Can quantum entanglement be simulated on a digital computer to any degree of precision?

Question

First principles modelling of physical phenomena has been very successful in physics. The largest limitation is perhaps the fact that many QM problems are NP hard so we would need really powerful computers if we want greater accuracy. But any QM model should be, in principle, still computable to any desired level of precision.

My question is: is this correct?

My problem with a positive answer would be that some local rule cellular automaton are Turing universal, which would imply that entanglement could be simulated by a model that uses a classical local rule. This seems wrong, doesn't it?

Answer 1

Here are some facts:

1. As others have said, the evolution of a quantum state, including entanglement, can be simulated arbitrarily well classically with sufficient resources. That said, it does generally require exponential resources due to the exponential growth of Hilbert space.

2. Of course, a (classical) computer can't deterministically predict the outcome of a particular measurement, only give correct probabilities. So it is very important to distinguish between simulating deterministic quantum state evolution classically (which is no problem) and actually replacing quantum mechanics with a classical model (which can't ever happen). The difference comes at the actual measurement.

3. A computer can also simulate any number of impossible things. You can make a computer simulation where energy disappears, objects travel faster than light, etc etc.

4. When you run a Bell test on your computer program, at some level what it will do is assign the outcome of one measurement, then communicate that to the other entangled particle so that they both have correllated outcomes in the right way. In other words, the whole program relies on the two "particles", however they are stored in the computer, being close enough to communicate with each other. As a result, a classical computer could never pass a loophole free Bell inequality test. Specifically, if you load the same program onto two computers and send them far apart, they will never be able to give measurements with the same outcomes as measurements on two entangled particles would.

5. Notice once again that it's no problem for both the computers to know what state they're supposed to be in before you measure them. It's getting the two measurement outcomes to be properly correllated (in all measurement bases) that just isn't possible.

Answer 2

Quite inefficiently compared to "direct implementation" in quantum hardware, but yes, entanglement defintely can be simulated to arbitrary degree of precision give adequate memory and time resources in a conventional, entirely classical digital computer! Just go and solve multi-particle Schrodinger equation.

And this fact has nothing to do with Bohmian pilot wave or any other correct interpretation of quantum mechanics.

Answer 3

I will try my own answer, which is mostly thinking out loud, and am ready to delete it if it is too non-sensical.

For starter, we need to agree on what we mean by theory; in particular a theory of what and with which logical rules.

Let's roughly say that a theory is set of true statements about objects of interest for the theory with rules that enables one to relate some true statements to others and hence discover new true statements from existing ones.

• A classical theory would be a theory whose logical structure follows propositional 2-valued (classical) logic

• A quantum theory would be a theory whose logical structure follows instead quantum logic (and where for instance there is non-distributivity of the disjunction relation)

Here is what I think is true:

• If the objects of the theory are usual physical observables like position, momentum, angular momentum etc..., then it is not possible to construct a classical theory based only on propositions about these observables that eventually does not contradict outcomes of some experiments on quantum systems.

• With the same set of objects, it is however possible to construct a quantum theory (in the previously defined sense) that gives rise to the right quantitative description of experiments of any kind on quantum systems (in virtue of Gleason's theorem).

Bell's theorem essentially acts at that level of description (for that type of ontology) and contributes to the first point essentially.

The twist comes from the fact that if the objects of the theory are the quantum state and probabilities of outcome of experiments of a particular system then it is possible to construct a classical theory of propositions about these objects that is in perfect agreement with experiments on quantum systems; and this is the Copenhagen Quantum Mechanics that we use everyday (equivalently the Everettian formulation is fine as well, although the space of objects is a bit smaller).

Simulations of quantum systems by a classical computer essentially compute and interpret the last theory which is again a (logically classical) theory of propositions about quantum state and probabilities of the system of interest. So this does not contradict, I think, Bell's theorem that does not rule out the same type of theories.

Answer 4

In short, yes it can be done but it isn't going to be exactly precise as quantum mechanics goes. People doing research in lattice QCD are doing this very sort of thing; however, the approaches that I've seen have been to program in known laws of physics and use vast amounts of computational power to simulate single particles. Now, this seems a bit counter-intuitive. After all, we see complex simulations all the time using several graphics engines for gaming and the detail of these simulations are becoming quite remarkable.

So back to quantum simulations and in particular quantum entanglement. I'm not going to sit here and argue (As many people like to do) classical physics and the finer points of quantum mechanics equations because I don't believe it's necessary here. There's no point because we're talking about processors that can simulate anything you want with any behavior you desire. Parallelization can simulate two things happening simultaneously communicating with each-other on a classical computer as two sets of instructions can be computed in parallel. As for measurement, since you're the programmer you get to determine what you can and cannot measure.

To my mind this isn't so much of a physics question as it is a programming question depending on what you expect to get out of it. Of course if you have a quantum computer then the methodology changes as there isn't any (Or perhaps only one layer of) abstraction between the physical process and the simulation. As for the actual measurements and/or two computers being separated at a distance, you would probably need a quantum computer (Or two of them) for the kind of precision you'd want but you can still simulate entanglement. I'll give some examples.

This doesn't have to be such a hard problem as in programming simulations. For instance, reference: List of QC_simulators where libraries can be found for simulating quantum entanglement (Just search the page for "Entanglement") in several languages.

I don't see it being non-computable on a classical computer in the sense that the simulation will simulate the quantum mechanical effect but to be clear, you won't actually be entangling two particles on a classical computer and be able to get the same measured results (Bell's inequality does apply here). Sure, you might program your simulation on a classical computer so that the particles are in their proper states but the act of measurement technically won't yield the same results as an actual act of measurement in a real life experiment.

To actually simulate entangled particles you would need a real quantum computer. Otherwise, Bell's inequality, well, wouldn't compute. There are various resources on quantum computing algorithms and programming quantum computers. I haven't researched them that much as I don't have access to a quantum computer but Google has setup a webGL based simulator using the gate method of QC which can be found here: Quantum Computing Playground which I've experimented with and is an interesting resource as are the libraries I previously linked.

In essence, the answer is both yes and no depending on what you're trying to get out of the simulation. Emulating quantum computations isn't like emulating a Nintendo. In that case they're both classical computers.

As a side-note, I would mention that the popular D-Wave computer doesn't use the gate model of quantum computing (Probably why there's been a lot of controversy surrounding their computer) but rather uses adiabatic quantum computing.

Answer 5

If the emission of two entangled photons from nonlinear optical crystals does satisfy your definition of any QM model, then yes, it is possible to simulate it on a digital computer to any degree of precision. And I believe that it is more a question about how good undetermined are two randomization processes on the computer, first for the polarization of one of the two photons (the second photon has then simply the opposite polarisation state) and second for the destabilization of this states on the way from the source to the detectors.

We know that if we place the detectors close enough to the source we reach near to 100% the possible detection of 50% of the right state. In other words, placing a horizontal polarizer in front of the first detector and a vertical polarizer in front of the second detector in nearby 50% of all cases we get the photons on the detectors. This happens, if the way ist short. Then longer the way, then more outer influences destroy the directions of polarization of the two photons and the 50% get down. If we reach 40% for each detector, then we have a success of 80% compared with the ideal situation.

In both cases, the real process and the computer simulation, the states of the two photon is unknown until we measure it or until the computer displays the results. The mathematical probability function of our knowledge about this states collapses in the moment of seeing the result. But in the case of computer model one has to agree with the fact that the two photon states were generated before and once generated this has nothing to do with any probability. Coming to this point I ask, could it be, that the pair production works the same way. Once produced the states are fixed and only outer disturbances wash out the clear entanglement of these fixed states.

P.S.: I see clear the statements of well educated scientists that this could not be because it is never teached by this way. But this could not be a counter-argument to my answer. The answer only could be wrong if any of the steps in my answer could be falsificated. But be carefully, for the real process this is not possible since the process in the time between the photons emission and the photons detection is not viewable and my explanation is as good as the old one.

Answer 6

My problem with a positive answer would be that some local rule cellular automaton are Turing universal, which would imply that entanglement could be simulated by a model that uses a classical local rule. This seems wrong, doesn't it?

There is a classical, local, deterministic and realistic equivalent of QM, called bohmian mechanics. And no, it is NOT non-local, e.g. supports instantanious actions. Bell's locality condition is just non-sense. See this Link.

There is a nice playlist on youtube considering this topic: https://youtu.be/_6TNF854Xmo?list=PL7LbfRoKBR5OpRjt8toBOmzqGjH7zaM1m

Also, currently there are experiments done that support this interpretation of QM: https://www.quantamagazine.org/20140624-fluid-tests-hint-at-concrete-quantum-reality/

Furthermore there is a deep correspondance between classical statistical mechanics and the operator formulation of QM via Koopman-von-Neumann classical (statistical) mechanics: http://en.wikipedia.org/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics

Strictly speaking, the only difference between usual classical (statistical) mechanics and QM is the fact that the particle's position and its momentum do not commute, that is a measurment on the former can change the later and vice versa. In ordinary classical mechanics this fact is blurred because one always interacts with heavy objects, for which this change is negligible.