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We can simulate the motion of atoms using a classical computer. Therefore, it seems we should be able to simulate quantum mechanical systems on classical computers. However, there still seems to be some mystery surrounding the collapse of the wave function upon measurement.

Suppose we wanted to simulate the measurement of a qubit using a classical computer. We could, in theory, simulate the time evolution of the many body (many qubit) Schrodinger equation, perhaps for a small system even in practice.

What would we see if we would simulate the measurement? In particular, would we see the mysterious wave function collapse?

It would be nice if an answer could point out what would be the simplest (physical) system that we would need to simulate in order to see (in detail) what happens when we measure a qubit (when a the function collapses).

DanielSank
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jhegedus
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  • Wave-function collapse by measurement and its simulation https://journals.aps.org/pra/abstract/10.1103/PhysRevA.44.39 – jhegedus May 08 '17 at 11:24
  • Simulating causal wavefunction collapse models http://iopscience.iop.org/article/10.1088/0264-9381/21/12/011 – jhegedus May 08 '17 at 11:25
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    Have a look at the answer to this question – By Symmetry May 08 '17 at 12:17
  • There are seven individual questions in this post. That's six more than there should be. I'd like to answer this post, but I don't want to answer seven things! Please narrow the post down to one focused question. It's ok to make multiple posts if you have multiple questions. – DanielSank May 08 '17 at 12:35
  • @DanielSank the questions are related, the main question is to figure out if a detailed simulation of wave function collapse would explain the mystery of the wave function collapse. Would that tell which QM interpretation is "correct" ? – jhegedus May 08 '17 at 16:33
  • Yeah, they're related, but the post still asks way more than anyone is going to want to answer. You might get an answer from someone who chooses to focus on one of the seven questions, but it would be a lot better if you could clean up the post so that in the end we get one question with one answer. Frankly, I could probably answer this but I'm not going to until the post has a single clear question. – DanielSank May 08 '17 at 16:37
  • I've edited the post quite a bit to make it more focused. If you disagree with the edits, you can easily roll them back :-) – DanielSank May 08 '17 at 16:42
  • Thanks, indeed the main question was to know if a simulation would allow to shed some light on the wave function collapse. If it could take the mystery/ hand waving and postulates out of it but if not, why not ? Could we actually see the Born rule emerging from the simulation ? https://en.wikipedia.org/wiki/Born_rule – jhegedus May 08 '17 at 16:55
  • Wave function collapse isn't all that much of a mystery anymore. It's only tricky because, when taken to the extreme with an archetypal perfect classical observer, some of the interpretations start to feel a bit metaphysical. Once you realize that you and I may not actually be perfect classical observers, but merely statistical approximations of a perfect classical observer a lot of the mystery goes away. – Cort Ammon May 08 '17 at 17:07
  • @CortAmmon ok, but how ? is there simple example for this mystery going away ? – jhegedus May 08 '17 at 17:26
  • If you do not start from the assumption that your observer is classical, and instead start from the assumption that your observer is a quantum system whose state is basically a random variable, the expectation of the result of the interaction at the quatum level approaches the classical solution as the size of the observer relative to they system approaches infinity. – Cort Ammon May 08 '17 at 17:35
  • Yes, kinda makes sense but I would love to see what this means, for a model system that I can comprehend. Maybe in a simulation or in some calculation. Observer's state is a random variable ? In what sense ? Unitary evolution has no random variables and I would like to see if a simulation could show how the Born rule (the random variables) emerge from the unitary evolution (SCH equation). – jhegedus May 08 '17 at 18:22

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Suppose we wanted to simulate the measurement of a qubit using a classical computer. We could, in theory, simulate the time evolution of the many body (many qubit) Schrodinger equation, perhaps for a small system even in practice.

Yes, we do this all the time as a matter of routine. If you look at most superconducting qubit papers, you'll see that the experimental data are compared against computer simulations.

Simulating a few interacting qubits can be done on a conventional CPU in a laptop. Using the world's largest supercomputing resources, we can simulate up to around 49 interacting qubits. Beyond that, present day classical computing resources cannot do simulations in a reasonable amount of time.

Note the comment about time: simulating a large quantum system isn't impossible in principle, it's just inefficient and becomes intolerably slow (and requires astronomical amounts of memory) as the quantum system's size increases. For a rough estimate, note that an $N$-qubit system's state requires $2^N$ real numbers. If we use 1-byte floats, then a 64GB system can hold the state of only $$ N = \log_2(64 \times 10^9) = 32$$ qubits, and we haven't even talked about the operators that act on those qubits.

What would we see if we would simulate the measurement? In particular, would we see the mysterious wave function collapse?

We don't need to speculate, this has already been done! Wave function collapse is already pretty well understood theoretically, and there have been experiments showing exactly what happens during the measurement-induced collapse, along with computer simulations. Here are a few examples:

There was even an experiment showing collapse during simultaneous measurement of non-commuting observables!

It would be nice if an answer could point out what would be the simplest (physical) system that we would need to simulate in order to see (in detail) what happens when we measure a qubit (when a the function collapses).

All of the experiments and simulations linked above used a single qubit. That's all you need in order to study wave function collapse.

Now, having said all of this, I think we still haven't really answered your question. I think you might be trying to ask how we get wave function collapse if we're just simulating the system by numerically solving the Schrodinger equation. That's a good question, and the answer is that we don't. The Schrodinger equation is not a complete description of quantum mechanics. In order to understand measurement and wave function collapse, we need a more complete and more modern expansion of the theory. This theory is very well studied and understood. It's not new or secret or magic, it just hasn't really made it into standard textbooks, university courses, or public knowledge yet.

In order to understand measurement and wave function collapse, we have to extend quantum mechanics to the case where we only have access to a sub-part of the total system. You see, when we measure, we're interacting something, photons, electrons, or whatever else our measurement apparatus uses, with the system under study. An oscilloscope, for example, contains a lot of atoms, which all become entangled with the observed system. We don't have access to the states of all those atoms, so in the end we have to find a way to describe a sub-part of the whole thing. This is done with the density matrix, which is nowadays a crucial part of quantum theory.

DanielSank
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  • "I think you might be trying to ask how we get wave function collapse if we're just simulating the system by numerically solving the Schrodinger equation. That's a good question, and the answer is that we don't."

    Hmm... why not ? If we simulate both the observer and the observed then we could see how the observer changes in response to the interaction with state of the observed. This could even show how the Born rule emerges. For this, a model system would be needed to be simulated, 1 qubit for the observed and lot of qubits for the observer and follow the time evolution of the observer.

     – jhegedus May 08 '17 at 17:18
  • "We don't have access to the states of all those atoms, so in the end we have to find a way to describe a sub-part of the whole thing. " - yes, we don't, but if we would, what would we see ? – jhegedus May 08 '17 at 17:20
  • @jhegedus Ah yes, if you simulate both the observer (including all of the atoms etc. in the measurement apparatus) then you find that after a measurement, the system is in an entangled state where each state of the system under study corresponds to a sensible state of the measurement apparatus. Schlosshauer's book shows this in detail. See also this paper by him. – DanielSank May 09 '17 at 00:46
  • @jhegedus please let me know what exactly I can add to this answer so that it addresses your questions. I look forward to completing this Q&A because it's a good question :-) – DanielSank May 09 '17 at 00:47
  • @jhegedus I should have said: once you have the entangled state between system and measurement apparatus, you can ask yourself what the system looks like from the point of view of the measurement apparatus. To do this, you have to compute the reduced density matrix of the system. Why you should do this (and what a reduced density matrix is), is explained in other posts that I can link to if you want. Once you compute this reduced density matrix, you find that it's diagonal in the basis of whatever operator couples the system and apparatus. This is, essentially, wave function collapse. – DanielSank May 09 '17 at 00:49
  • "you can ask yourself what the system looks like from the point of view of the measurement apparatus" -> exactly this is what I am after, to clarify what this mean, the "look like", what does it mean to "look like" ? The simulation could give a detailed answer what this means. For example if I think about the Sch-Cat experiment, what does the isotope look like from the cat's point of view ? What is the cat ? Is it a superposition of two cats ? Each having it's own perception of the isotope ? Would something like that come out of a simulation ? If we could simulate the Sch-Cat experiment ? – jhegedus May 09 '17 at 03:35
  • " Why you should do this (and what a reduced density matrix is), is explained in other posts that I can link to if you want." -> yes, I would like to know why and how the density matrix is related to the "what does the measured system looks like from the oberserver's point of view"-question. – jhegedus May 09 '17 at 03:36
  • "...Schlosshauer's book..." -> hmmm I might need to read his book and come back again after that. – jhegedus May 09 '17 at 03:40
  • Hmmm...this seems to be pretty nice indeed https://hsm.stackexchange.com/questions/3631/when-was-the-measurement-problem-solved – jhegedus May 09 '17 at 03:57
  • "Most people agree, that the measurement problem is partially(!) solved by invoking a relative-state formulation for Observer+System+Environment and combining it with a suitable interpretation, mostly many worlds or consistent histories. The observer measures the system in his einselected basis, and the irreversibility comes from the information-loss in the environment." -> the real question is about the einselected basis... what a simulation could show... but maybe can never show - because it is so complicated ? – jhegedus May 09 '17 at 04:19
  • What is the simplest system that we could simulate and it would still show the einselection and the corresponding wave function collapse and how the einselected states are interacting with each other. – jhegedus May 09 '17 at 05:16