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Physical Review A

Tomographic completeness and robustness of quantum reservoir networks

Tanjung Krisnanda, Huawen Xu, Sanjib Ghosh, T. C. H. Liew

Published Apr 4, 2023• 11 citations
Publisher

Abstract

Quantum reservoir processing offers an option to perform quantum tomography of input objects by postprocessing quantities, obtained from local measurements, from a quantum reservoir network that has interacted with the former. We develop a method to assess a tomographic completeness criterion for arbitrary quantum reservoir architectures. Furthermore, we propose a figure of merit that quantifies their robustness against imperfections. Measured quantities from the reservoir nodes correspond to effective observables acting on the input objects, and we provide a way to retrieve them. Finally, we present examples of quantum tomography for demonstration. Our general method offers guidance in optimizing implementations of quantum reservoir processing.

DOI
10.1103/physreva.107.042402

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We present a novel quantum tomographic reconstruction method based on Bayesian inference via the Kalman filter update equations. The method not only yields the maximum likelihood/optimal Bayesian reconstruction but also a covariance matrix expressing the measurement uncertainties in a complete way. From this covariance matrix the error bars on any derived quantity can be easily calculated. This is a first step towards the broader goal of devising an omnibus reconstruction method that could be adapted to any tomographic setup with little effort and that treats measurement uncertainties in a statistically well-founded way. In this first part, we restrict ourselves to the important subclass of tomography based on measurements with discrete outcomes (as opposed to continuous ones), and we also ignore any measurement imperfections (dark counts, less than unit detector efficiency, etc), which will be treated in a follow-up paper. We illustrate our general theory on real tomography experiments of quantum optical information processing elements.
Quantum state tomography is currently the standard tool for verifying that a state prepared in the lab is close to an ideal target state, but up to now there have been no rigorous methods for evaluating the precision of the state preparation in tomographic experiments. We propose a new estimator for quantum state tomography, and prove that the (always physical) estimates will be close to the true prepared state with a high probability. We derive an explicit formula for evaluating how high the probability is for an arbitrary finite-dimensional system and explicitly give the one- and two-qubit cases as examples. This formula applies for any informationally complete sets of measurements, arbitrary finite number of data sets, and general loss functions including the infidelity, the Hilbert-Schmidt, and the trace distances. Using the formula, we can evaluate not only the difference between the estimated and prepared states, but also the difference between the prepared and target states. This is the first result directly applicable to the problem of evaluating the precision of estimation and preparation in quantum tomographic experiments.