Conditioning through indifference in quantum mechanics
Keano De Vos, Gert de Cooman
TL;DR
The paper develops a decision-theoretic, imprecise-probability approach to conditioning quantum states on measurement outcomes by modeling beliefs with coherent sets of desirable measurements. It shows that the induced updating rules are consistent with density-operator formalisms and Lüders-like conditioning, and it provides a general mechanism to condition on a subspace $\mathcal{V}$ using a projection $\hat{P}_{\mathcal{V}}$, yielding $\hat{\rho}_{\mathcal{V}}=\dfrac{\hat{P}_{\mathcal{V}}\hat{\rho}\hat{P}_{\mathcal{V}}}{\mathrm{Tr}(\hat{P}_{\mathcal{V}}\hat{\rho}\hat{P}_{\mathcal{V}})}$. The authors extend the framework to conditioning on unions of orthogonal subspaces $\mathcal{V}_k$, obtaining $\hat{\rho}_{\mathcal{S}}=\dfrac{\sum_k\hat{P}_{\mathcal{V}_k}\hat{\rho}\hat{P}_{\mathcal{V}_k}}{\mathrm{Tr}(\sum_k\hat{P}_{\mathcal{V}_k}\hat{\rho}\hat{P}_{\mathcal{V}_k})}$, thereby combining Lüders-style updating with a law-of-total-probability perspective. The approach unifies precise and imprecise belief updates, clarifies the role of indifference to measurements induced by known subspaces, and points toward extensions to POVMs and conservative inference in quantum information tasks.
Abstract
We can learn (more) about the state a quantum system is in through measurements. We look at how to describe the uncertainty about a quantum system's state conditional on executing such measurements. We show that by exploiting the interplay between desirability, coherence and indifference, a general rule for conditioning can be derived. We then apply this rule to conditioning on measurement outcomes, and show how it generalises to conditioning on a set of measurement outcomes.