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January 16, 2007

Aaronson on the Nature of Quantum Mechanics

Posted by Urs Schreiber

Scott Aaronson is a remarkable thinker and expositor, as you can convince yourself of for instance by following his blog.

He is among the few who managed to pull off something of genuine intellectual interest from a meme that is currently haunting the high energy physics community, known as the “anthropic principle”.

If you want to see a lot of confusion among smart people, google for the “anthropic principle”. Then go back and read this piece by Scott Aaronson to see the difference.

And if you happen to have followed the stringy part of the blogosphere in the last months and are in need of some great entertainment, you should not miss the blog entry accompanying this.

But I am writing this here not to talk about the anthropic principle, but about quantum mechanics.

On this blog here, we enjoy, from time to time, to muse about the nature of quantum mechanics in the light of general abstract nonsense. Lately for instance in the entry Common Applications and also in the discussion starting here.

Scott Aaronson is a complexity theorist, thinking about quantum computation. Accordingly, he has his views on the nature of quantum mechanics. In his latest transcript of a lecture he is giving, he explains to his students why there are various reasons that we should not be surprised about the nature of quantum mechanics. An intellectual treat. Even - and maybe especially - for the layman.

In essence, he explains that if we are going to consider any generalization of ordinary classical probability theory, then quantum probability is the most natural of all alternatives.

Personally, I believe that if we are ever going to really understand “Why quantum mechanics?”, it will involve considerations considerably beyond what Scott Aaronson mentions there, namely such more along the lines of John’s Quantum Quandaries. But he certainly mentions some noteworthy points.

Among them, somewhat vague but intriguing, is a relation to Fermat’s last theorem that he points out.

But the most powerful insight he mentions is probably that nonlinear deformations of quantum mechanics would allow to solve NP problems in polynomial time.

In order to appreciate this, you may want to read Aaronson’s Reasons to Believe.

P.S.

John mentioned many related things in TWF 235.

Posted at January 16, 2007 2:13 PM UTC

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Re: Aaronson on the Nature of Quantum Mechanics

Aaronson’s piece was certainly the most intelligent and enlightened thing I’ve read about the anthropic principle in a long time. It actually made me smile and ponder, which I don’t think I had done in this context since I had read Carl Sagan’s words on the subject:

There is something stunningly narrow about how the Anthropic Principle is phrased. Yes, only certain laws and constants of nature are consistent with our kind of life. But essentially the same laws and constants are required to make a rock. So why not talk about a Universe designed so rocks could one day come to be, and strong and weak Lithic Principles? If stones could philosophize, I imagine Lithic Principles would be at the intellectual frontiers.

This comes from Pale Blue Dot (1994), for the curious.

I think it also relates to the brain-teaser I heard from David Brin at ICCS 2006, namely that it’s a thousand times more likely that we’re living in a simulation than in the real world (because the post-Singularity folks can run a thousand simulations at will), but I haven’t had enough caffeine this morning to figure out exactly how.

Posted by: Blake Stacey on January 16, 2007 4:04 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

Nick Bostrom, director of Oxford University’s impressive sounding ‘Future of Humanity Institute’ wrote a paper - Are You Living In a Computer Simulation? - on a similar theme.

Abstract:

This paper argues that at least one of the following propositions is true: (1) the human species is very likely to go extinct before reaching a “posthuman” stage; (2) any posthuman civilization is extremely unlikely to run a significant number of simulations of their evolutionary history (or variations thereof); (3) we are almost certainly living in a computer simulation. It follows that the belief that there is a significant chance that we will one day become posthumans who run ancestor-simulations is false, unless we are currently living in a simulation. A number of other consequences of this result are also discussed.

A synopsis appeared in the New Scientist in November.

Posted by: David Corfield on January 16, 2007 5:17 PM | Permalink | Reply to this

Bostrom years late; Re: Aaronson on the Nature of Quantum Mechanics

Dear David,

I also enjoyed Aaronson’s talk transcrpt very much. But as comment to Blake’s comment, I politely differ.

David Brin gave that talk because I invited him, as chair of that one of the three sessions I chaired at ICCS-2006. He agrees that I beat Bostrom to print by several years on that concept, in “Human Destiny and the End of Time” [Quantum, No.39, Winter 1991/1992?, Thrust Publications, 8217 Langport Terrace, Gaithersburg, MD 20877; ISSN 0198-6686. My article was also acknowledged by Dr. Gregory Benford as a source for his far future electron-positron intelligences, whom I’d specifically predicted were the ones simulating us. Brin has a more jaundiced view of Bostrom than I do, too, as do the founders of the Transhumanist movement that he somehow claims. As a scientist, I support priority by year of publication, not by degree of PR and degree the popular press is fooled.

Posted by: Jonathan Vos Post on January 18, 2007 3:20 AM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

[…] it’s a thousand times more likely that we’re living in a simulation than in the real world […]

My problem with statements like that is that I first seem to know what they mean, but once I think about it a little harder I realize that maybe I canot give them any precise meaning at all.

The great thing about what Scott Aaronson does is that he also thinks similar thoughts, but then manages to deduce from them a theorem that equates two complexity classes, namely

(1)PostBQP=PP \mathbf{PostBQP} = \mathbf{PP}

As an easy consequence, he gets the corollary that PP\mathbf{PP} is “closed under intersection”.

(this is explained in the last three paragraphs)

This is an honest theorem, which is now part of our body of mathematical knowledge.

Posted by: urs on January 16, 2007 5:38 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

This, then, is also my main complaint about the current fashion of mentioning the anthropic principle in high energy physics:

all too rarely do people go through the effort of sufficiently formalizing what they want to understand under the “anthropic principle”.

As Scott writes:

Usually the question is framed as, “Is the Anthropic Principle a legitimate part of science?” But that’s the wrong question. I hope to convince you that there are cases where we’d all be willing to apply the Anthropic Principle, and other cases where none of us would be willing to apply it […]

Scott reminds us that there are consistency checks on what we should count as a meaningful interpretation of the anthropic principle:

No application of the Anthropic Principle can be valid, if its validity would give us a means to solve NP\mathbf{NP}-complete problems in polynomial time.

Posted by: urs on January 16, 2007 5:46 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

urs wrote:

My problem with statements like that is that I first seem to know what they mean, but once I think about it a little harder I realize that maybe I canot give them any precise meaning at all.

This is the same complaint I have about such puzzlers. How do we define, for example, a Bayesian prior in such a situation?

Posted by: Blake Stacey on January 16, 2007 6:41 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

How do we define, for example, a Bayesian prior in such a situation?

Worse than that: how do we even decide what it means to “live in a simulation”.

Without having a red pill available, that is.

Posted by: urs on January 16, 2007 7:14 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

Something that struck me today with regard to quantum mechanics in light of the comment made by John about Wick rotation converting between thermal statics (statistical mechanics) and quantum dynamics (quantum mechanics) is how differently they get treated philosophically.

As Lawrence Sklar explained in the introduction to Physics and Chance (CUP, 1993), there are four physical theories which receive philosophical attention. GR is seen as having settled principles, and the philosophically relevant questions (e.g., substantivalism vs. relativism) agreed upon even if not settled. QM is also a well-established theory, but the philosophical questions (e.g., measurement process as contrasted with dynamic evolution) are less clear (i.e., we’re less sure what would constitute a satisfactory answer). The theory of elementary particles is taken to be not so well delineated, and not so thoroughly philosophically treated (but issues of reductionism considered). Finally, statistical mechanics is described as a ‘hodgepodge of approaches’, and the theory as not having been given much philosophical scrutiny, but to the extent it has not even which questions to pose are clear.

But I’m still unclear about the question I asked about quantum thermodynamics.

Posted by: David Corfield on January 16, 2007 6:26 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

David Corfield wrote:

Finally, statistical mechanics is described as a ‘hodgepodge of approaches’, and the theory as not having been given much philosophical scrutiny, but to the extent it has not even which questions to pose are clear.

Perhaps some of the questions which pertain to statistical mechanics are hiding in the discussions about weak vs. strong emergence?

Posted by: Blake Stacey on January 16, 2007 6:39 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

I’ve been working on the idea that quantum systems are inherently concurrent systems. They can be examined under directed algebraic topology (or ditopology) tools (e.g., see here, or google about it).

The most interesting problem to model is quantum entanglement: underlying this phenomenon one abstracts out a quantum state as fundamentally corresponding to a set of concurrent processes. It would be interesting to examine how to restore local realism under this general hypothesis.

One can go further into this and ask whether quantization arises from the idea that, given that quantum systems are seen as concurrent systems, there are naturally forbidden regions in the multi-dimensional state space, where each dimension represents a concurrent process, defined by the fact that these processes locally share “resources” and no two processes can “act” on the shared resources at the same “time”. Forbidden regions correspond to the discretization of nature in the quantum limit.

So nature would fundamentally be a huge deadlock avoidance system.

Feel free to delete this comment if you like… I just wanted to share these general ideas I’ve been thinking about. But even if one cannot take these ideas too far, at least it seems to me they could serve as a framework for computer simulations of quantum systems.

Thanks
Christine

Posted by: Christine Dantas on January 16, 2007 10:17 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

Christine, I assume the application of directed algebraic topology to concurrent systems is an outgrowth of the work of Herlihy and Shavit, which was done in the late 90s and won the 2004 Gödel Prize.

You might want to communicate with Prakash Panangaden at McGill about your work. He appears to have worked, as a theoretical computer scientist, on concurrent and distributed systems and also on quantum computing, and has a strong interest in the causal structure of general relativity and its connection to abstract notions arising in computer science.

Posted by: Chris W. on January 17, 2007 1:58 AM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

Chris W. wrote:

I assume the application of directed algebraic topology to concurrent systems is an outgrowth of the work of Herlihy and Shavit, which was done in the late 90s and won the 2004 Gödel Prize.

In terms of applications of general geometrical methods to concurrency, the idea goes back to the 70’s. See Goubault, E. Geometry and Concurrency: A User’s Guide. But you are right to cite the work of Herlihy and Shavit as an evidence of how this subject is getting a lot of attention recently, and giving rise to important developments in the field. It looks a beautiful paper, but I have never studied it in detail.

Panangaden also published in gr-qc, and this is one of his most interesting papers, I guess… I don’t know how far this is getting attention, but sounds an original and promissing line of research.

Best regards,
Christine

Posted by: Christine Dantas on January 17, 2007 11:51 AM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

Hi Christine, you probably already know that Carl Petri, one of the founders of Concurrent Systems as a field of study, used to propound the view that an adequate semantics for concurrency would need to be sufficiently broad to also encompass physics.

From what I’ve been reading recently on Anima ex Machina blog he’s updated this position somewhat to a more “out-there” Universe-is-a-Petri-net kind of stance. He might be right! (as might Smolin – I saw your Amazon review!) On the same page I linked you can also find thoughts of Seth Lloyd and some others on this topic from a conference in Berlin last year.

You’re probably also familiar with Vaughan Pratt who clearly thinks very deeply on this sort of topic (although his publications can be frighteningly dense).

When I was a CS student as Glasgow they used to ask us questions like the “big simulation” argument, just to freak us out. I was never able to see why this would confuse a hardy theoretical physicist though — surely they’d be interested (in theory) in the turtle at the bottom of the tower, ie the real simulation that presumably has to simulate itself?!

I’m looking forward to seeing where the quantum lambda-calculus course goes with this kind of thing. Quantum computation seems to rely on arbitrary-precision complex amplitudes — but having gone to all the trouble of defining True and False as morphisms in the course, surely we won’t now be allowed to pull high-precision complex numbers out of the hat? If you were designing a programming language they’d need to be defined and computed themselves somehow… who “computes” these amplitudes that they’re relying on?

Posted by: Allan E on January 17, 2007 2:09 AM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

Hi Allan,

you probably already know that Carl Petri, one of the founders of Concurrent Systems as a field of study, used to propound the view that an adequate semantics for concurrency would need to be sufficiently broad to also encompass physics.

And according to Nielsen and Chuang,

“Quantum computation and quantum information has taught us to think physically about computation (…) we can also learn to think computationally about physics.”

This view is attractive and I believe physics is heading towards this broad idea. I do not have a clue how far this will lead us.

Concerning Petri nets, these and other classical concurrency models have been generalized to the concept of po-spaces (“po” from “partial order”). See Sokolowski, S., Directed topology and concurrency a short overview.

Best,
Christine

Posted by: Christine Dantas on January 17, 2007 12:21 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

And concerning the question in your last paragraph, I don’t know how to answer to that… There is a huge gap between a broad idea and the technical details that one has to face in order to make the idea actually work or make sense! But thanks for pointing that out, I’ll have to think about it.

Posted by: Christine Dantas on January 17, 2007 12:31 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

Let me see if I understand what you’re getting at. If there are correlations among a set of concurrent processes, they must be constrained to avoid deadlocks. Perhaps the necessary constraints induce correlations and anti-correlations among states and state transitions in the system that resemble correlations among quantum states. These correlations also imply that the system avoid parts of its state space.

You seem to be assuming discretization at the outset, insofar as the number of concurrent processes is finite, and the associated state space is finite. Should I assume that you have in mind a continuous analog of a set of concurrent processes?

Here is a simple and fairly concrete model you might want to examine. Consider the state space to include a set of n Boolean variables, and the concurrent processes to be n Boolean transition functions of two variables that yield a “subsequent” state for each of the variables. The processes (transition functions) share resources in the significant sense that each of the functions operates on two variables, at least one of which might be used as an input by another function. What would constitute a deadlock in such a system, and what is required to avoid it? Is there necessarily a stochastic component to the system’s behavior? That is, does the transition structure have to rearrange itself every so often, in a way that can’t be described deterministically? I realize this is a very sketchy problem formulation.

By the way, with respect to the relevance of algebraic topology, the set of transition functions in this simple model can be considered as defining an abstract simplicial complex; the Boolean variables are the vertices and the functions define the edges. (Remember however that these functions have some internal structure, since we have 16 distinct, albeit interrelated, Boolean functions to choose from.) My previous questions can then be posed as questions about the “moves” that can and must occur within this complex, in order to avoid certain “pathologies” in the evolution of the system. (By the way, in this light, the collection of transition functions can be loosely regarded as a “gas” of edges [1-D!] which is evolving in parallel with and necessarily coupled to a “gas” of Boolean state variables.)

One more point: The Boolean variables are of course undergoing transitions with successive “time steps”. One might ask if one can define a causal order on these transitions or “events”. That is, given one such event, can one say that it was preceded in a well-defined way by another set of events, and followed accordingly by yet another set of events? It is not immediately evident how such a description is to be derived from the transition functions, although one might plausibly expect that it should be possible.

(I have in mind here a complementary description in terms of causal sets. By the way, regarding the compatibility of discreteness based on causal sets with Lorentz invariance, see Discreteness without symmetry breaking: a theorem [1 May 2006].)

Posted by: Chris W. on January 17, 2007 4:26 AM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

Dear Chris W.,

Thanks for this elaborate comment. Your first paragraph summarizes well the broad idea.

And yes, one could assume a continuous “scheduling” of the processes, and also the space could be assumed to be infinite dimensional. What I think is important here is to assume a local partial order – a “po-space”… (I’m interested in ergodic moves…) Examples of such spaces are found here:

* Sokolowski, S., Classifying holes of arbitrary dimensions in partially ordered cubes. Tech. Rep. 2000-1, Kansas State University, Computing and Information Sciences, Aug. 2000. linke here.

* Raussen, M., Geometric investigations of fundamental categories of dipaths. Unpublished, 2001. (sorry, can’t find the link now).

* Gaucher, P., About the globular homology of higher dimensional automata. Cahiers de Topologie et Geometrie Differentielle Categoriques XLIII-2 (2002), 107156. link here.

And thanks for proposing a problem. I guess we all have sketchy ideas for the moment! No problem about that, in fact it is great to know that this is quite an open field for research.

Your comment has given me months to think over! Don’t have much valuable to add for the moment, but thanks a lot.

Christine

Posted by: Christine Dantas on January 17, 2007 12:49 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

kill yourself if the measurement doesn’t yield a desired outcome, and then consider only those branches of the wavefunction in which you remain alive.

Arguments making use of such an idea are reminiscent of some formulations of the Sleeping Beauty problem. Reasoning about events when there are multiple people who you could be is tricky stuff. (And it’s interesting that the sleeping beauty problem describes such a situation without having to literally have multiple copies of yourself.)

Presumably halfers and thirders disagree over the equality of different complexity classes.

Posted by: Dan Piponi on January 17, 2007 12:25 AM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

kill yourself if the measurement doesn’t yield a desired outcome, and then consider only those branches of the wavefunction in which you remain alive.

Arguments making use of such an idea are reminiscent of some formulations of the Sleeping Beauty problem.

I've always found the kill-yourself-if-you-lose-the-quantum-lottery gambit obviously wrong, and (after reading the Sleeping Beauty problem) the correct answer seemed to me to be clearly (but not obviously, I had to think about it) 1/2.

So I guess that I'm consistently taking the position allied with opponents of (at least strong forms of) the Anthropic Principle. That's good to know, since I never knew whether I believed in that principle or not!

Posted by: Toby Bartels on January 17, 2007 1:43 AM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

Well, as the link indicates, I’m a old halfer and I generally think that mediocrity principle (which goes beyond the anthropic principle which is tautological) arguments are meaningless. I also have a dim view of Bayesian statistics for whatever it’s worth.

Just to add a data point.

Posted by: Aaron Bergman on January 17, 2007 1:48 AM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

Well, upon further consideration, I’ve changed my mind! In my halfer conclusion, I didn’t properly take into account the information that I (Sleeping Beauty) had been awakened. So I am now a thirder.

But I still have no trust in the quantum suicide gambit! This is because the corresponding piece of information —that I will survive— is noticeably missing. (That I survive in some branch of the wave function is no help.)

Posted by: Toby Bartels on January 18, 2007 10:30 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

Reasoning about events when there are multiple people who you could be is tricky stuff.

Just for the record, it might be worth pointing out that when Aaronson talks about “kill yoursef if …” in his talk, he is entertaining his audience with a bizarre story involving people and their lives.

I think the point is that there is a precise statement which is supposed to be illustrated by this, but which does not involve people and their lives.

Like there is a complexity class NP\mathbf{NP} which was originally described as “assume you have a probabilistic computer that can guess the right choice of path each time”, Aaronson introduces a complexity class PostBQP\mathbf{PostBQP} which is described like: “assume you have a quantum quantum computer and you happen to know that you are doing a measurement on it with the desired outcome”.

Maybe one should rather address this as being inspired by the anthropic principle. It boils down to thinking about a posteriori probabilities.

Posted by: urs on January 17, 2007 4:33 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

Presumably halfers and thirders disagree over the equality of different complexity classes.

I would rather think that the difference is in what they think the problem means precisely. That’s my point here: as soon as we try to make precise what we actually mean by all these stories, all mystery disappears.

Here is one way to make the “Sleeping Beauty”-problem precise:

Write a computer program, AA, that first generates a random number in {0,1}\{0,1\} and stores that in the variable rnd\mathrm{rnd}.

If that number is 1 then the program sends a ping to a given port.

If the number is 0, then the program first sends a ping to the given port, then it goes into a for-loop of 100 cycles that does nothing else, and after that it sends another ping to the given port.

Now your task is to write another program, BB, which is run once for each ping.

BB may return either 0 or 1. Your task is to program BB such that the correlation of this return value with the random number rnd\mathrm{rnd} stored by AA is high (without reading it out).

Is there any ‘halfer’ around who would want to offer his alternative interpretation of the “sleeping beauty problem”?

Posted by: urs on January 17, 2007 6:40 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

No ‘halfer’ here, but for another ‘thirder’ reinterpretation and much more, including a criticism of Susskind’s arguments about the Landscape, see Radford Neal’s Puzzles of Anthropic Reasoning Resolved Using Full Non-indexical Conditioning. The reinterpretation occurs in section 3.4, page 17.

Posted by: David Corfield on January 25, 2007 7:20 PM | Permalink | Reply to this

Re: Aaronson on the Nature of Quantum Mechanics

see Radford Neal’s Puzzles of Anthropic Reasoning Resolved Using Full Non-indexical Conditioning.

Thanks for this reference! I was not aware that there is such stuff about anthropic reasoning in the math arXiv.

Posted by: urs on January 25, 2007 7:32 PM | Permalink | Reply to this

Aaronson on the anthropic principle

Lest we run into the danger of being smart and still be confused after hearing the words “anthropic principle”, it might be worth pointing out that behind all the imagery of suicidal experimentors, there is a precise fact that Scott Aaronson discusses:

On p. 3 of Quantum Computing, Postselection, and Probabilistic Polynomial-Time the definition of the complexity class PostBQP\mathbf{PostBQP} is given:

Definition 1 PostBQP\mathbf{PostBQP} is the class of languages L{0,1} *L \subseteq \{0,1\}^* for which there exists a uniform family of polynomial-size quantum circuits {C n} n1\{C_n\}_{n \geq 1} such that for all inputs xx,

(i) After C nC_n is applied to the states |00|x|0 \cdots 0\rangle \otimes | x \rangle the first qubit has a nonzero probability of being measured to be |1|1\rangle.

(ii) If xLx \in L, then conditioned on the first qubit being |1|1\rangle, the second qubit is |1|1\rangle with probability at least 2/32/3.

(iii) If xLx \notin L, then conditioned on the first qubit being |1|1\rangle, the second qubit is |1|1\rangle with probability at most 1/31/3.

No suicides or sleeping beauties in this statement. All one sees is an a posteriori probability being mentioned.

Posted by: urs on January 17, 2007 6:08 PM | Permalink | Reply to this

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