Suppose I have a particle whose quantum state is known to be exactly spin-up along the z-axis. Then suppose I measure its spin along the z-axis. Quantum mechanics (QM) states that the probability of measuring spin-up is 100% and that of spin-down is 0%.

Does this 0% mean that measuring spin-down is impossible (forbidden and cannot occur), or does it mean that measuring spin-down is "possible but infinitely unlikely" (similar to, say, the probability of hitting the exact center of a perfect, mathematical dart board)?

Side notes:

On the one hand, some may argue that the question is moot because in practice, we cannot know the exact quantum state of a system nor measure exactly along the z-axis due to experimental limitations. My counter point would be that nothing in the mathematical formalism of QM prevents us from talking about this situation, and so we should expect the mathematical theory to make an exact prediction with a clear physical meaning. The formalism should provide an unambiguous answer for any ideal case, and then we can worry about experimental imprecision as an afterthought.

This point was never raised during any of my QM training, and I have always thought of zero probability as "impossible" in this context. But given that probability is not well understood philosophically, and that there are situations where "X has probability zero" does not imply "X cannot occur," I think this question deserves some thought.

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    This depends on the sample space, whether it is classical, quantum, or has nothing to do with physics is moot. If the sample space is finite then 0 probability means impossibility, if it is infinite there can be possible outcomes with 0 probability. The spin space is finite (up or down), so yes, it does imply impossibility. – Conifold Aug 4 at 6:33
  • @Conifold What justifies the claim that in a finite sample space, 0 probability implies impossibility? The only justification I can think of is that on finite sample spaces, we would not assign 0 probability to outcome X unless X were impossible. But QM gives us probabilities based on the Born rule, not based on our knowledge of whether an event is possible, so I'm not convinced this argument holds. – WillG Aug 4 at 7:02
  • If I flip a trick coin with heads on both sides, then getting tails has probability 0 because it is impossible. But QM just hands us probability 0 without reference to whether anything is possible or not. – WillG Aug 4 at 7:06
  • Are you familiar with the concept of probability density at a point, basically just the limit of (probability the sample lies in a given region)/(volume of region) as the size of the region surrounding the point approaches zero? In your dartboard example, the probability density at the center wouldn't normally be zero, but say we imagine the center has a "repulsive" property giving a probability density func. that approaches zero as you approach the center, and is exactly zero at the center. In this case would you say hitting the center is "possible but infinitely unlikely" or "impossible"? – Hypnosifl Aug 4 at 7:19
  • Either event occurs or it does not. With finitely many outcomes even a single occurrence will give you a non-zero fraction. And the Born rule is a surmise of what is and is not possible from multiple experiments. – Conifold Aug 4 at 7:26

Impossibility, like nothingness, is an intensive and, especially for metaphysics, highly significant notion of philosophy. Sensu stricto, it is not a mathematical notion. In the context of probability theory, impossibility is defined, if at all, by zero probability (not vice versa) on pedagogical reasons. A simple illustrative example is choosing a number uniformly at random from the unit interval [0,1] on the real number line: The probability of choosing a number from any finite subset of [0, 1] is zero, however, it is not an impossible event.

Hence in general, it is a good practice to reserve the word 'impossible' to describe the events that are beyond the confines of probability measure, definition, experience and the like.

Returning to the question, whether the event is impossible in the philosophical sense, or of zero probability measure is a matter to be deemed by physics. Sure, it can be discussed for philosophical elucidation; see, for instance, M. Hemmo and O. Shenker's 2013 paper "Probability Zero in Bohm's Theory" (Philosophy of Science 80, pp. 1148-1158), but cannot be judged relying on philosophical contemplation. Hegel's notorious case of "On the Orbit of the Planets" ought to be kept in mind (though it has been claimed that Hegel's motivation might be different than it has been perceived, see B. Beaumont's "Hegel and the Seven Planets," Mind 63, pp. 246-248, 1954, it is a faux pas, anyway).

As the upshot of these considerations, I recommend to migrate this question to Physics SX.

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  • I think the issue is that physics says "the probability is zero," but doesn't tell us anything beyond that (i.e., what probability means). – WillG Aug 10 at 6:46
  • The Hemmo and Shenker paper was interesting—their take is that in traditional QM, probability zero = impossible. However, they arrive at this by interpreting the collapse postulate as "first project the state vector onto the 'direction' corresponding to the measurement, then rescale the new vector to length one." The zero vector vector cannot simply be "scaled up" to unit length, so they deem this impossible. – WillG Aug 10 at 6:50
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    But this seems ad hoc to me. We could instead phrase the collapse postulate as "replace the state vector with a unit vector in the 'direction' corresponding to the measurement outcome," and then it remains sensible to talk about measuring one of the 0-probability outcomes. – WillG Aug 10 at 6:53

Your example doesn't work because you can't know the precise state (within any uncertainity) unless you just measured the system. Then until you measure it again, you can't say precisely which state it's in. You can't know you will measure any property with arbitrary certainity ahead of time.

There are more aspects to your question but I want to make sure this is clear first.

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  • Please read my first side-note, as this partly addresses your point. But additionally, some of your claims aren't correct. Between measurements the state evolves deterministically according to the Schrödinger equation, so you can say precisely what state the system is in after you measure it, provided you measure it perfectly. You can argue that perfect measurement is practically infeasible, but quantum theory does not forbid it. – WillG Aug 4 at 19:12
  • Yes the Schrodinger equation is completely deteterminstic. BUT what we experience (or measure, or "where we are in the universal wavefunction), or however you want to interpret the "wave function collapse" problem/interpretarion of qm, is NOT predictable ahead of time except for propabilties – J Kusin Aug 4 at 19:20
  • Ah, ok. So I agree in the case where the Born rule gives 0 < p < 1—then the wave function collapse is based on "pure randomness" and cannot be predicted (beyond probabilities), per standard interpretations of QM. So my question is, what are we to make of the "edge" cases p = 0 and p = 1? – WillG Aug 4 at 19:25
  • It ate comment so: p=0 to observe a particle past its light cone but p=? to observe a photon at an infinitely small volume of continuous space. Maybe we can only speak of probability as in what we can subjectively measure and preform. Objectivism or frequentism requires infinite trialing, something we can never do. Subj vs obj nature of probability ties directly in which qm interpretations to believe. – J Kusin Aug 4 at 19:44

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