# Emergent Complexity In Probability Distributions

I originally posted this in Physics SE but was told it's more appropriate here...

In quantum mechanics we describe waves of probability or probability distributions. Does this have to or should it be conceptualized as more of an abstract principal or can we intuit that the probability distributions exist in an already computed (So to speak) form that is superposition?

That then leads to the main point of the question which is: Is it possible for there to exist complexity or is it even implied such that there could (Or does) also exist computation (Or even thought) in this form?

Please forgive me if I'm thinking of this in the wrong way (I'm not a physicist). I got the idea from science fiction (Doctor Who) in which there is a race of beings that are "Quantum locked" meaning that they were inanimate until observed. That got me thinking about the nature of probability distributions in terms of computation as far as our Universe can be said to be computed (I'm a computer scientist).

• Para. 2, sentence 2 -- I'd suggest that you pick one of "does this have to..." or "should this be..." the sentence is hard to parse with the conjunction clause in there, and it affects the meaning, and thus answers, to the question
– Dave
Jan 16, 2017 at 11:35
• Para. 2 as a whole is hard to understand. I (and alanf) don't understand what options you are trying to choose between
– Dave
Jan 16, 2017 at 11:40
• Lots of people in the Physics forum don't know what subjects belong where, so, don't be concerned about what they think. Jan 16, 2017 at 15:36
• Whether we intuit them as "abstract principle" (conceptualism) or as existing "in an already computed form" (Platonism), or even as useful fictions (nominalism) does not alter the mathematics or its predictions, it is therefore a matter of personal taste and philosophical preference. Whether "complexity", "thought" or "computation" exist in "this" form equally depends on the adopted position, Platonists like Penrose say yes to all, but they are in a small minority philosophy.stackexchange.com/questions/39993/… Jan 17, 2017 at 2:19

Quantum complexity theory exists and it is one of the fastest growing fields in computer science. The answer to how this occurs will come out of an elaboration on your first question, the question of how do we treat the ontological status of probability amplitudes.

In quantum mechanics we deal with what are called probability amplitudes. Probability amplitudes differ from probability distributions; amplitudes can be negative and complex numbers while distributions cannot. The reason this works is because the absolute value of the amplitude is squared before it is treated like a normal probability; this assures us that we will never end up with a negative probability. All of the much discussed "weird" or "counterintuitive" aspects of quantum mechanics are due to the fact that probability amplitudes can cancel each other out. This is just how the formalism works, it's a property of complex numbers.

The question you're asking is one of the most central questions in the philosophy of physics; it is the most central question in the philosophy of quantum mechanics without a doubt. The general discussion of the question "how should we understand the ontological status of probability amplitudes" falls under the label of interpretations of quantum mechanics.

More or less, all interpretations of quantum mechanics share two qualities:

1. They interpret a formalism—a set of equations and principles to generate predictions via input of initial conditions
2. They interpret a phenomenology—a set of observations, including those obtained by empirical research and those obtained informally, such as humans' experience of an unequivocal world

Two qualities vary among interpretations:

1. Ontology—claims about what things, such as categories and entities, exist in the world
2. Epistemology—claims about the possibility, scope, and means toward relevant knowledge of the world

The reason interpretations of quantum mechanics are more often associated with philosophy rather than science (truly, some scientists think it is all just metaphysical questions with no verifiable answer so it's nonsense to talk about them) is due to the fact that the interpretations focus on discussing the ontological and epistemological status on quantum mechanics, not the formalism. Science is the cycle of collecting data, analyzing the data, coming up with predictive models and formulas that allow us to make accurate predictions, and then preforming experiments that verify or contradict those predictions. Quantum mechanics, so far, has provided the most well tested physical theory we have ever achieved and it has done so without settling the ontological status of wave function collapses, so many scientists think the answer to this question genuinely does not matter.

There are many different interpretations but there are two that stand out as the most popular today. One is referred to as the Copenhagen interpretation and the other is the many-worlds interpretation. From the Stanford Encyclopedia of Philosophy:

Today the Copenhagen interpretation is mostly regarded as synonymous with indeterminism, Bohr's correspondence principle, Born's statistical interpretation of the wave function, and Bohr's complementarity interpretation of certain atomic phenomena.

The Copenhagen interpretation treats quantum mechanical objects as purely probabilistic objects. It treats the wave function (the collection of amplitudes) as a truly probabilistic object. Generally, the Copenhagen school of thought is described as being a system that accepts nondeterminism in physical theories. They say we have absolutely no idea what the outcome of an experiment will be and neither does the particle until the experiment happens and it then instantly makes a "choice" based off of probability.

The fundamental idea of the MWI, going back to Everett 1957, is that there are myriads of worlds in the Universe in addition to the world we are aware of. In particular, every time a quantum experiment with different possible outcomes is performed, all outcomes are obtained, each in a different world, even if we are only aware of the world with the outcome we have seen.

The many-worlds interpretation says that there is no randomness, per se, of the outcome of wave function collapse. It says that there are many parallel universes that exist and each time a measurement is made the outcomes are all experienced by a different world. The reason we do not experience all of the outcomes is because our conscious experience is limited to one world, the world in which we exist.

The answer as to "how should we interpret the probability amplitude" is not yet agreed upon. There exist many different interpretations and all of them provide different answers. The issue is that, for the most part it is believed, there is absolutely no experiment we can do that will prove one or the other. We are not able to access the parallel worlds from the many-world interpretation so we will never be able to verify that the collapses are not actually random. This means that there will be no way for us to assure that the ontological probabilities of the Copenhagen interpretation are incorrect. Much of the literature of the philosophy of physics is devoted to these questions.

A Snapshot of Foundational Attitudes Toward Quantum Mechanics is a very interesting paper describing the current consensus and divides on this question in the current physics community. The results of their polling shows that the majority of modern physicists believe that randomness is inherent in physics and that the Copenhagen is the most believed interpretation. It is conjectured that this is due to that interpretation being the "traditional" or "canonical" interpretation and thus most physicists learned it as the default idea from their professors. That being said, most of the most prominent physicists push for the many-worlds interpretations (Lenny Susskind has a somewhat technical paper about why they might both be correct).

Now, on to complexity. Scott Aaronson is generally regarded as the most authoritative voice on quantum computational complexity, at least as far as theory goes. He has also written extensively on the philosophical side of quantum computation and what philosophers, physicists, and computer scientists should all focus on in regards to these questions. His work, along with many, many others, has shown that quantum computation is theoretically a well founded idea.

However, we have not yet built fully functional quantum computers. So far, there have been no physical results that prove that quantum computation, and therefore everything predicted by quantum complexity theory, cannot be physically realized. The largest problem we have faced is the problem of decoherence, which is the tendency for quantum objects to interact destructively with their environment. Decoherence is one of the main research topics in quantum complexity theory and it is the main source of trouble for our current design of actual quantum computers.

The way that computer scientists and physicists deal docoherence is by what are called quantum error correction. This is a process whereby quantum objects are allowed to remain coherent by introducing new objects into the environment that take the decoherence onto themselves and then are removed. It is an open question as to whether or not nature experiences quantum error correct by itself, however many specialists believe that it does.

So, from a philosophical perspective, some people remain skeptical that we will ever be able to have fully functional quantum computers. Others, however, are fully committed to the idea and believe that they are just around the corner. Further study into the true ontological nature of probability amplitudes will surely shed light on some of the questions the field faces, such as the true nature of a measurement. Where we stand, however, is that quantum complexity surely exists epistemologically. We have (almost) all of the math formulated. What remains are the actual machines that prove the mathematics is correct.

As for the Doctor Who question, it seems unlikely that a being such as that would exist, because if it was a being that had consciousness it would be able to experience itself and therefore would always be able to exist. The thought process behind the show is probably more sympathetic to interpretations that rely on the human mind playing a conscious role in quantum mechanics. This interpretation is generally disfavored and discredited by most working physicists and a lot of philosophers, however it does have its proponents.

In quantum mechanics we describe waves of probability or probability distributions. Does this have to or should it be conceptualized as more of an abstract principal or can we intuit that the probability distributions exist in an already computed (So to speak) form that is superposition?

What you are touching upon here is the issue of scientific realism vs scientific anti-realism (See here and here). We frequently resort to unobservable entities to explain our best scientific theories (gravity, electrons, genes, etc...). What is the status of such entities? Are they real objects in the same way that rocks, cheeseburgers and trees are? or are they conceptual tools similar in status to a Fourier Transform or to the Matrix representation of a process?

• A scientific realist holds that such entities have a real existence.
• A scientific anti-realist holds that we cannot know whether such entities are real or not, or stronger still that it is meaningless all together to discuss the reality of such entities. They are merely useful tools for predicting the outcome of experiments.

In particular, you are asking whether one should be a realist about quantum wave functions and quantum states (as you said "that the probability distributions exist in an already computed") or an anti-realist (as you said "an abstract principle")?

There is no definitive answer to this (just as there is no definitive answer to the interpretation of quantum mechanics in general).

The Copenhagen interpretation is definitely anti-realist, and as far as I know anti-realism with regards to wave functions is the dominant position (Feynman's famous "Shut up and calculate!"), but you can find those that argue for the opposite (see here, and here), and DeBroglie-Bohm pilot wave theory.

That then leads to the main point of the question which is: Is it possible for there to exist complexity or is it even implied such that there could (Or does) also exist computation (Or even thought) in this form?

This is quite difficult to parse, you should at least give a clearer definition of what you mean by "complexity". My reading of your question is the following:

"If quantum wave functions have no real existence and are just mere conceptual tools for calculating probability distributions, how can they give rise to complex structures and computations?"

Keep in mind that conceptual tools and mathematical models tend to be oversimplifications of reality. If a physical process is such that it has to be modeled by a Quantum Computer of a Quantum Turing Machine, then the underlying reality it is referring to is most likely something even more complex than what even quantum mechanical model is being used to describe it.

Consider a single classical computing NOT gate: From a computational point of view we have a "simple" process that can be represented by "0" and "1"- but in reality you have a much more complex analogue electronic process that be much more accurately described by a large number of differential equations.

I originally posted this in Physics SE but was told it's more appropriate here...

Physics is supposed to be about explaining how the world works, which is what this question is about.

In quantum mechanics we describe waves of probability or probability distributions.

Quantum mechanics doesn't describe waves of probability. For a start, the square amplitudes often don't obey the calculus of probability, and so the wave function can't be a wave of probability:

Does this have to or should it be conceptualized as more of an abstract principal or can we intuit that the probability distributions exist in an already computed (So to speak) form that is superposition?

I don't know what this means.

Sometimes physicists use mathematical tricks to make a calculation easier, e.g. - the centre of mass. In those cases, the calculations can be done without the trick: all physical quantities computed by centre of mass calculations can be computed in other ways.

There is no alternative to using the wave function, or Heisenberg picture observables or path integrals or whatever for quantum mechanical calculations. And when you take those seriously as explanations of how the world actually works they have implications that many physicists want to deny. For example, quantum mechanics entails that reality is a multiverse: a structure that is approximately similar to a collection of parallel universes under the right circumstances:

That then leads to the main point of the question which is: Is it possible for there to exist complexity or is it even implied such that there could (Or does) also exist computation (Or even thought) in this form?

Computation is possible using quantum mechanical phenomena such as interference and entanglement: that's what the theory of quantum computation is about. And studying quantum mechanics in terms of the flow of quantum information solves many problems with standard deeply confused accounts of quantum mechanics, e.g. - it eliminates quantum non-locality:

Thought involves knowledge creation, which is an evolutionary process that involves producing variations on existing ideas and then selecting among the variations. Both the variation and selection involve being able to read the existing ideas. Making copies of information prevents interference:

So thought doesn't require quantum computation. And there are papers about quantum mechanics applied to the brain, such as:

Please forgive me if I'm thinking of this in the wrong way (I'm not a physicist). I got the idea from science fiction (Doctor Who) in which there is a race of beings that are "Quantum locked" meaning that they were inanimate until observed.

The beings in question were inanimate when observed in Doctor Who. But this is nonsense, since quantum mechanics doesn't ditinguish between observation and interactions that record information even if nobody observes it. For example, light reflected from a moon such as Hyperion may go out into space without being observed, but it will still cause the Hyperion to be unable to undergo interference, just like a measurement:

@Xendi - I do not think the probability distributions "exist" - but of course we have to be very careful about what "exists" and what does not, and what we even mean by "exist". Remember though, that regardless of whether they "exist" or not, and whatever sense that would have, the physicist does not, a priori, care. That is because the physicist will formulate/accept any model of Nature, as long as it "works", i.e. makes correct predictions, doesn't contradict other observations... Physics is only a game of making up models for Nature. If those models are literally what happens in Nature... who knows.

As for "complexity" and "computation", and "thought", we should define those more carefully first. But I don't think any of that is implied by probability distributions in QM. Again, wave functions are just one convenient model at this point.