Measurement Strategies and Estimation Precision in Quantum Network Tomography

Abstract

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.

Figures (4)

• Figure 1: State distribution and measurement schemes. Red-outlined nodes denote monitors, and Werner states $\rho(w_i)$ are distributed along each link. The perfect Bell state $\phi^{+}$ is distributed in PEM. In JBM and PEM, links connected to monitor are measured directly (JBM Direct and PEM Direct), while others are estimated indirectly (JBM Indirect and PEM Indirect). LZM requires monitors at both end nodes of a link or path to perform LZM Direct and LZM Indirect measurements.
• Figure 2: Quantum Network Tomography of star networks. (a) two-monitor configuration using only JBM with two direct and one indirect JBM. (b) three-monitor configuration using only JBM with three direct JBMs. (c) two-monitor configuration combining JBM and LZM with one direct JBM, one indirect JBM, and one indirect LZM. (d) three-monitor configuration combining JBM and LZM with one direct JBM and two indirect LZMs.
• Figure 3: Numerical QCRB analysis of a single-link network. (a) Variation of the QCRB with the Werner parameter $w$ for LZM, JBM, and PEM. The inset enlarges the $w$ region where the curves overlap. (b) Ratio between LZM and JBM across varying Werner parameter values.
• Figure 4: Numerical QCRB analysis for a four-node star network involving two and three monitors. (a) JBM-only and hybrid (JBM+LZM) configurations with $w_1, w_2, w_3$ varying from 0.01 to 0.99. (b) JBM and hybrid configurations with fixed $w_0 = w_1 = 0.99$ and $w_2$ varying from 0.01 to 0.99.