Evaluation of Estimation Precision in Quantum Tomography

Takanori Sugiyama

Takanori Sugiyama, Peter S. Turner, Mio Murao

Quantum state tomography is currently the standard tool for verifying that a state prepared in the lab is close to an ideal target state, but up to now there have been no rigorous methods for evaluating the precision of the state preparation in tomographic experiments. We propose a new estimator for quantum state tomography, and prove that the (always physical) estimates will be close to the true prepared state with a high probability. We derive an explicit formula for evaluating how high the probability is for an arbitrary finite-dimensional system and explicitly give the one- and two-qubit cases as examples. This formula applies for any informationally complete sets of measurements, arbitrary finite number of data sets, and general loss functions including the infidelity, the Hilbert-Schmidt, and the trace distances. Using the formula, we can evaluate not only the difference between the estimated and prepared states, but also the difference between the prepared and target states. This is the first result directly applicable to the problem of evaluating the precision of estimation and preparation in quantum tomographic experiments.

Yanran Cheng, Zhihui Lou

Quantum tomography is one of the essential tasks in quantum information technology. Much effort has been devoted to increasing the precision and efficiency of quantum tomography. In this paper, we only focus on a specific tomography method, i.e., the linear regression estimation (LRE). We give a brief review on how to employ LRE for quantum state tomography, adaptive quantum state tomography and quantum Hamiltonian identification.

Giacomo Mauro D’Ariano, Chiara Macchiavello, Matteo G. A. Paris

Athira Kalavampara Raghunadhan, Matheus Guedes De Andrade, Don Towsley +3 more

This work investigates measurement strategies for link parameter estimation in Quantum Network Tomography (QNT), where network links are modeled as depolarizing quantum channels distributing Werner states. Three distinct measurement schemes are analyzed: local Z-basis measurements (LZM), joint Bell-state measurements (JBM), and pre-shared entanglement-assisted measurements (PEM). For each scheme, we derive the probability distributions of measurement outcomes and examine how noise in the distributed states influences estimation precision. Closed-form expressions for the Quantum Fisher Information Matrix (QFIM) are obtained, and the estimation precision is evaluated through the Quantum Cramer-Rao Bound (QCRB). Numerical analysis reveals that the PEM scheme achieves the lowest QCRB, offering the highest estimation accuracy, while JBM provides a favorable balance between precision and implementation complexity. The LZM method, although experimentally simpler, exhibits higher estimation error relative to the other schemes; however, it outperforms JBM in high-noise regimes for single-link estimation. We further evaluate the estimation performance on a four-node star network by comparing a JBM-only configuration with a hybrid configuration that combines JBM and LZM. When two monitors are used, the JBM-only strategy outperforms the hybrid approach across all noise regimes. However, with three monitors, it achieves a lower QCRB only in low-noise regimes with heterogeneous links. The results establish a practical basis for selecting measurement strategies in experimental quantum networks, enabling more accurate and scalable link parameter estimation under realistic noise conditions.

Shao-Ming Fei