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Quantum Test Found For Mathematical Undecidability
KentuckyFC writes "Philosophers have long wondered at the profound link between mathematics and physics, but how deep does this connection go? Pretty deep according to the results of a quantum experiment exploring the nature of mathematical undecidability. Here's how: any logical system must be based on axioms, which are propositions that are defined to be true. A proposition is logically independent from these axioms if it can neither be proved nor disproved from them; mathematicians say it is undecidable. In the experiment, researchers encoded a set of axioms as quantum states. A particular measurement on this system can then be thought of as a proposition which, if undecidable, yields a random result — which is what they found. 'This sheds new light on the (mathematical) origin of quantum randomness in these measurements,' say the researchers (abstract)."
It's a bit hard to explain all this stuff in few words. I could refer you to about half a dozen Wikipedia and Wolfram articles on the subjects and you'd still be in the dark. Instead I'll suggest you read GÃdel, Escher, Bach by Douglas Hofstadter, who tackles many of those subjects in an amusing and educational way.
If he explores all forms and substances Straight homeward to their symbol-essences; He shall not die.
From what I understood, they use qubits to encode facts about finite boolean functions. For example, they can use a number of qubits to encode a situation where f:{0,1}->{0,1} and f(0) = 0. Sure enough, the proposition f(1) = 0 is undecidable from the given information, and they claim that they can measure this fact, which, imho, is really cool.
However, those people who wanted to use qubits to establish consistency results should not hold their breath. For a finite structure, decidability of any statement can be checked by going through a long table. To do anything ineteresting, one would have to use infinitely many qubits, which I do not see happening.
They found a way to physically encode a mathematical "axiom" into quantum states. They set up a particular axiom as a quantum state machine, then measure the system. The measurement is done in such a way that it is equivalent to asking "is X true given this axiom?" where X is any mathematical "proposition". The answer to that question can be "yes", "no", or "not enough information". If the latter is the case, the results from the physical quantum experiment will show a random distribution.
So, if I have a mathematical proposition and I'm not sure if it is supported by a certain axiom, I could actually build the axiom into a quantum state machine and measure it in a way that tests my particular proposition. If the results after multiple runs are distributed randomly, then it means that the axiom can not prove or disprove the proposition.
No.
This is a method to determine whether or statements are part of a system, not whether they are true or false within the system.
So, it can tell you whether or not there is an answer, but not what the answer is.
Furthermore, it can only truly prove that something is not a member of the system, because then you get different answers when you query the system. But if you keep getting the same answers, well, that could just be coincidence. Hence, you can be fairly certain, but it is not the same thing as a proof.
When things get complex, multiply by the complex conjugate.
First, let me say this is extremely subtle stuff. I won't claim to understand it with even passing familiarity. But the summary and the article (which is a summary of a research paper) give enough clues to provide an educated guess.
Part of quantum mechanics involves the idea that some kinds of measurements are incompatible. For example, the famous Heisenberg principle says you can't make a measurement on a particle's position and velocity and get accurate measurements for each. If you make a measurement on position you'll get a result, and a physicist would then say that the particle is in a quantum state that has a well-defined position operator (actually he'd say that the particle is in an eigenstate of the position operator). You could make the measurement a second time, and you'd get the same position. Ditto for the third, fourth, etc time as well.
If you now go and try and measure velocity (momentum actually), you will also get a result. A physicist would write that particle is now in a quantum state with a well-defined momentum operator. Here's the catch: if you then go back and try to measure the particle's position again, you'll get a random result. It isn't possible to get a quantum state that has both position and momentum operators being well-defined.
Some kinds of operators are compatible, though. For those with some quantum mechanics knowledge, it would be possible to simultaneously measure the total magnetic spin of a particle (S^2) and the spin component along one axis (Sz). The mathies would talk about Hilbert spaces and diagonalizable matrices, but for our purposes we'll just say that the quantum state has several well defined operators.
So...my (limited) understanding of the paper is that the authors propose encoding a set of mathematical axiom by setting a particle into a quantum eigenstate that admits multiple well-defined operators, with each separate operator corresponding to a particular mathematical axiom.
If a particular mathematical proposition is compatible with the given set of axioms, it will then be associated with a well-defined quantum operator of the particle. Making a measurement would then give the same answer each time (like measuring position over and over). But, if the proposition were undecidable, then the quantum operator would not be well-defined, and the measurement would produce a different (random) result each time.
Actually implementing such a system would be another question entirely but, like so much of quantum mechanics, it does pose interesting thought experiments.