Quantum wave amplitudes behave in some ways like probabilities and in other ways unlike probabilities. Because of this, some concepts have analogues, while others don’t.
But no concepts are exactly equivalent. For example, evidence isn’t integrally linked to complex numbers, while entanglement is.
Nonetheless, it is instructive (imho) to consider how (assigned) probability is a property of the observer, and not an inherent property of the system. If a qubit is (|0> + |1>)/sqrt(2), and I measure it and observe 0, then I’m entangled with it so relative to me it’s now |0>. But what’s really happened is that I became (|observed 0> + |observed 1>)/sqrt(2), or rather, that the whole system became (|0,observed 0> + |1,observed 1>)/sqrt(2). This is closely analogous to the Law of Conservation of Probability; if you take Expectations conditional on the observation, then take Expectation of the whole thing, you get the original expectation back. This is because observing the system doesn’t change the system, it just changes you. This is obvious in Bayesian probability in the classical-mechanics world; the only reason it doesn’t seem obvious in the quantum realm is that we’ve been told over and over that “observing a quantum system changes it”.
Quite honestly, I don’t see how a Bayesian can possibly be a Copenhagenist. Quantum probability is Bayesian probability, because quantum entanglement is just the territory updating itself on an observation, in the same way that Bayesian ‘evidence entanglement’ is updating one’s map on an observation.
Um, but isn’t that just a convention? Why should we treat the “amplitude” of a classical probability as being the probability?
Does the problem have something to do with the extra directionality quantum probabilities have by virtue of the amplitude being in C? (so that |0> and (-1*|0>) can cancel each other out)
Classical probability transformations preserve amplitude and quantum ones preserve |amplitude|^2. That’s not a whole reason, but it’s part of one.
Yes, that’s part of the difference. Quantum transformations are linear in a two-dimensional wave amplitude but preserve a 1-dimensional |amplitude|^2. Classical transformations are linear in one-dimensional probability and preserve 1-dimensional probability.
Quantum wave amplitudes behave in some ways like probabilities and in other ways unlike probabilities. Because of this, some concepts have analogues, while others don’t.
But no concepts are exactly equivalent. For example, evidence isn’t integrally linked to complex numbers, while entanglement is.
Nonetheless, it is instructive (imho) to consider how (assigned) probability is a property of the observer, and not an inherent property of the system. If a qubit is (|0> + |1>)/sqrt(2), and I measure it and observe 0, then I’m entangled with it so relative to me it’s now |0>. But what’s really happened is that I became (|observed 0> + |observed 1>)/sqrt(2), or rather, that the whole system became (|0,observed 0> + |1,observed 1>)/sqrt(2). This is closely analogous to the Law of Conservation of Probability; if you take Expectations conditional on the observation, then take Expectation of the whole thing, you get the original expectation back. This is because observing the system doesn’t change the system, it just changes you. This is obvious in Bayesian probability in the classical-mechanics world; the only reason it doesn’t seem obvious in the quantum realm is that we’ve been told over and over that “observing a quantum system changes it”.
Quite honestly, I don’t see how a Bayesian can possibly be a Copenhagenist. Quantum probability is Bayesian probability, because quantum entanglement is just the territory updating itself on an observation, in the same way that Bayesian ‘evidence entanglement’ is updating one’s map on an observation.
Classical probability preserves amplitude, quantum preserves |amplitude|^2.
They’re different things, and they could, potentially, be even more different.
Um, but isn’t that just a convention? Why should we treat the “amplitude” of a classical probability as being the probability?
Does the problem have something to do with the extra directionality quantum probabilities have by virtue of the amplitude being in C? (so that |0> and (-1*|0>) can cancel each other out)
Classical probability transformations preserve amplitude and quantum ones preserve |amplitude|^2. That’s not a whole reason, but it’s part of one.
Yes, that’s part of the difference. Quantum transformations are linear in a two-dimensional wave amplitude but preserve a 1-dimensional |amplitude|^2. Classical transformations are linear in one-dimensional probability and preserve 1-dimensional probability.
Ah, I get it now, thanks!
(Copenhagen is still wrong though ;)