A semidefinite programming analysis of quantum bayesian solutions to the coarse-graining problem

Abstract

Coarse-grained descriptions of quantum systems can be used to address processes where information about the system is lost or when it is not possible to access all its degrees of freedom. There are some efforts in the literature to connect these descriptions with the conditional states formalism for quantum theory, approaching the necessary and sufficient conditions for the existence of an emergent dynamics through a Bayesian perspective, such as the emergent dynamics proposed by Brugger et al. However, this dynamics has been shown to be analytically limited: it is proportional to the Petz recovery map, depends on a initial state, and solves the problem only on a state-by-state case. With this analytical limitation in mind, in this work we propose a numerical benchmark of this solution across different coarse-graining scenarios. Moreover, by means of semidefinite programming techniques, we investigate the existence of an effective dynamics to the coarse-graining problem within the conditional state formalism that circumvent the state-dependence limitation of the previous approach. We also pursue additional investigations of the coarse-graining problem within the conditional state formalism, independently of this previous solution, such as proposing a new robustness measure to quantify how much noise can be added to a microscopic dynamics compatible with a given coarse-grained description, as well as formulating a semidefinite program that searches for compatible unitary dynamics in non-compatible coarse-graining scenarios. Our findings indicate that, in general, the solution proposed by Brugger et al. performs better when generated by the maximally mixed state, and that, through the proposed semidefinite programs, we identify the scenarios in which it is possible to find an effective dynamics that solves the problem for arbitrary initial states.