theory salon

12:54 AM

I once wrote "... So classical probability theory and statistics normally try to be careful..." in the context of justifications for Bayesian probabilities. I do have to work with probabilities in my daily job, and I try indeed to be careful with probabilities. It's not really the interpretation which is responsible for this, but keeping in mind that probabilities can be subtle is important.

I found your links to probability interpretations interesting. The propensity interpretation is actually closely linked to the classical (symmetry based) interpretation, because we often "know" the propensities because of symmetry reasons. Which is very interesting for me, because I once considered probability theory to be a powerful linearization technique.

What I realize now is that probability techniques can be used to introduce (additional) symmetries into problems which would lack them as deterministic problems. And being linear is just one quite strong symmetry property. But this also goes the other way. In an appropriate basis, some symmetries can turn into independence properties, i.e. some coupling coefficients become exactly zero.

Especially for photolithography, I think of the probabilities as actual zeros instead, i.e. that wave-packets don't really overlap in time or space. I may not know exactly when the wave-packet starts, but this is not really important as long as the wave-packet is isolated anyway. Of course, because of the statistics, the wave packets would even be allowed to overlap, and the mathematics should still work. However, this is only true if the used materials are not too non-linear.

I think I rarely (if ever) use the Bayesian interpretation. I'm not really opposed to it, it's just that I never needed it, and that I understand it less well than the more classical interpretations.

vzn

1:31 AM

re "density matrix" formulation of qm. not too familiar with it but am guessing it naturally hints at an LHV. ie as in, density of what?

do have one neat/ very serious ref on qm that refers to "probability fluid." a rare different pov. it is also very strict to point out the concept is only to be regarded as a pedagogical crutch, albeit interesting/ worthwhile.

re neumaier, have a question inspired by our dialog. you are a member on the theoretical physics site right? maybe you could ask it?

probability seems to be showing up in increasing ways in TCS. some of this ofc is due to its (TCSs) leading role in analyzing qm computing, but it shows up in a lot of other areas under very active research.

re your math blog post. worth mentioning in the blog room. maybe can get some hits/ feedback.

thermodynamics is an amazing example in some ways of LHV theory. for example consider that einstein gave evidence for the particle nature of matter via analyzing Brownian motion. one wonders if there is a 2nd order reality "generating" qm probability in the same way. aka bohms "implicate order".

vzn

2:01 AM

fyi, maybe you know this, bayesian methods are now quite accepted in machine-learning. there is even a new "deep learning" version recently.

maybe a place for some of these probability interpretation questions?

oh! lol you have 2k there. :o

that explains a lot :p

dont get too many philosophers around here. have you ever read aaronson's paper on philosophy? you might enjoy it.

-1

A: Algorithmic lens in the social sciences

re applications of complexity theory in social science-- scott aaronson has a daring & amusing-at-times essay tying complexity theory to deep century-old questions in philosophy that I ran across recently reading his blog. Why Philosophers Should Care About Computational Complexity http://arxiv....

Thomas Klimpel

10:06 AM

Density matrix

A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states. This should be contrasted with a single state vector that describes a quantum system in a pure state. The density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics. Explicitly, suppose a quantum system may be found in state with probability p1, or it may be found in state with probability p2, or it may be found in state with probability p3, and so on...

It is called density matrix in analogy to a classical system. If you work with mass densities instead of point particles, then you can also use probability densities to describe your uncertainty about the location of the point masses. If you want to describe a quantum system in a way that can directly be compared to the description of classical mechanics in term of (mass or probability) densities, then you have to use the density matrix formulation.

The density matrix formulation also converges better when certain parameters become small, i.e. sometimes the limit doesn't exist for the pure state formulation, but exists and is trivial to compute for the density matrix formulation.

Thomas Klimpel

10:29 AM

For practical computations, one can just decompose the density matrix into a sum of pure states. The optimal way to do this (i.e. that you get the least error for the number of pure states that you use) is the optimal coherent decomposition, where you compute the eigenvalue decomposition of the density matrix. The dynamics of Schrödinger equations is such that any such decomposition stays valid during time propagation, i.e. you can just propagate each individual pure state.

I have read Aaronson's paper on philosophy, very recommendable.

@vzn Yes, I can ask that question there. What is the question?

vzn

3:59 PM

ok yes have long felt the density matrix formulation might be the key to understanding a new interpretation of qm...

the question is re extending the neumaian framework to multiple particles, n-particle systems, n>=2.

am wondering how this can be formulated as a purely mathematical question, if possible.

presumably this also touches on the bell type analysis involving "separability".

ie is the same problem with extending the neumaian framework (nearly?) the same mathematical difficulty in finding separable forms of entangled systems...? seems likely...

which reminds me, aaronson had a paper on looking for LHV type systems in qm formulations... need to look that up again...

vzn

4:14 PM

need to scan the paper you cited.

vzn

4:53 PM

it appears from the abstract he feels hes already sketched it all out...

but a big question is how he/ it deals with entangled systems and nonlocality.

Thomas Klimpel

5:08 PM

I will start by asking how the construction alude to on page 43 is supposed to work. Entanglement already occurs in the context of finite dimensional Hilbert spaces, so there seems to be a way to deal with entangled systems.

vzn

5:29 PM

p43 of the slides

> Thus it is possible to simulate arbitrary quantum systems which have a finite number of levels by the Maxwell equations, and hence by a classical model.

(!) but that seems to be everything right there.

surely every quantum system can be approximated to an arbitrary degree of accuracy as levels increase...? almost analogously to the error terms on a Taylor expansion/ approximation etc.

or maybe that is exactly what needs to be proven? seems Neumaier maybe already has most or all of it...

think he needs to focus on entanglement/ separability which are widely thought (conventional wisdom) to be the "can of worms" for any unconventional/ classical interpretation/ picture of QM. not sure if/ where he has done that.

although he does have the paper cited where he reinterprets bell nonlocality.

Thomas Klimpel

7:11 PM

physicsoverflow.org/25484/…

vzn

ah. aaronson has been looking into that very issue quite a bit. he has a huge ambitious paper on that.

it appears you may have been too cautious in summarizing neumaiers views.

on p47 he sticks with single photon claims which would not be considered too controversial. (a single photon system is nearly a toy model.)

> We conclude that
> classical second-order stochastic optics is precisely the quantum mechanics of a single photon, with all its phenomenological bells and whistles.
> As a consequence, it is possible (at least in principle) to simulate with classical electromagnetic waves and suitable classical linear optical networks any quantum system that can be embedded into the single photon quantum system.

but then p48 it gets a bit wilder...

> Since all Hilbert spaces arising in applications of quantum physics are separable, they have a countable basis, and can be embedded into the single photon quantum system, at least in principle.
> Thus it appears that, all quantum systems can be simulated by classical electromagnetic waves!
> Of course, a practical realization may be difficult.

?!? afaik not all hilbert spaces are separable. not sure what he means by that. eg isnt a bell experiment modelled by a hilbert space?

Thomas Klimpel

7:30 PM

separable hilbert space just means that the hilbert space has a countable basis. All practically relevant hilbert spaces are either finite or have a countable basis.

vzn

then p49, a rather radical assertion that is definitely utterly anti- copenhagen!

he seems at least not to be using the word "separable" as it is used in some physics papers.

> We are now nearly ready to put the puzzle together whose pieces I have shown to you.
> We have looked at quantum mechanics from many different perspectives, everywhere uncovering its classical aspects.
> All this suggests the existence of a classical theory underlying quantum mechanics.

Thomas Klimpel

@vzn Do you have a name or reference for Aaronson's paper?

vzn

←even crazier, agree with him, & have conjectured this for decades... (ofc its an old idea going back to crazy... bohm/ einstein/ schroedinger/ debroglie etc)

note that nobel-winning t'hooft has been going in very LHV directions for many years, many of his cohorts think hes gone off the deep end...

re aaronson its is now nearly-notorious boson sampling

quantum computing with noninteracting bosons ppt / aaronson

he is quite an entertainer/ showman at times. even reminiscent/ flashes of feynman.

eg also his P vs NP slides are quite amusing.

youve read his blog of course?

vzn

9:08 PM

the boson sampling idea now already has some experimentalist tests, quite a feat...

Transcript for