## A new derivation of the Born rule

This post is an explanation of a recent paper coauthored by Sean Carroll and Charles Sebens, where they propose a derivation of the Born rule in the context of the Many World approach to quantum mechanics. While the attempt itself is not fully successful, it contains interesting ideas and it is thus worthwhile to know.

A note to the reader: here I will try to enlighten the preconditions and give only a very general view of their method, and for this reason you won’t find any equation. It is my hope that if after having read this you’re still curious about the real math, you will point your browser to the preceding link and read the paper for yourself.

If you are not totally new to LessWrong, you should know by now that the preferred interpretation of quantum mechanics (QM) around here is the Many World Interpretation (MWI), which negates the collapse of the wave-function and postulates a distinct reality (that is, a branch) for every base state composing a quantum superposition.

MWI historically suffered from three problems: the absence of macroscopic superpositions, the preferred basis problem, the Born rule derivation. The development of decoherence famously solved the first and, to a lesser degree, the second problem, but the role of the third still remains one of the most poorly understood side of the theory.

Quantum mechanics assigns an amplitude, a complex number, to each branch of a superposition, and postulates that the probability of an observer to find the system in that branch is the (squared) norm of the amplitude. This, very briefly, is the content of the Born rule (for pure states).

Quantum mechanics remains agnostic about the ontological status of both amplitudes and probabilities, but MWI, assigning a reality status to every branch, demotes ontological uncertainty (which branch will become real after observation) to indexical uncertainty (which branch the observer will find itself correlated to after observation).

Simple indexical uncertainty, though, cannot reproduce the exact predictions of QM: by the Indifference principle, if you have no information privileging any member in a set of hypothesis, you should assign equal probability to each one. This leads to forming a probability distribution by counting the branches, which only in special circumstances coincides with amplitude-derived probabilities. This discrepancy, and how to account for it, constitutes the Born rule problem in MWI.

There have been of course many attempts at solving it, for a recollection I quote directly the article:

One approach is to show that, in the limit of many observations, branches that do not obey the Born Rule have vanishing measure. A more recent twist is to use decision theory to argue that a rational agent should act as if the Born Rule is true. Another approach is to argue that the Born Rule is the only well-deﬁned probability measure consistent with the symmetries of quantum mechanics.

These proposals have failed to uniformly convince physicists that the Born rule problem is solved, and the paper by Carroll and Sebens is another attempt to reach a solution.

Before describing their approach, there are some assumptions that have to be clarified.

The first, and this is good news, is that they are treating probabilities as rational degrees of belief about a state of the world. They are thus using a Bayesian approach, although they never call it that way.

The second is that they’re using self-locating indifference, again from a Bayesian perspective.

Self-locating indifference is the principle that you should assign equal probabilities to find yourself in different places in the universe, if you have no information that distinguishes the alternatives. For a Bayesian, this is almost trivial: self-locating propositions are propositions like any other, so the principle of indifference should be used on them as it should on any other prior information. This is valid for quantum branches too.

The third assumption is where they start to deviate from pure Bayesianism: it’s what they call Epistemic Separability Principle, or ESP. In their words:

the outcome of experiments performed by an observer on a speciﬁc system shouldn’t depend on the physical state of other parts of the universe.

This is a kind of a Markov condition: the request that the system is such that it screens the interaction between the observer and the system observed from every possible influence of the environment.

It is obviously false for many partitions of a system into an experiment and an environment, but rather than taking it as a Principle, we can make it an assumption: an experiment is such only if it obeys the condition.

In the context of QM, this condition amounts to splitting the universal wave-function into two components, the experiment and the environment, so that there’s no entanglement between the two, and to consider only interactions that can factors as a product of an evolution for the environment and an evolution for the experiment. In this case, environment evolution act as the identity operator on the experiment, and does not affect the behavior of the experiment wave-function.

Thus, their formulation requires that the probability that an observer finds itself in a certain branch after a measurement is independent on the operations performed on the environment.

Note though, an unspoken but very important point: probabilities of this kind depends uniquely on the superposition structure of the experiment.

A probability, being an abstract degree of belief, can depend on all sorts of prior information. With their quantum version of ESP, Carroll and Sebens are declaring that, in a factored environment, probabilities of a subsystem does not depend on the information one has about the environment. Indeed, in this treatment, they are equating factorization and lack of logical connection.

This is of course true in quantum mechanics, but is a significant burden in a pure Bayesian treatment.

That said, let’s turn to their setup.

They imagine a system in a superposition of base states, which first interacts and decoheres with an environment, then gets perceived by an observer. This sequence is crucial: the Carroll-Sebens move can only be applied when the system already has decohered with a sufficiently large environment.

I say “sufficiently large” because the next step is to consider a unitary transformation on the “system+environment” block. This transformation needs to be of this kind:

- it respects ESP, in that it has to factor as an identity transformation on the “observer+system” block;

- it needs to equally distribute the probability of each branch in the original superposition on a different branch in the decohered block, according to their original relative measure.

Then, by a simple method of rearranging labels of the decohered base, one can show that the correct probabilities comes out by the indifference principle, in the very same way that the principle is used to derive the uniform probability distribution in the second chapter of Jaynes’ Probability Theory.

As an example, consider a superposition of a quantum bit, and say that one branch has a higher measure with respect to the other by a factor of square root of 2. The environment needs in this case to have at least 8 different base states to be relabeled in such a way to make the indifference principle work.

In theory, in this way you can only show that the Born rule is valid for amplitudes which differ one another by the square root of a rational number. Again I quote the paper for their conclusion:

however, since this is a dense set, it seems reasonable to conclude that the Born Rule is established.

Evidently, this approach suffers from a number of limits: the first and the most evident is that it works only in a situation where the system to be observed has already decohered with an environment. It is not applicable to, say, a situation where a detector reads a quantum superposition directly, e.g. in a Stern-Gerlach experiment.

The second limit, although less serious, is that it can work only when the system to be observed decoheres with an environment which has sufficiently base states to distribute the relative measure in different branches. This number, for a transcendental amplitude, is bound to be infinite.

The third limit is that it can only work if we are allowed to interact with the environment in such a way as to leave the amplitudes of the interaction between the system and the observer untouched.

All of these, which are understood as limits, can naturally be reversed and considered as defining conditions, saying: the Born rule is valid only within those limits.

I’ll leave it to you to determine if this constitutes a sufficient answers to the Born rule problem in MWI.

## Common sense quantum mechanics

**Related to: Quantum physics sequence.**

*TLDR: Quantum mechanics can be derived from the rules of probabilistic reasoning. The wavefunction is a mathematical vehicle to transform a nonlinear problem into a linear one. The Born rule that is so puzzling for MWI results from the particular mathematical form of this functional substitution.*

This is a brief overview a recent paper in *Annals of Physics* (recently mentioned in Discussion):

Quantum theory as the most robust description of reproducible experiments (arXiv)

by Hans De Raedt, Mikhail I. Katsnelson, and Kristel Michielsen. Abstract:

It is shown that the basic equations of quantum theory can be obtained from a straightforward application of logical inference to experiments for which there is uncertainty about individual events and for which the frequencies of the observed events are robust with respect to small changes in the conditions under which the experiments are carried out.

In a nutshell, the authors use the "plausible reasoning" rules (as in, e.g., Jaynes' *Probability Theory*) to recover the quantum-physical results for the EPR and Stern–Gerlach experiments by adding a notion of experimental reproducibility in a mathematically well-formulated way and *without any "quantum" assumptions*. Then they show how the Schrodinger equation (SE) can be obtained from the nonlinear variational problem on the probability *P* for the particle-in-a-potential problem when the classical Hamilton-Jacobi equation holds "on average". The SE allows to transform the nonlinear variational problem into a linear one, and in the course of said transformation, the (real-valued) probability *P* and the action *S* are combined in a single complex-valued function ~*P*^{1/2}exp(*iS*) which becomes the argument of SE (the wavefunction).

This casts the "serious mystery" of Born probabilities in a new light. Instead of the observed frequency being the square(d amplitude) of the "physically fundamental" wavefunction, the wavefunction is seen as a mathematical vehicle to convert a difficult nonlinear variational problem for inferential probability into a manageable linear PDE, where it so happens that the probability enters the wavefunction under a square root.

Below I will excerpt some math from the paper, mainly to show that the approach actually works, but outlining just the key steps. This will be followed by some general discussion and reflection.

**1. Plausible reasoning and reproducibility**

The authors start from the usual desiderata that are well laid out in Jaynes' *Probability Theory* and elsewhere, and add to them another condition:

There may be uncertainty about each event. The conditions under which the experiment is carried out may be uncertain. The frequencies with which events are observed are reproducible and robust against small changes in the conditions.

Mathematically, this is a requirement that the probability *P*(*x*|*θ*,*Z*) of observation *x* given an uncertain experimental parameter *θ* and the rest of out knowledge *Z*, is maximally *robust* to small changes in *θ* and independent of *θ*. Using log-probabilities, this amounts to minimizing the "evidence"

for any small *ε* so that |Ev| is not a function of *θ* (but the probability is).

**2. The Einstein****–****Podolsky****–****Rosen****–****Bohm experiment**

There is a source S that, when activated, sends a pair of signals to two routers R_{1,2}. Each router then sends the signal to one of its two detectors *D _{i}*

_{+,–}(

*i*=1,2). Each router can be rotated and we denote as

*θ*the angle between them. The experiment is repeated

*N*times yielding the data set {

*x*

_{1},

*y*

_{1}}, {

*x*

_{2},

*y*

_{2}}, ... {

*x*,

_{N}*y*} where

_{N}*x*and

*y*are the outcomes from the two detectors (+1 or –1). We want to find the probability

*P*(

*x*,

*y*|

*θ*,

*Z*).

After some calculations it is found that the single-trial probability can be expressed as *P*(*x*,*y*|*θ*,*Z*) = (1 + *xyE*_{12}(*θ*) ) / 4, where *E*_{12}(*θ*) = Σ_{x,y=+–1} *xy**P*(*x*,*y*|*θ*,*Z*) is a periodic function.

From the properties of Bernoulli trials it follows that, for a data set of *N* trials with *n _{xy}* total outcomes of each type {

*x*,

*y*},

and expanding this in a Taylor series it is found that

The expression in the sum is the Fisher information *I _{F}* for

*P*. The maximum robustness requirement means it must be minimized. Writing it down as

*I*= 1/(1 –

_{F}*E*

_{12}(

*θ*)

^{2}) (

*dE*

_{12}(

*θ*)/

*d*

*θ*)

^{2}one finds that

*E*

_{12}(

*θ*) = cos(

*θ*

*I*

_{F}^{1/2}+

*φ*), and since

*E*

_{12}must be periodic in angle,

*I*

_{F}^{1/2}is a natural number, so the smallest possible value is

*I*= 1. Choosing

_{F}*φ*=

*π*it is found that

*E*

_{12}(

*θ*) = –cos(

*θ*), and we obtain the result that

which is the well-known correlation of two spin-1/2 particles in the singlet state.

Needless to say, our derivation did not use any concepts of quantum theory. Only plain, rational reasoning strictly complying with the rules of logical inference and some elementary facts about the experiment were used

**3. The Stern****–****Gerlach experiment**

This case is analogous and simpler than the previous one. The setup contains a source emitting a particle with magnetic moment **S**, a magnet with field in the direction **a**, and two detectors *D*_{+} and *D*_{–}*.*

Similarly to the previous section, *P*(*x*|*θ*,*Z*) = (1 + *xE*(*θ*) ) / 2, where *E*(*θ*) = *P*(+|*θ*,*Z*) – *P*(–|*θ*,*Z*) is an unknown periodic function. By complete analogy we seek the minimum of *I _{F}* and find that

*E*(

*θ*) = +–cos(

*θ*), so that

In quantum theory,

[this]equation is in essence just the postulate (Born’s rule) that the probability to observe the particle with spin up is given by the square of the absolute value of the amplitude of the wavefunction projected onto the spin-up state. Obviously, the variability of the conditions under which an experiment is carried out is not included in the quantum theoretical description. In contrast, in the logical inference approach,[equation]is not postulated but follows from the assumption that the (thought) experiment that is being performed yields the most reproducible results, revealing the conditions for an experiment to produce data which is described by quantum theory.

To repeat: there are no wavefunctions in the present approach. The only assumption is that a dependence of outcome on particle/magnet orientation is observed with robustness/reproducibility.

**4. Schrodinger equation**

A particle is located in unknown position *θ* on a line segment [–*L*, *L*]. Another line segment [–*L*, *L*] is uniformly covered with detectors. A source emits a signal and the particle's response is detected by one of the detectors.

After going to the continuum limit of infinitely many infinitely small detectors and accounting for translational invariance it is possible to show that the position of the particle *θ* and of the detector *x* can be interchanged so that *dP*(*x*|*θ*,*Z*)/*d**θ* = –*dP*(*x*|*θ*,*Z*)/*dx*.

In exactly the same way as before we need to minimize Ev by minimizing the Fisher information, which is now

However, simply solving this minimization problem will not give us anything new because nothing so far accounted for the fact that the particle moves in a potential. This needs to be built into the problem. This can be done by requiring that the classical Hamilton-Jacobi equation holds *on average*. Using the Lagrange multiplier method, we now need to minimize the functional

Here *S*(*x*) is the action (Hamilton's principal function). This minimization yields solutions for the two functions *P*(*x|**θ*,*Z*) and *S*(*x*). It is a difficult nonlinear minimization problem, but it is possible to find a matching solution in a tractable way using a mathematical "trick". It is known that standard variational minimization of the functional

yields the Schrodinger equation for its extrema. On the other hand, if one makes the substitution combining two real-valued functions *P* and *S* into a single complex-valued *ψ*,

*Q* is immediately transformed into *F*, concluding the derivation of the Schrodinger equation. Incidentally, *ψ* is constructed so that *P*(*x|**θ*,*Z*) = |*ψ*(*x*|*θ*,*Z*)|^{2}, which is the Born rule.

Summing up the meaning of Schrodinger equation in the present context:

Of course, a priori there is no good reason to assume that on average there is agreement with Newtonian mechanics ... In other words, the time-independent Schrodinger equation describes the collective of repeated experiments ... subject to the condition that the averaged observations comply with Newtonian mechanics.

The authors then proceed to derive the time-dependent SE (independently from the stationary SE) in a largely similar fashion.

**5. What it all means**

Classical mechanics assumes that everything about the system's state and dynamics can be known (at least in principle). It starts from axioms and proceeds to derive its conclusions deductively (as opposed to inductive reasoning). In this respect quantum mechanics is to classical mechanics what probabilistic logic is to classical logic.

Quantum theory is viewed here not as a description of what really goes on at the microscopic level, but *as an instance of logical inference*:

in the logical inference approach, we take the point of view that a description of our knowledge of the phenomena at a certain level is independent of the description at a more detailed level.

and

quantum theory does not provide any insight into the motion of a particle but instead describes all what can be inferred (within the framework of logical inference) from or, using Bohr’s words,

saidabout the observed data

Such a treatment of QM is similar in spirit to Jaynes' *Information Theory and Statistical Mechanics* papers (I, II). Traditionally statistical mechanics/thermodynamics is derived bottom-up from the microscopic mechanics and a series of postulates (such as ergodicity) that allow us to progressively ignore microscopic details under strictly defined conditions. In contrast, Jaynes starts with minimum possible assumptions:

"The quantity

xis capable of assuming the discrete valuesx... all we know is the expectation value of the function_{i}f(x) ... On the basis of this information, what is the expectation value of the functiong(x)?"

and proceeds to derive the foundations of statistical physics from the maximum entropy principle. Of course, these papers deserve a separate post.

This community should be particularly interested in how this all aligns with the many-worlds interpretation. Obviously, any conclusions drawn from this work can only apply to the "quantum multiverse" level and cannot rule out or support any other many-worlds proposals.

In quantum physics, MWI does quite naturally resolve some difficult issues in the "wavefunction-centristic" view. However, we see that the concept wavefunction is not really central for quantum mechanics. This removes the whole problem of wavefunction collapse that MWI seeks to resolve.

The Born rule is arguably a big issue for MWI. But here it essentially boils down to "*x* is quadratic in *t* where *t =* sqrt(*x*)". Without the wavefunction (only probabilities) the problem simply does not appear.

Here is another interesting conclusion:

if it is difficult to engineer nanoscale devices which operate in a regime where the data is reproducible, it is also difficult to perform these experiments such that the data complies with quantum theory.

In particular, this relates to the decoherence of a system via random interactions with the environment. Thus decoherence becomes not as a physical intrinsically-quantum phenomenon of "worlds drifting apart", but a property of experiments that are not well-isolated from the influence of environment and therefore not reproducible. Well-isolated experiments are robust (and described by "quantum inference") and poorly-isolated experiments are not (hence quantum inference *does not apply*).

In sum, it appears that quantum physics when viewed as inference does not require many-worlds any more than probability theory does.

## Quantum Decisions

CFAR sometimes plays a Monday / Tuesday game (invented by palladias). Copying from the URL:

On Monday, your proposition is true. On Tuesday, your proposition is false. Tell me a story about each of the days so I can see how they are different. Don't just list the differences (because you're already not doing that well). Start with "I wake up" so you start concrete and move on in that vein, naming the parts of your day that are identical as well as those that are different.

So my question is (edited on 2014/05/13):

On Monday, I make my decisions by rolling a normal die. Example: should I eat vanilla or chocolate ice-cream? I then decide that if I roll 4 or higher on a 6-sided die, I'll pick vanilla. I roll the die, get a 3, and so proceed to eat chocolate ice-cream.

On Tuesday, I use the same procedure, but use a quantum random number generator instead. (For the purpose of this discussion, let's assume that I can actually find a true/reliable generator. May be I'm shooting a photon through a half-silvered mirror.)

What's the difference? (Relevant discussion pointed out Pfft.)

## Another problem with quantum measure

Let's play around with the quantum measure some more. Specifically, let's posit a theory T that claims that the quantum measure of our universe is increasing - say by 50% each day. Why could this be happening? Well, here's a quasi-justification for it: imagine there are lots and lots of of universes, most of them in chaotic random states, jumping around to other chaotic random states, in accordance with the usual laws of quantum mechanics. Occasionally, one of them will partially tunnel, by chance, into the same state our universe is in - and then will evolve forwards in time exactly as our universe is. Over time, we'll accumulate an ever-growing measure.

That theory sounds pretty unlikely, no matter what feeble attempts are made to justify it. But T is observationally indistinguishable from our own universe, and has a non-zero probability of being true. It's the reverse of the (more likely) theory presented here, in which the quantum measure was being constantly diminished. And it's very bad news for theories that treat the quantum measure (squared) as akin to a probability, without ever renormalising. It implies that one must continually sacrifice for the long-term: any pleasure today is wasted, as that pleasure will be weighted so much more tomorrow, next week, next year, next century... A slight fleeting smile on the face of the last human is worth more than all the ecstasy of the previous trillions.

One solution to the "quantum measure is continually diminishing" problem was to note that as the measure of the universe diminished, it would eventually get so low that that any alternative, non-measure diminishing theory, not matter how initially unlikely, would predominate. But that solution is not available here - indeed, that argument runs in reverse, and makes the situation worse. No matter how initially unlikely the "quantum measure is continually increasing" theory is, eventually, the measure will become so high that it completely dominates all other theories.

## Quantum versus logical bombs

Child, I'm sorry to tell you that the world is about to end. Most likely. You see, this madwoman has designed a doomsday machine that will end all life as we know it - painlessly and immediately. It is attached to a supercomputer that will calculate the 10^{100th} digit of pi - if that digit is zero, we're safe. If not, we're doomed and dead.

However, there is one thing you are allowed to do - switchout the logical trigger and replaced it by a quantum trigger, that instead generates a quantum event that will prevent the bomb from triggering with 1/10th measure squared (in the other cases, the bomb goes off). You ok paying €5 to replace the triggers like this?

If you treat quantum measure squared exactly as probability, then you shouldn't see any reason to replace the trigger. But if you believed in many worlds quantum mechanics (or think that MWI is possibly correct with non-zero probability), you might be tempted to accept the deal - after all, everyone will survive in one branch. But strict total utilitarians may still reject the deal. Unless they refuse to treat quantum measure as akin to probability in the first place (meaning they would accept all quantum suicide arguments), they tend to see a universe with a tenth of measure-squared as exactly equally valued to a 10% chance of a universe with full measure. And they'd even do the reverse, replace a quantum trigger with a logical one, if you paid them €5 to do so.

Still, most people, in practice, would choose to change the logical bomb for a quantum bomb, if only because they were slightly uncertain about their total utilitarian values. It would seem self evident that risking the total destruction of humanity is much worse than reducing its measure by a factor of 10 - a process that would be undetectable to everyone.

Of course, once you agree with that, we can start squeezing. What if the quantum trigger only has 1/20 measured-squared "chance" of saving us? 1/000? 1/10000? If you don't want to fully accept the quantum immortality arguments, you need to stop - but at what point?

## Are coin flips quantum random to my conscious brain-parts?

Hello rationality friends! I have a question that I bet some of you have thought about...

I hear lots of people saying that classical coin flips are not "quantum random events", because the outcome is very nearly determined by thumb movement when I flip the coin. More precisely, one can stay that the state of my thumb and the state of the landed coin are strongly entangled, such that, say, 99% of the quantum measure of the coin flips outcomes my post-flip thumb observes all land heads.

First of all, I've never actually seen an order of magnitude estimate to support this claim, and would love it if someone here can provide or link to one!

Second, I'm not sure how strongly entangled my thumb movement is with my subjective experience, i.e., with the parts of my brain that consciously process the decision to flip and the outcome. So even if the coin outcome is almost perfectly determined by my thumb, it might not be almost perfectly determined by my decision to flip the coin.

For example, while the thumb movement happens, a lot of calibration goes on between my thumb, my motor cortex, and my cerebellum (which certainly affects but does not seem to directly process conscious experience), precisely because my motor cortex is unable to send, on its own, a precise and accurate enough signal to my thumb that achieves the flicking motion that we eventually learn to do in order to flip coins. Some of this inability is due to small differences in environmental factors during each flip that the motor cortex does not itself process directly, but is processed by the cerebellum instead. Perhaps some of this inability also comes directly from quantum variation in neuron action potentials being reached, or perhaps some of the aforementioned environmental factors arise from quantum variation.

Anyway, I'm altogether not *that* convinced that the outcome of a coin flip is sufficiently dependent on my decision to flip as to be considered "not a quantum random event" by my conscious brain. Can anyone provide me with some order of magnitude estimates to convince me either way about this? I'd really appreciate it!

**ETA:** I am not asking if coin flips are "random enough" in some strange, undefined sense. I am *actually* asking about quantum entanglement here. In particular, when your PFC decides for planning reasons to flip a coin, does the evolution of the wave function produce a world that is in a superposition of states (coin landed heads)⊗(you observed heads) + (coin landed tails)⊗(you observed tails)? Or does a monomial state result, either (coin landed heads)⊗(you observed heads) or (coin landed tails)⊗(you observed tails) depending on the instance?

At present, despite having been told many times that coin flips are not "in superpositions" relative to "us", I'm not convinced that there is enough mutual information connecting my frontal lobe and the coin for the state of the coin to be entangled with me (i.e. not "in a superposed state") before I observe it. I realize this is somewhat testable, e.g., if the state amplitudes of the coin can be *forced to have complex arguments differing in a predictable way so as to produce expected and measurable interference patterns*. This is what we have failed to produce at a macroscopic level in attempts to produce visible superpositions. But I don't know if we fail to produce messier, less-visibly-self-interfering superpositions, which is why I am still wondering about this...

Any help / links / fermi estimates on this will be greatly appreciated!

## False vacuum: the universe playing quantum suicide

Imagine that the universe is approximately as it appears to be (I know, this is a controversial proposition, but bear with me!). Further imagine that the many worlds interpretation of Quantum mechanics is true (I'm really moving out of Less Wrong's comfort zone here, aren't I?).

Now assume that our universe is in a situation of false vacuum - the universe is not in its lowest energy configuration. Somewhere, at some point, our universe may tunnel into true vacuum, resulting in a expanding bubble of destruction that will eat the entire universe at high speed, destroying all matter and life. In many worlds, such a collapse need not be terminal: life could go one on a branch of lower measure. In fact, anthropically, life *will* go on somewhere, no matter how unstable the false vacuum is.

So now assume that the false vacuum we're in is highly unstable - the measure of the branch in which our universe survives goes down by a factor of a trillion every second. We only exist because we're in the branch of measure a trillionth of a trillionth of a trillionth of... all the way back to the Big Bang.

None of these assumptions make any difference to what we'd expect to see observationally: only a good enough theory can say that they're right or wrong. You may notice that this setup transforms the whole universe into a quantum suicide situation.

The question is, how do you go about maximising expected utility in this situation? I can think of a few different approaches:

- Gnaw on the bullet: take the quantum measure as a probability. This means that you now have a discount factor of a trillion every second. You have to rush out and get/do all the good stuff as fast as possible: a delay of a second costs you a reduction in utility of a trillion. If you are a negative utilitarian, you also have to rush to minimise the bad stuff, but you can also take comfort in the fact that the potential for negative utility across the universe is going down fast.
- Use relative measures: care about the relative proportion of good worlds versus bad worlds, while assigning zero to those worlds where the vacuum has collapsed. This requires a natural zero to make sense, and can be seen as quite arbitrary: what would you do about entangled worlds, or about the non-zero probability that the vacuum-collapsed worlds may have worthwhile life in them? Would the relative measure user also put zero value to worlds that were empty of life for other reasons than vacuum collapse? For instance, would they be in favour of programming an AI's friendliness using random quantum bits, if it could be reassured that if friendliness fails, the AI would kill everyone immediately?
- Deny the measure: construct a meta ethical theory where only classical probabilities (or classical uncertainties) count as probabilities. Quantum measures do not: you care about the sum total of all branches of the universe. Universes in which the photon went through the top slit, went through the bottom slit, or was in an entangled state that went through both slits... to you, there are three completely separate universes, and you can assign totally unrelated utilities to each one. This seems quite arbitrary, though: how are you going to construct these preferences across the whole of the quantum universe, when forged your current preferences on a single branch?
- Cheat: note that nothing in life is certain. Even if we have the strongest evidence imaginable about vacuum collapse, there's always a tiny chance that the evidence is wrong. After a few seconds, that probability will be dwarfed by the discount factor of the collapsing universe. So go about your business as usual, knowing that most of the measure/probability mass remains in the non-collapsing universe. This can get tricky if, for instance the vacuum collapsed more slowly that a factor of a trillion a second. Would you be in a situation where you should behave as if you believed vacuum collapse for another decade, say, and then switch to a behaviour that assumed non-collapse afterwards? Also, would you take seemingly stupid bets, like bets at a trillion trillion trillion to one that the next piece of evidence will show no collapse (if you lose, you're likely in the low measure universe anyway, so the loss is minute)?

## The ongoing transformation of quantum field theory

Quantum field theory (QFT) is the basic framework of particle physics. Particles arise from the quantized energy levels of field oscillations; Feynman diagrams are the simple tool for approximating their interactions. The "standard model", the success of which is capped by the recent observation of a Higgs boson lookalike, is a quantum field theory.

But just like everything mathematical, quantum field theory has hidden depths. For the past decade, new pictures of the quantum scattering process (in which particles come together, interact, and then fly apart) have incrementally been developed, and they presage a transformation in the understanding of what a QFT describes.

At the center of this evolution is "N=4 super-Yang-Mills theory", the maximally supersymmetric QFT in four dimensions. I want to emphasize that from a standard QFT perspective, this theory contains nothing but scalar particles (like the Higgs), spin-1/2 fermions (like electrons or quarks), and spin-1 "gauge fields" (like photons and gluons). The ingredients aren't something alien to real physics. What distinguishes an N=4 theory is that the particle spectrum and the interactions are arranged so as to produce a highly extended form of supersymmetry, in which particles have multiple partners (so many LWers should be comfortable with the notion).

In 1997, Juan Maldacena discovered that the N=4 theory is equivalent to a type of string theory in a particular higher-dimensional space. In 2003, Edward Witten discovered that it is also equivalent to a different type of string theory in a supersymmetric version of Roger Penrose's twistor space. Those insights didn't come from nowhere, they explained algebraic facts that had been known for many years; and they have led to a still-accumulating stockpile of discoveries about the properties of N=4 field theory.

What we can say is that the physical processes appearing in the theory can be understood as taking place in either of two dual space-time descriptions. Each space-time has its own version of a particular large symmetry, "superconformal symmetry", and the superconformal symmetry of one space-time is invisible in the other. And now it is becoming apparent that there is a third description, which does not involve space-time at all, in which both superconformal symmetries are manifest, but in which space-time locality and quantum unitarity are not "visible" - that is, they are not manifest in the equations that define the theory in this third picture.

I cannot provide an authoritative account of how the new picture works. But here is my impression. In the third picture, the scattering processes of the space-time picture become a complex of polytopes - higher-dimensional polyhedra, joined at their faces - and the quantum measure becomes the volume of these polyhedra. Where you previously had particles, you now just have the dimensions of the polytopes; and the fact that in general, an *n*-dimensional space doesn't have *n* special directions suggests to me that multi-particle entanglements can be something more fundamental than the separate particles that we resolve them into.

It will be especially interesting to see whether this polytope combinatorics, that can give back the scattering probabilities calculated with Feynman diagrams in the usual picture, can work solely with ordinary probabilities. That was Penrose's objective, almost fifty years ago, when he developed the theory of "spin networks" as a new language for the angular momentum calculations of quantum theory, and which was a step towards the twistor variables now playing an essential role in these new developments. If the probability calculus of quantum mechanics can be obtained from conventional probability theory applied to these "structures" that may underlie familiar space-time, then that would mean that superposition does not need to be regarded as ontological.

I'm talking about this now because a group of researchers around Nima Arkani-Hamed, who are among the leaders in this area, released their first paper in a year this week. It's very new, and so arcane that, among physics bloggers, only Lubos Motl has talked about it.

This is still just one step in a journey. Not only does the paper focus on the N=4 theory - which is not the theory of the real world - but the results only apply to part of the N=4 theory, the so-called "planar" part, described by Feynman diagrams with a planar topology. (For an impressionistic glimpse of what might lie ahead, you could try this paper, whose author has been shouting from the wilderness for years that categorical knot theory is the missing piece of the puzzle.)

The N=4 theory is not reality, but the new perspective should generalize. Present-day calculations in QCD already employ truncated versions of the N=4 theory; and Arkani-Hamed et al specifically mention another supersymmetric field theory (known as ABJM after the initials of its authors), a deformation of which is holographically dual to a theory-of-everything candidate from 1983.

When it comes to *seeing reality* in this new way, we still only have, at best, a fruitful chaos of ideas and possibilities. But the solid results - the mathematical equivalences - will continue to pile up, and the end product really ought to be nothing less than a new conception of how physics works.

## If MWI is correct, should we expect to experience Quantum Torment?

If the many worlds of the Many Worlds Interpretation of quantum mechanics are real, there's at least a good chance that Quantum Immortality is real as well: All conscious beings should expect to experience the next moment in at least one Everett branch even if they stop existing in all other branches, and the moment after that in at least one other branch, and so on forever.

However, the transition from life to death isn't usually a binary change. For most people it happens slowly as your brain and the rest of your body deteriorates, often painfully.

Doesn't it follow that each of us should expect to keep living in this state of constant degradation and suffering for a very, very long time, perhaps forever?

I don't know much about quantum mechanics, so I don't have anything to contribute to this discussion. I'm just terrified, and I'd like, not to be reassured by well-meaning lies, but to know the truth. How likely is it that Quantum Torment is real?

## Question on decoherence and virtual particles

Doing some insomniac reading of the Quantum Sequence, I think that I've gotten a reasonable grasp of the principles of decoherence, non-interacting bundles of amplitude, etc. I then tried to put that knowledge to work by comparing it with my understanding of virtual particles (whose rate of creation in any area is essentially equivalent to the electromagnetic field), and I had a thought I can't seem to find mentioned elsewhere.

If I understand decoherence right, then quantum events which can't be differentiated from each other get summed together into the same blob of amplitude. Most virtual particles which appear and rapidly disappear do so in ways that can't be detected, let alone distinguished. This seems as if it could potentially imply that the extreme evenness of a vacuum might have to do more with the overall blob of amplitude of the vacuum being smeared out among all the equally-likely vacuum fluctuations, than it does directly with the evenness of the rate of vacuum fluctuations themselves. It also seems possible that there could be some clever way to test for an overall background smear of amplitude, though I'm not awake enough to figure one out just now. (My imagination has thrown out the phrase 'collapse of the vacuum state', but I'm betting that that's just unrelated quantum buzzword bingo.)

Does anything similar to what I've just described have any correlation with actual quantum theory, or will I awaken to discover all my points have been voted away due to this being complete and utter nonsense?

## Debugging the Quantum Physics Sequence

This article should really be called "Patching the argumentative flaw in the Sequences created by the Quantum Physics Sequence".

There's only one big thing wrong with that Sequence: the central factual claim is wrong. I don't mean the claim that the Many Worlds interpretation is correct; I mean the claim that the Many Worlds interpretation is *obviously* correct. I don't agree with the ontological claim either, but I especially don't agree with the epistemological claim. It's a strawman which reduces the quantum debate to Everett versus Bohr - well, it's not really Bohr, since Bohr didn't believe wavefunctions were physical entities. Everett versus Collapse, then.

I've complained about this from the beginning, simply because I've also studied the topic and profoundly disagree with Eliezer's assessment. What I would like to see discussed on this occasion is not the physics, but rather how to patch the arguments in the Sequences that depend on this wrong sub-argument. To my eyes, this is a highly visible flaw, but it's not a deep one. It's a detail, a bug. Surely it affects nothing of substance.

However, before I proceed, I'd better back up my criticism. So: consider the existence of single-world retrocausal interpretations of quantum mechanics, such as John Cramer's transactional interpretation, which is descended from Wheeler-Feynman absorber theory. There are no superpositions, only causal chains running forward in time and backward in time. The calculus of complex-valued probability amplitudes is supposed to arise from this.

The existence of the retrocausal tradition already shows that the debate has been represented incorrectly; it should at least be Everett versus Bohr versus Cramer. I would also argue that when you look at the details, many-worlds has no discernible edge over single-world retrocausality:

- Relativity isn't an issue for the transactional interpretation: causality forwards and causality backwards are both local, it's the existence of loops in time which create the appearance of nonlocality.
- Retrocausal interpretations don't have an exact derivation of the Born rule, but neither does many-worlds.
- Many-worlds finds hope of such a derivation in a property of the quantum formalism: the resemblance of density matrix entries to probabilities. But single-world retrocausality finds such hope too: the Born probabilities can be obtained from the product of
*ψ*with*ψ**, its complex conjugate, and*ψ**is the time reverse of*ψ.* - Loops in time just fundamentally bug some people, but splitting worlds have the same effect on others.

I am not especially an advocate of retrocausal interpretations. They are among the possibilities; they deserve consideration and they get it. Retrocausality may or may not be an element of the real explanation of why quantum mechanics works. Progress towards the discovery of the truth requires exploration on many fronts, that's happening, we'll get there eventually. I have focused on retrocausal interpretations here just because they offer the clearest evidence that the big picture offered by the Sequence is wrong.

It's hopeless to suggest rewriting the Sequence, I don't think that would be a good use of anyone's time. But what I *would* like to have, is a clear idea of the role that "the winner is ... Many Worlds!" plays in the overall flow of argument, in the great meta-sequence that is Less Wrong's foundational text; and I would also like to have a clear idea of how to patch the argument, so that it routes around this flaw.

In the wiki, it states that "Cleaning up the old confusion about QM is used to introduce basic issues in rationality (such as the technical version of Occam's Razor), epistemology, reductionism, naturalism, and philosophy of science." So there we have it - a synopsis of the function that this Sequence is supposed to perform. Perhaps we need a working group that will identify each of the individual arguments, and come up with a substitute for each one.

## Utility functions and quantum mechanics

Interpreting quantum mechanics throws an interesting wrench into utility calculation.

Utility functions, according to the interpretation typical in these parts, are a function of the state of the world, and an agent with consistent goals acts to maximize the expected value of their utility function. Within the many-worlds interpretation (MWI) of quantum mechanics (QM), things become interesting because "the state of the world" refers to a wavefunction which contains all possibilities, merely in differing amounts. With an inherently probabilistic interpretation of QM, flipping a quantum coin *has* to be treated linearly by our rational agent - that is, when calculating expected utility, they have to average the expected utilities from each half. But if flipping a quantum coin is just an operation on the state of the world, then you can use *any function you want* when calculating expected utility.

And all coins, when you get down to it, are quantum. At the extreme, this leads to the possible rationality of quantum suicide - since you're alive in the quantum state *somewhere*, just claim that your utility function non-linearly focuses on the part where you're alive.

As you may have heard, there have been several papers in the quantum mechanics literature that claim to recover ordinary rules for calculating expected utility in MWI - how does that work?

Well, when they're not simply wrong (for example, by replacing a state labeled by the number a+b with the state |a> + |b>), they usually go about it with the Von Neumann-Morgenstern axioms, modified to refer to quantum mechanics:

- Completeness: Every state can be compared to every other, preferencewise.
- Transitivity: If you prefer |A> to |B> and |B> to |C>, you also prefer |A> to |C>.
- Continuity: If you prefer |A> to |B> and |B> to |C>, there's some quantum-mechanical measure (note that this is a change from "probability") X such that you're indifferent between (1-X)|A> + X|C> and |B>.
- Independence: If you prefer |A> to |B>, then you also prefer (1-X)|A> + X|C> to (1-X)|B> + X|C>, where |C> can be anything and X isn't 1.

In classical cases, these four axioms are easy to accept, and lead directly to utility functions with X as a probability. In quantum mechanical cases, the axioms are harder to accept, but the only measure available is indeed the ordinary amplitude-squared measure (this last fact features prominently in Everett's original paper). This gives you back the traditional rule for calculating expected utilities.

For an example of why these axioms are weird in quantum mechanics, consider the case of light. Linearly polarized light is actually the same thing as an equal superposition of right-handed and left-handed circularly polarized light. This has the interesting consequence that even when light is linearly polarized, if you shine it on atoms, those atoms will change their spins - they'll just change half right and half left. Or if you take circularly polarized light and shine it on a linear polarizer, half of it will go through. So anyhow, we can make axiom 4 read "If you are indifferent between left-polarized light and right-polarized light, then you must also be indifferent between linearly polarized light (i.e. left+right) and circularly polarized light (right+right)." But... can't a guy just want circularly polarized light?

Under what sort of conditions does the independence axiom make intuitive sense? Ones where something more complicated than a photon is being considered. Something like *you*. If MWI is correct and you measure the polarization of linearly polarized light vs. circularly polarized light, this puts your brain in a superposition of linear vs. circular. But nobody says "boy, I really want a circularly polarized *brain*."

A key factor, as is often the case when talking about recovering classical behavior from quantum mechanics, is decoherence. If you carefully prepare your brain in a circularly polarized state, and you interact with an enormous random system (like by breathing air, or emitting thermal radiation), your carefully prepared brain-state is going to get *shredded*. It's a fascinating property of quantum mechanics that once you "leak" information to the outside, things are qualitatively different. If we have a pair of entangled particles and a classical phone line, I can send you an *exact* quantum state - it's called quantum teleportation, and it's sweet. But if one of our particles leaks even the tiniest bit, even if we just end up with three particles entangled instead of two, our ability to transmit quantum states is gone completely.

In essence, the states we started with were "close together" in the space where quantum mechanics lives (Hilbert space), and so they could interact via quantum mechanics. Interacting with the outside even a little scattered our entangled particles farther apart.

Any virus, dust speck, or human being is constantly interacting with the outside world. States that are far enough apart to be perceptibly different to us aren't just "one parallel world away," like would make a good story - they are cracked wide open, spread out in the atmosphere as soon as you breathe it, spread by the Earth as soon as you push on it with your weight. If we were photons, one could easily connect with their "other selves" - if you try to change your polarization, whether you succeed or fail will depend on the orientation of your oppositely-polarized "other self"! But once you've interacted with the Earth, this quantum interference becomes negligible - so negligible that we seem to neglect it. When we make a plan, we don't worry that our nega-self might plan the opposite and we'll cancel each other out.

Does this sort of separation explain an approximate independence axiom, which is necessary for the usual rules for expected utility? Yes.

Because of decoherence, non-classical interactions are totally invisible to unaided primates, so it's expected that our morality neglects them. And if the states we are comparing are noticeably different, they're never going to interact, so independence is much more intuitive than in the case of a single photon. Taken together with the other axioms, which still make a lot of sense, this defines expected utility maximization with the Born rule.

So this is my take on utility functions in quantum mechanics - any living thing big enough to have a goal system will also be big enough to neglect interaction between noticeably different states, and thus make decisions as if the amplitude squared was a probability. With the help of technology, we can create systems where the independence axiom breaks down, but these systems are things like photons or small loops of superconducting wire, not humans.

## E.T. Jaynes and Hugh Everett - includes a previously unpublished review by Jaynes of a published short version of Everett's dissertation

E.T. Jaynes had a brief exchange of correspondence with Hugh Everett in 1957. The exchange was initiated by Everett, who commented on recently published works by Jaynes. Jaynes responded to Everett's comments, and finally sent Everett a letter reviewing a short version of Everett's thesis published that year.

Jaynes reaction was extremely positive at first: "It seems fair to say that your theory is the logical completion of quantum theory, in exactly the same sense that relativity was the logical completion of classical theory." High praise. But Jaynes swiftly follows up the praise with fundamental objections: "This is just the fundamental cause of Einstein's most serious objections to quantum theory, and it seems to me that the things that worried Einstein still cause trouble in your theory, but in an entirely new way." His letter goes on to detail his concerns, and insist, wtih Bohm, that "Einstein's objections to quantum theory have never been satisfactorily answered.

The Collected Works of Everett has some narrative about their interaction:

http://books.google.com/books?id=dowpli7i6TgC&lpg=PA261&dq=jaynes%20everett&pg=PA261#v=onepage&q&f=false

Hugh Everett marginal notes on page from E. T. Jaynes' "Information Theory and Statistical Mechanics"

http://ucispace.lib.uci.edu/handle/10575/1140

Hugh Everett handwritten draft letter to E.T. Jaynes, 15-May-1957

http://ucispace.lib.uci.edu/handle/10575/1186

Hugh Everett letter to E. T. Jaynes, 11-June-1957

http://ucispace.lib.uci.edu/handle/10575/1124

E.T. Jaynes letter to Hugh Everett, 15-October-1957 - **Never before published**

https://sites.google.com/site/etjaynesstudy/jaynes-documents/Jaynes-Everett_19571015.pdf?

Directory at Google site with all the links and docs above. Also links to Washington University at St. Louis copyright form for this doc, Everett's thesis, long and short forms, and Jaynes' paper (the papers they were discussing in their correspondence). I hope to be adding the final letter in this exchange, Jaynes to Hewitt 17-June-1957, within a couple of weeks. , and maybe some documents from the Yahoo Group ETJaynesStudy as well.

https://sites.google.com/site/etjaynesstudy/jaynes-documents

For perspective on Jaynes more recent thoughts on quantum theory:

Jaynes paper on EPR and Bell's Theorem: http://bayes.wustl.edu/etj/articles/cmystery.pdf

Jaynes speculations on quantum theory: http://bayes.wustl.edu/etj/articles/scattering.by.free.pdf

## Timeless Physics Question

Timeless physics is what you end up with if you take MWI, assume the universe is a standing wave, and remove the extraneous variables. From what I understand, for the most part you can take a standing wave and add a time-reversed version, you end up with a standing wave that only uses real numbers. The problem with this is that the universe isn't quite time symmetric.

If I ignore that complex numbers ever were used in quantum physics, it seems unlikely that complex numbers is the correct solution. Is there another one? Should I be reversing charge and parity as well as time when I make the standing real-only wave?

## Personal research update

**Synopsis:** The brain is a quantum computer and the self is a tensor factor in it - or at least, the truth lies more in that direction than in the classical direction - and we won't get Friendly AI right unless we get the ontology of consciousness right.

*Followed by: Does functionalism imply dualism? *

Sixteen months ago, I made a post seeking funding for personal research. There was no separate Discussion forum then, and the post was comprehensively downvoted. I did manage to keep going at it, full-time, for the next sixteen months. Perhaps I'll get to continue; it's for the sake of that possibility that I'll risk another breach of etiquette. You never know who's reading these words and what resources they have. Also, there has been progress.

I think the best place to start is with what orthonormal said in response to the original post: "I don't think anyone should be funding a Penrose-esque qualia mysterian to study string theory." If I now took my full agenda to someone out in the real world, they might say: "I don't think it's worth funding a study of 'the ontological problem of consciousness in the context of Friendly AI'." That's my dilemma. The pure scientists who might be interested in basic conceptual progress are not engaged with the race towards technological singularity, and the apocalyptic AI activists gathered in this place are trying to fit consciousness into an ontology that doesn't have room for it. In the end, if I have to choose between working on conventional topics in Friendly AI, and on the ontology of quantum mind theories, then I have to choose the latter, because we need to get the ontology of consciousness right, and it's possible that a breakthrough could occur in the world outside the FAI-aware subculture and filter through; but as things stand, the truth about consciousness would never be discovered by employing the methods and assumptions that prevail *inside* the FAI subculture.

Perhaps I should pause to spell out *why* the nature of consciousness matters for Friendly AI. The reason is that the value system of a Friendly AI must make reference to certain states of conscious beings - e.g. "pain is bad" - so, in order to make correct judgments in real life, at a minimum it must be able to tell which entities are people and which are not. Is an AI a person? Is a digital copy of a human person, itself a person? Is a human body with a completely prosthetic brain still a person?

I see two ways in which people concerned with FAI hope to answer such questions. One is simply to arrive at the right computational, functionalist definition of personhood. That is, we assume the paradigm according to which the mind is a computational state machine inhabiting the brain, with states that are coarse-grainings (equivalence classes) of exact microphysical states. Another physical system which admits the same coarse-graining - which embodies the same state machine at some macroscopic level, even though the microscopic details of its causality are different - is said to embody another instance of the same mind.

An example of the other way to approach this question is the idea of simulating a group of consciousness theorists for 500 subjective years, until they arrive at a consensus on the nature of consciousness. I think it's rather unlikely that anyone will ever get to solve FAI-relevant problems in that way. The level of software and hardware power implied by the capacity to do reliable whole-brain simulations means you're already on the threshold of singularity: if you can simulate whole brains, you can simulate part brains, and you can also modify the parts, optimize them with genetic algorithms, and put them together into nonhuman AI. Uploads *won't* come first.

But the idea of explaining consciousness this way, by simulating Daniel Dennett and David Chalmers until they agree, is just a cartoon version of similar but more subtle methods. What these methods have in common is that they propose to outsource the problem to a computational process using input from cognitive neuroscience. Simulating a whole human being and asking it questions is an extreme example of this (the simulation is the "computational process", and the brain scan it uses as a model is the "input from cognitive neuroscience"). A more subtle method is to have your baby AI act as an artificial neuroscientist, use its streamlined general-purpose problem-solving algorithms to make a causal model of a generic human brain, and then to somehow extract from that, the criteria which the human brain uses to identify the correct scope of the concept "person". It's similar to the idea of extrapolated volition, except that we're just extrapolating concepts.

It might sound a lot simpler to just get *human* neuroscientists to solve these questions. Humans may be individually unreliable, but they have lots of cognitive tricks - heuristics - and they *are* capable of agreeing that something is verifiably true, once one of them does stumble on the truth. The main reason one would even consider the extra complication involved in figuring out how to turn a general-purpose seed AI into an artificial neuroscientist, capable of extracting the essence of the human decision-making cognitive architecture and then reflectively idealizing it according to its own inherent criteria, is shortage of time: one wishes to develop friendly AI before someone else inadvertently develops unfriendly AI. If we stumble into a situation where a powerful self-enhancing algorithm with arbitrary utility function has been discovered, it would be desirable to have, ready to go, a schema for the discovery of a friendly utility function via such computational outsourcing.

Now, jumping ahead to a later stage of the argument, I argue that it is extremely likely that distinctively quantum processes play a fundamental role in conscious cognition, because the model of thought as distributed classical computation actually leads to an outlandish sort of dualism. If we don't concern ourselves with the merits of my argument for the moment, and just ask whether an AI neuroscientist might somehow overlook the existence of this alleged secret ingredient of the mind, in the course of its studies, I do think it's possible. The obvious noninvasive way to form state-machine models of human brains is to repeatedly scan them at maximum resolution using fMRI, and to form state-machine models of the individual voxels on the basis of this data, and then to couple these voxel-models to produce a state-machine model of the whole brain. This is a modeling protocol which assumes that everything which matters is physically localized at the voxel scale or smaller. Essentially we are asking, is it possible to mistake a quantum computer for a classical computer by performing this sort of analysis? The answer is definitely yes if the analytic process intrinsically assumes that the object under study is a classical computer. If I try to fit a set of points with a line, there will always be a line of best fit, even if the fit is absolutely terrible. So yes, one really can describe a protocol for AI neuroscience which would be unable to discover that the brain is quantum in its workings, and which would even produce a specific classical model on the basis of which it could then attempt conceptual and volitional extrapolation.

Clearly you can try to circumvent comparably wrong outcomes, by adding reality checks and second opinions to your protocol for FAI development. At a more down to earth level, these exact mistakes could also be made by *human* neuroscientists, for the exact same reasons, so it's not as if we're talking about flaws peculiar to a hypothetical "automated neuroscientist". But I don't want to go on about this forever. I think I've made the point that wrong assumptions and lax verification can lead to FAI failure. The example of mistaking a quantum computer for a classical computer may even have a neat illustrative value. But is it plausible that the brain is *actually* quantum in any significant way? Even more incredibly, is there really a valid apriori argument against functionalism regarding consciousness - the identification of consciousness with a class of computational process?

I have previously posted (here) about the way that an abstracted conception of reality, coming from scientific theory, can motivate denial that some basic appearance corresponds to reality. A perennial example is time. I hope we all agree that there is such a thing as the appearance of time, the appearance of change, the appearance of time flowing... But on this very site, there are many people who believe that reality is actually timeless, and that all these appearances are *only* appearances; that reality is fundamentally static, but that some of its fixed moments contain an illusion of dynamism.

The case against functionalism with respect to conscious states is a little more subtle, because it's not being said that consciousness is an illusion; it's just being said that consciousness is some sort of property of computational states. I argue first that this requires dualism, at least with our current physical ontology, because conscious states are replete with constituents not present in physical ontology - for example, the "qualia", an exotic name for very straightforward realities like: the shade of green appearing in the banner of this site, the feeling of the wind on your skin, really every sensation or feeling you ever had. In a world made solely of quantum fields in space, there are no such things; there are just particles and arrangements of particles. The truth of this ought to be especially clear for color, but it applies equally to everything else.

In order that this post should not be overlong, I will not argue at length here for the proposition that functionalism implies dualism, but shall proceed to the second stage of the argument, which does not seem to have appeared even in the philosophy literature. If we are going to suppose that minds and their states correspond solely to combinations of mesoscopic information-processing events like chemical and electrical signals in the brain, then there must be a mapping from possible exact microphysical states of the brain, to the corresponding mental states. Supposing we have a mapping from mental states to coarse-grained computational states, we now need a further mapping from computational states to exact microphysical states. There will of course be borderline cases. Functional states are identified by their causal roles, and there will be microphysical states which do not stably and reliably produce one output behavior or the other.

Physicists are used to talking about thermodynamic quantities like pressure and temperature as if they have an independent reality, but objectively they are just nicely behaved averages. The fundamental reality consists of innumerable particles bouncing off each other; one does not need, and one has no evidence for, the existence of a separate entity, "pressure", which exists in parallel to the detailed microphysical reality. The idea is somewhat absurd.

Yet this is analogous to the picture implied by a computational philosophy of mind (such as functionalism) applied to an atomistic physical ontology. We do know that the entities which constitute consciousness - the perceptions, thoughts, memories... which make up an experience - actually exist, and I claim it is also clear that they do not exist in any standard physical ontology. So, unless we get a very different physical ontology, we must resort to dualism. The mental entities become, inescapably, a new category of beings, distinct from those in physics, but systematically correlated with them. Except that, if they are being correlated with coarse-grained neurocomputational states which do not have an exact microphysical definition, only a functional definition, then the mental part of the new combined ontology is fatally vague. It is impossible for fundamental reality to be objectively vague; vagueness is a property of a concept or a definition, a sign that it is incomplete or that it does not need to be exact. But reality itself is necessarily exact - it is *something* - and so functionalist dualism cannot be true unless the underdetermination of the psychophysical correspondence is replaced by something which says for all possible physical states, exactly what mental states (if any) should also exist. And that inherently runs against the functionalist approach to mind.

Very few people consider themselves functionalists *and* dualists. Most functionalists think of themselves as materialists, and materialism is a monism. What I have argued is that functionalism, the existence of consciousness, and the existence of microphysical details as the fundamental physical reality, together imply a peculiar form of dualism in which microphysical states which are borderline cases with respect to functional roles must all nonetheless be assigned to precisely one computational state or the other, even if no principle tells you how to perform such an assignment. The dualist will have to suppose that an exact but arbitrary border exists in state space, between the equivalence classes.

This - not just dualism, but a dualism that is necessarily arbitrary in its fine details - is too much for me. If you want to go all Occam-Kolmogorov-Solomonoff about it, you can say that the information needed to specify those boundaries in state space is so great as to render this whole class of theories of consciousness not worth considering. Fortunately there is an alternative.

Here, in addressing this audience, I may need to undo a little of what you may think you know about quantum mechanics. Of course, the local preference is for the Many Worlds interpretation, and we've had that discussion many times. One reason Many Worlds has a grip on the imagination is that it looks easy to imagine. Back when there was just one world, we thought of it as particles arranged in space; now we have many worlds, dizzying in their number and diversity, but each *individual world* still consists of just particles arranged in space. I'm sure that's how many people think of it.

Among physicists it will be different. Physicists will have some idea of what a wavefunction is, what an operator algebra of observables is, they may even know about path integrals and the various arcane constructions employed in quantum field theory. Possibly they will understand that the Copenhagen interpretation is not about consciousness collapsing an actually existing wavefunction; it is a positivistic rationale for focusing only on measurements and not worrying about what happens in between. And perhaps we can all agree that this is inadequate, as a final description of reality. What I want to say, is that Many Worlds serves the same purpose in many physicists' minds, but is equally inadequate, though from the opposite direction. Copenhagen says the observables are real but goes misty about unmeasured reality. Many Worlds says the wavefunction is real, but goes misty about exactly how it connects to observed reality. My most frustrating discussions on this topic are with physicists who are happy to be vague about what a "world" is. It's really not so different to Copenhagen positivism, except that where Copenhagen says "we only ever see measurements, what's the problem?", Many Worlds says "I say there's an independent reality, what else is left to do?". It is very rare for a Many World theorist to seek an exact idea of what a world is, as you see Robin Hanson and maybe Eliezer Yudkowsky doing; in that regard, reading the Sequences on this site will give you an unrepresentative idea of the interpretation's status.

One of the characteristic features of quantum mechanics is entanglement. But both Copenhagen, and a Many Worlds which ontologically privileges the position basis (arrangements of particles in space), still have atomistic ontologies of the sort which will produce the "arbitrary dualism" I just described. Why not seek a quantum ontology in which there are complex natural unities - fundamental objects which aren't simple - in the form of what we would presently called entangled states? That was the motivation for the quantum monadology described in my other really unpopular post. :-) [**Edit:** Go there for a discussion of "the mind as tensor factor", mentioned at the start of this post.] Instead of saying that physical reality is a series of transitions from one arrangement of particles to the next, say it's a series of transitions from one set of entangled states to the next. Quantum mechanics does not tell us which basis, if any, is ontologically preferred. Reality as a series of transitions between overall wavefunctions which are partly factorized and partly still entangled is a possible ontology; hopefully readers who really are quantum physicists will get the gist of what I'm talking about.

I'm going to double back here and revisit the topic of how the world seems to look. Hopefully we agree, not just that there is an appearance of time flowing, but also an appearance of a self. Here I want to argue just for the bare minimum - that a moment's conscious experience consists of a set of things, events, situations... which are simultaneously "present to" or "in the awareness of" something - a conscious being - you. I'll argue for this because even this bare minimum is not acknowledged by existing materialist attempts to explain consciousness. I was recently directed to this brief talk about the idea that there's no "real you". We are given a picture of a graph whose nodes are memories, dispositions, etc., and we are told that the self is like that graph: nodes can be added, nodes can be removed, it's a purely relational composite without any persistent part. What's missing in that description is that bare minimum notion of a perceiving self. Conscious experience consists of a subject perceiving objects in certain aspects. Philosophers have discussed for centuries how best to characterize the details of this phenomenological ontology; I think the best was Edmund Husserl, and I expect his work to be extremely important in interpreting consciousness in terms of a new physical ontology. But if you can't even notice that there's an observer there, observing all those parts, then you won't get very far.

My favorite slogan for this is due to the other Jaynes, Julian Jaynes. I don't endorse his theory of consciousness at all; but while in a daydream he once said to himself, "Include the knower in the known". That sums it up perfectly. We *know* there is a "knower", an experiencing subject. We know this, just as well as we know that reality exists and that time passes. The adoption of ontologies in which these aspects of reality are regarded as unreal, as appearances as only, may be motivated by science, but it's false to the most basic facts there are, and one should show a little more imagination about what science will say when it's more advanced.

I think I've said almost all of this before. The high point of the argument is that we should look for a physical ontology in which a self exists and is a natural yet complex unity, rather than a vaguely bounded conglomerate of distinct information-processing events, because the latter leads to one of those unacceptably arbitrary dualisms. If we can find a physical ontology in which the conscious self can be identified directly with a class of object posited by the theory, we can even get away from dualism, because physical theories are mathematical and formal and make few commitments about the "inherent qualities" of things, just about their causal interactions. If we can find a physical object which is absolutely isomorphic to a conscious self, then we can turn the isomorphism into an identity, and the dualism goes away. We can't do that with a functionalist theory of consciousness, because it's a many-to-one mapping between physical and mental, not an isomorphism.

So, I've said it all before; what's new? What have I accomplished during these last sixteen months? Mostly, I learned a lot of physics. I did not originally intend to get into the details of particle physics - I thought I'd just study the ontology of, say, string theory, and then use that to think about the problem. But one thing led to another, and in particular I made progress by taking ideas that were slightly on the fringe, and trying to embed them within an orthodox framework. It was a great way to learn, and some of those fringe ideas may even turn out to be correct. It's now abundantly clear to me that I really could become a career physicist, working specifically on fundamental theory. I might even *have to* do that, it may be the best option for a day job. But what it means for the investigations detailed in this essay, is that I don't need to skip over any details of the fundamental physics. I'll be concerned with many-body interactions of biopolymer electrons *in vivo*, not particles in a collider, but an electron is still an electron, an elementary particle, and if I hope to identify the conscious state of the quantum self with certain special states from a many-electron Hilbert space, I should want to understand that Hilbert space in the deepest way available.

My only peer-reviewed publication, from many years ago, picked out pathways in the microtubule which, we speculated, might be suitable for mobile electrons. I had nothing to do with noticing those pathways; my contribution was the speculation about what sort of physical processes such pathways might underpin. Something I did notice, but never wrote about, was the unusual similarity (so I thought) between the microtubule's structure, and a model of quantum computation due to the topologist Michael Freedman: a hexagonal lattice of qubits, in which entanglement is protected against decoherence by being encoded in topological degrees of freedom. It seems clear that performing an ontological analysis of a topologically protected coherent quantum system, in the context of some comprehensive ontology ("interpretation") of quantum mechanics, is a good idea. I'm not claiming to know, by the way, that the microtubule is the locus of quantum consciousness; there are a number of possibilities; but the microtubule has been studied for many years now and there's a big literature of models... a few of which might even have biophysical plausibility.

As for the interpretation of quantum mechanics itself, these developments are highly technical, but revolutionary. A well-known, well-studied quantum field theory turns out to have a bizarre new nonlocal formulation in which collections of particles seem to be replaced by polytopes in twistor space. Methods pioneered via purely mathematical studies of this theory are already being used for real-world calculations in QCD (the theory of quarks and gluons), and I expect this new ontology of "reality as a complex of twistor polytopes" to carry across as well. I don't know which quantum interpretation will win the battle now, but this is *new information*, of utterly fundamental significance. It is precisely the sort of altered holistic viewpoint that I was groping towards when I spoke about quantum monads constituted by entanglement. So I think things are looking good, just on the pure physics side. The real job remains to show that there's such a thing as quantum neurobiology, and to connect it to something like Husserlian transcendental phenomenology of the self via the new quantum formalism.

It's when we reach a level of understanding like that, that we will truly be ready to tackle the relationship between consciousness and the new world of intelligent autonomous computation. I don't deny the enormous helpfulness of the computational perspective in understanding unconscious "thought" and information processing. And even conscious states are still *states*, so you can surely make a state-machine model of the causality of a conscious being. It's just that the reality of how consciousness, computation, and fundamental ontology are connected, is bound to be a whole lot deeper than just a stack of virtual machines in the brain. We will have to fight our way to a new perspective which subsumes and transcends the computational picture of reality as a set of causally coupled black-box state machines. It should still be possible to "port" most of the thinking about Friendly AI to this new ontology; but the differences, what's new, are liable to be crucial to success. Fortunately, it seems that new perspectives are still possible; we haven't reached Kantian cognitive closure, with no more ontological progress open to us. On the contrary, there are still lines of investigation that we've hardly begun to follow.

## Problems of the Deutsch-Wallace version of Many Worlds

The subject has already been raised in this thread, but in a clumsy fashion. So here is a fresh new thread, where we can discuss, calmly and objectively, the pros and cons of the "Oxford" version of the Many Worlds interpretation of quantum mechanics.

This version of MWI is distinguished by two propositions. First, there is no definite number of "worlds" or "branches". They have a fuzzy, vague, approximate, definition-dependent existence. Second, the probability law of quantum mechanics (the Born rule) is to be obtained, not by counting the frequencies of events in the multiverse, but by an analysis of rational behavior in the multiverse. Normally, a prescription for rational behavior is obtained by maximizing expected utility, a quantity which is calculated by averaging "probability x utility" for each possible outcome of an action. In the Oxford school's "decision-theoretic" derivation of the Born rule, we somehow *start* with a ranking of actions that is deemed rational, then we "divide out" by the utilities, and obtain probabilities that were implicit in the original ranking.

I reject the two propositions. "Worlds" or "branches" can't be vague if they are to correspond to observed reality, because vagueness results from an object being dependent on observer definition, and the local portion of reality does not owe its existence to how we define anything; and the upside-down decision-theoretic derivation, if it ever works, must implicitly smuggle in the premises of probability theory in order to obtain its original rationality ranking.

Some references:

"Decoherence and Ontology: or, How I Learned to Stop Worrying and Love FAPP" by David Wallace. In this paper, Wallace says, for example, that the question "how many branches are there?" "does not... make sense", that the question "how many branches are there in which it is sunny?" is "a question which has no answer", "it is a non-question to ask how many [worlds]", etc.

"Quantum Probability from Decision Theory?" by Barnum et al. This is a rebuttal of the original argument (due to David Deutsch) that the Born rule can be justified by an analysis of multiverse rationality.

## A case study in fooling oneself

**Note: This post assumes that the Oxford version of Many Worlds is wrong, and speculates as to why this isn't obvious. For a discussion of the hypothesis itself, see Problems of the Deutsch-Wallace version of Many Worlds. **

smk asks how many worlds are produced in a quantum process where the outcomes have unequal probabilities; Emile says there's no exact answer, just like there's no exact answer for how many ink blots are in the messy picture; Tetronian says this analogy is a great way to demonstrate what a "wrong question" is; Emile has (at this writing) 9 upvotes, and Tetronian has 7.

My thesis is that Emile has instead provided an example of how to dismiss a question and thereby fool oneself; Tetronian provides an example of treating an epistemically destructive technique of dismissal as epistemically virtuous and fruitful; and the upvotes show that this isn't just their problem. [**edit**: Emile and Tetronian respond.]

I am as tired as anyone of the debate over Many Worlds. I don't expect the general climate of opinion on this site to change except as a result of new intellectual developments in the larger world of physics and philosophy of physics, which is where the question will be decided anyway. But the mission of Less Wrong is supposed to be the refinement of rationality, and so perhaps this "case study" is of interest, not just as another opportunity to argue over the interpretation of quantum mechanics, but as an opportunity to dissect a little bit of irrationality that is not only playing out here and now, but which evidently has a base of support.

The question is not just, what's wrong with the argument, but also, how did it get that base of support? How was a situation created where one person says something irrational (or foolish, or however the problem is best understood), and a lot of other people nod in agreement and say, that's an excellent example of how to think?

On this occasion, my quarrel is not with the Many Worlds interpretation as such; it is with the version of Many Worlds which says there's no actual number of worlds. Elsewhere in the thread, someone says there are uncountably many worlds, and someone else says there are two worlds. At least those are meaningful answers (although the advocate of "two worlds" as the answer, then goes on to say that one world is "stronger" than the other, which is meaningless).

But the proposition that there is no definite number of worlds, is as foolish and self-contradictory as any of those other contortions from the history of thought that rationalists and advocates of common sense like to mock or boggle at. At times I have wondered how to place Less Wrong in the history of thought; well, this is one way to do it - it can have its own chapter in the history of intellectual folly; it can be known by its mistakes.

Then again, this "mistake" is not original to Less Wrong. It appears to be one of the defining ideas of the Oxford-based approach to Many Worlds associated with David Deutsch and David Wallace; the other defining idea being the proposal to derive probabilities from rationality, rather than vice versa. (I refer to the attempt to derive the Born rule from arguments about how to behave rationally in the multiverse.) The Oxford version of MWI seems to be very popular among thoughtful non-physicist advocates of MWI - even though I would regard *both* its defining ideas as nonsense - and it may be that its ideas get a pass here, partly because of their social status. That is, an important faction of LW opinion believes that Many Worlds is the explanation of quantum mechanics, and the Oxford school of MWI has high status and high visibility within the world of MWI advocacy, and so its ideas will receive approbation without much examination or even much understanding, because of the social and psychological mechanisms which incline people to agree with, defend, and laud their favorite authorities, even if they don't really understand what these authorities are saying or why they are saying it.

However, it is undoubtedly the case that many of the LW readers who believe there's no definite number of worlds, believe this because the idea genuinely makes sense to them. They aren't just stringing together words whose meaning isn't known, like a Taliban who recites the Quran without knowing a word of Arabic; they've actually thought about this themselves; they have gone through some subjective process as a result of which they have consciously adopted this opinion. So from the perspective of analyzing how it is that people come to hold absurd-sounding views, this should be good news. It means that we're dealing with a genuine failure to reason properly, as opposed to a simple matter of reciting slogans or affirming allegiance to a view on the basis of something other than thought.

At a guess, the thought process involved is very simple. These people have thought about the wavefunctions that appear in quantum mechanics, at whatever level of technical detail they can muster; they have decided that the components or substructures of these wavefunctions which might be identified as "worlds" or "branches" are clearly approximate entities whose definition is somewhat arbitrary or subject to convention; and so they have concluded that there's no definite number of worlds in the wavefunction. And the failure in their thinking occurs when they don't take the next step and say, is this at all consistent with reality? That is, if a quantum world is something whose existence is fuzzy and which doesn't even have a definite multiplicity - that is, we can't even say if there's one, two, or many of them - if those are the properties of a quantum world, then is it possible for the real world to be one of those? It's the failure to ask that last question, and really think about it, which must be the oversight allowing the nonsense-doctrine of "no definite number of worlds" to gain a foothold in the minds of otherwise rational people.

If this diagnosis is correct, then at some level it's a case of "treating the map as the territory" syndrome. A particular conception of the quantum-mechanical wavefunction is providing the "map" of reality, and the individual thinker is perhaps making correct statements about what's on their map, but they are failing to check the properties of the map against the properties of the territory. In this case, the property of reality that falsifies the map is, the fact that it definitely exists, or perhaps the corollary of that fact, that something which definitely exists definitely exists at least once, and therefore exists with a definite, objective multiplicity.

Trying to go further in the diagnosis, I can identify a few cognitive tendencies which may be contributing. First is the phenomenon of bundled assumptions which have never been made distinct and questioned separately. I suppose that in a few people's heads, there's a rapid movement from "science (or materialism) is correct" to "quantum mechanics is correct" to "Many Worlds is correct" to "the Oxford school of MWI is correct". If you are used to encountering all of those ideas together, it may take a while to realize that they are not linked out of logical necessity, but just contingently, by the narrowness of your own experience.

Second, it may seem that "no definite number of worlds" makes sense to an individual, because when they test their own worldview for semantic coherence, logical consistency, or empirical adequacy, it seems to pass. In the case of "no-collapse" or "no-splitting" versions of Many Worlds, it seems that it often passes the subjective making-sense test, because the individual is actually relying on ingredients borrowed from the Copenhagen interpretation. A semi-technical example would be the coefficients of a reduced density matrix. In the Copenhagen interpetation, they are probabilities. Because they have the mathematical attributes of probabilities (by this I just mean that they lie between 0 and 1), and because they can be obtained by strictly mathematical manipulations of the quantities composing the wavefunction, Many Worlds advocates tend to treat these quantities as inherently being probabilities, and use their "existence" as a way to obtain the Born probability rule from the ontology of "wavefunction yes, wavefunction collapse no". But just because something is a real number between 0 and 1, doesn't yet explain how it manages to be a probability. In particular, I would maintain that if you have a multiverse theory, in which all possibilities are actual, then a probability must refer to a *frequency*. The probability of an event in the multiverse is simply how often it occurs in the multiverse. And clearly, just having the number 0.5 associated with a particular multiverse branch is not yet the same thing as showing that the events in that branch occur half the time.

I don't have a good name for this phenomenon, but we could call it "borrowed support", in which a belief system receives support from considerations which aren't legitimately its own to claim. (Ayn Rand apparently talked about a similar notion of "borrowed concepts".)

Third, there is a possibility among people who have a capacity for highly abstract thought, to adopt an ideology, ontology, or "theory of everything" which is only expressed in those abstract terms, and to then treat that theory as the whole of reality, in a way that reifies the abstractions. This is a highly specific form of treating the map as the territory, peculiar to abstract thinkers. When someone says that reality is made of numbers, or made of computations, this is at work. In the case at hand, we're talking about a theory of physics, but the ontology of that theory is incompatible with the definiteness of one's own existence. My guess is that the main psychological factor at work here is intoxication with the feeling that one understands reality totally and in its essence. The universe has bowed to the imperial ego; one may not literally direct the stars in their courses, but one has known the essence of things. Combine that intoxication, with "borrowed support" and with the simple failure to think hard enough about where on the map the imperial ego itself might be located, and maybe you have a comprehensive explanation of how people manage to believe theories of reality which are flatly inconsistent with the most basic features of subjective experience.

I should also say something about Emile's example of the ink blots. I find it rather superficial to just say "there's no definite number of blots". To say that the number of blots depends on definition is a lot closer to being true, but that undermines the argument, because that opens the possibility that there is a right definition of "world", and many wrong definitions, and that the true number of worlds is just the number of worlds according to the right definition.

Emile's picture can be used for the opposite purpose. All we have to do is to scrutinize, more closely, what it actually is. It's a JPEG that is 314 pixels by 410 pixels in size. Each of those pixels will have an exact color coding. So clearly we can be entirely objective in the way we approach this question; all we have to do is be precise in our concepts, and engage with the genuine details of the object under discussion. Presumably the image is a scan of a physical object, but even in that case, we can be precise - it's made of atoms, they are particular atoms, we can make objective distinctions on the basis of contiguity and bonding between these atoms, and so the question will have an objective answer, if we bother to be sufficiently precise. The same goes for "worlds" or "branches" in a wavefunction. And the truly pernicious thing about this version of Many Worlds is that it prevents such inquiry. The ideology that tolerates vagueness about worlds serves to protect the proposed ontology from necessary scrutiny.

The same may be said, on a broader scale, of the practice of "dissolving a wrong question". That is a gambit which should be used sparingly and cautiously, because it easily serves to instead justify the dismissal of a legitimate question. A community trained to dismiss questions may never even *notice* the gaping holes in its belief system, because the lines of inquiry which lead towards those holes are already dismissed as invalid, undefined, unnecessary. smk came to this topic fresh, and without a head cluttered with ideas about what questions are legitimate and what questions are illegitimate, and as a result managed to ask something which more knowledgeable people had already prematurely dismissed from their own minds.

## How Many Worlds?

How many universes "branch off" from a "quantum event", and in how many of them is the cat dead vs alive, and what about non-50/50 scenarios, and please answer so that a physics dummy can maybe kind of understand?

(Is it just 1 with the live cat and 1 with the dead one?)

## [Link] New paper: "The quantum state cannot be interpreted statistically"

From a recent paper that is getting non-trivial attention...

"Quantum states are the key mathematical objects in quantum theory. It is therefore surprising that physicists have been unable to agree on what a quantum state represents. There are at least two opposing schools of thought, each almost as old as quantum theory itself. One is that a pure state is a physical property of system, much like position and momentum in classical mechanics. Another is that even a pure state has only a statistical significance, akin to a probability distribution in statistical mechanics. Here we show that, given only very mild assumptions, the statistical interpretation of the quantum state is inconsistent with the predictions of quantum theory. This result holds even in the presence of small amounts of experimental noise, and is therefore amenable to experimental test using present or near-future technology. If the predictions of quantum theory are confirmed, such a test would show that distinct quantum states must correspond to physically distinct states of reality."

From my understanding, the result works by showing how, if a quantum state is determined only statistically by some true physical state of the universe, then it is possible for us to construct clever quantum measurements that put statistical probability on outcomes for which there is literally zero quantum amplitude, which is a contradiction of Born's rule. The assumptions required are very mild, and if this is confirmed in experiment it would give a lot of justification for a phyicalist / realist interpretation of the Many Worlds point of view.

More from the paper:

"We conclude by outlining some consequences of the result. First, one motivation for the statistical view is the obvious parallel between the quantum process of instantaneous wave function collapse, and the (entirely non-mysterious) classical procedure of updating a probability distribution when new information is acquired. If the quantum state is a physical property of a system -- as it must be if one accepts the assumptions above -- then the quantum collapse must correspond to a real physical process. This is especially mysterious when two entangled systems are at separate locations, and measurement of one leads to an instantaneous collapse of the quantum state of the other.

In some versions of quantum theory, on the other hand, there is no collapse of the quantum state. In this case, after a measurement takes place, the joint quantum state of the system and measuring apparatus will contain a component corresponding to each possible macroscopic measurement outcome. This is unproblematic if the quantum state merely reflects a lack of information about which outcome occurred. But if the quantum state is a physical property of the system and apparatus, it is hard to avoid the conclusion that each marcoscopically different component has a direct counterpart in reality.

On a related, but more abstract note, the quantum state has the striking property of being an exponentially complicated object. Specifically, the number of real parameters needed to specify a quantum state is exponential in the number of systems n. This has a consequence for classical simulation of quantum systems. If a simulation is constrained by our assumptions -- that is, if it must store in memory a state for a quantum system, with independent preparations assigned uncorrelated states -- then it will need an amount of memory which is exponential in the number of quantum systems.

For these reasons and others, many will continue to hold that the quantum state is not a real object. We have shown that this is only possible if one or more of the assumptions above is dropped. More radical approaches[14] are careful to avoid associating quantum systems with any physical properties at all. The alternative is to seek physically well motivated reasons why the other two assumptions might fail."

On a related note, in one of David Deutsch's original arguments for why Many Worlds was *straightforwardly* obvious from quantum theory, he mentions Shor's quantum factoring algorithm. Essentially he asks any opponent of Many Worlds to give a real account, not just a parochial calculational account, of why the algorithm works when it is using exponentially more resources than could possibly be classically available to it. The way he put it was: "where was the number factored?"

I was never convinced that regular quantum computation could really be used to convince someone of Many Worlds who did not already believe it, except possibly for bounded-error quantum computation where one must accept the fact that there are different worlds to find one's self in after the computation, namely some of the worlds where the computation had an error due to the algorithm itself (or else one must explain the measurement problem in some different way as per usual). But I think that in light of the paper mentioned above, Deutsch's "where was the number factored" argument may deserve more credence.**Added: **Scott Aaronson discusses the paper here (the comments are also interesting).

## That cat: not dead and alive

I've read through the Quantum Physics sequence and feel that I managed to understand most of it. But now it seems to me that the Double Slit and Schrodinger's cat experiments are not described quite correctly. So I'd like to try to re-state them and see if anybody can correct any misunderstandings I likely have.

With the Double Slit experiment we usually hear it said the particle travels through both slits and then we see interference bands. The more precise explanation is that there is an complex valued amplitude flow corresponding to the particle moving through the left slit and another for the right slit. But if we could manage to magically "freeze time" then we would find ourselves in one position in configuration space where the particle is unambiguously in one position (let's say the left slit). Now any observer will have no way of knowing this at the time, and if they did detect the particle's position in any way it would change the configuration and there would be no interference banding.

But the particle really is going through the left slit right now (as far as we are concerned), simply because that is what it means to be at some point in configuration space. The particle is going through the right slit for other versions of ourselves nearby in configuration space.

The amplitude flow then continues to the point in configuration space where it arrives at the back screen, and it is joined by the amplitude flow via the right slit to the same region of configuration space, causing an interference pattern. So this present moment in time now has more than one past, now we can genuinely say that it did go through both. Both pasts are equally valid. The branching tree of amplitude flow has turned into a graph.

So far so good I hope (or perhaps I'm about to find out I'm completely wrong). Now for the cat.

I read recently that experimenters have managed to keep two clouds of caesium atoms in a coherent state for a hour. So what would this look like if we could scale it up to a cat?

The problem with this experiment is that a cat is a very complex system and the two particular types of states we are interested in (i.e. dead or alive) are very far apart in configuration space. It may help to imagine that we could rearrange configuration space a little to put all the points labelled "alive" on the left and all the dead points on the right of some line. If we want to make the gross simplification that we can treat the cat as a very simple system then this means that "alive" points are very close to the "dead" points in configuration space. In particular it means that there are significant amplitude flows between the two sets of points, that is significant flows across the line in both directions. Of course such flows happen all the time, but the key point is here the direction of the complex flow vectors would be aligned so as to cause a significant change in the magnitude of the final values in configuration space instead of tending to cancel out.

This means that as time proceeds the cat can move from alive to dead to alive to dead again, in the sense that in any point of configuration space that we find ourselves will contain an amplitude contribution both from alive states and from dead states. In other words two different pasts are contributing to the present.

So sometime after the experiment starts we magically stop the clock on the wall of the universe. Since we are at a particular point the cat is either alive or dead, let's say dead. So the cat is not alive and dead at the same time because we find ourselves at a single point in configuration space. There are also other points in the configuration space containing another instance of ourselves along with an alive cat. But since we have not entangled anything else in the universe with the cat/box system as time ticks along the cat would be buzzing around from dead to alive and back to dead again. When we open the box things entangle and we diverge far apart in configuration space, and now the cat remains completely dead or alive, at least for the point in configuration space we find ourselves in.

How to sum up? Cats and photons are never dead or alive or going left or right at the same moment from the point of view of one observer somewhere in configuration space, but the present has an amplitude contribution from multiple pasts.

If you're still reading this then thanks for hanging in there. I know there's some more detail about observations only being from a set of eigenvalues and so forth, but can I get some comments about whether I'm on the right track or way off base?

## Schroedinger's cat is always dead

Suppose you believe in the Copenhagen interpretation of quantum mechanics. Schroedinger puts his cat in a box, with a device that has a 50% chance of releasing a deathly poisonous gas. He will then open the box, and observe a live or dead cat, collapsing that waveform.

But Schroedinger's cat is lazy, and spends most of its time sleeping. Schroedinger is a pessimist, or else an optimist who hates cats; and so he mistakes a sleeping cat for a dead cat with probability P(M) > 0, but never mistakes a dead cat for a living cat.

So if the cat is dead with probability P(D) >= .5, Schroedinger *observes* a dead cat with probability P(D) + P(M)(1-P(D)).

If observing a dead cat causes the waveform to collapse such that the cat is dead, then P(D) = P(D) + P(M)(1-P(D)). This is possible only if P(D) = 1.

## Does quantum mechanics make simulations negligible?

I've written a prior post about how I think that the Everett branching factor of reality dominates that of any plausible simulation, whether the latter is run on a Von Neumann machine, on a quantum machine, or on some hybrid; and thus the probability and utility weight that should be assigned to simulations in general is negligible. I also argued that the fact that we live in an apparently quantum-branching world could be construed as weak anthropic evidence for this idea. My prior post was down-modded into oblivion for reasons that are not relevant here (style, etc.) If I were to replace this text you're reading with a version of that idea which was more fully-argued, but still stylistically-neutral (unlike my prior post), would people be interested?

## Polarized gamma rays and manifest infinity

Most people (not all, but most) are reasonably comfortable with infinity as an ultimate (lack of) limit. For example, cosmological theories that suggest the universe is infinitely large and/or infinitely old, are not strongly disbelieved a priori.

By contrast, most people are fairly uncomfortable with *manifest* infinity, actual infinite quantities showing up in physical objects. For example, we tend to be skeptical of theories that would allow infinite amounts of matter, energy or computation in a finite volume of spacetime.

## 2011 Buhl Lecture, Scott Aaronson on Quantum Complexity

I was planning to post this in the main area, but my thoughts are significantly less well-formed than I thought they were. Anyway, I hope that interested parties find it nonetheless.

In the Carnegie Mellon 2011 Buhl Lecture, Scott Aaronson gives a remarkably clear and concise review of P, NP, other fundamentals in complexity theory, and their quantum extensions. In particular, beginning around the 46 minute mark, a sequence of examples is given in which the intuition from computability theory would have accurately predicted physical results (and in some cases this actually happened, so it wasn't just hindsight bias).

In previous posts we have learned about Einstein's arrogance and Einstein's speed. This pattern of results flowing from computational complexity to physical predictions seems odd to me in that context. Here we are using physical computers to derive abstractions about the limits of computation, and from there we are successfully able to intuit limits of physical computation (e.g. brains computing abstractions of the fundamental limits of brains computing abstractions...) At what point do we hit the stage where individual scientists can rationally know that results from computational complexity theory are more fundamental than traditional physics? It seems like a paradox wholly different than Einstein rationally knowing (from examining bits of theory-space evidence rather than traditional-experiment-space evidence) that relativity would hold true. In what sort of evidence space can physical brain computation yielding complexity limits count as bits of evidence factoring into expected physical outcomes (such as the exponential smallness of the spectral gap of NP-hard-Hamiltonians from the quantum adiabatic theorem)?

Maybe some contributors more well-versed in complexity theory can steer this in a useful direction.

## Quantum computing for the determined (Link)

21 videos, which cover subjects including the basic model of quantum computing, entanglement, superdense coding, and quantum teleportation.

To work through the videos you need to be comfortable with basic linear algebra, and with assimilating new mathematical terminology. If you’re not, working through the videos will be arduous at best! Apart from that background, the main prerequisite is determination, and the willingness to work more than once over material you don’t fully understand.

In particular, you don’t need a background in quantum mechanics to follow the videos.

The videos are short, from 5-15 minutes, and each video focuses on explaining one main concept from quantum mechanics or quantum computing. In taking this approach I was inspired by the excellent Khan Academy.

**Link:** michaelnielsen.org/blog/quantum-computing-for-the-determined/

**Author:** Michael Nielsen

### The basics

- The qubit
- Tips for working with qubits
- Our first quantum gate: the quantum NOT gate
- The Hadamard gate
- Measuring a qubit
- General single-qubit gates
- Why unitaries are the only matrices which preserve length
- Examples of single-qubit quantum gates
- The controlled-NOT gate
- Universal quantum computation

### Superdense coding

- Superdense coding: how to send two bits using one qubit
- Preparing the Bell state
- What’s so special about entangled states anyway?
- Distinguishing quantum states
- Superdense coding redux: putting it all together

### Quantum teleportation

- Partial measurements
- Partial measurements in an arbitrary basis
- Quantum teleportation
- Quantum teleportation: discussion

### The postulates of quantum mechanics (TBC)

## States of knowledge as amplitude configurations

I am reading through the sequence on quantum physics and have had some questions which I am sure have been thought about by far more qualified people. If you have any useful comments or links about these ideas, please share.

Most of the strongest resistance to ideas about rationalism that I encounter comes not from people with religious beliefs per se, but usually from mathematicians or philosophers who want to assert arguments about the limits of knowledge, the fidelity of sensory perception as a means for gaining knowledge, and various (what I consider to be) pathological examples (such as the zombie example). Among other things, people tend to reduce the argument to the existence of proper names a la Wittgenstein and then go on to assert that the meaning of mathematics or mathematical proofs constitutes something which is fundamentally not part of the physical world.

As I am reading the quantum physics sequence (keep in mind that I am not a physicist; I am an applied mathematician and statistician and so the mathematical framework of Hilbert spaces and amplitude configurations makes vastly much more sense to me than billiard balls or waves, yet connecting it to reality is still very hard for me) I am struck by the thought that all thoughts are themselves fundamentally just amplitude configurations, and by extension, all claims about knowledge about things are also statements about amplitude configurations. For example, my view is that the color red does not exist in and of itself but rather that the experience of the color red is a statement about common configurations of particle amplitudes. When I say "that sign is red", one could unpack this into a detailed statement about statistical properties of configurations of particles in my brain.

The same reasoning seems to apply just as well to something like group theory. States of knowledge about the Sylow theorems, just as an example, would be properties of particle amplitude configurations in a brain. The Sylow theorems are not separately existing entities which are of themselves "true" in any sense.

Perhaps I am way off base in thinking this way. Can any philosophers of the mind point me in the right direction to read more about this?

## the Universe, Computability, and the Singularity

EDIT at Karma -5: Could the next "good citizen" to vote this down leave me a comment as to why it is getting voted down, and if other "good citizens" to pile on after that, either upvote that comment or put another comment giving your different reason?

Original Post:

Questions about the computability of various physical laws recently had me thinking: "well of course every real physical law is computable or else the universe couldn't function." That is to say that in order of the time-evolution of anything in the universe to proceed "correctly," the physical processes themselves must be able to, and in real-time, keep up with the complexity of their actual evolution. This seems to me a proof that every real physical process is computable by SOME sort of real computer, in the degenerate case that real computer is simply an actual physical model of the process itself, create that model, observe whichever features of its time-evolution you are trying to compute, and there you have your computer.

Then if we have a physical law whose use in predicting time evolution is provably uncomputable, either we know that this physical law is NOT the only law that might be formulated to describe what it is purporting to describe, or that our theory of computation is incomplete. In some sense what I am saying is consistent with the idea that quantum computing can quickly collapse down to plausibly tractable levels the time it takes to compute some things which, as classical computation problems, blow up. This would be a good indication that quantum is an important theory about the universe, that it not only explains a bunch of things that happen in the universe, but also explains how the universe can have those things happen in real-time without making mistakes.

What I am wondering is, where does this kind of consideration break with traditional computability theory? Is traditional computability theory limited to what Turing machines can do, while perhaps it is straightforward to prove that the operation of this Universe requires computation beyond what Turing machines can do? Is traditional computability theory limited to digital representations whereas the degenerate build-it-and-measure-it computer is what has been known as an analog computer? Is there somehow a level or measure of artificiality which must be present to call something a computer, which rules out such brute-force approaches as build-it-and-measure-it?

At least one imagining of the singularity is absorbing all the resources of the universe into some maximal intelligence, the (possibly asymptotic) endpoint of intelligences desiging greater intelligences until something makes them stop. But the universe is already just humming along like clockwork, with quantum and possibly even subtler-than-quantum gears turning in real time. What does the singularity add to this picture that isn't already there?

## Quantum Joint Configuration article: need help from physicists

EDIT: 1:19 PM PST 22 December 2010 I completed this post. I didn't realize an uncompleted version was already posted earlier.

I wanted to read the quantum sequence because I've been intrigued by the nature of measurement throughout my physics career. I was happy to see that articles such as joint configuration use beams of photons and half and fully silvered mirrors to make its points. I spent years in graduate school working with a two-path interferometer with one moving mirror which we used to make spectrometric measurements on materials and detectors. I studied the quantization of the electromagnetic field, reading and rereading books such as Yariv's Quantum Electronics and Marcuse's Principles of Quantum Electronics. I developed with my friend David Woody a photodetector ttheory of extremely sensitive heterodyne mixers which explained the mysterious noise floor of these devices in terms of the shot noise from detecting the stream of photons which are the "Local Oscillator" of that mixer.

My point being that I AM a physicist, and I am even a physicist who has worked with the kinds of configurations shown in this blog post, both experimentally and theoretically. I did all this work 20 years ago and have been away from any kind of Quantum optics stuff for 15 years, but I don't think that is what is holding me back here.

So when I read and reread the joint configuration blog post, I am concerned that it makes absolutely no sense to me. I am hoping that someone out there DOES understand this article and can help me understand it. Someone who understands the more traditional kinds of interferometer configurations such as that described for example here and could help put this joint configuration blog post in terms that relate it to this more usual interferometer situation.

I'd be happy to be referred to this discussion if it has already taken place somewhere. Or I'd be happy to try it in comments to this discussion post. Or I'd be happy to talk to someone on the phone or in primate email, if you are that person email me at mwengler at gmail dot com.

To give you an idea of the kinds of things I think would help:

1) How might you build that experiment? Two photons coming in from right angles could be two radio sources at the same frequency and amplitude but possibly different phase as they hit the mirror. In that case, we get a stream of photons to detector 1 proportional to sin(phi+pi/4)^2 and a stream of photons to detector 2 proportional to cos(phi+pi/4)^2 where phi is the phase difference of the two waves as they hit the mirror, and I have not attempted to get the sign of the pi/4 term right to match the exact picture. Are they two thermal sources? In which case we get random phases at the mirror and the photons split pretty randomly between detector 1 and detector 2, but there are no 2-photon correlations, it is just single photon statistics.

2) The half-silvered mirror is a linear device: two photons passing through it do not interact with each other. So any statistical effect correlating the two photons (that is, they must either both go to detector 1 or both go to detector 2, but we will never see one go to 1 and the other go to 2) must be due to something going in the source of the photons. Tell me what the source of these photons is that gives this gedanken effect.

3) The two-photon aspect of the statistical prediction of this seems at least vaguely EPR-ish. But in EPR the correlations of two photons come about because both photons originate from a single process, if I recall correctly. Is this intending to look EPRish, but somehow leaving out some necessary features of the source of the two photons to get the correlation involved?

I remaing quite puzzled and look forward to anything anybody can tell me to relate the example given here to anything else in quantum optics or interferometers that I might already have some knowledge of.

Thanks,

Mike

## Help: Is there a quick and dirty way to explain quantum immortality?

I had an incredibly frustrating conversation this morning trying to explain the idea of quantum immortality to someone whose understanding of MWI begins and ends at pop sci fi movies. I think I've identified the main issue that I wasn't covering in enough depth (continuity of identity between near-identical realities) but I was wondering whether anyone has ever faced this problem before, and whether anyone has (or knows where to find) a canned 5 minute explanation of it.

## Deep Structure Determinism

Sort of a response to: Collapse Postulate

Abstract: There are phenomena in mathematics where certain structures are distributed "at random;" that is, statistical statements can be made and probabilities can be used to predict the outcomes of certain totally deterministic calculations. These calculations have a deep underlying structure which leads a whole class of problems to behave in the same way statistically, in a way that appears random, while being entirely deterministic. If quantum probabilities worked in this way, it would not require collapse or superposition.This is a post about physics, and I am not a physicist. I will reference a few technical details from my (extremely limited) research in mathematical physics, but they are not necessary to the fundamental concept. I am sure that I have seen similar ideas somewhere in the comments before, but searching the site for "random + determinism" didn't turn much up so if anyone recognizes it I would like to see other posts on the subject. However my primary purpose here is to expose the name "Deep Structure Determinism" that jasonmcdowell used for it when I explained it to him on the ride back from the Berkeley Meetup yesterday.

Again I am not a physicist; it could be that there is a one or two sentence explanation for why this is a useless theory--of course that won't stop the name "Deep Structure Determinism" from being aesthetically pleasing and appropriate.

For my undergraduate thesis in mathematics, I collected numerical evidence for a generalization of the Sato-Tate Conjecture. The conjecture states, roughly, that if you take the right set of polynomials, compute the number of solutions to them over finite fields, and scale by a consistent factor, these results will have a probability distribution that is precisely a semicircle.

The reason that this is the case has something to do with the solutions being symmetric (in the way that y=x^{2} if and only if y=(-x)^{2} is a symmetry of the first equation) and their group of symmetries being a circle. And stepping back one step, the conjecture more properly states that the numbers of solutions will be roots of a certain polynomial which will be the minimal polynomial of a random matrix in SU_{2}.

That is at least as far as I follow the mathematics, if not further. However, it's far enough for me to stop and do a double take.

A "random matrix?" First, what does it mean for a matrix to be random? And given that I am writing up a totally deterministic process to feed into a computer, how can you say that the matrix is random?

A sequence of matrices is called "random" if when you integrate of that sequence, your integral converges to integrating over an entire group of matrices. Because matrix groups are often smooth manifolds they are designed to be integrated over, and this ends up being sensible. However a more practical characterization, and one that I used in the writeup for my thesis, is that if you take a histogram of the points you are measuring, the histogram's shape should converge to the shape of the group--that is, if you're looking at the matrices that determine a circle, your histogram should look more and more like a semicircle as you do more computing. That is, you can have a probability distribution over the matrix space for where your matrix is likely to show up.

The actual computation that I did involved computing solutions to a polynomial equation--a trivial and highly deterministic procedure. I then scaled them, and stuck them in place. If I had not know that these numbers were each coming from a specific equation I would have said that they were random; they jumped around through the possibilities, but they were concentrated around the areas of higher probability.

So bringing this back to quantum physics: I am given to understand that quantum mechanics involves a lot of random matrices. These random matrices give the impression of being "random" in that it seems like there are lots of possibilities, and one must get "chosen" at the end of the day. One simple way to deal with this is postulate many worlds, wherein there no one choice has a special status.

However my experience with random matrices suggests that there could just be some series of matrices, which satisfies the definition of being random, but which is inherently determined (in the way that the Jacobian of a given elliptic curve is "determined.") If all quantum random matrices were selected from this list, it would leave us with the subjective experience of randomness, and given that this sort of computation may not be compressible, the expectation of dealing with these variables as though they are random forever. It would also leave us in a purely deterministic world, which does not branch, which could easily be linear, unitary, differentiable, local, symmetric, and slower-than-light.

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