量子态、量子过程和量子网络层析成像中的可辨识性
Identifiability in Quantum State, Process, and Network Tomography
- CONNECT Research Centre, School of Engineering, Trinity College Dublin（都柏林三一学院工程学院连接研究中心）
- Manning College of Information and Computer Science, University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校曼宁信息与计算机科学学院）
- Department of Computing, School of Science, South East Technological University（东南理工大学理学院计算系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过费舍尔信息矩阵统一研究量子态、过程和网络层析成像的可辨识性，证明QNT的秩等于路径-链路矩阵秩，并揭示IC设置下QST/QPT满秩而QNT可能失秩。
AI中文摘要：
量子态层析成像（QST）、量子过程层析成像（QPT）和量子网络层析成像（QNT）是相关的参数估计问题，旨在重建不同的物理量。QST 根据测量结果估计由密度矩阵表示的未知量子态。QPT 利用已知的输入态和对应输出的测量来表征未知的量子信道。相比之下，QNT 旨在从可访问的监测节点收集的端到端探测测量中推断与各个链路相关的参数。这三个层析成像问题之间的一个关键区别在于实现可辨识性所需的条件，即可从可用测量统计中唯一确定未知参数的能力。在 QST 和 QPT 中，实验者可以选择信息完备（IC）测量集。QNT 将可达测量限制为拓扑和监测节点布置所允许的范围，因此可允许的探测路径决定了关于链路参数的可用信息。本研究通过共同的费舍尔信息矩阵（FIM）研究所有这三个层析成像问题。我们对 QNT 的 FIM 进行因式分解，并证明其秩在每个内部参数值处等于路径-链路关联矩阵的秩，因此局部和全局可辨识性一致。然后我们证明，在 IC 设置下 QST 和 QPT 达到满秩，当探测路径使链路参数不可区分时 QNT 失去秩，而增加副本数量会缩放 FIM 的特征值但保持其秩不变。
英文摘要：
Quantum State Tomography (QST), Quantum Process Tomography (QPT), and Quantum Network Tomography (QNT) are related parameter-estimation problems that aim to reconstruct different physical quantities. QST estimates an unknown quantum state, represented by its density matrix, from the measurement outcomes. QPT characterises an unknown quantum channel using known input states and measurements of the corresponding outputs. QNT, in contrast, aims to infer parameters associated with individual links from end-to-end probe measurements collected at accessible monitor nodes. A key distinction among the three tomography problems lies in the conditions required to achieve identifiability, the ability to determine unknown parameters uniquely from the available measurement statistics. In QST and QPT, the experimenter can choose an Informationally Complete (IC) measurement set. QNT limits the reachable measurements to what topology and monitor placement allow, so the admissible probe paths fix the information available about the link parameters. This work studies all three tomography problems through a common Fisher Information Matrix (FIM). We factorise QNT FIM and show that its rank equals the rank of the path-link incidence matrix at every interior parameter value, so local and global identifiability coincide. We then show that QST and QPT attain full rank under IC settings, QNT loses rank when the probe paths leave link parameters indistinguishable, and increasing the number of copies scales the FIM eigenvalues while leaving its rank fixed.
