[howling]

In some sense, classical probability is just the study of category of Markov Kernels [0]. We can gain much of the insight without a lot of measure theory/functional analysis machineries by restricting to its full subcategory of finite states, which gets us to the category of stochastic matrix.

In this view, quantum probability (of finite states) is just about study of category of classical quantum maps described in Picturing Quantum Processes by Bob Coecke and Aleks Kissinger [1]. I highly recommend the book, which requires hardly any prerequisites other than linear algebra and mathematical maturity. Once we get into this framework, no philosophical interpretation is required. E.g. we can't reproduce a quantum state by classical data just means any morphism from a (non-trivial) purely quantum state to a purely classical state does not have a left inverse.

[0]: https://en.wikipedia.org/wiki/Markov_kernel [1]: https://www.cambridge.org/core/books/picturing-quantum-proce...

[klysm]

Idk anything about measure theory, but I do know some things about markov chains so I was excited about [0]. Unfortunately measure theory is invoked directly in the definition so I’ll have to go down that wiki rabbit hole

[atomflunder]

I worked through large parts of PQP ([1]) with only the prerequisites that you state, as part of a Uni course taught by Aleks Kissinger. I got quite comfortable with the formalism and could solve most problems without too much issue, but at the end I still knew nothing at all about quantum physics. YMMV