In ordinary probability theory the conditional probability/likelihood is defined in terms of the joint and marginal likelihoods. Specifically, if the joint probability of two variables is $\mathcal{L}(x,y)$, the marginal likelihood of $x$ is $\mathcal{L}(x) \equiv \int \mathcal{L}(x,y) \operatorname{d}y$, and the conditional likelihood is then defined by $$\mathcal{L}(y|x) = \frac{\mathcal{L}(x,y)}{\mathcal{L}(x)}.$$

In quantum mechanics, the probabilities/likelihoods are defined in terms of probability amplitudes by the absolute square of the amplitudes. Is there an analogous relation for a conditional probability amplitude? Concretely, say we have a joint wave function of the variables $x$ and $y$, $\Psi(x,y)$. Can we define a conditional amplitude by the following relations: \begin{align} \psi(x) & \equiv \sqrt{ \int |\Psi(x,y)|^2 \operatorname{d}y} \\ \psi_c(y|x) & \equiv \frac{\Psi(x,y)}{\psi(x)}. \end{align}

Admittedly, there is a phase function ambiguity that is, strictly, allowable. That is, transforming the marginal and conditional amplitudes by $\psi(x) \rightarrow \psi(x) \mathrm{e}^{i \alpha(x)}$ and $\psi_c(x) \rightarrow \psi_c(x) \mathrm{e}^{-i \alpha(x)}$ for any function $\alpha(x)$ does not change the predicted probabilities.

I guess the question is less whether such quantities can be defined, because I obviously just defined them. I'm more curious about whether they've been used.

  • 2
    $\begingroup$ It is done using projectors and traces, not joint wave functions, see Bobo's paper. Classical conditional probability can also be stated in terms of projectors, but representation theorems for commutative algebras that lead to representations of projectors in terms of subsets and quotients. That does not work in the non-commutative case. $\endgroup$ – Conifold Jul 13 '17 at 2:17

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