# Classical/Quantum Coin Toss

I am having a brainfreeze moment and have confused myself, help appreciated!

Quantum Coin: Superposition Heads AND Tails.

Classical Mechanics: Deterministic (in principle, if not in practice) if I repeat the same experiment I get the same result.

Quantum Mechanics: Non-deterministic no way that I can predict if I get heads or tails.

Now think of some physical implementation of a quantum coin perhaps I send some electron to mirror, afterwards it is on a superposition on both sides of the mirror. Perhaps reflected (heads) with probability 0.9 and transmitted (tails) with probability 0.1.

My question is does a classical analogy exist here? It can't be both deterministic and agree with the probabilities predicted by quantum mechanics right? Is the problem just that I should not be applying classical physics at all here? Does this question even make any sense?

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1) Classical is not equivalent to determinism. You could use probabilities in classical problems (Statistical mechanics, for instance)

2) It makes more sense, in fact, to consider the difference between a classical probabilistical problem, and a quantum probabilistical problem.

3) Quantum correlations are stronger than classical correlations, this is because, in quantum mechanics, we work with probabilites complex amplitudes $\psi$, instead of working directly with probabilites $p$ (The relation is $p = |\psi|^2$). Some experiment results cannot be explained by classical correlations.

4) If you consider, for instance, a superposition 1-spin quantum state like $\psi = |+_z>$ + $|-_z>$, a measurement of the spin on the $z$ axis will gives you always $+1$ OR $-1$. So, from the point of view of the measurement, it is an OR, it is not a AND. You will have 50% probability to measure +1, and 50% probability to measure -1.

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Thank you for your reply. Yes you can use probability in classical problems but this is because in statistical mechanics it is just unrealistic to compute all the different forces, in principle you could do it. If I sent the electron into the mirror with the same velocity, same angle etc then I should always get the same answer (classically). –  Gus Jun 7 '13 at 11:11
Electron, like photon, etc. are quanta (of fields). For instance, if you know perfectly the momentum (so idem for the velocity) of the electron, the uncertainty about the position is infinite . So you cannot consider an electron as classical. –  Trimok Jun 7 '13 at 11:54
The evolution of the electron wave function is also completely deterministic - Schrodinger's equation. It's just that your observables on a wave function are not deterministic because the instruments you use to observe have a random phase with respect the electron under study. Why random phase? It's just statistical mechanics again, but quantum version. –  Nanite Dec 4 '13 at 20:59

There is a key difference between probabilities in classical mechanics and quantum mechanics. In classical mechanics we result in indeterminate answers when the maximal amount of information is not given about the state of the system, whereas in quantum mechanics, even with maximal information (for example the case of state being $\frac{1}{\sqrt{2}}(|+_z>+|-_z>)$) our measurement of the system can result in a spread of answers, each with different associated probabilities. The result of physical coin flips is not completely random, and the 'randomness' is given by the variation of initial conditions (how we flip the coin). If we built a robot precise enough, we could get it to flip coins with the same initial condition and result in the same end result every time. (This is due to the determinism of classical mechanics you spoke of.)

Classically, the initial conditions determine the final result, so if we were to flip a coin with a range of initial positions, momenta and spin of the coin, the resulting 'probability' distribution of flips, will in general depend on which range of each of these quantities you take (and of course the distribution on these values).

I don't completely understand your question, but hopefully with this added information you can see the answer you are looking for. Please comment if it doesn't and I'll try to be more helpful.

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Once you measure a single electron's property, it collapses. It is a quantum randomness. Thus there is no classical analogy.

There are two kinds of randomness:

1. classical randomness, unknown because we are lazy. (contribute to entropy)
2. quantum randomness, unknown because the God plays dice. (does not contribute to entropy)

Classical randomness happens during the whole time interval; quantum randomness happens only at the moment when wave function collapses.

Pure State Non-zero Entropy Paradox (See more, if you are interested in entropy.)

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