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Optics and Spectroscopy

Mathematical modeling of the accuracy characteristics in problems of precision quantum tomography of biphoton states

Yu. I. Bogdanov, R. F. Galeev, S. P. Kulik, Ekaterina V. Moreva

Published Jul 1, 2007• 2 citations
Publisher

Abstract

A model that approximately takes into account instrumental errors in problems of precision reconstruction of quantum states is considered. The model is based on the notion of coherence volume, which characterizes the quality of the experimental and technological realization of the measurement protocol of a quantum state. Various sources of instrumental errors that affect the reconstruction accuracy of quantum states are mathematically modeled. It is shown that, for precision tomography of quantum states, one should take into account random coincidences that occur because the operating time of the photon detection system is finite. Among other possible sources of instrumental errors of the measurement protocol, one should take into account errors of the rotation angles and optical thicknesses of polarization plates, a possible drift in the parameters of the pump laser, and so on.

DOI
10.1134/s0030400x0707017x

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A model that approximately takes into account instrumental errors in problems of precision reconstruction of quantum states is considered. The model is based on the notion of coherence volume, which characterizes the quality of the experimental and technological realization of the measurement protocol of a quantum state. Various sources of instrumental errors that affect the reconstruction accuracy of quantum states are mathematically modeled. It is shown that, for precision tomography of quantum states, one should take into account random coincidences that occur because the operating time of the photon detection system is finite. Among other possible sources of instrumental errors of the measurement protocol, one should take into account errors of the rotation angles and optical thicknesses of polarization plates, a possible drift in the parameters of the pump laser, and so on.
Abstract In this work we present an algorithm of building an adequate model of polarizing quantum state measurement. This model takes into account chromatic aberration of the basis change transformation caused by the parasitic dispersion of the wave plates crystal and finite radiation spectral bandwidth. We show that the chromatic aberration reduces the amount of information in the measurements results. Using the information matrix approach we estimate the impact of this effect on the qubit state reconstruction fidelity for different values of sample size and spectral bandwidth. We also demonstrate that our model outperforms the standard model of projective measurements as it could suppress systematic errors of quantum tomography even when one performs the measurements using wave plates of high order.
Quantum state tomography (QST) is a crucial technique for characterizing the state of a quantum system. However, the inherent limitations of measurement processes, including detector efficiency and photon counting statistics, introduce noise that affects the accuracy of the reconstructed quantum state. This work presents a rigorous information-theoretic framework for analyzing and quantifying the accuracy achievable in QST. We leverage concepts from information theory, specifically mutual information and channel capacity, to precisely determine the trade-offs between measurement resolution and the complexity of state reconstruction. The central argument is that the achievable accuracy is fundamentally limited by the information that can be reliably extracted from the measurement data, rather than relying solely on empirical estimates of error. This approach provides a theoretical benchmark for evaluating QST protocols and identifying optimal measurement strategies. The key contributions lie in establishing a quantifiable relationship between measurement parameters, noise levels, and the fidelity of the reconstructed quantum state. This offers a more robust and predictive method for assessing the quality of QST than traditional approaches.
Abstract We develop a new method for high-precision tomography of ion qubit registers under conditions of limited distinguishability of logical states. It is not always possible to achieve low error rates during the readout of the quantum states of ion qubits due to the finite lifetime of excited levels, photon scattering, detector dark counts, low numerical aperture, etc. However, the model of fuzzy quantum measurements makes it possible to ensure precise tomography of quantum states. To do this, we developed a fuzzy measurement model based on counting the number of fluorescent photons. A statistically adequate algorithm for the reconstruction of quantum states of ion qubit registers based on fuzzy measurement operators is proposed. The algorithm uses the complete information available in the experiment and makes it possible to account for systematic measurement errors associated with the limited distinguishability of the logical states of ion qubits. We show that the developed model, although computationally more complex, contains significantly more information about the state of the qubit and provides a higher accuracy of state reconstruction compared to the model based on the threshold algorithm.
An operational method has been proposed for estimating the efficiency of quantum cryptography protocols for quantum states with discrete variables. The method is based on the estimate of the quality of the protocol by means of a universal asymptotic distribution of the characteristic of the accuracy of the reconstruction of the quantum state, which is called fidelity. For a specially designed measurement matrix, the condition number, which is minimal for the optimal protocol, is introduced. The method has been tested in a numerical simulation and real experiments with polarization qubits. It has been shown that the optimal choice of a polarization transformer makes it possible to strongly improve the quality of the reconstruction of states.