🤖 AI Summary
This paper addresses the fundamental question: “Can global classical information be reconstructed solely from subsystem measurement statistics?” To this end, we introduce *virtual quantum Markov chains* (VQMCs)—a class of quantum states from which *any* global classical shadow information can be fully reconstructed via local operations and measurements, without access to the global system. We provide the first algebraic characterization of VQMCs and formulate a semidefinite programming (SDP) framework for their verification. We prove that the W state satisfies the VQMC condition while the GHZ state does not, thereby achieving an intrinsic distinction between these paradigmatic three-party entangled states. Furthermore, we refute “free-lunch”-style sampling optimization, establishing the additivity of optimal sampling overhead. Finally, we uncover a fundamental divergence in information recovery mechanisms between VQMCs and conventional quantum Markov chains, significantly extending measurement-statistics-based quantum information reconstruction theory.
📝 Abstract
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.
