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arXiv:2609.13584

Identifiability in Quantum State, Process, and Network Tomography

Athira Kalavampara Raghunadhan, Matheus Guedes De Andrade, Don Towsley, Indrakshi Dey, Daniel Kilper, Nicola Marchetti

quant-phSubmitted Sep 11, 2026• Announced Sep 15, 2026• 0 citations
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Abstract

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.

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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.
The characterization of the evolution of a quantum system is one of the main tasks to accomplish to achieve quantum information processing. The standard quantum process tomography (SQPT) has the unique property that it can be applied without introducing any additional quantum resources. In present work, we shall focus on the following two topics about the SQPT. At first, in the SQPT protocol for a ddd-dimensional system, one should encounter a problem in solving of a set of d4d^4d4 linear equations in order to get the matrix containing the complete information about the unknown quantum channel. Until now, the general form of the solution is unknown. And a long existed conviction is that the solutions are not unique. Here, we shall develop a self-consistent scheme, in which bounded linear operators are presented by vectors, to construct the set of linear equations. With the famous Cramer's rule for the set of linear equations, we are able to give the general form of the solution and prove that it is unique. In second, the central idea of the SQPT is to prepare a set of linearly independent inputs and measuring the outputs via the quantum state tomography (QST). Letting the inputs and the measurements be prepared by two sets of the rank-one positive-operator-valued measures [POVMs], where each POVM is supposed to be linearly independent and informationally complete (IC), we observe that SQPT now is equivalent to deciding a unknown state with a set of product IC-POVM in the d2d^2d2-dimensional Hilbert space. Following the general linear state tomography theory, we show that the product symmetric IC-POVM should minimize the mean-square Hilbert-Schmidt distance between the estimator and the true states. So, an optimal SQPT can be realized by preparing both the inputs and the measurements as the symmetric IC-POVM.
Quantum process tomography (QPT) methods aim at identifying a given quantum process. QPT is a major quantum information processing tool, since it especially allows one to characterize the actual behavior of quantum gates, which are the building blocks of quantum computers. The present paper focuses on the estimation of a unitary process. This class is of particular interest because quantum mechanics postulates that the evolution of any closed quantum system is described by a unitary transformation. Unitary processes have significantly fewer parameters than general quantum processes (22nqb vs. 24nqb−22nqb real independent parameters for nqb qubits). By assuming that the process is unitary we develop two methods that scale better with the size of the system. In the present paper, we stay as close as possible to the standard setup of QPT: the operator has to prepare copies of different input states. The properties those states have to satisfy in order for our method to achieve QPT are very mild. Therefore, we choose to operate with copies of 2nqb initially unknown pure input states. In order to perform QPT without knowing the input states, we perform measurements on half the copies of each state, and let the other half be transformed by the system before measuring them (each copy is only measured once). This setup has the advantage of removing the issue of systematic (i.e., same on all the copies of a state) errors entirely because it does not require the process input to take predefined values. We develop a straightforward analytical solution that first estimates the states from the averaged measurements and then finds the unitary matrix (representing the process) coherent with those estimates by using our analytical solution to an extended version of Wahba’s problem. This estimate may then be used as an initial point for a fine tuning algorithm that maximizes the likelihood of the measurements. Simulation results show the effectiveness of the proposed methods.
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 nnn coherent-state pulses augmented with continuous-variable (CV) squeezing (n=1n=1n=1) or weak temporal-mode entanglement (n>1n>1n>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.
Quantum Process Tomography (QPT) methods aim at identifying, i.e. estimating, a quantum process. QPT is a major quantum information processing tool, since it especially allows one to experimentally characterize the actual behavior of quantum gates, that may be used as the building blocks of quantum computers. We here consider unitary (possibly dense, i.e. without sparsity constraints) processes, which corresponds to isolated systems. Moreover, we develop a QPT method applicable to a significant number of qubits and hence to a high state space dimension, which allows one to tackle more complex problems. Using the unitarity of the process allows us to propose a method that first achieves part of QPT by performing an eigenanalysis of the estimated density matrix of a process output. We then complete the QPT by using another state differently. The relevance of our method is validated by means of simulations.
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.
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.