
Quantum mechanics as a generalization of probability (2007) - ascertain
http://www.scottaaronson.com/democritus/lec9.html
======
yummyfajitas
I'd disagree with the title of this a bit. Rather than describing it as
"probability to allow minus signs", I'd describe it as "probability in L^2
instead of L^1". The author actually discusses this a bit towards the end.

It makes sense to ask what is the distance between two continuous probability
distributions. It's given by:

\int | p1(x) - p2(x) | dx

L^1 (the space of all functions for which \int |f(x)| dx < \infty) is a weird
space, and does not admit concepts like "what is the angle between two
vectors".

Quantum mechanics changes this to:

\int |p1(x) - p2(x)|^2 dx

Functions like this are called L^2. Once you put the square in, you can
immediately derive a lot of geometry, inner products, angles between vectors,
etc.

So I'd argue that QM is probability in L^2.

~~~
tagrun
It is L^2 for infinite dimensional Hilbert spaces like position or momentum,
where quantum states become functions. For finite dimensional Hilbert spaces
such as spins, quantum state is a complex vector rather than a function, and
the "braket" is not an integral but vector product.

But I don't think that's even the point. The whole article is a
_rationalization_ of basic rules of quantum mechanics; when I look back at the
history of physics, rationalization of known --at the time-- physical laws
(which are often replaced by "better laws" after a while when get a better we
understanding of nature) is most often counter-productive.

~~~
yummyfajitas
It's L^2 in finite dimensional spaces also - you can just as well define || v
|| = \int |v_i| dC(i) (where C is the counting measure). Finite dimensional
spaces can also be interpreted as functions, i.e. C^3 is the same as {1,2,3}
=> C.

This is a rationalization but it's useful. We have a physical theory in L^2,
and we know that you can't have such a theory in L^2.1 or L^7. Our theories
might be totally wrong. But we are confident they can't be only a little
wrong.

~~~
tagrun
> It's L^2 in finite dimensional spaces also

The concern of "is this thing square-integrable" arises only in infinite
dimensional spaces. Inner product of all $\mathbb C^N$ where N in finite _is_
finite. There is no point in saying "L^2" when talking about spins etc.
(forcefully casting a finite sum of finite terms to integration and then
introducing the concept of "square integrable" isn't helpful, if not trivial).

> This is a rationalization but it's useful.

You can do that kind of rationalization when you "know" the answer ahead. It
has zero prediction power.

Newton had a rationalization about the independent nature of time. Descartes
also had a rationalization about how action-at-a-distance works for gravity.
Aether was also a popular rationalization once. None of those rationalizations
held any value at the end.

Such "rationalizations" are better cut by Occam's razor and left for
philosophers.

At least, I prefer not waste my time with such stuff as a physicist.

~~~
namlede
I also disagree; the article's rationalization is productive.

Your mention of Occam's razor is key. The point of this rationalization is
that it shows quantum mechanics (or at least a subset of the theory) is
mathematically simple, thus serving a threefold purpose:

1) Due to Occam's razor, a simpler theory is more likely to be correct than a
more complicated theory with the same explanatory power. If quantum mechanics
can be expressed more simply, then the estimated probability of its
correctness should be increased (by only a slight amount).

2) Quantum mechanics, explained this way, is interesting from a purely
mathematical standpoint. Even if we knew quantum mechanics doesn't describe
reality, mathematicians (and theoretical computer scientists like Aaronson)
would perhaps still investigate it. Of course, this may not interest
physicists.

3) The article presents a novel way to teach quantum mechanics. Mathematical
simplicity can (to some extent) replace intuition as a way for learners to
grasp the theory. As Aaronson remarks, quantum mechanics is often taught by
following the historical order in which the ideas discovered. Starting from
the "conceptual core" (if Aaronson is correct about the conceptual core) is
arguably a superior pedagogical technique.

~~~
tjradcliffe
The problem with Occam's razor is that there is no generally agreed-upon
notion of simplicity. Occam himself believed it proved the existence of God,
and only God. After all, what did you need material reality for when it could
be "explained" as God's dreams and imaginings? To Occam that was far "simpler"
than the messy reality of matter, and the messy reality we know of today would
have been anathema to him.

"Simpler" theories are not ab initio more likely correct. The world is full of
"simpler" theories that are wrong: the four elements, the caloric theory of
heat, Newtonian dynamics and gravity, and so on.

"Simpler" is a purely human notion, and a heavily culturally laden one at
that. For this reason, Occam's razor is best left in the dust-heap of
philosophy. It is never useful in doing actual science, except now and then by
accident.

~~~
TeMPOraL
Occam's razor doesn't say "pick the simplest theory", it says "pick the
simplest theory out of all that agree with all discussed experimental
evidence". We didn't throw away four elements because they weren't simple
enough; we did it because that theory didn't explain the things we saw.

As for the notion of simplicity itself, we have Kolmogorov complexity and
Solomonoff induction, they capture the essence pretty well (though I heard
that with some caveats).

------
richardjordan
Per the footnotes at the bottom this builds on the work of Hardy a few years
ago, who demonstrated that QM could easily have been derived by Victorian era
mathematicians as a generalization of probability theory available at the
time.

While not everything in the book is perfect, it also reminds me of Penrose's
Road to Reality, which painstakingly takes the question "what is a number?" to
build up a sophisticated understanding of what that means and then explains
how this impacts Physics in a very real way.

We've made so many advances in recent decades we sometimes forget to step back
and reformulate our teaching methodologies to incorporate what we know,
simplify the teaching, and make the ideas more accessible earlier, so we can
put more minds to work on extending them. That's a shame.

~~~
richardjordan
...also regarding his final point about the speed of light - of course special
relativity which explains this is similarly a generalization of older theories
such as Galilean relativity.

If you look at the work of Feigenbaum - e.g.
[http://arxiv.org/abs/0806.1234](http://arxiv.org/abs/0806.1234) \- you'll see
how similar to Hardy deriving QM purely by continuing older lines of thought,
the fact the universe has a maximum possible speed that anything can travel -
the speed of light (though it's not strictly about light) can be derived from
postulates as old as Galilean relativity. The speed of light cannot be
infinite - it must have some value.

~~~
lohankin
Much simpler derivation of Lorentz transform:
[http://www.jackpenkethman.com/speedoflight/speedoflight.html](http://www.jackpenkethman.com/speedoflight/speedoflight.html)

~~~
theoh
Is this the same argument that Minkowksi came up with in 1908, describing it
as "staircase wit" (as it would have been impressive had it been thought of
before Einstein's theory)?

------
Totient
I love this description of quantum. The paper Aaronson links to, "Quantum
Theory from Five Reasonable Axioms" ([http://arxiv.org/abs/quant-
ph/0101012](http://arxiv.org/abs/quant-ph/0101012)), is also a great read.

The "probability in L^2" cleared up a lot of confusion for me, although I
still have a very poor intuition for what the Born probabilities are
probabilities _of_. If you believe the MWI story, it seems like it's the
probability you will "find yourself in the universe where this outcome
happened" but even that sounds odd to me.

~~~
tim333
>I still have a very poor intuition for what the Born probabilities are
probabilities of

I'm not sure anyone knows at a deep level. Experimentally you can count how
many particles go a given way and it matches the calculation but how that
actually works I think remains a mystery.

------
antognini
There's an interesting article on negative probabilities by Gábor Székely in
which he shows that they can be generate by flipping a "half coin" \-- a coin
that, if flipped twice, gives the result from a single flip of an ordinary
coin.

[http://wilmott.com/pdfs/100609_gjs.pdf](http://wilmott.com/pdfs/100609_gjs.pdf)

------
javajosh
FYI Scott Aaronson wrote _Quantum Computing since Democritus_ [1] and it is an
amazing book. Highly recommended for anyone interested in math, physics,
theoretical computer science, quantum computing or just good science writing.

[1] [http://www.amazon.com/Quantum-Computing-since-Democritus-
Aar...](http://www.amazon.com/Quantum-Computing-since-Democritus-
Aaronson/dp/0521199565)

------
njackson5dW
As other people pointed out, it is probability using the L2 norm versus the
standard probability that preserves the L1 norm. This means the following:
Standard probability distributions are vectors that sum to 1. I.e. x=[0.5 0.2
0.3] is a probability distribution because each x(i)>=0 and sum x(i)=1. This
means that the l1 norm of distribution vectors is always 1.

The L2 norm of a vector is the square root of the sum of the entries squared.
Hence a distribution would be a vector x(i) such that sum x(i)^2 is one. This
allows x(i) to have both negative and even imaginary numbers.

This awesome talk by Scott
[http://www.scottaaronson.com/blog/?p=1345](http://www.scottaaronson.com/blog/?p=1345)
explains the difference between L1 and L2 probability using the Latke vs
Hamentaschen Farsical debate. See the video from the link.

------
drhodes
BTW, Stanford is offering an Intro to QM through their OpenEdx platform
starting Sep 30th.

[https://class.stanford.edu/courses/Engineering/QMSE-01/Fall2...](https://class.stanford.edu/courses/Engineering/QMSE-01/Fall2014/about)

------
carlob
Saying that most quantum physics courses are taught from the historical point
of view rather than starting with the math seems like a bit of a straw man.

My experience (Rome University) was that first we were taught the basic
mathematics (Hilbert spaces, functional analysis) for almost a year, before
being introduced to the physics which followed a rather axiomatic (not
dissimilar to this).

Sometimes our professor would casually drop bombs like the title of this
article or things like: "you know in the end quantum mechanics is just Markov
chains in imaginary time", but even those made sense in time (Wick rotation).

~~~
coliveira
The author probably has no experience with modern physics education. He
probably met somebody who learned these things 50 years ago and keeps
repeating this unfounded notion.

~~~
wetmore
Scott Aaronson teaches at MIT... I'm sure he has some experience with modern
physics education.

~~~
coliveira
He teaches CS, not Physics. What they are teaching in the CS department has
nothing to do with what physicists learn.

~~~
TeMPOraL
> _What they are teaching in the CS department has nothing to do with what
> physicists learn._

This line of thinking is a huge problem in science and the reason Scott has to
write articles like the one discussed. "Yeah, so those computer guys are so
full of themselves that they dare say they know something about how the
universe works". Well, the thing is, they actually do.

------
danohuiginn
picking up on a tangent in that article:

<i>Two other perfect examples of "obvious-in-retrospect" theories are
evolution and special relativity. Admittedly, I don't know if the ancient
Greeks, sitting around in their togas, could have figured out that these
theories were true. But certainly -- certainly! -- they could've figured out
that they were possibly true: that they're powerful principles that would've
at least been on God's whiteboard when She was brainstorming the world.</i>

To me, evolution is a perfect example of the need for practical knowledge.
Darwinian and Lamarckian evolution are both absolutely reasonable theories;
it's just that only one of them matches the world we live in.

~~~
duckingtest
Actually, both. Heritable epigenetics is Lamarckian evolution.

------
dmazin
Feynman also wrote about negative probability and quantum mechanics:
[http://cds.cern.ch/record/154856/files/pre-27827.pdf](http://cds.cern.ch/record/154856/files/pre-27827.pdf)

~~~
tagrun
Even the best minds were not always perfect (at some point, he thought
positron could as well be thought as an electron going backward in time
---which we today know isn't the case).

This is a quite similar story: ghosts have the wrong sign for the kinetic
term:
[http://en.wikipedia.org/wiki/Faddeev%E2%80%93Popov_ghost](http://en.wikipedia.org/wiki/Faddeev%E2%80%93Popov_ghost)

Note that he of course didn't try to introduce negative probabilities as basic
things.

In both cases it was just a matter of interpretation --just with "bad"
physical consequences.

------
infogulch
I like the tone.

> Basically, quantum mechanics is the operating system that other physical
> theories run on as application software (with the exception of general
> relativity, which hasn't yet been successfully ported to this particular
> OS).

~~~
themgt
Yeah, it's absolutely nuts to hear that the copenhagen interpretation is still
taught in universities, because it's such a desperate attempt to avoid
admitting this truth of how fundamental QM is to reality.

~~~
_delirium
The Copenhagen interpretation is no longer dominant among physicists, but I
wouldn't say it's widely agreed that QM is "fundamental to reality", at least
not in a strong philosophical sense of that phrase. Plenty hold a position
along the lines of: QM is a mathematical model that agrees with experiment, so
far.

~~~
xenophonf
_Every_ scientific theory is a model that agrees with experiment, so far.

~~~
_delirium
That is one position, yes, although surprisingly not all that common among
working scientists, because it's often seen as too skeptical of a position.
The strong version, "instrumentalism", boils down to roughly: science does not
discover any "truths" about the universe, but rather is just a process of
building mathematical models that correctly predict observations. And we can
say little else about these models except that they predict experimental data
accurately so far. In particular, in this view, we cannot say that any
components of the mathematical models are necessarily physically "real" or in
any strong sense "explain" reality, merely that they correctly predict
observed regularities.

Despite not being that popular a view among scientists generally, it _is_
however a fairly popular view among quantum physicists, many of whom aren't
that willing to commit to the physical reality of a good deal of the
mathematical apparatus of QM.

~~~
TheOtherHobbes
"A simplified but useful picture of the goal of scientific research is that
scientists obtain large amounts of data about the world via observation and
experiment, and then try to find regularities and patterns in that data.

But a regularity or pattern is nothing more or less than a method for
_compressing_ the data: if a particular pattern shows up in many places in a
data set, then we can create a compressed version of the data by describing
the pattern only once, and then specifying the different places that the
pattern shows up. The most compressed version of the data is in some sense the
ultimate scientific description.

There is a sense in which the goal of all science is finding theories that
provide ever more concise descriptions of data."

[http://arxiv.org/pdf/1312.4456v1.pdf](http://arxiv.org/pdf/1312.4456v1.pdf)

~~~
jxjdjr
This. Also like to add that when people talk about a theory being nice or
natural, they are saying that they find it easy to compress because it looks
in some respects like theories they already know.

------
mrestko
I don't have the math to understand this. If I wanted to, what should I seek
out on Coursera or similar?

~~~
Energy1
Linear Algebra -> Real Analysis -> Functional Analysis.

Linear Algebra and Analysis can be either applied or pure and of many
different levels of sophistication.

------
Scene_Cast2
For reading up on p-norms related to the article, I found the wikipedia page
on Lp spaces[1] to be fairly accessible as a learning tool.

Not sure if it's my learning style, but most math-related wikipedia pages read
as a reference and seem to assume prior knowledge (especially with notation).

[1]
[http://en.wikipedia.org/wiki/Lp_space](http://en.wikipedia.org/wiki/Lp_space)

~~~
coliveira
> most math-related wikipedia pages read as a reference

This is by design, which is why it is called an encyclopedia. The fact that it
assumes prior knowledge just means that you need to refer to other entries
when in doubt.

~~~
jev
Nope, because the math articles are mainly made up of circular references
which only use generalized, abstract examples to illustrate.

------
infinity0
"As a direct result of this "QWERTY" approach to explaining quantum mechanics
- which you can see reflected in almost every popular book and article, down
to the present -- the subject acquired an undeserved reputation for being
hard."

The same goes for cryptography. Most cryptography courses spend at least the
first hour talking about historical irrelevance like substitution ciphers etc.
Crypto I [1] (Dan Boneh) follows the latter approach, i.e. starting from
modern theoretical principles, defining security properties in terms of
computational complexity and games.

I quite liked the Quantum Computing course [2] (Anuj Dawar) from the Cambridge
CST, which also followed that approach, though it didn't present this stuff as
a "generalisation of probability". No-cloning theorem in 3rd or 4th lecture,
IIRC.

edit: After reading this article fully, I think it would have made for a good
"lecture 0" in the above course, bridging the gap between more elementary
maths and it.

[1] [http://coursera.org/course/crypto](http://coursera.org/course/crypto) [2]
[https://www.cl.cam.ac.uk/teaching/1415/QuantComp/](https://www.cl.cam.ac.uk/teaching/1415/QuantComp/)

------
kazinator
> _We 've talked about why the amplitudes should be complex numbers, and why
> the rule for converting amplitudes to probabilities should be a squaring
> rule._

The squaring rule is actually a special case of multiplying a number by its
complex conjugate, which the article doesn't mention, unfortunately.

That is to say, if we have a number z = x + iy, we can obtain its norm from
sqrt(x _x + y_ y). But another way to express this is simply sqrt(z z _). The
product z z_ is just (x + iy)(x - iy). That of course is just x^2 - (iy)^2
which goes to x^2 - (-1y^2) -> x^2 + y^2.

Geometrically, the conjugate of a complex number has the opposite angle. If z
is 20 degrees from the real axis, z* is -20 degrees. Since multiplication of
complex numbers is additions of their arguments (i.e. angle components), the
two cancel out and the result is on the real number line.

------
TeMPOraL
Between Scott Aaronson and Eliezer Yudkowsky, who both wrote articles
explaining QM "directly from the conceptual core", is there any textbook that
follows this path further and with a lot more of details?

~~~
selimthegrim
John Bell's book of lecture notes, "Speakable and Unspeakable in Quantum
Mechanics"

------
malexw
I'm having trouble understanding his explanation of interference. I understand
applying the 45 degree counter-clockwise rotation twice would transform the
qubit from |0> to |1>. I don't understand how this implies that there are two
paths to state |0>. How could those two rotations could get you anywhere
besides |1>?

~~~
gizmo686
> How could those two rotations could get you anywhere besides |1>

Obviously they can't, which is why the math shows that they don't.

After the first rotation, you are the state (|0>+|1>)/sqrt(2). The physical
interpretation of this this state is that it represents a 50% chance of being
in |0>, and a 50% chance of being in |1>. If you apply this rotation to either
of those two possibilities, you arrive back at (|0>+|1>)/sqrt(2), which still
has a 50% chance of being in the state |0>. The two paths leading to this are
when the intermediate state is |0> or |1>.

When you actually do the math (in which "rotation" is just a name we give to
multiplying by a unitary matrix, U". You find that you end up in the state
(.5-.5)|0>+(.5+.5)|1> = .5|0>-.5|0>+.5|1>+.5|1> = |1>.

Here, we can again see the semblance of 2 paths leading to zero (the two |0>
terms), however they have opposite signs, so cancel out.

~~~
malexw
Aha, thanks to you and ufo for the explanation. I think the problem was that I
was thinking about it in complex exponential notation. If I had done it in
matrix form, the cancellation of the two amplitudes of the |0> state would
have been much more obvious.

------
javajosh
I'll play! What necessitates the requirement that probability amplitude varies
continuously? (This requirement is assumed in the section "Real vs. Complex
Numbers", the so-called "continuity assumption").

------
grondilu
I was a bit bugged by the section about the density matrix. He writes:

> Then you compute the outer product of the vector with itself

I'm not sure what he means by outer product here. Isn't the outer product of a
vector by itself always nul?

~~~
nshepperd
The outer product of vectors `u, v` is the matrix `A_ij = u_i · v_j`. That is,
a matrix containing the product of each component of u with each component of
v. You may have been thinking of the cross product.

------
coliveira
So basically he is complaining that physicists go to the trouble of learning
physical phenomena that support quantum theory, instead of learning directly
the mathematics of the theory. Does he really know the meaning of science and
how it works? Of course the math is important, but the math will do nothing
for you if you don't understand the evidence for the theory, and how it can be
falsifiable. These are things that you can only grasp from the history of how
physics got here.

------
tomrod
What an excellent write up.

------
sjg007
I wish I'd seen this is 2007.

------
thewarrior
I read this and didn't understand anything. Can anyone ELI 5 ?

~~~
gone35
Sure! To quote Scott's own TL/DR:

 _Quantum mechanics is what you would inevitably come up with if you started
from probability theory, and then said, let 's try to generalize it so that
the numbers we used to call "probabilities" can be negative numbers._

That's it. So in QM you can model events whose probabilities 'interfere' with
each other by canceling each other out ( _eg_ independent events A and B have
'probabilities' -20% and +20% respectively, but you want the 'probability' of
either A or B occurring to be 0%), and do all other sorts of weird stuff.

Now _why_ would you want to do such a thing is a whole different matter. But
at least this should get you started.

~~~
thewarrior
Hmm that makes sense in a weird sort of way. Positive and Negative
probablities canceling each other out out.

------
ThisIBereave
If anyone else was curious about Scott's usage of "God" in this lecture, he
talks about it here[1]. TLDR: they are "tongue-in-cheek references to an
Einsteinian God."

1:
[http://www.scottaaronson.com/blog/?p=189](http://www.scottaaronson.com/blog/?p=189)

~~~
lutusp
I wish people would use the word "nature" instead of "god" in public lectures
and writings, to avoid seeming to lend credence to revealed religion. Insiders
know exactly what Einstein meant by the word, but the problem is not insiders.

~~~
michaelsbradley
Can you expand a little bit on that line of reasoning?

It seems to me reasonable to say that the physical sciences can, for example,
lend very little or no credence to the claim that Jesus of Nazareth was the
long expected Messiah of the Jews, i.e. because a judgment on the matter seems
rather outside their scope.

But let's consider a different claim: that the existence of God, the origin
and end of all things, can be known with certainty by the natural light of
reason.

To develop the idea we might reasonably consider whether any of the loftier
domains of the physical sciences, e.g. physical cosmology, can tell us
anything about God's existence. Fr. Robert Spitzer, S.J.[1] (among others),
has been writing[2] and speaking articulately on the subject for a number of
years, and a fair bit of his material is freely available online[3].

[1]
[http://en.wikipedia.org/wiki/Robert_Spitzer_(priest)](http://en.wikipedia.org/wiki/Robert_Spitzer_\(priest\))

[2] [http://www.amazon.com/New-Proofs-Existence-God-
Contributions...](http://www.amazon.com/New-Proofs-Existence-God-
Contributions/dp/0802863833)

[3] [http://www.magiscenter.com/video-clips-and-
more/](http://www.magiscenter.com/video-clips-and-more/)

~~~
lutusp
> Can you expand a little bit on that line of reasoning?

Yes, certainly. Einstein was called out on his frequent allusions to god in
his public talks and writings, and under some pressure he finally described
how he saw god and religion.

Einstein said that his references to god were in fact with respect to
Spinoza's god, an abstract god who played no part in human affairs and that
bore no resemblance to the god religious believers picture. In other words,
nature -- not a judge, but a morally neutral environment.

> But let's consider a different claim: that the existence of God, the origin
> and end of all things, can be known with certainty by the natural light of
> reason.

But that's not possible without evidence. Let me explain the difference
between a scientist's attitude toward issues of fact, and a religious
believer's attitude.

To a religious believer, a claim is assumed to be true until evidence proves
it false. To a scientist, a claim is assumed to be _false until evidence
proves it true_ \-- the exact opposite.

Why do scientists take this position, formally known as the _null hypothesis_?
Because it's the only rational way to address issues of evidence. Let's take
Bigfoot as an example -- to a nonscientist, Bigfoot exists until his
nonexistence is proven. But disproving Bigfoot's existence requires _proof of
a negative_ , which is an impossible evidentiary burden.

Bigfoot could be hiding under some rock on a distant planet, therefore proving
his nonexistence is not possible, therefore Bigfoot exists. Therefore
everything exists -- UFOs, fairies, a teapot orbiting out in space in Bertrand
Russell's famous argument on this issue
([http://en.wikipedia.org/wiki/Russell's_teapot](http://en.wikipedia.org/wiki/Russell's_teapot)),
and god -- all without a shred of evidence.

Imagine if law adopted a religious outlook -- people would be guilty of any
crimes they were _unable to prove they didn 't commit_. But law (at least in
modern times) adopts an approximately scientific attitude toward evidence,
usually codified as "innocent until proven guilty."

This is the real meaning of the chasm between religion and science, and it's
not a trivial one.

> To develop the idea we might reasonably consider whether any of the loftier
> domains of the physical sciences, e.g. physical cosmology, can tell us
> anything about God's existence.

Very easy to answer -- without evidence, no such claim can be sustained. Full
stop.

------
maximumoverload
The good thing about quantum mechanics is that you don't have to know anything
about it and still say stuff with it that sounds incredibly profound.
"Everything is just a probability! We are all waves, maaan."

(Sort of like Freud's psychoanalysis. Everything is a penis, or your mother.)

This has probably nothing to do with the article though.

