We study Quantum Network Tomography (QNT) for end-to-end link-error characterization under imperfect Bell-state measurements (BSMs), where multiplicative coupling between link and measurement parameters makes identifiability non-trivial. For an n-node star network, we design probes that ensure unique identifiability and derive closed-form expressions for the Fisher Information Matrix (FIM) and Maximum Likelihood Estimators (MLEs), and characterize estimation precision through the Cramer-Rao Bound (CRB). The results show that BSM imperfections degrade estimation precision, while the proposed probes maintain nearly stable precision for individual link parameters as the network size increases. Monte Carlo simulations further confirm that the Mean Squared Error (MSE) approaches the CRB with increasing sample size.
We study Quantum Network Tomography (QNT) for end-to-end link-error characterization under imperfect Bell-state measurements (BSMs), where multiplicative coupling between link and measurement parameters makes identifiability non-trivial. For an n-node star network, we design probes that ensure unique identifiability and derive closed-form expressions for the Fisher Information Matrix (FIM) and Maximum Likelihood Estimators (MLEs), and characterize estimation precision through the Cramer-Rao Bound (CRB). The results show that BSM imperfections degrade estimation precision, while the proposed probes maintain nearly stable precision for individual link parameters as the network size increases. Monte Carlo simulations further confirm that the Mean Squared Error (MSE) approaches the CRB with increasing sample size.
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.
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.
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 fragile nature of quantum information makes it practically impossible to completely isolate a quantum state from noise under quantum channel transmissions. Quantum networks are complex systems formed by the interconnection of quantum processing devices through quantum channels. In this context, characterizing how channels introduce noise in transmitted quantum states is of paramount importance. Precise descriptions of the error distributions introduced by non-unitary quantum channels can inform quantum error correction protocols to tailor operations for the particular error model. In addition, characterizing such errors by monitoring the network with end-to-end measurements enables end-nodes to infer the status of network links. In this work, we address the end-to-end characterization of quantum channels in a quantum network by introducing the problem of Quantum Network Tomography. The solution for this problem is an estimator for the probabilities that define a Kraus decomposition for all quantum channels in the network, using measurements performed exclusively in the end-nodes. We study this problem in detail for the case of arbitrary star quantum networks with quantum channels described by a single Pauli operator, like bit-flip quantum channels. We provide solutions for such networks with polynomial sample complexity. Our solutions provide evidence that pre-shared entanglement brings advantages for estimation in terms of the identifiability of parameters.
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.