Schrödinger equation evolution, not an extra thing that needs to be added. It is linear because the Schrödinger equation itself is linear. The Schrödinger equation evolution is unitary because the time derivative of the wavefunction is equal to the same function multiplied by an antihermitian operator.

• To answer your first question, it's because those "two separate final outcomes" result in the exact same experimental signature. The two beams of atoms are combined using S-G and end up at a single particle detector. The two outcomes I listed were for clarity of what was going on -- the +Z and -Z are combined by S-G into a single beam. Regarding "I want to find a flaw in the MWI", that is a misunderstanding of my intention. I want to find an example that intuitively exemplifies the common complaint that the "branch counting measure" is non-intuitive basically due to the square in the Born rule – user1247 Mar 24 '15 at 19:29
• Regarding your second paragraph, isn't it a very common and mostly agreed-upon complaint of the MWI that the Born rule implies a non-intuitive non-uniform measure for branch counting? The point is not to show that MWI is internally inconsistent. The point is just that ultimately the Born rule is not "derived from simple branch counting", since you ultimately need to axiomatically assume a weird counting measure in order to agree with the Born rule. – user1247 Mar 24 '15 at 19:32
• Regarding your third paragraph, I understand that what you say is correct if a S-G truly makes a measurement, but my operating assumption is that the system can be made coherent until the final particle detectors measure the presence of an out-going particle. In other words I'm assuming that each S-G is isolated from the outside environment. So each S-G does not represent a "measurement", just a pathway for the schrodinger evolution to divede up the wave function and cause interference before measurement. Does that make sense? – user1247 Mar 24 '15 at 19:38
• Fourth paragraph, my understanding is that the Born rule is built into the definition of a Hilbert space through definition of wave function normalization. Basically if you define the norm of your wave function of "1" to correspond to 100% probability, then the Born rule is unavoidable given that the norm is the sqrt of sum of squared amplitudes. This can be seen in Gleason's thoerem or Everett's "derivation" of Born in his thesis. – user1247 Mar 24 '15 at 19:44
• The rest of your paragraphs I think extend from the misinterpretation of my question explained above. Thank you so much for engaging though! I really hope that you will understand my question better given the above comments and help me understand how to figure out the expected probabilities for these kinds of situations. Assuming the wave function stays coherent until the end, I still don't know how to handle this situation, because like I said in the OP, for example when the particle detectors register different observables are being measured simultaneously depending on the outcome. – user1247 Mar 24 '15 at 19:50