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.

翻译：量子马尔可夫链将随机变量的经典马尔可夫链推广到量子领域，并展现出独特的固有性质，使其成为量子信息论中的一个重要特征。本文提出了虚拟量子马尔可夫链（VQMCs）的概念，重点关注子系统从测量统计中保留关于全局系统经典信息的场景。作为量子马尔可夫链的推广，VQMCs刻画了这样一种状态：通过局部量子操作和测量，可以从子系统中恢复任意的全局阴影信息。我们给出了虚拟量子马尔可夫链的代数表征，并证明了虚拟量子恢复完全由量子态在其子系统上的分块矩阵决定。值得注意的是，我们发现两类三方纠缠态之间存在区别：W态是VQMC，而GHZ态则不是。此外，我们建立了半定规划来确定虚拟量子马尔可夫链的最优采样开销和鲁棒性。我们证明最优采样开销是可加的，这表明在并行调用VQMC状态时，不存在进一步降低恢复采样成本的“免费午餐”。我们的研究结果阐明了量子马尔可夫链与虚拟量子马尔可夫链之间的区别，将我们对量子恢复的理解扩展到优先考虑测量统计中经典信息的场景。