# Does the uncertainty principle make simulation of systems impossible?

Is it possible to fully define a system, then be incapable of simulating or calculating its future states due to the Uncertainty Principle? If it can be done, how?

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Yes, because 'fully define' probably doesn't mean what you think it means. – Danu Dec 1 '13 at 17:45
Then start with some definition(s) you will use for 'fully define'. – Luke Burgess Dec 1 '13 at 17:55
Wavefunction, or density matrix. If these are all there is to know about the system, and evolve deterministically, then the HUP is not an issue. – lionelbrits Dec 1 '13 at 23:52
I think the question derives from comments I made on another post (now removed). My original comment was that even with infinite resources such that computational cost and precision was not a problem, it would not be possible to simulate exactly a collection of particles because one could not know the exact momentum and exact position for initial conditions to make it match any given observation exactly. But I agree entirely that the solution of probabilities is deterministic, but that only gives a good (or great) idea of position/momentum, not exactly where things are. – tpg2114 Dec 2 '13 at 0:47

The Uncertainty Principle will never, as far as we know, prevent you from simulating any physical system. The reason for this is that quantum mechanics is - except for that little problem with measurements - completely deterministic.

To be more precise, say you want to simulate a given system within quantum mechanics. You begin by describing your preparation procedure of the initial state, you describe the hamiltonian which drives the evolution of the system, and you describe any measurements you will do at any given point. Then quantum mechanics allows you to calculate, at least in principle, the evolution of the system's state via the quantum Liouville equation, $$i\hbar\frac{\partial\rho}{\partial t}=[H,\rho].$$ When you perform measurements, the formalism will tell you the probabilities of each outcome and the state you should use to continue the unitary evolution. The whole thing is completely simulatable. (On the other hand, there is no guarantee on you being able to find a computer that will do this in less than the age of the universe.)

Even in classical mechanics, this is not an issue. Say you have a classical particle which you want to simulate using some hamiltonian mechanics, but you're worried that you can never have full information about both position and momentum. The Uncertainty Principle does limit your precision to a patch of area $\hbar$ in phase space. However, your preparation procedure will produce some sort of definite probability distribution over phase space which determines what positions and momenta are more likely than others. This probability density can then be propagated deterministically in time using liouvillian mechanics. This formalism will give you, at any given time, the probability distribution over the position and momentum of the particle; if you repeat the experiment over your ensemble then you can simulate the distribution of final values.

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+1, Loved the second sentence. However, "Even in classical mechanics, this is not an issue" - I got a little bit lost here: did you mean to say "this is also an issue"? i.e. you're saying that in any simulation one should propagate uncertainties? – WetSavannaAnimal aka Rod Vance Dec 1 '13 at 23:39
What I'm saying is that, if uncertainties are a problem that could potentially mess up a hamiltonian/lagrangian/newtonian mechanics simulation, then the uncertainties themselves can be simulated, to predict the distribution of the final outcome of the experiment. Simulation of uncertainties is not necessary but it can be done if needed. – Emilio Pisanty Dec 2 '13 at 0:43
Thanks: that's what I thought you were saying. – WetSavannaAnimal aka Rod Vance Dec 2 '13 at 1:03
Thanks for the answer Emilio, though I didn't follow one point. Say a classical evolution contains bifurcations, such as the orbit switching in the Lorenz attractor. Then the neighbourhood of a point in the flow is not preserved during the evolution, so it would seem that to propagate a density function (representing your uncertainty) you would need to propagate each point on the support of your density function independently, requiring an infinite amount of computation. This would seem to run counter to the suggestion that uncertainty is not a problem in simulating classical systems. – ComptonScattering Dec 2 '13 at 21:44
@ComptonScattering Not really, or yes, depending on exactly what you mean. My claim is that if you want to estimate the probability density of some observable to a given finite accuracy at some finite future time, and are given sufficiently accurate information about the initial distribution, then it is in principle possible to run such a simulation, even if chaos or bifurcations are present. I make no claim, however, about such a computation being doable within the age of the universe, or about the scaling of the needed precision on the initial state. But it is in principle doable. – Emilio Pisanty Dec 2 '13 at 23:40

As Emilio pointed out, the uncertainty principle is not a limiting factor. However, as for simulating or calculating future states, this is not really generally possible for classical systems, because of chaotic behaviour.

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Most chaotic behavior is actually deterministic if we had enough information. Turbulence in fluids for example only appears to be chaotic -- given good enough initial conditions, it is entirely deterministic. But we'll be unlikely to actually generate accurate enough IC's to be deterministic. – tpg2114 Dec 2 '13 at 1:13
Well, all of known physics is deterministic if we had enough information. Chaotic systems happen to exhibit exponential sensitivity to initial conditions. For a system where the Lyupanov exponent is positive, you need exponentially more precision in initial conditions for a linear increase in predictive power. Things that require exponential resources soon run into fundamental physical limits of computation. Sure, you can predict the turbulence in a coffee cup days into the future, but how many suns worth of heat did that calculation produce? – lionelbrits Dec 2 '13 at 2:21
@tpg2114 Look up topological transitivity (heaven knows why it is called this BTW): some iterations are qualitatively fundamentally different from others; topological transitivity is often taken to be a hallmark of chaos and makes it impracticable very quickly to foretell behavior accurately. Lionel is not exaggerating when he speaks of suns worth of energy! – WetSavannaAnimal aka Rod Vance Dec 2 '13 at 10:26

Let's start off by removing the restriction of computational resources such that we're not limited by computing power and by finite precisions. Let's also use the word exact to mean absolute certainty (ie. probability is precisely 1) about a quantity.

Take a real group of particles at an initial state. We may or may not be able to derive a set of governing equations that is completely deterministic in the sense that given the exact initial momentum and exact initial position of each particle in the system, we could solve for the exact momentum and position at any other time. But even if such a governing equation existed, we could never take a real system and translate it into initial conditions because we can never measure the exact momentum and exact position to start the simulation. We are limited by the Uncertainty Principle, so no matter how deterministic the equations may be, we cannot be more precise than our uncertainty in the initial conditions.

Now, we can change variable such that we're using a wavefunction or probability density as our variables. These may be solved deterministically, and we may know them exactly from our real system to use as initial conditions. That's awesome, but it means all we can do in the limit of infinite resources is come up with the exact probability of the state of the system at a future time.

But we cannot turn those probabilities into an exact momentum and exact position, no matter how exact the probabilities are. We can be certain to any reasonable amount that our particle is at a position and momentum, but we can never be exactly certain.

It's being pedantic of course, but much of the limiting cases tend to be. In a practical world, we can get good enough or better than we could ever manufacture real equipment to measure. But in theory, it is fundamentally impossible to simulate exact momentums and exact positions of real systems because we could never know the initial state exactly.

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