  Quantum Markov chains generalize classical Markov chains for random variables
to the quantum realm and exhibit unique inherent properties, making them an
important feature in quantum information theory. In this work, we propose the
concept of virtual quantum Markov chains (VQMCs), focusing on scenarios where
subsystems retain classical information about global systems from measurement
statistics. As a generalization of quantum Markov chains, VQMCs characterize
states where arbitrary global shadow information can be recovered from
subsystems through local quantum operations and measurements. We present an
algebraic characterization for virtual quantum Markov chains and show that the
virtual quantum recovery is fully determined by the block matrices of a quantum
state on its subsystems. Notably, we find a distinction between two classes of
tripartite entanglement by showing that the W state is a VQMC while the GHZ
state is not. Furthermore, we establish semidefinite programs to determine the
optimal sampling overhead and the robustness of virtual quantum Markov chains.
We demonstrate the optimal sampling overhead is additive, indicating no free
lunch to further reduce the sampling cost of recovery from parallel calls of
the VQMC states. Our findings elucidate distinctions between quantum Markov
chains and virtual quantum Markov chains, extending our understanding of
quantum recovery to scenarios prioritizing classical information from
measurement statistics.
