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What is the origin of probabilities in quantum theory if it is not postulated? I mean, how can we interpret Born's Rule as a probability and how does it enter in quantum theory? More than this, I want to understand the differene between classical probabilities and quantum probabilities.

Classicaly, statistics and probabilities comes in when we consider huge sets of particles and we cannot solve all the differential equations that rises up, not in a finite human time. So we work with statistics etc. But statistics in quantum theory are intrinsec in some way to the theory. I do not understand how one set is larger (quantum set) than the other (classical set), speaking in terms of a possible larger set of probabilities that contains both of them. If this set exists, I can try to understand how quantum probabilities emerges and why nature follows this and not any of others general theories that is mathematically possible. Does it make sense? Are there any good reference on these topics?

Following this line I can define repeatable outcomes in quantum theory as being "a measurement j that every time it is performes and an outcome k is obtained, a subsequent measurement of j in the same system gives outcome k with probability 1". From this, I have the definition of a set of measurements that are repeatable-outcomes:

Definition: "A set of outcome-repeatable measurements {$j_1 , j_2 , ... , j_n$} is said to be compatible if there is another measurement j with outcomes {1 , ... , m} and functions $f_1, ... , f_n$ such that the possible outcomes of each $j_s$ are $f_s(\{1, ... , m\})$ and

$p_{j_s} = \Sigma_{k \in f_s^-1(l)} p_j(k)$

The measurement j is called a refinement of each $j_i$, and each $j_i$ is called a coarse-grainning of j"

Well, this definition states that measurements are compatible if I can reconstruct (???) the probabilities of a set of measurements from the probabilities of another measurement? I do not understand exactly this concept of coarse-grainning and I am looking for some reference that can discuss this in more details in this context. Also, are there any good reference that links this two questions: about the nature of probabilities in quantum theory and the coarse-graining concept?

This jointly performation possibility of measurements is well-known essential in quantum theory. From this definition we can see that if a set of measurements is compatible than I can measured them jointly without disturbing the system/the results of each other; or, I can perform a measurement in j and them use the functions f to reconstruct the outcomes of eache $j_k$. But what is the nature of these functions? How can I define them?

At last, but not least, i am also trying to see if there is any way that connects Shannon's entropy for information with quantum systems? Are there any "larger information theory" that contains both classical and quantum information theories? Recently I saw some efforts in deriving quantum mechanics from some assumptions of information theory and Bayesian inference and now I am looking forward for more than this.

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  • $\begingroup$ The classical limiting procedure of von-Neumann quantum entropy yields Shannon's classical entropy, entailing forfeiture of quantum information. $\endgroup$ – Cosmas Zachos Dec 4 '19 at 17:57
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For an overview, you may wish to read the wikipedia articles about quantum probability and the book 'Quantum Logic' by Svozil.

As for the origin of probabilities in quantum theory if they are not postulated, you need to look at the appropriate interpretations of quantum theory. For example, Bohmian mechanics introduces additional position coordinates and a deterministic dynamics for them, and gets probabilities in the same way as in the classical statistical mechanics of equilibrium, by assuming some pre-ordained form of quantum equilibrium.

My thermal interpretation of quantum physics is also deterministic. It gets probabilities in the same way as one gets classical probabilities for throwing a coin from the laws of Newtonian mechanics: By coarse-graining the exact deterministic quantum dynamics (the Ehrenfest equations for q-expectations), one obtains a stochastic, nonlinear effective dynamics, which (for a binary measurement) leads, due to bistability, to an effectively random emergence of the pointer variable in one of the two stable basins of attraction.

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