## Quantum Computing Since Democritus Lecture 11: Decoherence and Hidden Variables

After a week of brainbreaking labor, here it is at last: My Grand Statement on the Interpretation of Quantum Mechanics.

Granted, I don’t *completely* solve the mysteries of quantum mechanics in this lecture. I didn’t see any need to — since to judge from the quant-ph arXiv, those mysteries are solved at least twenty times a week. Instead I merely *elucidate* the mysteries, by examining two very different kinds of stories that people tell themselves to feel better about quantum mechanics: decoherence and hidden variables.

“But along the way,” you’re wondering, “will Scott also touch on the arrow of time, the Second Law of Thermodynamics, Bell’s Inequality, the Kochen-Specker Theorem, the preferred-basis problem, discrete vs. continuous Hilbert spaces, and even the Max-Flow/Min-Cut Theorem?” Man oh man, is *someone* in for a treat.

I assume that, like Lecture 9, this will be one of the most loved and hated lectures of the course. So bring it on, commenters. You think I can’t handle you?

**Update (4/5):** Peter Shor just posted a delightful comment that I thought I’d share here, in the hope of provoking more discussion.* *

Interpretations of quantum mechanics, unlike Gods, are not jealous, and thus it is safe to believe in more than one at the same time. So if the many-worlds interpretation makes it easier to think about the research you’re doing in April, and the Copenhagen interpretation makes it easier to think about the research you’re doing in June, the Copenhagen interpretation is not going to smite you for praying to the many-worlds interpretation. At least I hope it won’t, because otherwise I’m in big trouble.

Comment #1 April 3rd, 2007 at 8:02 am

A guy named Michael Atiyah gave a rather nice talk on a rather vague idea for doing-away with quantum mechanics and/or reinterpreting it, at the Alain Connes birthday party conference. He intended the talk to be wildly speculative and as food for thought. I’m not going to be able to tell you everything, but one aspect of it caught me.

The infinite-dimensionality of quantum mechanics — the state space. He wanted some kind of intuitive reason for why it should exist, in the spirit of the intuitive arguments Einstein gave for the Lorentz transforms. Why should quantization of mass or energy, space or time, etc, force such a construct on us? Vaguely speaking, if time or space is quantized, there would be gaps between objects and delays between cause and effect. He then brought in an analogy with control systems from electrical engineering, and foisted the idea that perhaps there is a 1st order delay differential equation (which he calls a retarded differential equation) that could perhaps put it all together.

He gave an example like this, consider a DE f'(t) = kf(t-a) where a>0 is some constant. Then to determine the behavior of f(t) for t>a you need as an initial condition all the values of f in the interval [0,a]. Now you’re talking about an infinite-dimensional space. So a rather vague idea of quantizing time gives you a way to justify an infinite-dimensional state-space while retaining the continuum.

He went on to give a rather nice talk and flesh out the idea some more, giving some credit to a guy named Raju. Here is a reference to one of Raju’s recent papers on the topic:

The electrodynamic 2-body problem and the origin of quantum mechanics, Foundations of Physics 34, (June 2004), 937–962.

Back to your regularly scheduled program…

Comment #2 April 3rd, 2007 at 11:08 am

Hey, nice lecture!

I have a question that I suppose you have already answered so feel free to tell me I am an idiot if you don’t want to answer it again. You talk about non-relativistic quantm mechanics Has a QFT computer been definied and if it has, what sort of computational power does it have compared to an ordinary quantum computer?

Comment #3 April 3rd, 2007 at 11:22 am

Kitaev, Freedman, and Wang showed that a quantum computer can efficiently simulate topological quantum field theories, which implies that a QFT computer would be no more powerful than an ordinary quantum computer.

Comment #4 April 3rd, 2007 at 11:23 am

Johan: Most people expect that QFT computers will be no more powerful than ordinary quantum computers, for the same reasons why classical computers don’t suddenly become more powerful when you throw in relativity. But the truth is that there aren’t many rigorous results here — partly, as I understand it, because of the difficulty of even

definingQFT computers. (We’re dealing, after all, with a theory that’s valid only in a certain energy regime, and that’s known to break down if you go beyond it.)The one definitive result I can point you to is that of Freedman, Kitaev, and Wang, who showed that topological quantum field theories (a special class of (2+1)-dimensional QFT’s, where all the degrees of freedom are topological) yield no more power than ordinary quantum computers. See also Aharonov, Jones, and Landau, who reinterpreted Freedman et al.’s result in much more CS-friendly terms.

Comment #5 April 3rd, 2007 at 11:26 am

Sorry, Jim — just missed you! As I wrote above, Freedman et al. applies only to the special case of TQFT’s.

Comment #6 April 3rd, 2007 at 11:58 am

One more question on this issue of finite versus infinite dimensional Hilbert spaces: is it true that the uncertainty principle can only hold true in infinite-dimensional spaces?

It’s one striking feature of quantum mechanics, but strangely enough never seems to play a role in quantum computation (and information ?).

Comment #7 April 3rd, 2007 at 12:25 pm

No, Pascal, there are finite-dimensional analogues of the uncertainty principle. A simple example is this: you can’t measure a qubit in both the {|0⟩,|1⟩} basis and the {|+⟩,|-⟩} basis. Indeed, the product of your uncertainties in the two bases must be at least a constant.

Comment #8 April 3rd, 2007 at 12:30 pm

Pascal Koiran asks:

is it true that the uncertainty principle can only hold true in infinite-dimensional spaces?No. The coherent states that are the physical realization of minimum uncertainty states also exist in all finite-dimensional Hilbert spaces, and all the same uncertainty relations apply with minimal modification.

A good text is Perelomov’s

Generalized Coherent States and Their Applications(see BibTeX below).Caveat: the same is true of coherent states that Serge Lang famously said of elliptic curves: “It is possible to write endlessly on elliptic curves (this is not a threat).” 🙂

—————–

@book{Perelomov:86,

author = {A. Perelomov},

title = {Generalized Coherent States and Their Applications}, publisher = {Springer-Verlag},

year = 1986, }

Comment #9 April 3rd, 2007 at 12:47 pm

You lectures are simply amazing. I read them as an art viewer views paintings without bothering too much about the

rigor. It’s like bungi jumping without bothering too much about dynamics of how it works. You show limits of human understanding yet show pure scientific spirit of exploration for truth that too in most promising directions. Thanks a lot for choosing to be in TCS!:)Comment #10 April 3rd, 2007 at 3:34 pm

In regard to your thought experiment, about experiencing different colors of dots you say:

“Then what’s the probability that “you” (i.e. the quantum computer) would see the dot change color?”

But it seems to me that the main point is, for the person to have a memory about his previous experiences, his brain must have more than just the one qubit you mention in your example (3/5 |R> + 4/5 |B> etc.) He would have to have some ancilla qubits which record his previous state. If he has only a one bit brain then he can’t know whether the dot changed color even classically.

You could imagine that the quantum upload had some routine which every millisecond or whatever cnotted the content of the “what color dot am I looking at” qubit into some fresh |0> ancilla qubit (say |0> means blue |1> means red). Then after t timesteps he could check to see whether he has seen the dot change color by performing the following unitary operation: if all t memory qubits are in the |0000….> state or the |1111…> state do nothing to an ancilla qubit initialized to zero, and otherwise apply a NOT to it. This ancilla qubit is then the “have I seen the dot change color” qubit.

Now the conditional probability question has an answer, equal to the absolute square of the amplitude for the “have I seen the dot change color” qubit to be |1> (which I think is the same as the classical answer.)

One might argue that this is essentially equivalent to John Preskill’s response since the person’s memory is decohering his “which color dot am I looking at qubit”, but I am reluctant to call this decoherence because the uploaded person still can coherently manipulate his memory qubits.

Comment #11 April 3rd, 2007 at 3:56 pm

Stephen, thanks for your interesting comment!

Yes, it’s clear that you won’t

rememberthe color change. If someone asks you afterward whether you saw the dot change color or not, the only honest answer will be that you have no idea.Even so, it’s slightly unsettling that, conditioned on what you’re seeing at time t

_{1}, quantum mechanics can’t even give you a probabilistic prediction for what “you’ll” see at time t_{2}!I have a lot of sympathy for your response to this problem, which I do see as basically equivalent to Preskill’s (since even if the “decoherence” is only temporary, it still completely changes the nature of the experiment, making it possible to talk about the past by reference to memories in the present).

Comment #12 April 3rd, 2007 at 5:17 pm

You have time running down in your diagrams! That makes no sense at all.

Also, I understand that you were explaining the arrow of time in the context of decoherence, not proposing a theory for why there is an arrow of time in the first place. But from the perspective of this latter issue, the question is why we’re not in thermal equilibrium from the start; why did we begin in such a special state?

Comment #13 April 3rd, 2007 at 5:29 pm

Hi Sean,

(1) In computer science, time flows down! (Just like trees grow down.)

(2) I completely agree with you that the “deep” question about the arrow of time is why we started in such a low-entropy / unentangled state. There’s a reason I sidestepped that can-o-worms! (Still, I should at least mention it somewhere…)

Comment #14 April 3rd, 2007 at 6:38 pm

But is it a separate question about the initial conditions, is there an additional aspects of the boundary conditions to be explained, or is decoherence explained in terms of the thermodynamic arrow of time?

Comment #15 April 3rd, 2007 at 6:58 pm

Well, we can suppose that the universe has two terminal conditions: The Big Bang on one side, and the Big Rip on the other. If we define time as moving “forward” in the condition that systemic entropy increases, then since the Big Bang represents the point of minimum entropy and the Big Rip (=newfangled heat death) is the point of maximum entropy. In between these two points, time doesn’t need to have a particular “direction” for individual events, but the accumulation of entropy allows us to nominally define a direction for time’s arrow.

The blog On Philosophy has a decent explanation of this view.

Comment #16 April 4th, 2007 at 12:46 am

I asked the question about the uncertainty principle & infinite-dimensional spaces because I vaguely remember reading somewhere the following (wrong ?) statement: there’s no uncertainty principle in finite-dimensional spaces because the relation AB-BA = Identity cannot hold for (finite dimensional) matrices.

The linear algebra claim is certainly true (the left-hand side has trace 0 but not the right-hand side).

However I do not really understand the connection to the uncertainty principle…

Comment #17 April 4th, 2007 at 12:52 am

Another question: have you noticed that you can attach to the root of your multiverse tree another multiverse tree, with leaves pointing upwards instead of downwards?

(That would for sure make Sean Carroll happy!)

This suggests that before the big-bang, there existed perhaps another multiverse with a time arrow opposite to the time arrow of our multiverse…

I think that Sean Carroll had a blog post on a similar idea, and even an actual physics paper!

Perhaps he would care to comment on that?

Comment #18 April 4th, 2007 at 4:31 am

Pascal, I’m a mathematician, not a physicist, so take this with a grain of salt but…

The general mathematical form of the uncertainty principle is the Robertson-Schroedinger relation: if A and B are two Hermitian operators on a Hilbert space and phi is any nonzero vector in the Hilbert space then

Delta(A,phi) Delta(B,phi) >= (1/2) /

where Delta(A,phi), the standard deviation of A in state phi, is given by

Delta(A,phi)=( / )^{1/2}

where A’ is A normalized to have expected value zero, i.e.,

A’=A-/*Id.

In particular, the minimal possible product of the Deltas is norm of the smallest eigenvalue of [A,B]. The nice thing about taking [A,B]=Id, in this context, is that we know what the eigenvalues of the identity are, but we get a nontrivial bound whenever [A,B] is nondegenerate.

Comment #19 April 4th, 2007 at 4:35 am

> Close tag.

Oh, gosh, HTML (or at least Safari) hates my inner products. Well, if that doesn’t close my tag then I’ll have to wait for someone cannier in the ways of blogs to clean up my mess. Sorry!

I just checked, and all of my formulas are in Wikipedia’s entry for the Uncertainty Principle, except that they nromalize their state to have norm 1 from the beginning. So I’ll refer you there rather than trying to retype them in a way that will typeset correctly.

Comment #20 April 4th, 2007 at 4:39 am

Pascal Koiran:

There’s no uncertainty principle in finite-dimensional spaces because the relation AB-BA = Identity cannot hold for (finite dimensional) matrices.That’s formally correct, but there

isa very simple loophole, that quantum system engineers exploit to establish a connection between finite-dimensional coherent states and infinite-dimensional harmonic oscillator states.Consider spin operators {Sx,Sy,Sz} in the usual representation of 2j+1 (finite) dimensions, satisfying [Sx,Sy] = i S_z. Now restrict attention to states |&psi⟩ that are near the “north pole” of the Hilbert space, i.e., such that Sz ≈ j. Define rescaled operators p=Sx/sqrt(j), q = Sy/sqrt(j). Then [p,q]≈i when acting on those “north pole” states.

For quantum simulation purposes, this means that harmonic oscillators can be treated as large-j spins, provided that a control loop (or equivalently, a thermal bath) is present to restrict quantum trajectories to the neighborhood of the north pole. Then j need only be set to the (nondimensional) energy scale of the excitations of the system being simulated.

It is very convenient to introduce an energy cut-off in this way, because all the “nice” algebraic properties of the spin operators are preserved. And, it is pleasant to write simulation codes in which “everything is a finite-dimensional spin” — this keeps life simple.

On a more fundamental level, it would presumably be possible to use the above trick to cut-off high energy states in path integrals by treating x and p as operators rather than oscillator coordinates — it is not clear (to me) whether this is the approach that people are exploiting in what is called noncommutative geometry.

There is a lot of literature on path integrals defined over Lie groups — most of this literature is not engineer-friendly. But I get the impression that fundamental research in this area is definitely not out of reach of graduate students.

Perhaps someone can comment?

Comment #21 April 4th, 2007 at 4:51 am

Hi Scott,

I had two questions,

1. How would a hidden variable theory explain the correlation between the two qubits after measuring the EPR pair?

2. Why would the tree “run out of room to expand” if we start in an infinite dimensional Hilbert space?

Comment #22 April 4th, 2007 at 5:19 am

I will remark that David Speyer’s post and my post are saying pretty much the same thing about uncertainty relations, each in our own idiom.

Also Scott was right to foresee that “this will be one of the most loved and hated lectures of the course”, given that many people have formed strong opinions regarding the interface between classical and quantum reality.

Barry Mazur coined an aphorism that explains why: “Utter confidence is the gift of ignorance.” 🙂

Comment #23 April 4th, 2007 at 6:47 am

Ghaith:

(1) As the lecture explains, any hidden-variable theory has to invoke “instantaneous communication” between two entangled qubits in order to explain their correlations — that’s precisely the content of Bell’s theorem. This does

notformally violate special relativity, since even though one’s description of the hidden variables involves nonlocality (usually in a preferred reference frame), one can’t exploit that to send signals faster than light. Of course, many people use this as an argument against hidden-variable theories anyway.(2) Even in an infinite-dimensional Hilbert space, any “branch” has to be thought of as having a finite width ε, just because of quantum fluctuations within that branch. And assuming (as I am) that we’re talking about a bounded region of spacetime, that means we can only have O(1/ε) branches.

As a toy example, think of the possible positions of a particle along a 1-cm interval. Even though the particle’s position is a real number, if two positions differ by less than (say) 10

^{-33}cm then they can never be distinguished, and should therefore be thought of as belonging to the same branch. But that implies that there can be at most 10^{33}branches.Comment #24 April 4th, 2007 at 11:01 am

Scott,

Does working on QC make people seem crazier, or only seemingly crazier people work on QC?

Anyhow, thanks for a great series of lectures!

I have another lingering question which I’d love to hear comments from you and your readers: how do you guys manage your schedules? Is there a well-known algorithm to sort out the mess of reading, refereeing, teaching, grant writing, admin. work, networking, blogging, commenting on blogs (!), and a myriad of other stuff you do? If not a specific algorithm, is there a complexity class of such algo.?

The brute-force algorithm “take things as they come” has been ok for me, I’m just wondering about a more efficient one I am not aware of.

Comment #25 April 4th, 2007 at 11:21 am

HN: Are you implying that I come across as less than sane?

(

“You may be right / I may be crazy / but it just might be a LOOOO-natic you’re lookin’ for…”🙂 )Regarding your other question: if you ever figure out how to manage an academic schedule, please let me know! I’ve been wondering for years.

Comment #26 April 4th, 2007 at 12:01 pm

Are you implying that I come across as less than sane?No, you’re too sane for the sake of your own good.

Comment #27 April 4th, 2007 at 12:09 pm

🙂

Comment #28 April 4th, 2007 at 3:08 pm

scott, do you actually believe in the many worlds interpretation?

Comment #29 April 4th, 2007 at 3:28 pm

I believe in

everyinterpretation of quantum mechanics to the extent it points out the problem, and disbelieve in every interpretation to the extent it claims to have solved it.Comment #30 April 4th, 2007 at 4:20 pm

re: HN and Scott’s comments on time-management in academia: Knuth suggests batch-processing — keep blocks of time for single tasks, don’t do too much task-swapping. I, personally, have lots of room for improvement in time-mgmt, but have found this technique very helpful whenever i have followed it .. aravind

Comment #31 April 4th, 2007 at 5:11 pm

Scott,

I have two questions about your remark

“if two positions differ by less than (say) 10-33 cm then they can never be distinguished, and should therefore be thought of as belonging to the same branch.”

1) are you all of a sudden a physicist ?

2) since you refer to the Planck length, which has a meaning only in quantum gravity, do you suggest that the interpretation of quantum theory depends on (has to wait for) quantum gravity?

Comment #32 April 4th, 2007 at 5:34 pm

Wolfgang: If you have to be a physicist to have any definite belief about the Planck scale, then sure — I’m a physicist. 🙂

As for whether quantum gravity (and specifically, the holographic entropy bound) is relevant to the interpretation of quantum mechanics: sure it is,

ifpeople insist on talking about infinite-dimensional Hilbert spaces in the first place! In other words, if a believer in infinitely many branches objected to my quantum-gravity argument, on the grounds that I was bringing in something extraneous to quantum mechanics itself, my response would boil down to: “You started it!”Comment #33 April 4th, 2007 at 10:29 pm

Pascal: Here is the generalized uncertainty principle as stated in my (alas, unfinised) notes:

Let X and Y be real-valued quantum random variables and let Z = i[X,Y]; Z is another real-valued quantum random variable. Then

Var[X]Var[Y] ≥ Ex[Z]^2/4

with respect to any given state of the system |psi> or rho. In the event that the commutator Z is constant, then the right side is easy to compute.

Comment #34 April 4th, 2007 at 10:32 pm

I believe in every interpretation of quantum mechanics to the extent it points out the problem, and disbelieve in every interpretation to the extent it claims to have solved it.Maybe I am the thick one on this topic, but: Do we know that there is any real “problem” other than that we humans have trouble believing the truth, i.e., non-commutative probability?

Comment #35 April 5th, 2007 at 4:29 am

“It has not yet become obvious to me that there’s no real problem. I cannot define the real problem, therefore I suspect there’s no real problem, but I’m not sure there’s no real problem.”

Etc.

Comment #36 April 5th, 2007 at 5:47 am

Greg, I see the measurement problem as a “hard problem” like consciousness or the existence of the universe, not an “easy problem” like … uh, y’know … quantum gravity or P vs. NP. In other words, I have no idea what an answer would look like, and I’m unwilling to say whether or not one exists. What I know is that, absent some insight all of us are missing, people will continue to pose the problem for as long as there’s quantum mechanics, just as other people will continue to say that it’s meaningless. This view is not incompatible with yours.

Comment #37 April 5th, 2007 at 6:47 am

So far, this is very enjoyable discussion that has the added virtue of being very friendly and thought-provoking for students. So thank you Scott, yet again.

Just to mention, even if quantum mechanics had

neverbeen discovered, these same issues (determinism, computational complexity, origin and fate of the universe,etc.) would still be discussed, and surprisingly many of the same mathematical and philosophical arguments would apply.A student-friendly introduction to this literature is Lenore Blum’s highly readable

Computing over the Reals: where Turing meets Newton(which includes many further references). One point of view is that quantum information theory is simply the most natural “complexification” of the real mathematics that Blum’s article discusses.As usual in mathematics, complexification introduces new invariances and new insights that make links to other branches of mathematics (and physics) easier to see. On this grounds, we can expect a glorious mathematical future for quantum information theory.

An alternative point of view, which is particularly congenial to engineers, regards the fundamental equations of quantum mechanics as settled … maybe not perfectly, but well enough for practical work, in particular as the equations describe nonrelativistic dynamics and measurement. In other words, just accept Nielsen and Chuang Chapters 2 and 8 as gospel!

From this engineering point of view, quantum equations of motion are rather like fluid equations of motion, and quantum information theory is rather like computational fluid dynamics (CFD).

If we do a bit of historical digging, we find that biggest breakthroughs in CFD over the past twenty years have been partly associated with improved understanding of the CFD equations, but to an even greater extent, have been driven by better numerical techniques for solving them. Overset grid techniques in particular have assumed a central role—these techniques allow engineers to wrap numerical grids around the “lumpy” aerodynamic objects that arise in practice.

The article

Thirty years of development and application of CFD at Boeingreviews recent CFD history in a very student-friendly form. It is interesting that this review article, written in 2002 by Boeing’s own CFD experts, failed to foresee the further acceleration of CFD techniques to become the enabling technology behind the half-trillion dollar enterprise that is the Boeing 878 Dreamliner; this provides a not-too-common example of a technology that works far better in the real world than even its creators envisioned.In our own experience, the analog of overset grids in CFD appears to be Kähler manifolds in quantum simulation … a kind of all-purpose mathematical object on which it is particularly convenient to realize quantum equations of motion. It will be interesting to see how far these techniques can be pushed.

To bring this post to a point, we discern in the above examples that quantum information theory helps humanity to create ideas that work, to create technologies that work, and most important (IMHO), to create communities that work—meaning peaceful communities that create resources and jobs. It is desirable for students, especially, to appreciate that quantum mechanics is a big elephant that can be embraced from many different directions and in service of many different objectives: all of which are wonderful.

———

Note: I set out to post the least cynical, most cheerfully optimistic essay that I could .. others are encouraged to try too, and links to interesting literature are especially nice. 🙂

Comment #38 April 5th, 2007 at 8:19 am

Scott says

I believe in every interpretation of quantum mechanics to the extent it points out the problem, and disbelieve in every interpretation to the extent it claims to have solved it.Interpretations of quantum mechanics, unlike Gods, are not jealous, and thus it is safe to believe in more than one at the same time. So if the many-worlds interpretation makes it easier to think about the research you’re doing in April, and the Copenhagen interpretation makes it easier to think about the research you’re doing in June, the Copenhagen interpretation is not going to smite you for praying to the many-worlds interpretation. At least I hope it won’t, because otherwise I’m in big trouble.

Comment #39 April 5th, 2007 at 9:43 am

But if quantum mechanics is an ‘operating system’, or a ‘syntax’, does it make any sense to interpret quantum mechanics?

s.

Comment #40 April 5th, 2007 at 10:03 am

I just stole Peter Shor’s comment for a blog post much less scientific than this one, linked in the URL field.

Comment #41 April 5th, 2007 at 10:23 am

I like that, serafino: “The Interpretation of Windows XP.” 🙂

At the risk of sounding like some Continental philosopher smoking his pipe and uttering vacuous profundities: the basic goal with interpretations is start from the “syntax” of quantum mechanics, and connect it to the “semantics” of what we actually experience.

Comment #42 April 5th, 2007 at 11:26 am

I see the measurement problem as a “hard problem” like consciousness or the existence of the universe, not an “easy problem” likeWhy single out measurement from the rest of quantum probability, given that it is an unavoidable corollary of the theory? You know full well that if a quantum Alice entangles with i.i.d. quantum Bobs in repeated trials, then her state will concentrate at the perception that the Copenhagen interpretation is true.

I have no idea what an answer would look likeLike Feynman, I have no idea what the question should look like.

Comment #43 April 5th, 2007 at 11:54 am

Cool comment from Shor. Now, if you can get Deutsch and Feynman to comment, as would be possible in some worlds… I’m deliberately NOT putting on my Science Fiction Author hat, which would tempt me to babble about the Multiverse.

Would you be willing to comment on the below? It appeared in the past 2 days:

arXiv:quant-ph/0407008 (cross-list from quant-ph) [ps, pdf, other] :

Title: Classically-Controlled Quantum Computation

Authors: Simon Perdrix, Philippe Jorrand

Comments: 20 pages

Quantum computations usually take place under the control of the classical world. We introduce a Classically-controlled Quantum Turing Machine (CQTM) which is a Turing Machine (TM) with a quantum tape for acting on quantum data, and a classical transition function for a formalized classical control. In CQTM, unitary transformations and measurements are allowed. We show that any classical TM is simulated by a CQTM without loss of efficiency. The gap between classical and quantum computations, already pointed out in the framework of measurement-based quantum computation is confirmed. To appreciate the similarity of programming classical TM and CQTM, examples are given.

Comment #44 April 5th, 2007 at 12:00 pm

Another doofus question:

Does measurement based quantum computing (MBQC) have anything to do with the problem of measurement? (I always thought it did.) In fact I thought that the existence of MBQC pretty much refuted the big idea behind measurement-free or collapse-free interpretations. Am I wrong?

Comment #45 April 5th, 2007 at 12:05 pm

In computer science, time flows down! (Just like trees grow down.)And in all sorts of crazy directions when you call a procedure or function. And then there are digital circuit diagrams where time flows in many different directions.

Comment #46 April 5th, 2007 at 1:56 pm

Would you be willing to comment on the below?Jonathan: The paper in question shows that you can have a universal quantum Turing machine with a classical tape head, or in other words that it’s only the tape symbols that need to be in superposition. This looks to me like a correct result, also easy and unsurprising.

Comment #47 April 5th, 2007 at 2:11 pm

James: Despite the claims we sometimes hear to the contrary, no interpretation of quantum mechanics can

everbe “refuted” by measurement-based quantum computing or any other quantum-mechanical phenomenon. This is because, by definition, all interpretations lead to exactly the same predictions for all such phenomena.(If they don’t, then we should think of them not as interpretations but as rival physical theories.)

Having said that, MBQC really is a beautiful discovery, and it’s reasonable to hope that understanding it better might clarify some of the issues in quantum foundations.

Comment #48 April 5th, 2007 at 6:35 pm

James: in addition to what Scott says above, the “magic” of MBQC has more to do with the interplay between measurement and entanglement. To get non-trivial computation out of measurements, you also need the quantum correlations + ‘classical’ feed-forward; and it is not unusual to see MBQC compared to teleportation. There’s nothing in MBQC that isn’t already contained in much more popular (or more frequently popularized) instances of the oddness of quantum info.

Consider what a classical counterpart to MBQC would look like (with some fudging in order to get something which is only slightly trivial, instead of completely trivial). Replace the entanglement graph by bits which are either correlated between neighbors in a graph, or instructions to toggle your bit depending on an interaction with someone else. Just in establishing the correlations corresponding to the entanglement graph, you are performing a computation (with your measurements just corresponding to looking at the actual bit-values, rather than blindly copying or toggling the single bit you have).

Comment #49 April 5th, 2007 at 7:05 pm

Scott,

Thanks for answering my naïve questions.

I absolutely agree with what you said re interpretations vs. rival theories. What this implies to me is that every interpretation must have something essentially equivalent to collapse, (perhaps disguised). For many-worlds this has always seemed obvious to me, they just call it splitting, but it waddles like a collapse and it quacks like a collapse, so it’s a duck. I guess for Bohm, the collapse-equivalent is choosing or making apparent the previous choice of one of many, perhaps infinitely many, nearby trajectories. (If there really is an equivalent to collapse in an interpretation, but one says not by choosing other words, that seems like a fraudulent sophistry to me.)

In a parallel inference, if one interpretation, e.g. hidden variables, requires nonlocality, then they all do, no matter how much they try to disguise it. (Of course I recognize the distinction between Einstein-nonlocality and signal-nonlocality, although this is not easy to grasp, and even harder to believe in, sort of like the twin paradox.) It has always seemed to me that the nonlocality denying interpretations pretty much rely on “don’t ask, don’t tell” or “you can’t ask that question” which just means ducking the issue.

(I must confess that I have never been able to comprehend any of the “you can have locality, but you must give up realism” positions. To me they all seem to require either a terrible misuse of language, or they go back to “don’t go there.”)

I am eagerly hoping for something like the dialogue between “Axioms” and “You” in “Is P versus NP Formally Independent”, only between a skeptic and a believer in (advocate of?) this locality-yes, reality-no position. Maybe it would help me get my head around it.

Incidentally, the same argument above would imply that if one interpretation requires “rolling the dice”, they all do, including Bohm. I’ll buy that. The so called determinism is just another form of rolling one big die at the beginning of the universe, instead of lots of little dice all the time. I will admit the trajectories are a neat implementation of this idea, however.

Based on the above, I conclude that any reasonable interpretation of QM must include all three of randomness, Einstein–nonlocality and collapse. Strangely, this seems to be noncontroversial for randomness, but highly controversial for collapse and Einstein-nonlocality.

On the other hand, just because I call something an interpretation rather than a rival theory doesn’t mean this is true. There is already quite a literature of authors accusing Bohm of being a rival theory, rather than an interpretation. I have always thought the mathematical evidence for Bohm being identical to SQM, and hence an interpretation, was pretty airtight, modulo one division by zero issue. But I was surprised by your proof that Bohm does not work in finite dimensional Hilbert spaces. (I had certainly never heard that one before.) If that is true of Bohm and not true of other interpretations, I think that would be a strong argument for the rival-theory position.

I am tempted to ask what happens to Bohm if you try to coarse-grain it or project it down or in some other way reduce it to a finite dimension, but that sounds like too ill-formed a question even for me. Instead I will ask how Bohm deals with your counterexample if it is embedded in an infinite dimensional Hilbert space.

My guess is that it merely boils down to choosing one from a set of nearby trajectories.

Neil,

Thanks for trying to help. I will study your reply some more. I don’t know where the magic comes from, but to me at least the weirdness of entanglement is covered by the Einstein-nonlocal aspect.

Sorry for such a long post. I know this is not my blog.

Jim

Comment #50 April 5th, 2007 at 8:43 pm

At the risk of sounding like some Continental philosopher smoking his pipe and uttering vacuous profundities: the basic goal with interpretations is start from the “syntax” of quantum mechanics, and connect it to the “semantics” of what we actually experience.Yes, Scott, you’re at risk.

I really don’t understand what the problem is. I understand that people who are less used to quantum probability would perceive a fundamental problem here. But once you get used to the later chapters of Nielsen and Chuang, for example, then what is the real mystery? You can learn that if you dephase a qubit, it becomes a c-bit. You can learn that if you start with qubit A, entangle it with qubit B, then dephase B, then B has become classical and, voila, has measured A. So there measurement appears, modelled as a quantum operation. Measurement is not some tacked-on extra thing; it appears

insidethe game with unitary operators if you combine them properly.Of course, a realistic observer is more complicated than a dephased qubit and a realistic measurement is more complicated than creating entanglement between qubits. But why should realistic measurement be fundamentally different from this fairly simple special case, which is after all important in quantum algorithms?

I have the feeling that “the measurement problem” serves one of two purposes. It is either interesting for people who haven’t learned the above; or it is a “hook” to get people to study better questions in quantum probability. Otherwise, again, I just don’t know what people are trying to accomplish. I don’t want to accuse anyone of being a dolt, least of all any of the serious experts, but these discussions bother me more every time I see them. Certainly in my own notes (if I ever find time to revise them), I hope to debunk the measurement problem rather than philosophize over it.

Comment #51 April 5th, 2007 at 9:50 pm

James, it’s a pleasure to see you struggle intelligently with some of the issues all of us abyss-dwellers eventually face.

To answer your questions:

If you embed my counterexample into an infinite-dimensional Hilbert space, what you’ll get is a wavefunction with a discontinuity, which a Bohmian would reject as “pathological.”

You can define hidden-variable theories in finite-dimensional Hilbert spaces that are

closeto Bohmian mechanics; the only problem is that they won’t be perfectly “deterministic” (in the restricted sense that Bohmian mechanics is).So perhaps one should say: while the

ideaof hidden-variable theories doesn’t lead to any physical predictions, Bohm’sspecifichidden-variable theory does implicitly make a prediction — namely, that the right Hilbert space is that of the positions of point particles in R^{3}. And this is a prediction that, even in Bohm’s time, there were excellent reasons for thinking was wrong.Comment #52 April 5th, 2007 at 10:03 pm

Greg, I’m tempted to tell you what others have told me about music: if a particular analysis of the measurement problem doesn’t “do it” for you (i.e. doesn’t give you any new insights about quantum mechanics), then you shouldn’t bother studying it.

As for whether these analyses “do it” for anyone, the evidence we have is that many-worlds led Deutsch to quantum computing, Bohmian mechanics led Bell to Bell’s inequality, Copenhagen and many-worlds have helped Shor think about his research (as the man himself just told us), etc. etc.

Comment #53 April 5th, 2007 at 10:24 pm

Copenhagen and many-worlds have helped Shor think about his researchSure, I appreciate both of these for their intuitive or pedagogical value. I use them too. But I don’t think of them as answers to a problem. Granted, Copenhagen is an answer to a question: it’s the correct description of what an agent in a quantum world perceives. But that’s an empirical question, not a philosophical one.

Okay, I concede that pedagogy and intuition count as a third purpose to the measurement “problem”.

Comment #54 April 6th, 2007 at 12:11 am

Here’s a question, Greg: is there

anythingthat counts for you as a philosophical problem?Comment #55 April 6th, 2007 at 4:17 am

About those philosophical problems, I remember that Dirac gave a speech in Rome (April 14, 1972), talking about the development of QM. He pointed out the crucial role of the present quantum formalism, that he thought wasn’t the ultimate and definitive formalism. Since I was there, with my tape recorder, I can quote his words precisely.

“I must say that I also do not like indeterminism. I have to accept it because it is certainly the best that we can do with our present knowledge. One can always hope that there will be future developments which will lead to a drastically different theory from the present quantum mechanics and for which there may be a partial return of determinism. However, so long as one keeps to the present formalism, one has to have this indeterminism.”

Comment #56 April 6th, 2007 at 5:30 am

Serafino says:

“I must say that I also do not like indeterminism. I have to accept it because it is certainly the best that we can do with our present knowledge.”Just to link the above statement to information theory, it is a characteristic prediction of quantum mechanics, confirmed by real-world experience, that experiments can have many possible outcomes. And this obvious and seemingly boring statement has profound mathematical consequence.

Example: if we scatter 10^16 photons off a test mass (e.g., a LIGO or a nanoscale cantilever), and measure them with homodyne interferometry, then each photon yields a binary-valued data point, and therefore, each experiment yields a data record that is a binary number of 10^16 bits (note: the preceding examples are not abstract … they are the way that measurements are done in the real world).

Now comes the mathematical point: almost all members of the ensemble of 2^10^16 possible data records are algorithmically incompressible. And this statement is nothing more than the Kolmogorov-Chaitin definition of randomness.

So quantum mechanics is necessarily random,

and so is any other theory that predicts sufficiently large ensembles of possible data records.The preceding is what we teach engineering students about the origin of quantum randomness. The main virtue of this approach is that it encourages students to move on to practical applications, rather then stopping to “solve the mystery of quantum randomness.”

Would we really want to remove the randomness from quantum mechanics, if the price were that every data record would necessarily be algorithmically compressible? That would be much too high a price!

Of course, the mystery of quantum mechanics can be rescued by introducing hidden variables as “God’s crib sheet.” But for engineering purposes there is not much point in doing this, so long as we are not allowed to look at the crib sheet.

Of course, three quantum mysteries (or more) remain. Why is the quantum state space of the universe so large? Why is so little of this state space accessible to us, who are embedded within it? Why do the quantum equations of motion so carefully guard the classical-quantum boundary from direct observation?

Comment #57 April 6th, 2007 at 5:49 am

OK, a question now. Can anyone point me to a good description of one of these “you can have locality, but you must give up realism” interpretations? It seems to me that the real meaning of Bell’s theorem is that I am forced to give up on locality no matter what, so realism is kind of a red herring. (In terms of wave function like descriptions, Bell’s theorem says that I am really required to think of the wave function for two particles as a function on space^2, not two functions on space; in terms of Hilbert space descriptions it says that I can’t mimic an entangled state by a pure state in some larger Hilbert space.)

I’d be curious to see how these non-realist local theories get around this. Thanks in advance!

Comment #58 April 6th, 2007 at 6:13 am

David, it depends what you mean by “local.” You can think of the Copenhagen interpretation as “local but not realistic” in the following sense: nothing Alice can do to her half of an EPR pair can possibly affect Bob’s

density matrix, and in Copenhagen the density matrix is all there is.Of course, if you want a density matrix describing Alice’s and Bob’s systems

jointly, then it has to be entangled. But maybe that’s not so bad, since even in the classical world, we know that a joint probability distribution over two systems in general has to be correlated.Comment #59 April 6th, 2007 at 7:29 am

Here’s a question, Greg: is there anything that counts for you as a philosophical problem?No! I believe in philosophical implications, but not philosophical problems.

I don’t want to get too impolitic about philosophers at the moment, but I can say this about the boundary between science and philosophy. Historically, philosophy has been a repository for confusions in science for which no one had an answer, or for which there can be no answer. Until Kepler and Newton and those guys, the motion of the planets was a good philosophy problem. Now it mostly isn’t, it’s physics. It’s no longer fun to debate why Mercury chases Venus; instead, you just learn Newton’s answers.

I view quantum information theory, the whole arc of it from von Neumann to Holevo and Shor, as the same cure for the philosophy of quantum mechanics. It clears the air. I was thrilled at QIP 2004, because the whole conference made quantum philosophy trite.

Even so, philosophical implications are a good thing.

Actually I want to emphasize a specific point here. I am convinced that the much of the driving force of quantum philosophy is the old-fashioned language of separate unitary operators and measurements. If you only know the Copenhagen business as Max Born knew it, then it really is confusing and an invitation to philosophize. But once you get used to mixed states and quantum operations, and generally classical probability as a special case of quantum probability, then the measurement “problem” really seems like a pretense.

Comment #60 April 6th, 2007 at 7:38 am

Of course, if you want a density matrix describing Alice’s and Bob’s systems jointly, then it has to be entangled. But maybe that’s not so bad, since even in the classical world, we know that a joint probability distribution over two systems in general has to be correlated.I completely support this explanation, except for the word “maybe”. Entanglement is no more than a flavor of correlation.

Also one should note the bias in the fossilized term “density matrix”. For a lot of physicists, indeed for virtually all of them until the past few decades, density matrices are “just a formalism”. A much better name is

mixed state, whereas a vector state in a Hilbert space is apure state. Mixed states let you completely recover locality. When I learned about mixed states, they lifted a fog.Anyway, the short answer to your question, David, is that you don’t give up on locality at all, you just redefine it.

Comment #61 April 6th, 2007 at 7:49 am

Thanks, that helps a lot!

Comment #62 April 7th, 2007 at 1:24 am

This is a topic I can get angry about. Where might we be by now, if the dominant attitude in physics had always been:

of coursequantum mechanics is incomplete; the work of fundamental physics will remain unfinished so long as all we have are quantum theories. I suppose that, if nothing else had been invented, the prevailing theories might be Bohmian field theory and general relativity, or even a Bohmian string theory; and the big conceptual problem in physics would be to understand the relationship between Bohmian nonlocality and relativistic locality.A world in which Bohmian mechanics was the dominant paradigm would, I think, be intellectually much healthier. It may sound strange to say that even today, there is a prevailing complacency towards the meaning of quantum theory; but just see how many people there still are who feel their intellectual duty is to adapt themselves to quantum reality, become comfortable in a quantum universe, etc. So far as I can tell, this is mostly a matter of ceasing to ask questions such as, why does an observable take the particular value it does; did it have a value before the measurement; is the ‘quantum state’ the actual state of the object, or just an aid to calculation; and so forth. These are completely natural questions to ask, and they would be a lot harder to ignore if

Bohmianmechanics, with its classical determinism and objectivity, were the orthodoxy, and the Copenhagen interpretation was the minority viewpoint.Comment #63 April 7th, 2007 at 2:36 am

Mitchell, out of genuine curiosity, let me ask two questions to try and bridge the gap between your way of thinking and mine.

(1) Does it matter to you whether people adopt Bohmian mechanics or any of a dozen

othernonlocal hidden-variable theories that I could write down, with different guiding equations? As I mentioned earlier, the big problem I have with Bohmian mechanics is that it only works in infinite-dimensional Hilbert spaces.(2) Do you think a physicist would be wrong to say of a philosophical paradigm, “I’ll embrace this if, and only if, it leads to new insights into concrete problems that I’m trying to solve”?

Comment #64 April 7th, 2007 at 5:32 am

(1) Scott, if you mean Bohm-like theories based on

~~observables~~beables other than position, at least they all have equations of motion that don’t need “measurement” as a primitive concept. So in that sense, yes, any of them would be a refreshing return to objective physics. It might even be an enlightening switch to tackle the mind-body problem from the perspective of, say, momentum-space Bohmian mechanics. That said, I’m not so sure that they are all of a piece. Howard Wiseman has a curious unpublished theorem which says that a particular property, which I’ll call “WVC”, is true only for the position basis (I’ll see if he wants to discuss it here). Also, if you took the momentum observables of your quantum theory to be the classical beables of your Bohmian theory, I am somewhat skeptical that they would still warrant the name of “momenta”. It would be time for a rethink of nomenclature from first principles, something that’s not necessary for position-basis Bohmian mechanics.Objective-collapse theories like Ghirardi-Rimini-Weber also pass my basic objectivity test, as do sets of decoherent histories, although the formalism is getting a little mysterious there – e.g. you can have a coarse-graining in which observable X is only specified as taking some value in an interval (a,b); to interpret that as a beable, I think you’d have to regard X as

interval-valued. One of the incoherencies which has been tolerated in the quantum age is the idea of objectively indeterminate properties: the particle has a position, just no particular position. It’s easy to joke about, but I do think the effect has been to retard progress. If you don’t even notice the incoherence, you’re not likely to do anything about it.Infinite-dimensional Hilbert spaces… That shouldn’t be a problem for

you, all you have to do is dust off Blum, Shub, and Smale, right? OK, there’s lots of talk about finite-dimensional Hilbert spaces in quantum gravity, and I’m not sure how the holographic principle looks from a Bohmian perspective. As I too see the charm that a discrete fundamental physics would have, I suppose I would look for discrete approximations to continuum Bohmian gravity [algebraic-geometric hocus-pocus redacted here]… I’m getting a little too jolly here, let’s move on to(2) which I will answer with a question of my own: Do you think my objection to the notion of objectively indeterminate properties is just a “philosophical paradigm”? I think it’s more like a prerequisite of rational thought.

Comment #65 April 7th, 2007 at 9:04 am

Where might we be by now, if the dominant attitude in physics had always been: of course quantum mechanics is incompleteWe would only have gotten less done. When I see important constructions like the Lindblad equation — which is a mixture of the Schrodinger equation and classical Brownian motion — I see the philosophical dissatisfaction with quantum probability fade.

It is certainly faded among operator algebraists, for whom quantum probability is no more than non-commutative probability. Even Nielsen and Chuang does not convey the true message that classical and quantum probability satisfy virtually identical axioms; all you have to do is strike commutativity.

If the community had stuck forever to the sentiment that there has to be something wrong with quantum probability, it would not have found these great ideas that undermine that sentiment. Or I should say,

whenthe community stuck to that sentiment, it did not find most of these ideas. I concede that John Bell did not like quantum probability, and he found the Bell inequalities. But he was the last exception, and that was 40 years ago.As for “Bohmian mechanics”, it’s not a separate theory at all, it’s just a way of explaining quantum mechanics. I don’t want to say that it is never useful, but it is a very conservative, dogmatic explanation. If you want to put it above all other explanations, it reminds me of Minkowski’s description of certain skeptics of relativity: It’s like hearing a symphony with cotton in your ears.

Comment #66 April 7th, 2007 at 9:18 am

Scott, funny how your position regarding finite dimensional Hilbert spaces is completely orthogonal to mine, I guess it is what you get used to that matters. Almost all physical systems are described by infinite dimensional Hilbert spaces (by QFT rather than QM). Finite dimensional Hilbert spaces are an abstraction that works only in certain limits (e.g non-relativistic limit). I was always wondering if there is any substantial difference in interpretation issues, especially one invoking relativity, if the right framework is used.

As for the holographic bounds, finite entropy does not imply by itself a finite Hilbert space, I don’t see any strong argument for the latter. However, the scaling with the area rather than the volume requires some extreme level of bulk non-locality, I could imagine that being important.

Comment #67 April 7th, 2007 at 10:17 am

Moshe, why does finite entropy not imply a finite Hilbert space dimension? (This probably goes back to our different definitions of the word “entropy.”)

I thought (from reading Bousso’s papers and talking to Susskind) that even in string theory, the log of the Hilbert space dimension goes like the surface area over 4. I saw that as a great

virtueof the theory. If it needed an infinite-dimensional Hilbert space, then I’d feel like string theory couldn’t possibly be correct.Comment #68 April 7th, 2007 at 10:31 am

Yeah, it does go back to that notion of thermodynamic vs. Von Neumann entropy. An ideal gas in some finite volume has an infinite dimensional Hilbert space and a finite thermodynamic entropy (at any finite temperature). There is some hope that finite *maximum* entropy associated with a region of space imply finite Hilbert space associated with that region of space. Problem is I cannot figure out what does it mean to associate Hilbert space OR entropy with a finite region of space. Once gravity is involved the notion of finite part of space is not even well-defined.

I am not sure which argument you refer to, in AdS/CFT for example the entropy of any bulk configuration (say a black hole) is explained in terms of conventional quantum field theory (living on finite space). The number of degrees of freedom is finite, but the Hilbert space in infinite dimensional.

I’m having fun avoiding argument for a change, but let me just say that in my mind philosophical preconceptions are something to strongly avoid when searching for an unknown theory. I am wondering though why you have such a strong intuition that a finite Hilbert space is necessary, where there are all those well-understood examples of non-perturbative quantum gravity where things simply don’t work that way.

Comment #69 April 7th, 2007 at 11:00 am

More from the peanut gallery:

Scott,

I want to ask about decoherence as a rival theory, rather than merely an interpretation.

As I understand lecture 11, you treat decoherence as just another interpretation, not an alternate theory. Is that truly your view? Is that the general view? Would Zurek agree to that?

On the other hand, I had hoped that decoherence could be viewed as an actual rival theory that went beyond QM. (Not a rival theory of type A, that disagrees with SQM at some point, but a rival theory of type B, that agrees with SQM at all points where both make predictions, and then goes on to make additional testable predictions.)

Any how, if decoherence really is only an interpretation, they sure seem to be going through a lot of mathematics to make themselves feel better.

Of course, this idea that QM is not complete has a long history (smiley)

I always thought that QED and QFT, not to mention string theory put paid to the idea that there was nothing useful to be added to SQM, but most people don’t seem to interpret the question that way. Anyway, I would like to ask if you also hold that SQM is complete in some important sense. If so, I will need to ask for an explanation of what this means.

Tying this back to decoherence as more than just an interpretation:

As I understand it, decoherence (D) basically does the same thing as collapse (C).

However, the hope would be you could do a better job of predicting when and where D or C will occur by observing the environment. Perhaps you could even control or engineer the environment to delay or accelerate the D/C in order to benefit your quantum computer? (Maybe this is what D-Wave needs to do!)

Or does decoherence merely consist of solving the three body problem (system, apparatus, environment) in SQM?

David Speyer,

I really like your geometric interpretations of Bell’s theorem. They give me new clues to chew on.

Greg,

“Mixed states let you totally recover locality.”

This is totally new to me, could you please explain further, or point me to a preexisting explanation?

Thanks,

Jim

Comment #70 April 7th, 2007 at 11:11 am

Scott: Technically speaking, infinite dimensions need not imply infinite entropy. To understand what is going on, it helps to brush up on the spectral theorem for infinite-dimensional Hilbert spaces. (Let’s say countable-dimension for now. Yet bigger Hilbert spaces are pathological, at least for the fundamental laws of physics.)

The spectral theorem says that the spectrum of a self-adjoint operator, such as either a measurement or a density operator, has two parts: A point spectrum with honest eigenvectors, and a continuous spectrum with only approximate eigenvectors. A typical example of the latter is measuring position for a particle trapped in an interval. An eigenstate would be a delta function, which is not normalizable. (That is, <psi|psi> cannot be made finite.)

Nonetheless, a density operators is always pure-point; in fact, the set of eigenvalues is always an absolutely convergent series. (Whose sum is 1, of course.) Depending on how you choose this series, the total entropy may be finite or infinite. It is also possible to choose a Hamiltonian so that any bounded-temperature state has finite entropy.

However, string theorists could argue that this is splitting hairs, because if all finite-temperature states have finite entropy, then you could say that the Hilbert space of the universe is approximately finite-dimensional. They are certainly prepared to grasp the distinction between finite entropy and finitely many states, but my guess is that they don’t consider it important. In fact, my guess is that they view finite entropy as the more intrinsic notion. (At least, I would if I were one of them!)

Comment #71 April 7th, 2007 at 11:24 am

Greg, in my experience there is no universal agreement on that among string theorists (at least those who are at all interested in the question), we are all trying to figure out the rules of the game.

\

So far, one of the strongest points for me is that questions regarding gravitational entropy are always mapped via various dualities to similar questions in conventional physics (ideal gas and slight generalizations thereof). Systems with finite dimensional Hilbert spaces are few and far between, and as far as I can remember now are not utilized to describe gravitational entropy. I see no reason that utilizing only such systems (say spin systems) is an absolute necessity. It is entirely possible I am missing something obvious…

\

Also, I am not sure the distinction between finite entropy and finite Hilbert space has to do with the continuous spectrum. A simple Harmonic oscillator has finite entropy (for fixed temperature, or for a fixed energy) but infinite dimensional Hilbert space, and the spectrum is discrete.

Comment #72 April 7th, 2007 at 11:58 am

Also, I am not sure the distinction between finite entropy and finite Hilbert space has to do with the continuous spectrum.My only point is that a normalized density matrix cannot have a continuous spectrum; it is always diagonalizable, in fact trace class.

Also, I cannot resist a bit of snark for Scott. If you appreciate the distinctions among continuous-spectrum operators, point-spectrum operators, and trace-class operators, as you have to do to understand finite vs infinite entropy, then you are doing

operator algebras. (Gasp!)“Mixed states let you totally recover locality.”This is totally new to me, could you please explain further, or point me to a preexisting explanation?

Well, there is more than one thing to say, but the first step is really simple. In classical probability, there are states (distributions), then there are joint states, then there are marginal states. Taking a marginal is a one-sided inverse to lifting from a state to a joint state. In order to have any theory of locality, you have to have both joint states are marginals.

In the quantum case, there are pure states (vectors) and mixed states (density matrices). If you only learn about pure states, then you cannot have marginals, because in general the marginal of a pure state is a mixed state. Within the world of mixed states, everything is fine. The marginal of a mixed state is another mixed state.

It is easy to see that mixed states satisfy the first property of locality. Namely, if Alice and Bob share a joint state, then nothing that Alice can do at a distance can change Bob’s marginal state.

Nielsen and Chuang explain these matters in ample detail. They do not, however, pound the table the way that I like to. For example, they conventionally refer to the marginal of a mixed state as a partial trace. That makes it sound like it’s just a formalism, and not supporting leg of interpretation.

Comment #73 April 7th, 2007 at 12:30 pm

Jim,

As I understand lecture 11, you treat decoherence as just another interpretation, not an alternate theory. Is that truly your view?Decoherence is not an interpretation — it’s a

phenomenonpredicted by quantum mechanics, which no one really disputes. It’s often presented as a recent discovery, but it actually goes back to Schrödinger, von Neumann, etc. It’s just working out the details that’s the hard part.Decoherence makes no predictions — none whatsoever — beyond those of standard quantum mechanics. Indeed, that’s precisely the point of it.

The “philosophical” question that people debate is whether decoherence is enough, by itself, to banish the interpretive problems, or whether you also need something else.

Is that the general view?Yes, what I said above is the general view.

Would Zurek agree to that?Yes, I think so. 🙂

Comment #74 April 7th, 2007 at 12:42 pm

Scott and Greg,

Thanks very much.

I’ll start on N&C.

It looks like it will take a while,

But it should be fun.

Comment #75 April 7th, 2007 at 12:47 pm

The “philosophical” question that people debate is whether decoherence is enough, by itself, to banish the interpretive problems, or whether you also need something else.There is a legitimate physics problem here, namely to describe the actual process by which quantum rules degenerate to a classical limit. It is fair to say that it is far from completely understood; and since it isn’t, there is in principle room for a radical new extension of quantum probability.

However, just because such an extension is conceivable, that doesn’t make it likely. In an idealized situation, it is commonplace to make a classical limit from a quantum system, with decoherence alone. A dephased qubit is a classical bit, case closed. So there is no positive evidence that decoherence is inadequate.

This discussion is analogous to searches for a fifth fundamental force, besides gravity, weak, strong, and electromagnetism. Since no one understands how the world is composed of the first four, it’s reasonable to look for a fifth one. Just as long as you don’t wishfully suppose that it exists, because there is no positive evidence that it does.

Comment #76 April 7th, 2007 at 12:49 pm

Moshe, since the distinctions between entropy,

maximumentropy, and log(dim(H)) can get extremely confusing, let me tell you the quantity that interests me in purely operational terms. I’m interested in the maximum number of bits that can in principle be stored in a given region, such that any one of those bits, of our choice, can later be reliably retrieved. Call that quantity N.So, if N can be finite even with an infinite-dimensional Hilbert space (which is something I’d have to understand better), then maybe I

amOK with infinite-dimensional Hilbert spaces after all.On the other hand, if N can be infinite then I’m

notOK, since then I’d worry about an infinite amount of computation being performed with a finite amount of resources. And Idohave a preconception that that’s impossible, in the same way I have preconceptions that causality isn’t violated and the Second Law is true. That is, these are all things I’m willing to give up if forced to, but my price is exorbitantly high.Comment #77 April 7th, 2007 at 1:03 pm

So, if N can be finite even with an infinite-dimensional Hilbert space (which is something I’d have to understand better), then maybe I am OK with infinite-dimensional Hilbert spaces after all.The point is that if the Hilbert space is finite-dimensional, then the information capacity of the system is limited

algebraically. But you could also limit the capacity thermally or dynamically. To take Moshe’s example, the nth state of a simple harmonic oscillator has energy n+½. So to double the entropy of a state of the oscillator, you have tosquareits expected energy or its temperature. Anyone can imagine, and physicists can sometimes derive, reasons that heating a system to a googol degrees is unphysical.Part of the subtext of this is Strominger’s famous derivation of the Beckenstein-Hawking entropy of a black hole (well, certain idealized black holes) versus the unpersuasive calculations in loop quantum gravity. As I understand it, Strominger did a dynamical calculation of entropy, whereas the LQG paper that I saw imposed a capacity limit algebraically. I think that it misses the point to replace dynamical solutions with algebraic fiat, and it’s something that I have seen elsewhere in anti-string-theory work.

Comment #78 April 7th, 2007 at 5:26 pm

Scott says:

Decoherence is not an interpretation … it’s just working out the details that’s the hard part …… which is why few textbooks on quantum mechanics start out with a discussion of measurement as a decoherent process — Chapter 2 of Nielsen and Chuang is a prominent exception. Howard Carmichael has a new textbook coming out that will likely do justice to this topic. It’s not so easy to answer the question “what is it that avalanche photodiodes measure, exactly?”

Quantum measurement is IMHO such an inexhaustibly rich subject that students who begin by studying measurement are at-risk of never studying quantum dynamics at all. Doh! We mustn’t risk that! 🙂

Comment #79 April 7th, 2007 at 7:19 pm

Greg, as an explanation, “noncommutative probability” is up there with Moliere’s “dormitive virtue”. It is one of the things that needs explaining! If a theory features fundamental probabilities a la Kolmogorov, I can interpret them as counterfactual relative frequencies. Such an interpretation is not possible for complex-valued probability amplitudes or Wigner’s quasiprobabilities, and I doubt that noncommutative probability spaces provide an explanation either. Classical probability theory has at least one natural ontological interpretation, its nonclassical formal generalizations do not. With Bohmian mechanics, it should be possible to

explainwhere noncommutative phenomenological probabilities come from,precisely because it has a mechanism. (And the same goes for why quantum computing is more powerful than classical computing, by the way.)Comment #80 April 7th, 2007 at 7:40 pm

If a theory features fundamental probabilities a la Kolmogorov, I can interpret them as counterfactual relative frequencies. Such an interpretation is not possible for complex-valued probability amplitudes or Wigner’s quasiprobabilitiesBut it is possible with density matrices, which is the way that things are done in standard noncommutative probability. That is, a density matrix is the correct summary of all of the probabilities. It doesn’t “explain” in the sense of revealing underlying determinism, since that isn’t really possible.

Basically words like “explain” and “ontological” are loaded. You are using them as a request for determinism. But there is a theorem that there isn’t determinism in any natural form. Bohm’s view was, even if we can’t have natural determinism, let’s describe it as an artificial kind of determinism. So okay, you can do that, but it isn’t a different theory, just a different explanation, and it’s only occassionally useful.

Comment #81 April 7th, 2007 at 8:00 pm

Mitchell: You might not

likesaying that a particle is objectively in superposition, but I don’t see how this view is logically incoherent, or how rejecting it is a “prerequisite for rational thought.” In debates about quantum foundations, saying “anyone who disagrees with me is irrational” is sort of the nuclear option… 🙂Comment #82 April 7th, 2007 at 8:06 pm

I’m not requesting determinism, I’m requesting something much more basic, namely that theories should be clear about what it is that they allege to exist. You’ve just said, in effect, that the universal density matrix is the bottom line for you, and that it can be read as a big tabulation of classical probabilities, over pure states I guess. That would mean that the actual state of things is made up of those pure states. But which pure states? Unless you have an answer to that, you only have a phenomenological theory.

Comment #83 April 7th, 2007 at 8:13 pm

Unless you have an answer to that, you only have a phenomenological theory.Ah, now we come to the heart of the matter. Modern physics was born when Galileo suggested that, instead of seeking the “true nature of motion,” we should just try to describe it phenomenologically. And it’s been a pretty successful strategy for the last 400 years, wouldn’t you say?

Comment #84 April 7th, 2007 at 8:13 pm

Scott, I don’t mind if someone says that state vectors are the actual states. I just demand clarity regarding what I am being asked to entertain as a description of reality – classical beables, state vectors, both, neither.

Comment #85 April 7th, 2007 at 8:32 pm

No, mitchell, I’m saying that density matrices are the actual states.

Comment #86 April 7th, 2007 at 9:01 pm

Greg: you just made me happy! Thank you for saying what your candidate for actuality is. Can we pursue this a little further, in several directions?

1. Cosmological dynamics. You mentioned Lindblad equations. Do you have any opinion as to whether the evolution of the

universaldensity matrix is unitary? (Or it might even be stationary, if you’re an Huniverse=0 guy.)2. If you say that ρX is the actual state of entity X, can I take that statement at face value? It won’t turn out to actually be a probability distribution or a dispositional description or some other less-than-ultimate characterization of X?

3. States of subsystems. At what point do I get back the phenomenal world, with its particular experimental outcomes? Is this a multiverse theory?

Comment #87 April 7th, 2007 at 10:37 pm

Scott, indeed I would not generally relate the dimension of the Hilbert space and the number N you mention. Those two quantities have direct relation only if your system is physically made of bits, localized two state systems. For other systems, for example an ideal gas, I take it that N is the number of yes/no questions that completely specify the microstate, which I would probably just call the entropy, I agree that N has to be finite.

Incidentally, just to repeat the above, the main conceptual difficulty in my mind is defining what you mean by associating entropy, or computation, or anything else, with a finite part of space. Defining such regions only makes sense using a fixed metric, and therefore is ill-defined in quantum gravity. The known cases where entropy is counted and comes out right always count the total entropy associated with the whole spacetime.

Comment #88 April 8th, 2007 at 5:03 am

Mitchell Porter:

I just demand clarity regarding what I am being asked to entertain as a description of reality – classical beables, state vectors, both, neither.Mitchell, suppose a mathematics student asked the seemingly reasonable question: “I just demand clarity regarding what I am being asked to entertain as a description of

mathematicalreality – Peano integers, Cantor reals, ZFC sets, Godel propositions, … all, none.”What is the best way to respond to this student’s demand?

Isn’t there plenty of evidence that physical reality is at least as conceptually flexible as mathematical reality? Even though we might not wish it to be so?

Wittgenstein was among the very few modern philosophers to achieve a transition, in his personal thinking, from the “early Wittgenstein”‘s insistence upon logical clarity to the “later Wittgenstein”‘s embrace of conceptual flexibility.

This achievement is admired particularly because it is uncommon … most people retain for life the ontology that they embrace when young.

So it helps to pick a good one! And there is IMHO no one right answer. Robust ecosystems require many species. 🙂

Comment #89 April 8th, 2007 at 6:22 am

John, the question about mathematics is just as legitimate. Do Peano integers exist? Do Cantor reals exist? The answer is yes or no. And if they do exist, then they have some particular relationship to the rest of reality. Conceptual flexibility is a human attribute, not an attribute of physical or mathematical reality per se, necessitated by the radically limited nature of what we actually

know. Since we truly know so little, it is a potentially useful thing to be able to entertain diverse possibilities. But don’t miss the tree for the forest. The point of the diversity, as far as I am concerned, is to give us a chance at eventually knowing the truth. That cause is not helped by being vague about whether there actuallyareanswers. To be is to besomething, even if we don’t or can’t know what.Comment #90 April 8th, 2007 at 6:53 am

Mitchell Porter says:

Do Peano integers exist? Do Cantor reals exist? The answer is yes or no.Respectfully, many people would disagree. This same point was poignantly expressed in a recent issue of the

Journal of the History of Philosophy:——–

@article{Look:02,

author = {B. Look},

title = {“{R}adical {E}nlightenment: {P}hilosophy and the {M}aking of

{M}odernity, 1650-1750” by {J}onathan {I}. {I}srael is

reviewed.},

journal = {Journal of the History of Philosophy},

year = 2002,

volume = 40,

issue = {3},

pages = {399–400},

jasnote = {As someone trained in a philosophy department to work on the history of philosophy, I felt uneasy in one respect with this book. While Israel emphasizes the battles between philosophies and ideas, he does not concern himself so much with the process of doing philosophy. That is, if there is a failing in this book, it is that we are presented with descriptions of philosophical views without always being given an adequate account of why such views were held by individual thinkers or how the theses of the radical Enlightenment, say, are related to each other. (For example, how did radicals see the relation between naturalism and republicanism?) Philosophical views, one used to believe at least, were held for reasons and because of the results of arguments, but these arguments and reasons do not play a central role in the historical narrative. To be fair, had Israel attempted the kind of detailed philosophical account of the arguments of particular works, his book would have been a genuinely mammoth and nearly unreadable tome. Yet it remains perhaps ironic that, in leaving open the possibility that Enlightenment views were advanced for self-interested motives, for the acquisition of political power, and not out of a commitment to “Truth,” a historian who seems so sympathetic to the ideals of the Enlightenment has produced a work that, at first glance, could be used by the new opponents of the Enlightenment and its legacy: post-modernists. Be that as it may, there is no denying that this book is a very important addition to the field and will doubtless alter the way we view the intellectual history of Europe.},}

Comment #91 April 8th, 2007 at 7:27 am

Do you have any opinion as to whether the evolution of the universal density matrix is unitary?There are dilation theorems to the effect that if the evolution of the universe is non-unitary (a Lindblad equation, say), then that model lifts to unitary evolution in a bigger universe. This is a generalization of state purification: Any mixed state is a marginal of a pure state on a bigger system. So you might as well say that the state of the entire universe is pure and evolves unitarily.

If you say that ρX is the actual state of entity X, can I take that statement at face value?Well, that is what I do!

It won’t turn out to actually be a probability distribution or a dispositional description or some other less-than-ultimate characterization of X?A density matrix is the non-commutative generalization of a probability distribution. But see, I am a Bayesian: I believe that the actual state of a classical object is a probability distribution. Even before I did quantum information theory, I believed that probability distributions

areultimate. Since you didn’t demand determinism, I’m allowed to say this.At what point do I get back the phenomenal world, with its particular experimental outcomes?Immediately. If ρ is the state of a system and x is a quantum Boolean (that is, a self-adjoint projection), then Tr(ρx) is the probability that x is true. Actually, I am attracted to the operator algebraist’s notation, in which ρ is a dual vector on the algebra of quantum random variables. So they would write ρ(x).

Is this a multiverse theory?It is not a multiverse

description. All of these descriptions — Bohm, multiverse, mixed-Copenhagen — describe the same scientific theory.Comment #92 April 8th, 2007 at 7:47 am

mitchell: Okay, one more comment, so that I won’t sound evasive. But, in order to issue this clarification, I am going to pretend to be a philosopher for the moment. Again, re my answer to Scott, I don’t really believe in philosophical problems, but I am fine with philosophical implications.

Just like a probability distribution, a density matrix is an epistemological object. (I.e., it is a description of what an observer might know.) In order to be a Bayesian, and especially in order to be a quantum Bayesian like me, you have to accept ontological relativism. (I.e., what actually “is” is observer-dependent.) The ontological absolute in my working understanding of quantum probability is that different epistemological stories, by observers who are in a position to confer, are always consistent.

In particular, people generally are in a position to confer, so we can assemble an ontology which is absolute — for us people.

Now to step back to science and away from philosophical hot air, what I am saying is that all of human society is essentially one classical physical system: all human perceptions are commuting random variables. So there exists a common density matrix or even a probability distribution to describe what all people see. The truth is relative in principle, but not in practice for different people.

Comment #93 April 8th, 2007 at 8:43 am

Greg:

I am a Bayesian: I believe that the actual state of a classical object is a probability distribution.Is this just a way of saying that you think probabilistically about possibilities, or are you really asserting (for example) that when you think a coin has a 50-50 chance of showing heads or tails, you think that its

actualstate is neither heads upnortails up, but the probability function itself?In other words: the state of the object is some element (you don’t know which) of the set of possibilities over which the probability distribution is defined. If the distribution is the state of anything, it is the state of your belief about the object, not the state of the object itself.

If I do assume that this is what you meant, then I am back to asking what set of possibilities your density matrix is a distribution over, because the actuality will be some element of that set. It looks like the answer ought to be: the possibilities represented by those projection operators. But then we are also back to the preferred basis problem. I would say that your theoretical work is not done until you specify some particular set of mutually orthogonal projectors, as the final statement of what might be real according to your theory. I don’t care if it’s position at one moment and momentum the next, or even if you propose an instantaneous probability distribution over sets of projectors, regarding which observables get to be actual. But noncommutative probability simply leaves the theory ungrounded and therefore unfinished.

Comment #94 April 8th, 2007 at 8:53 am

Could gambling solve the persistent problem of so many interpretations?

http://hanson.gmu.edu/gamble.html

http://backreaction.blogspot.com/2007/04/could-gambling-save-science.html

-serafino

The question of whether the waves are something “real” or a function to describe and predict phenomena in a convenient way is a matter of taste. I personally like to regard a probability wave, even in 3N-dimensional space, as a real thing, certainly as more than a tool for mathematical calculations … Quite generally, how could we rely on probability predictions if by this notion we do not refer to something real and objective?

-M. Born, Dover publ., 1964, Natural Philosophy of Cause and Chance, p. 107

Comment #95 April 8th, 2007 at 9:01 am

Greg, I posted that comment before I saw your addendum, but I don’t think it changes anything, except to confirm that you were talking about states of belief. But eventually you have to talk about the possibilities and alleged actualities to which these states of belief refer, and that is what I am talking about.

John, it will take more than a reminder of human intellectual frailty to get me to give up the law of the excluded middle.

Comment #96 April 8th, 2007 at 9:24 am

Mitchell Porter says:

“It will take more than a reminder of human intellectual frailty to get me to give up the law of the excluded middle.”As Douglas Adams or maybe Terry Pratchett might say: “It’s not really a law, it’s really more of a suggestion or guideline.”

I’d be grateful for a citation by either! 🙂

Comment #97 April 8th, 2007 at 9:29 am

Is this just a way of saying that you think probabilistically about possibilities, or are you really asserting (for example) that when you think a coin has a 50-50 chance of showing heads or tails, you think that its actual state is neither heads up nor tails up, but the probability function itself?It is certainly the former, but it’s not just that. The latter is roughly correct, except that I have reservations about the word “actual”, because it connotes an ontological absolute that cannot be entirely true. It is a description of a state of knowledge, which for some purposes is as actual as it gets.

If I do assume that this is what you meant, then I am back to asking what set of possibilities your density matrix is a distribution over,Your question presupposes that a density matrix is a kind of probability distribution, i.e., a

special caseof a probability distribution. But it isn’t, it’s ageneralizationof a probability distribution. If I were to tell you that a citrus fruit is like an orange, but more general, then you simply wouldn’t understand what I am saying if you clung to the idea that an orange is as general as it gets, and that a citrus fruit must therefore be a special case. You could keep asking, “How can an orange be yellow, unless it is not yet ripe?” Your question, “How can a density matrix not be a distribution over THE set of states, unless it is not complete?” is very similar.Remember, this thread is you asking me how I view quantum probability. When I say “non-commutative probability”, I really mean “not necessarily commutative probability”. Commutative probability is a special case in which a density matrix just becomes a probability distribution over a set of possibilities. But in the non-commutative case, a density matrix is a more general object that isn’t, or more precisely isn’t uniquely, a distribution over a set of possibilities. As you seem to have studied the matter, yes, the set of orthogonal projectors isn’t unique.

Nonetheless, the density matrix is the epistemological reality. As a rule, mathematical models are open to generalization. If you can’t accept the idea of generalizing probability distributions to something else, then you can’t understand what I’m saying.

Comment #98 April 8th, 2007 at 9:39 am

But eventually you have to talk about the possibilities and alleged actualities to which these states of belief refer, and that is what I am talking about.Well you say belief, while I say knowledge. The distinction is important, because knowledge is that part of belief which is reliable.

Otherwise your assertion is at least on the same terms as my remark, but I don’t agree with it. The knowledge of two different thinkers does not have to be consistent unless they can confer. When it isn’t consistent, there does not exist a common actuality. But by the rules of quantum probability, when two different thinkers can confer, then their knowledge is always consistent.

This is a counterintuitive conclusion, because in human society, people can always confer. There is never a need for one person to assume that another is in a quantum superposition. But in a society of qubits, that is exactly how things would stand. If there were sentient quantum computers — conceivably they will exist one day — I’m sure that they wouldn’t philosophize about the incompleteness of quantum probability. Instead, they would reject absolute ontology, which is it what it sounds like you are digging for.

Comment #99 April 8th, 2007 at 10:59 am

Greg and Mark, your dialog seems to be converging onto one-page 1963 article by Edmund Gottier titled

Is Justified True Belief Knowledge?Gottier’s article actually

settled a philosophical question… an event that is almost unique in the annals of philosophy.The answer, by the way, is “no”. But as Hemingway’s Jake Barnes says, ““Isn’t it pretty to think so?” 🙂

Comment #100 April 8th, 2007 at 1:15 pm

I’m familiar with the Gottier examples, and to me it seems like all they point out is that if we stick with the familiar formulation that “knowledge is a true, justified belief” then we have to be careful what sorts of things we accept as “justified.” His examples are all ones in which the purported knowledge is true and has a justification, but the justification has no connection to the truth of the proposition. I think that there’s no reason we can’t keep the traditional definition of knowledge as long as we are sure to specify that “justified” means justified in a relevant sense and by a process whose general application will also reliably produce truth.

Comment #101 April 8th, 2007 at 1:18 pm

(Oops, I copied you on the spelling, but thought, “Isn’t it pronounced ‘Gettier’?” Should have gone with the gut: Gettier, not Gottier.)

Comment #102 April 8th, 2007 at 1:45 pm

Hey, I’m the one who got the spelling of “Gettier” wrong! It was only 35 years ago that I first read that article. 🙂

Your phrase “`justified’ means justified in a relevant sense” is a surely a darn tricky standard to apply, whether the context is math, science, engineering, politics, … or even marriage, as Dave Bacon is no doubt finding out! 🙂

Lance Fortnow’s thread on the Continuum Hypothesis illustrates how subtle these issues can be, even when physics is (seemingly) not involved.

Comment #103 April 8th, 2007 at 1:46 pm

Scott,

I very recently discovered your excellent blog. I know very little about quantum computing, or the related advanced physics, and I’m looking forward to using your blog as a learning resource.

One of my personal interests is to understand the mathematics that underlies such things. I just read lecture 1 in this series, and I’m wondering what prerequisite math subjects would you recommend to help me get started with this quantum stuff, and to follow the lectures with a bit more understanding? I’ve got an engineering background and made it to introductory linear algebra years ago. I enjoy studying and reading about math as a hobby now, so no matter how daunting the task, I won’t run away with my tail between my legs!

Scott

Comment #104 April 8th, 2007 at 3:13 pm

Scott, the nice thing about the lectures being online is that you can start reading them now, and then if there’s anything you don’t understand, look it up on Wikipedia or Mathworld or some other online resource. By and large, all the math I use is

extremelyelementary — I can’t think of anything you’d need beyond complex numbers and linear algebra, maybe a wee bit of programming and discrete math. That’s not to say the ideas aren’t hard, but hopefully they’re hard in a self-contained sort of way.Comment #105 April 10th, 2007 at 10:11 pm

A few days pass and the blog caravan moves on. I suspect I’ve lost my chance of changing anyone’s mind. Nonetheless:

Greg, I think I understand well enough what you are saying but I reject the philosophy of it as pernicious and retrograde. Specification of an “absolute ontology” is a

minimalstandard for any theory which has pretensions to finality. The point is not to be able to declare dogmatically that the theory is correct, the point is to have an exact specification of the way the world might be. The focus on epistemic states rather than possible states of the world is pernicious because it allows this point to be obscured. I do not understand why theoretical physicists, who have more reason than anyone to think that the world might be completely knowable (in outline if not in all its particulars), would settle for such a thing, but noncommutative probability combined with the epistemic focus offers a way to do it.As a generalization of the concept of probability, the significant thing about noncommutative probability is precisely that it abandons the view that the probabilities are associated with a determinate (not determinist) set of possibilities. If one’s focus is ontological, this is immediately perceived as a problem, because you want to know what the actual states

of the worldare supposed to be in a given theory. If states are epistemic, this is apparently not so clear, which is why I wish Bohm had prevailed over Bohr.John, I agree with Carl regarding Gettier; he did not falsify the definition of knowledge as justified true belief; he simply exposed (though everyone should already have known this, at least since Hume) that many beliefs which we are accustomed to thinking of as justified are not so, strictly speaking. From the perspective of philosophical skepticism, there is very little knowledge, and it seems that it’s the difficulty of justification which mostly makes it so – there are always too many other empirically indistinguishable possibilities.

And Scott A., I owe some sort of response to your mention of Galileo. Galileo may have expelled a host of baseless apriorisms from the scientific method, but I have to wonder whether he would endorse a theoretical approach which apriori makes a

virtueof incomplete descriptions.Greg, you also say:

The knowledge of two different thinkers does not have to be consistent unless they can confer.If it isn’t even consistent, then it was never knowledge! – unless you think that theworldcan be self-inconsistent. The priors, guesses about the state of the world, or epistemic strategies of rational agents can be inconsistent-until-conferral as you describe, without impugning their rationality, butthose things are not knowledge.As for sentient quantum computers rejecting absolute ontology on the grounds that they must be able to conceive of their interlocutors as being in superposed states – so long as something like the Bohmian option exists, they will have no

needto reject absolute ontology. Again, I want to distinguish between epistemic uncertainty and ontological indeterminacy. A rationality based on quantum priors is not a problem. But an anti-ontology is.Comment #106 April 11th, 2007 at 8:47 am

[…] Why is many-worlds winning the foundations debate? April 11, 2007 at 11:47 am | In Quantum, Philosophy, Uncategorized | Almost every time the foundations of quantum theory are mentioned in another science blog, the comments contain a lot of debate about many-worlds. I find it kind of depressing the extent to which many people are happy to jump on board with this interpretation without asking too many questions. In fact, it is almost as depressing as the fact that Copenhagen has been the dominant interpretation for so long, despite the fact that most of Bohr’s writings on the subject are pretty much incoherent. […]

Comment #107 April 11th, 2007 at 8:56 am

Greg, I think I understand well enough what you are saying but I reject the philosophy of it as pernicious and retrograde.You are free to do that. Although I said all along, I’m not interested in philosophy for its own sake. My real work is mathematics (with elements of physics and computer science), and my philosophy is simply the way that I explain the ideas to myself and to other people. My defense of my philosophy is “it works for me”, that is, I find it helpful for my own research.

If I can at least correctly explain my viewpoint to you, then that’s good enough.

The point is not to be able to declare dogmatically that the theory is correct, the point is to have an exact specification of the way the world might be.On the contrary, in my view, what has an exact specification is the rules that the world follows, and not necessarily its state.

As a generalization of the concept of probability, the significant thing about noncommutative probability is precisely that it abandons the view that the probabilities are associated with a determinate (not determinist) set of possibilities.Yes it does. That’s why I love it.

The priors, guesses about the state of the world, or epistemic strategies of rational agents can be inconsistent-until-conferral as you describe, without impugning their rationality, but those things are not knowledge.I concede that there is a legitimate wrinkle about what one ever might have meant by knowledge. Beliefs that are rational, predictive, and reliable are at least hard to distinguish from absolute knowledge. Maybe I would concede that absolute knowledge does not exist, although as an expedient, I’m happy to call “RBR beliefs” knowledge.

Again, it’s important to remember that if you have a whole society of rational agents whose perceptions all commute, then there will be an appearance of absolute knowledge.

So long as something like the Bohmian option existsAgain, the Bohmian option is no different as a predictive theory, it’s only a different pedagogy. It does serve to show that the actual theory of quantum probability might well be complete. After that, the only question is which pedagogy you like best.