A non-commutative Bayes’ theorem. (English) Zbl 1500.46052
Over the last 20 years, there has been a lot of interest in generalizing Bayesian inference from classical probability to quantum probability, see for example [H. Barnum and E. Knill, J. Math. Phys. 43, No. 5, 2097–2106 (2002; Zbl 1059.81027); M. S. Leifer, AIP Conf. Proc. 889, 172–186 (2007; Zbl 1138.81334); B. Coecke and R. W. Spekkens, Synthese 186, No. 3, 651–696 (2012; Zbl 1275.60006); B. Jacobs, Electron. Proc. Theor. Comput. Sci. (EPTCS) 287, 225–238 (2019; Zbl 1486.60004)]. Among the main challenges are to find a well-behaved generalization of Bayesian updating to the quantum setting and to develop methods for computing it.
Together with other recent preprints by the same authors, the present paper contributes to this line of investigation. It studies a particularly natural, but restrictive notion of Bayesian inversion in the Heisenberg picture: given finite-dimensional C*-algebras \(\mathcal{A}\) and \(\mathcal{B}\) with states \(\omega : \mathcal{A} \to \mathbb{C}\) and \(\xi : \mathcal{B} \to \mathbb{C}\), the authors define a quantum channel \(G : \mathcal{A} \to \mathcal{B}\) to be a Bayesian inverse of a quantum channel \(F : \mathcal{B} \to \mathcal{A}\) if \[ \xi(G(A) B) = \omega(A F(B)) \qquad \forall A \in \mathcal{A}, \: B \in \mathcal{B}. \] The main result of the paper is a necessary and sufficient condition for when the Bayesian inverse \(G\) exists (Theorems 5.62 and 6.22). The obvious necessary condition \(\xi = \omega \circ F\) is assumed throughout, but is found not to be sufficient. Much of the subtlety with the problem of existence of \(G\) is due to the difficulties that arise in the case where \(\xi\) does not have full support, and the authors emphasize that their careful treatment of this aspect improves upon other works significantly.
Prior to solving the existence question, Section 3 discusses some basics of the above notion of Bayesian inversion, and in particular explains the sense in which the Bayesian inverse \(G\) is unique up to almost sure equality. Section 4 provides a good selection of special cases of the above problem, and provides a number of more concrete criteria for the existence of a Bayesian inverse. These criteria take the form of commutativity conditions, which suggests that Bayesian inverses in the above sense may be relatively rare.
Together with other recent preprints by the same authors, the present paper contributes to this line of investigation. It studies a particularly natural, but restrictive notion of Bayesian inversion in the Heisenberg picture: given finite-dimensional C*-algebras \(\mathcal{A}\) and \(\mathcal{B}\) with states \(\omega : \mathcal{A} \to \mathbb{C}\) and \(\xi : \mathcal{B} \to \mathbb{C}\), the authors define a quantum channel \(G : \mathcal{A} \to \mathcal{B}\) to be a Bayesian inverse of a quantum channel \(F : \mathcal{B} \to \mathcal{A}\) if \[ \xi(G(A) B) = \omega(A F(B)) \qquad \forall A \in \mathcal{A}, \: B \in \mathcal{B}. \] The main result of the paper is a necessary and sufficient condition for when the Bayesian inverse \(G\) exists (Theorems 5.62 and 6.22). The obvious necessary condition \(\xi = \omega \circ F\) is assumed throughout, but is found not to be sufficient. Much of the subtlety with the problem of existence of \(G\) is due to the difficulties that arise in the case where \(\xi\) does not have full support, and the authors emphasize that their careful treatment of this aspect improves upon other works significantly.
Prior to solving the existence question, Section 3 discusses some basics of the above notion of Bayesian inversion, and in particular explains the sense in which the Bayesian inverse \(G\) is unique up to almost sure equality. Section 4 provides a good selection of special cases of the above problem, and provides a number of more concrete criteria for the existence of a Bayesian inverse. These criteria take the form of commutativity conditions, which suggests that Bayesian inverses in the above sense may be relatively rare.
Reviewer: Tobias Fritz (Innsbruck)
MSC:
| 46L53 | Noncommutative probability and statistics |
| 81P16 | Quantum state spaces, operational and probabilistic concepts |
| 81P45 | Quantum information, communication, networks (quantum-theoretic aspects) |
| 62F15 | Bayesian inference |
| 18M40 | Dagger categories, categorical quantum mechanics |
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