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.
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.
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.
With steady progress in the development of quantum networks, the question on how to best provide end-to-end characterization of such networks (Quantum Network Tomography) is quickly becoming more pressing. Initial results demonstrated how we can utilize multipartite entanglement distribution to determine error probabilities of single-Pauli channels and depolarizing channels. In this work, we show how the analysis of quantum capacity regions can be used as a powerful new tool in quantum network tomography. As a first application of the proposed method, we demonstrate how we can characterize the loss on quantum channels in the network directly from quantum capacity region diagrams, even in the presence of bit-flip errors. Our results indicate that quantum capacity regions are not only valuable for network design, resource allocation, and protocol benchmarking, but also show promise for applications in quantum network tomography, particularly in loss tomography.
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.
The goal of quantum network tomography (QNT) is the characterization of internal quantum channels in a quantum network from external peripheral operations. Prior research has primarily focused on star networks featuring bit-flip and depolarizing channels, leaving the broader problem -- such as QNT for networks with arbitrary topologies and general Pauli channels -- largely unexplored. Moreover, establishing channel identifiability remains a significant challenge even in simplified quantum star networks.
In the first part of this paper, we introduce a novel network tomography method, termed Mergecast, in quantum networks. We demonstrate that Mergecast, together with a progressive etching procedure, enables the unique identification of all internal quantum channels in networks characterized by arbitrary topologies and Pauli channels. As a side contribution, we introduce a subclass of Pauli channels, termed bypassable Pauli channels, and propose a more efficient unicast-based tomography method, called BypassUnicast, for networks exclusively comprising these channels. In the second part, we extend our investigation to a more realistic QNT scenario that incorporates state preparation and measurement (SPAM) errors. We rigorously formulate SPAM errors in QNT, propose estimation protocols for such errors within QNT, and subsequently adapt our Mergecast approaches to handle networks affected by SPAM errors. Lastly, we conduct NetSquid-based simulations to corroborate the effectiveness of our proposed protocols in identifying internal quantum channels and estimating SPAM errors in quantum networks. In particular, we demonstrate that Mergecast maintains good performance under realistic conditions, such as photon loss and quantum memory decoherence.
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.
With steady progress in the development of quantum networks, the question on how to best provide end-to-end characterization of such networks (Quantum Network Tomography) is quickly becoming more pressing. Initial results demonstrated how we can utilize multipartite entanglement distribution to determine error probabilities of single-Pauli channels and depolarizing channels. In this work, we show how the analysis of quantum capacity regions can be used as a powerful new tool in quantum network tomography. As a first application of the proposed method, we demonstrate how we can characterize the loss on quantum channels in the network directly from quantum capacity region diagrams, even in the presence of bit-flip errors. Our results indicate that quantum capacity regions are not only valuable for network design, resource allocation, and protocol benchmarking, but also show promise for applications in quantum network tomography, particularly in loss tomography.