Monitoring the performance of internal network elements is crucial for network management and optimizing network operations. Quantum Network Tomography (QNT) focuses on the characterization of quantum channels through end-to-end measurements between selected subsets of network nodes. In this work, we examine optimal strategies for monitor placement in QNT. Our study specifically targets a four-node quantum star network in which links are modelled as depolarizing channels. We investigate QNT methods that utilize a single monitor in the four-node star network, and evaluate estimation performance for all possible monitor placements in the star network. We perform a theoretical analysis of the estimators by calculating the Quantum Fisher information matrix (QFIM) and the Quantum Cramer-Rao Bound (QCRB). The QCRB provides a lower bound on the variance of any estimator, thereby offering a benchmark for their efficiency. We identify the monitor placement at the hub as a baseline to analyze the performance of end-to-end estimation given that the monitor has direct access to every link of the network. Through further investigation, we find that the best performance is achieved by placing the monitor in the leaf with the least noisy link for the 4- node star network.
Monitoring the performance of internal network elements is crucial for network management and optimizing network operations. Quantum Network Tomography (QNT) focuses on the characterization of quantum channels through end-to-end measurements between selected subsets of network nodes. In this work, we examine optimal strategies for monitor placement in QNT. Our study specifically targets a four-node quantum star network in which links are modelled as depolarizing channels. We investigate QNT methods that utilize a single monitor in the four-node star network, and evaluate estimation performance for all possible monitor placements in the star network. We perform a theoretical analysis of the estimators by calculating the Quantum Fisher information matrix (QFIM) and the Quantum Cramer-Rao Bound (QCRB). The QCRB provides a lower bound on the variance of any estimator, thereby offering a benchmark for their efficiency. We identify the monitor placement at the hub as a baseline to analyze the performance of end-to-end estimation given that the monitor has direct access to every link of the network. Through further investigation, we find that the best performance is achieved by placing the monitor in the leaf with the least noisy link for the 4- node star network.
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.
Reliable quantum communication requires accurate characterization of the quantum links. This paper studies Quantum Network Tomography (QNT) under limited monitoring resources, where unknown link parameters are inferred from path-based measurements performed at monitor nodes. We introduce a cyclic sequential QNT protocol (CSQP) for arbitrary network topologies and develop a fixed-point learnability framework with an explicit algorithm for estimating link-level Werner parameters. We characterize identifiability through the rank of the path-link incidence matrix and show that the CSQP learnability conditions guarantee full rank and a nonsingular Quantum Fisher Information Matrix (QFIM). Building on this framework, we formulate monitor placement and measurement assignment as an optimization problem whose constraints enforce learnability and identifiability without topology-specific reformulation. Two Integer Linear Programming (ILP) formulations are introduced: Unconstrained QFIM-based formulation (QF), which maximizes QFIM-trace, and monitoring-overhead constrained QFIM formulation (QMF), which maximizes QFIM-trace subject to a per-monitor overhead constraint. Both formulations are evaluated on star and tree networks to compare monitor placements and measurement assignments. The results show that QMF distributes monitoring load evenly across all monitors and provides greater potential for parallel monitoring under resource constraints, while QF is more suitable when estimation information is prioritized, particularly in practical networks with non-uniform link noise.
Errors are the fundamental barrier to the development of quantum systems. Quantum networks are complex systems formed by the interconnection of multiple components and suffer from error accumulation. Characterizing errors introduced by quantum network components becomes a fundamental task to overcome their depleting effects in quantum communication. Quantum Network Tomography (QNT) addresses end-to-end characterization of link errors in quantum networks. It is a tool for building error-aware applications, network management, and system validation. We provide an overview of QNT and its initial results for characterizing quantum star networks. We apply a previously defined QNT protocol for estimating bit-flip channels to estimate depolarizing channels. We analyze the performance of our estimators numerically by assessing the Quantum Cramèr-Rao Bound (QCRB) and the Mean Square Error (MSE) in the finite sample regime. Finally, we provide a discussion on current challenges in the field of QNT and elicit exciting research directions for future investigation.
Network tomography refers to the use of inference techniques for inferring internal network states from end-to-end probes. Quantum probes, implemented by sending blocks of n coherent-state pulses augmented with continuous-variable (CV) squeezing (n=1) or weak temporal-mode entanglement (n>1) over a lossy channel to a receiver with homodyne detection capabilities, are known to carry information about the channel transmissivity. Assuming a subset of nodes in an optical network is capable of sending and receiving such probes through intermediate nodes with all-optical switching capabilities, we leverage these quantum probes to estimate link transmissivities.
To determine how to route the probes in a network, we propose a probe construction algorithm that guarantees link identifiability, while maximizing the number of information orthogonal sets of transmissivities. A set of probes induces a Fisher information matrix (FIM). We then derive two metrics, the determinant of the FIM and the trace of its inverse, to evaluate the performance of the probes. In particular, our results can be used to characterize the quantum improvement in estimating link transmissivities in a general optical network.
The experimental realization of quantum information systems will be difficult due to how sensitive quantum information is to noise. Overcoming this sensitivity is central to designing quantum networks capable of transmitting quantum information reliably over large distances. Moreover, the ability to characterize communication noise in quantum networks is crucial in developing network protocols capable of overcoming the effects of noise in quantum networks. In this context, quantum network tomography refers to the characterization of channel noise in a quantum network through end-to-end measurements. In this work, we propose network tomography protocols for quantum star networks formed by quantum channels characterized by a single, non-trivial Pauli operator. Our results further the end-to-end characterization of quantum bit-flip star networks by introducing tomography protocols where state distribution and measurements are designed separately. We build upon previously defined quantum network tomography protocols, as well as provide novel methods for the unique characterization of bit-flip probabilities in stars. We introduce a theoretical benchmark based on the Quantum Fisher Information matrix to compare the efficiency of quantum network protocols. We apply our techniques to the protocols proposed, and perform an initial analysis on the potential benefits of entanglement for Quantum Network Tomography. Furthermore, we simulate the protocols using N etSquid to assess the convergence properties of the estimators obtained for particular parameter regimes. Our findings show that the efficiency of protocols depend on parameter values and motivate the search for adaptive quantum network tomography protocols.
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.