Four Verses from the Dàodéjīng by Jochen Szangolies

Dear Eckard,

the details of DQC1 are, of course, completely public---see for instance the publications here and https://arxiv.org/abs/0709.0548.

The problem considered in DQC1 (Deterministic Quantum Computation with One pure qubit) is the evaluation of the (normalized) trace of a unitary matrix. This is generally believed to be hard classically, but a quantum algorithm can be formulated that provides an estimate in a number of trials that does not scale exponentially. The algorithm's answer is obtained by Pauli measurements on a single qubit whose state is separable from those upon which the unitary acts. Consequently, there's no entanglement present.

However, there is another kind of quantum correlation, known as quantum discord, that is often postulated to be the reason behind this algorithm's advantage over classical resources. Discord involves quantum states that are probabilistic mixtures of 'pure' quantum states; only such mixtures serve as 'resource' states for DQC1.

But all of this is still very much an open area of research. Still, it's an exciting idea!

Dear Peter,

thanks for your kind comments. I'm not sure why you think that my QM background is at odds with the themes of my essay---quite to the contrary, I think it is my background there that opened up the possibility of viewing the world rather as covered by perhaps even superficially inconsistent pictures, rather than as being wholly cut from the same marble (because of the notion of complementarity).

I will try and have a look at your essay---although one cautionary note upfront is that I think it's generally not a good idea to try and produce a classical explanation for CHSH > 2, since that's easily done via data-rejection, timing issues, or similar such tricks. In my opinion, you should instead focus your efforts on inequalities that take only actually observed events into consideration, such as the Clauser-Horne one. But more on that probably later.

Hi Satyavarapu Naga Parameswara Gupta,

thanks for your comment and your appreciation. I'll try to have a look at your essay, but I can't promise I'll find something interesting to say. I will have a look, though.

(Sorry for just copy-pasting your name, I was unsure which address to use.)

Dear Francesco,

thanks for your comment! I'll have a look at your essay, 'absolute relativism' is a nice turn of phrase and indicates that you're already aware of the main problems relativist approaches face---as phrased by Connor Oberst, "if you swear that there's/ no truth, and who cares/ how come you say it,/ like you're right". (But that just as a diversion.)

Regarding my definition of information, I essentially appeal to algorithmic information theory---there, the information content of an object is defined by the shortest program necessary in order to make a computer output that object (or a description of it). So a string like 'aaaaaaa....a' can be reduced to 'na', where n is an integer, while a jumble of random characters in generally can't be reduced much.

This captures the intuition that information content ought to have something to do with the redundancy in some object: what is highly redundant can't tell you anything new, and thus, contains little information; if there is little redundancy, however, there's new information at each step. In other words, low-information objects can be highly compressed, high-information objects have little room for compression.

Does that help clear things up?

@All, I'm sorry if I'm a bit tardy in my replies---work has been keeping me a little busy lately. This should clear up in February, though, so I hope to be quicker with my answers then.

In the meantime, why don't you have a look at the discussion of my essay over at the fantastic blog on all things consciousness, Conscious Entities where Peter Hankins provides his insightful commentary on the issues I discuss in the essay!

Hi Luca,

thanks for your nice comments (in particular in regard to the Mary's room argument)! As for your skepticism, I think there's many different ways of approaching this question. One, for instance, is that I think Gödel's two incompleteness theorems establish a limitation of the human capacity to formalize mathematics, not a limitation on mathematics itself, as it's usually phrased---in other words, it's not that 'mathematics is incomplete', but rather, that the axiom systems human mathematicians can formulate---being necessarily finitely specifiable, effective, and so on, which ultimately boils down to being computable---don't suffice to capture all of mathematics.

But to say that this implies the existence of something beyond our modeling capacity would require a belief in the preexistence of mathematical objects, i.e. some form of Platonism---which is not something I myself would be willing to commit to.

But things get clearer once one things instead of abstract formal systems in terms of computers. The halting problem is an instance of the same phenomenon that also gives rise to incompleteness (Lawvere's fixed-point theorem, which Noson Yanofsky, who also has an entry in this contest, discusses in a very accessible way here), and the two are equivalent in some ways. Now, with the math we know how to use, we can't in general predict whether a given TM halts; nevertheless, one might justifiably hold that there's still a fact of the matter regarding whether it will or won't. But Turing machines are themselves abstract entities, and would require an infinite tape or something equivalent to build; so again, it's not immediately clear if that has any real-world significance.

So let's now turn to physics. You mention Breuer's results, which I think are very interesting; but there's many more indications of a connection between QM and logical independence (in fact, John Wheeler managed to get himself thrown out of Gödel's office for suggesting a connection). For instance, Caslav Brukner and colleagues could show that if they encode a set of axioms into a quantum state, measurements related to propositions that are not derivable from those axioms yield a random outcome. However, this is not specific to Gödel's incompleteness, as the axiom systems were too simple for Gödel's construction to apply---Gödel's theorems then simply mean that such phenomena never 'go away', so to speak.

A more qualitative line of reasoning is the following: you can represent any non-computable function with a computation plus a source of randomness; consequently, if we are beings using computational methods to model a non-computable world, we should expect the world to look like a computation with intermittent random events. That's of course exactly what we have in QM. An obvious counterexample here might seem to be Bohmian mechanics: but actually, it turns out that a model that reproduces quantum mechanics by introducing nonlocal influences (as BM does) cannot be computable---otherwise, one could use these influences to transmit information, contradicting the tenets of special relativity.

There are many more results pointing in a similar direction scattered about the literature (and of course, there's my own argument that I cite in the essay); but I think it's as yet unclear what the picture is they paint. Zwick in the 1970s, and later on Peres and Zurek, proposed that QM's "inability to completely describe the measurement process appears to be not a flaw of the theory but a logical necessity which is analogous to Gödel's undecidability theorem". I think that it's again as with the incompleteness theorems in mathematics: they're not a limitation on mathematics itself, but on human mathematicians; likewise, the limitation we see is not one on physics, but on physicists: using computable models, which we must if I'm right, to model a non-computable world, will inevitable lead to something like a quantum description.

But this is an argument that needs to be made much more carefully, and which I'm in the agonizingly slow process of working out. We'll see what comes of it!

Dear Tejinder,

thanks for your kind words! I'll have a look at what you're doing with the 'phase transition'-imagery. In some ways, I wonder if one couldn't develop that metaphor in a more rigorous way---Tegmark, of course, has already proclaimed consciousness to be a 'state of matter'. So perhaps we're just one early pocket of self-awareness percolating in a universe about to transform. Ah well, that might make a good plot for a science fiction story, at least...

Dear Heinrich,

thanks for your interesting comments! I'll have to have a look at your 'fundamental universe'. And you're right, I did read Gabriel's "Warum es die Welt nicht gibt" exactly because I also perceived some kinship in his ideas, but found it ultimately rather disappointing.

Regarding models and strong emergence, well, it depends what stance you take. The traditional view would be that models possess some immediate correspondence with the things they model, at least up to some suitable approximation, and that thus we should consider the entities they posit, and the explanations they provide, as giving us some insight into what really goes on in the world. Here, I think, strong emergence is troubling: in some sense, there would be certain things that violate Leibniz' principle of sufficient reason---facts about the world that would obtain without any answer to the question of why they should.

In some sense, though, this is also true in classical reductionist ideas---after all, if everything can be reduced to some set of base facts, then those facts themselves admit no further justification. From this point of view, strong emergence maybe doesn't seem that much worse---it merely adds additional 'fundamental' facts that obtain at some higher level of coarse-graining, for instance.

But in my view, all models are inaccurate to some degree---every model is incomplete by necessity, just as all axiomatic systems (of sufficient power) are. The world admits no more of a single model than mathematics admits of a single axiom system.

Are then facts that fail to fall under the purview of a given model strongly emergent? From the point of the view of the model, you might say so: after all, within that model, there is no way of reducing them to fundamental facts; they're true for no reason.

But this tells us something about the model, in my opinion, not something about the world. In other words, it's our problem, not that of the world---just as the fact that there is no single axiom system for all of mathematics ultimately may be considered a limitation of human mathematicians; if we weren't limited to finitely specifiable, effective systems, these problems wouldn't exist. So I think that if my thoughts entail strong emergence in some sense, then one also should think that mathematics contains strong emergence. But then it seems to me that's just fine, after all, mathematicians have still been able to make great strides in mathematical understanding post Gödel. (In that manner I, too, am optimistic about the progress of science---our understanding of the world will continually increase, but I don't think it will come to a true end point, although for all practical purposes, it eventually may.)

(By the way, have you seen Sabine Hossenfelder's take on strong emergence in this contest? I think it's a clever idea, although ultimately, I don't believe it really buys the necessary elbow room for things like free will.)

You're right also to point out the difference between consciousness, the sense of self, and so on. I wasn't intending to suggest an equivalence here, but I think there is a sort of progression---in order to be properly called 'conscious', you need to have a sense of self, and in order to have that, you need to be able to conceive of yourself as an entity distinct from the rest of the world, and in order to do so, you need to be able to model the world as sort of a container with you in it. Each lower rung is necessary, but not sufficient, for the next one. (As for literature on the subject of consciousness, I can't really claim to be an expert, but if you haven't read it, I still think that Jaegwon Kim's 'Philosophy of Mind' is one of the best overviews of the subject, and contains many references for further digging into the topics you find particularly interesting.)

There's also an issue regarding what constitutes a model that you highlight. In some sense, a brick is a model for our solar system---we could use it, say, in a model of the nearest couple of star systems. That empty beer bottle over there is Alpha Centauri, the football is Epsilon Eridani, and so on.

Undoubtedly, however, it would make for a very bad model of the solar system. That's because only its most elementary structure---the quantity, its being 'one thing'---is represented. So it's not clear in how far my talk about 'the world without models' is itself something of a model.

In particular, one also may talk about 'the round square cupola of Berkeley college', but does that necessitate a model of it? Can one model impossible objects? I think that one instead has models of its parts, or understands its properties individually, without really being able to put things together---so one knows (by model) what 'round' means, what 'square' means, what 'cupola' means and so on; but one can't really put these together, since that would be an impossible object. I think one understands 'the world without models' in a similar way.

Grüße zurück aus Köln,

Jochen

Dear Narendra,

thanks for your comment. My essay isn't really all that concerned with quantum mechanics, but regarding your question, well, it depends. QM is certainly a different kind of theory than classical mechanics---after all, it generally can only give probabilistic predictions, while classical mechanics comes with a (to some) reassuring certainty (at least in principle).

But that doesn't mean it's not a full fledged theory. If by this you mean a mathematical apparatus, and a way to connect the mathematics with observation (sometimes called the 'minimal interpretation' that makes the difference between physical theories and purely mathematical frameworks), then QM is just as reputable a theory as classical mechanics.

If you want further to know 'what it all means', however, quantum mechanics seems much less amenable to intuition. But is that the fault of the theory, or of the human mind that isn't used to it? I don't have an answer.