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I understand uncertainty from a combinatorial and game theoretic perspective, as functions of incomplete or imperfect information, or intractability which is a type of inaccessible information in that the complete game tree cannot be expressed.

This answer on quantum uncertainty was helpful because it raises what may be understood as computational issues. (For instance, running the game tree for Conway's Game of Life is indistinguishable to running the game--there is only a single node for each ply--and the only way to predict outcomes of novel, non-trivial starting configurations is to run the game. This condition may, among other things, be taken as a commentary on Laplace's demon.)

Part of my question relates to recent interest in "quantum games", such as Quantum Tic-Tac-Toe and Quantum Chess. There is also a type of finite game that is initially intractable but "collapses" into a condition of tractability, which is to say from an indefinite to a definite state, for which the "critical mass", or threshold, is the recognition that the outcome may no longer be affected. (This threshold, notably, is subjective both for humans and automata;)

In the paper On the Importance of being Quantum, Dr. Akl states:

"Furthermore, bringing quantum physics into Chess should be understood as being significantly different from merely introducing to the game elements of chance, the latter manifesting itself, for example, in games involving dice or playing cards where all the possible outcomes and their odds are known in advance."

Part of what I'm trying to gauge is do we need to be thinking about random number generation for certain types of combinatorial problems?

A quote lifted from Joe Fitzsimons in an answer to a question quantum indeterminacy on Theoretical Computer Science clarifies this point:

"Quantum mechanics allows for true randomness in the sense that the outcome of certain experiments is independent of the prior state of both the system and the environment."

Which would seem to be a different type of indeterminacy than those arising out of observational limitations.


On a related subject, from a terminological standpoint, I'd be very interested in people's thoughts on this question:

Are games involving uncertainty truly quantum games, or merely representation of ideas arising out of quantum mechanics?

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    Definitely important to note is that the final quote you gave is stated from the perspective of assuming indeterminacy (i.e. The Copenhagen Interpretation). Other interpretations actually state that the quantum universe is indeed deterministic. They are interpretations rather than theories because it is currently believed that there is no way to create an experiment which distinguishes which interpretation is right. – Cort Ammon - Reinstate Monica Jun 17 '17 at 22:50
  • @CortAmmon Very useful! (I was originally going to ask "Is the Jury still out on randomness?" ;) – DukeZhou Jun 17 '17 at 22:51
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    All of the interpretations trade off desirable characteristics: realism, determinism, local realism, and completeness (Einstein's terms for the categories). It's been shown that you can't get your way on all four categories. Copenhagen, which is the most popular, gives up determinism, and creates this concept of "waveform collapse." deBroglie-Bohem pilot wave is deterministic, and actually really easy to understand, but it is unabashedly non-local, relying heavily on non-local effects. This was undesirable in the era when these interpretations were coming forth, so Copenhagen "won." – Cort Ammon - Reinstate Monica Jun 17 '17 at 22:57
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    However, later it was shown that even Copenhagen interpretations had to admit "spooky action at a distance," so pilot wave has some recognition still (but the momentum of the community cemented Copenhagen's interpretation's popuilarity). Then there's Many Worlds, which gives up realism. It argues that once an observation occurs, there is no way to objectively describe the state of the subject, one can only describe it subjectively w.r.t the observer. – Cort Ammon - Reinstate Monica Jun 17 '17 at 22:59
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    There's others, but those, to me, are the ones I would recommend to someone who is interested in stepping beyond the famous "uncertainty" and trying to better capture how these interpretations behave. – Cort Ammon - Reinstate Monica Jun 17 '17 at 23:00
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Quantum indeterminacy is inextricable from observation because that which we consider to be "quantum indeterminacy" is related to the interpretations of QM rather than the mathematical model behind QM.

(I expect this question to get migrated to Physics.SE, because it is really more of a physics question than a philosophy question. When it gets there, I know it will be seen by those who want technical accuracy. This answer is written from a very high level perspective to tie into the more philosophical aspects of QM. Please forgive any technical errors which arise from this process)

When you really get down to it, QM is a model. Its a set of mathematical equations which someone argues describes the behavior of something in the real world. These mathematical equations are very well specified in the unyielding language of mathematics. If you can set up an experiment in the language of this mathematical model, everyone will reach a consensus as to what the result of the experiment will be (also in the language of this mathematical model). The tricky part is in attaching real life meanings to these mathematical devices.

Consider a classical example first. Take the model x=1/2*a*t^2 + v_0^*t + x_0, which we will call a "kinematic model." Everyone will agree that if you take a=2, t=3, v_0=1, and x_0=0, the model will predict that x=12. That's easy. But what does it mean to say x=12. That's the interpretation part of the problem. Once I say that "a is the acceleration of an object," "t is how long the object moves," "v_0 is the initial velocity of the object," and "x_0 is the initial position of the object", I can claim "x is the final position of the object." Until I add that interpretation, it's simply math. Once the interpretation is added, it becomes a model of the physical world around us which can be tested.

So what's an "initial velocity of an object?" That's a lot of words. To describe what I mean, I'll need other words. We'll have to come to agree on a meaning of velocity, for instance. For most of physics, it is assumed that we have a consensus on the meanings of these words, so we use them casually. If I'm studying the reentry of a NASA spacecraft, I'll throw around terms like "freespace energy flux" with the assumption that everyone will agree on what that term means.

With QM, it gets harder. The connection between the world we can observe empirically and the mathematics of QM is more strained than it is in other aspects of physics. We talk of things like "measurements," which are very natural concepts on the scale of human empirical interaction, but they don't quite behave intuitively when you get really small. Thus, the scientists have to take more time explaining their interpretations of how we should think of the model's reflection of reality. In fact, the entire concept of indeterminacy arises in the interpretation layer of QM. The underlying wavefunction equations are straight forward deterministic mathematical equations. Its in the application of these mathematical equations that we find the need for indeterminacy. The famous Heisenburg uncertainty principle only applies to a very particular set of states. It just so happens that, when you attempt to apply QM to reality, those states occur over and over and become very important.

One of the challenges QM faced in the early days was captured in four attributes of an interpretation which were desirable. Using Einstein's terms for them, they were:

  • Realism - The idea that things objectively exist, even when they're not being measured. If you believe the universe is real, then you believe that you can look at the moon, then look away, and be confident that the moon did not cease to exist the instant you looked away from it. It persists whether or not it is being observed.
  • Completeness - An ideal for many is to have a complete interpretation, which means it accounts for everything observable in the physical world. A theory which predicts the exact behavior of everything in the world, except at one central point of the universe where Loki decided what happens would not be complete. Generally speaking, physicists seek completeness.
  • Determinism - The idea that there is no randomness built into the model. When you're trying to explain why people should spend a few years learning your theory of how the universe works, they like to know that they're learning something worthwhile. That's easier to argue if your theory is deterministic.
  • Local realism (aka Locality) - This is a strange requirement but very important for QM. Local realism is the idea that all of our measurements are measuring something real, and the effects of measurement don't exceed the speed of light. So much suggests that light is a "speed limit" for our universe, that theories which depend on exceeding it are treated with suspicion. This also matters because if your interpretation has a speed limit, it provides a hard and fast guarantee that Loki (this time a hyper-massive black hole, rather than a Norse god) can't influence your results any faster than you can observe Loki's influence yourself.

As things go, they have found it is not possible to map the mathematics of the quantum mechanics model into reality without sacrificing one of these attributes. Something has to go.

The Copenhagen interpretation, by far the most famous, sacrificed determinism. It stated that an observation would cause a "waveform collapse" which turns a superposition of quantum states into a single classical state. Which state it ends up with is defined randomly, using probability theory. Most people's understanding of quantum uncertainty derives from the Copenhagen interpretation because it won the political fight and because enshrined as the most popular interpretation. However, its full of confusion. Every student worth their salt eventually asks "what is an observation anyways?" It seems like a pretty important thing, since it causes waveforms to collapse. It turns out that the naive answers to this question rapidly turn into dead ends. The double-slit experiment series of experiments typically frustrates new students indoctrinated into the Copenhagen interpretation, and they become enamored with this concept of 'uncertainty' until they develop enough math to move beyond this phase.

Other interpretations give up different things. One interpretation is, in my opinion, highly helpful for you in understanding the issues you are exploring: The Many Worlds Interpretation (MWI). MWI is a complete deterministic interpretation, meaning the state of the current universe fully describes the future state of the universe with no uncertainty. How does it accomplish this? It gives up something dear: realism. Under MWI, one cannot say that any object has an objective state which is measurable. Instead, it is assumed that all states are subjective with respect to an observer. In MWI, you cannot say "the moon exists," but must rather say "whether the moon exists is consistent with my observations of whether the moon exists."

By making this sacrifice, rather than determinism, MWI actually makes it much simpler to stick to the actual mathematics. Take Schrodinger's cat. For Copenhagen enthusiasts, this is a very exacting study in word choice. There's nothing in the Copenhagen interpretation which misrepresents what happens in the Schrodinger's cat thought experiment, but most people start to break out into hives at the idea of a cat that is in a superposition of states of both dead and alive (colloquially termed "half alive and half dead," though that is actually not technically precise wording, and leads people astray). Now consider MWI. Like Copenhagen, MWI starts by describing the cat in a superposition of states entangled with the state of the radioactive isotope in the box. However, once the observation happens, instead of having some complicated wave form collapse occur, MWI takes a simpler approach. The cat's state (alive or dead), cannot be talked about objectively. It can only be discussed with respect to the observer. Thus, we say "The cat's state is consistent with the observer's observed state. Iff the cat is alive, we can be certain the observer measured it as alive. Iff the cat is dead, we can be certain the observer measured it as dead." Case closed. It's elegant really... if you can stomach the possibility that the universe may not be "real."

So this is why I say quantum indeterminacy is inextricable from observation. The indeterminacy only comes into the story once one begins to use an interpretation to tie the mathematical model into having empirical meaning in our universe. Different interpretations have different ways of handling this, including some which permit complete determinism in the universe. If you can permit a deterministic interpretation of QM, then indeterminism must be tied into the interpretation, not anything in the mathematics below.

  • Thank you for taking the time to post such a thoughtful answer! This jibes well with my combinatorial conception of the universe, and realism, completeness, determinism and local realism resonate. MWI seems to be a very good analogue for automata (reality is subjective and comprised of information) and game trees (many worlds). Part of what I do involves generation of uncertainty, and it's helpful from a design standpoint to be able to separate the Copenhagen from other conceptions. – DukeZhou Jun 18 '17 at 1:09
  • PS There is a very good play called Copenhagen about the Bohrs and Heisenberg. :) – DukeZhou Jun 18 '17 at 1:12

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