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This question may be a little subjective, but I would like to understand, from a geometric perspective, how the structure of quantum theory differs from that of classical probability theory.

I have a good understanding of classical information geometry, and a reasonable understanding of quantum theory, for finite Hilbert spaces at least. (I am mostly only interested in finite spaces for now.)

I understand information geometry mostly from the perspective of Amari, who essentially derives everything from the "Pythagorean theorem". This says that we can treat the Kullback-Leibler divergence as a kind of squared distance. That is, if we have classical probability distributions $P$, $Q$ and $R$ such that $$ D_{KL}(R\|Q) + D_{KL}(Q\|P) = D_{KL}(R\|P) $$ then we may interpret $P$, $Q$ and $R$ as points that lie on the corners of a right triangle, with $Q$ at the right angle. I like to think of this as a definition of orthogonality rather than a theorem, and much of the rest of information geometry (mixture families and exponential families, the Fisher metric, dually flat spaces, and so on) can be seen as following from it.

Unfortunately Amari's book doesn't cover quantum information geometry. Here we have the quantum relative entropy, $\mathrm{Tr}\,\rho(\log \rho - \log \sigma)$, which generalises the Kullback-Leibler divergence. Presumably a similar Pythagorean interpretation can be made, with presumably gives rise to a similar dually flat structure, with quantum analogs of the Fisher metric, mixture and exponential families, and so on.

Here is my question: looking at these two objects purely as geometric spaces, is it possible to point at one or a few key defining differences between them?

Here is one observation as a possible starting point. In the quantum case, if we fix a basis and only consider distributions that are mixtures of the eigenstates, then we have a manifold that is exactly equivalent to the classical probability simplex. However, this works for any unitary basis, and so the quantum case has a kind of rotational symmetry that's lacking in the classical case.

I suspect this is not the only difference, however, and so I am looking for a characterisation of the differences between the classical probability simplex and its quantum analog, in terms of their properties as geometric manifolds.

(I would also appreciate pointers to good introductions to quantum information geometry, ideally pitched at a similar level to Amari's book and available online.)

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  • $\begingroup$ It may be helpful to compare the computational complexity differences between the classes BPP and BQP. When I studied quantum computation, the big difference I noticed between that and probabilistic computation was cancellation/rotation. Quantum bits have both magnitude and phase, where probability bits only have magnitude of probability. $\endgroup$ – Larry B. Jun 21 '18 at 18:10

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