How do we expand Bayes theorem to account for probability amplitudes? The Bayes theorem simply states:
$$ P(B | A) P(A) = P(A | B) P(B) $$
I wonder if there is something that can be meaningfully said as generalization of this relationship when the probabilities in question arise from applying the Born rule to a quantum system. Is there an expression akin to Bayes that applies directly to probability amplitudes?
 A: Let us rephrase the Bayes theorem in a slightly different way. Given a classical channel $P(B|A)$ (a conditional probability distribution of $B$ given $A$) and a probability distribution $P(B)$, Bayes theorem defines another classical channel (the posterior conditional probability distribution of $A$ given $B$) via the Bayesian inverse, as $$Q(A|B) := P(B|A)\frac{P(A)}{P(B)},$$ where $P(B)=\sum_A P(B|A) P(A)$. Importantly, the posterior distribution satisfies  $\sum_B Q(A|B) P(B)=P(A)$. In other words, if $P(B)$ is the "output" of distribution $P(A)$ passed through the channel $P(B|A)$, then $P(A)$ is the "output" of the distribution $P(B)$ passed through the channel $Q(A|B)$.
There is a quantum analogue of this version of Bayes rule. Consider a quantum channel $\mathcal{N}$ which maps some density matrix $\sigma_A$ to another density matrix $\sigma_B = \mathcal{N}(\sigma_A)$. Then, the "Petz recovery map" $\mathcal{R}$ is another quantum channel, which is defined via its action on an arbitrary density matrix $\omega_B$ as
$$\mathcal{R}(\omega_B) := \sigma_A^{1/2}\mathcal{N}^\dagger(\sigma_B^{-1/2} \omega_B \sigma_B^{-1/2}) \sigma_A^{1/2}.$$
The recovery channel obeys $\mathcal{R}(\mathcal{N}(\sigma_A))=\sigma_A$. If $\mathcal{N}$ is a classical channel, then the recovery channel is its classical Bayesian inverse, as expected. Note that $\mathcal{R}$ depends on both $\mathcal{N}$ and $\sigma_A$ (just like the classical Bayesian inverse $Q(A|B)$ depends both on the choice of $P(B|A)$ and $P(A)$).
For more:

*

*Wilde, Quantum information theory, 2017, section 12.3.

*Leifer and Spekkens, "Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference", PRA, 2013, especially section "IV. Quantum Bayes Theorem" (note that they call $\mathcal{R}$ the "Barnum-Knill
recovery map", instead of the Petz recovery map)

