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arXiv:2001.07255 · Laser Physics Letters

Precise tomography of optical polarisation qubits under conditions of chromatic aberration of quantum transformations

B. I. Bantysh, Yu. I. Bogdanov, Nadezhda A. Bogdanova, Yu A Kuznetsov

Published Feb 5, 2020• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1612-202x/ab6e72

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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.
The wave plate is a basic device for transforming and measuring the polarization states of light. It is known that the transformation of light by means of two wave plates makes it possible to measure the state of polarization in an arbitrary basis. The finite spectral width of the light, however, leads to a chromatic aberration of the polarization quantum transformation caused by the parasitic dispersion of the birefringence of the plate material. This causes systematic errors in the tomography of quantum polarization states and significantly reduces its accuracy. This study is a development of our work1, in which an adequate model for quantum measurements of polarization qubits under chromatic aberration was first formulated. This work includes a generalization of the results obtained earlier for the cases of two-qubit states. Along with examples of random states those uniformly distributed over the Haar measure are considered. Using a matrix of complete information, it is quantitatively traced how the presence of chromatic aberrations under conditions of a finite spectral width of light leads to the loss of information in quantum measurements. It is shown that the use of the developed model of fuzzy measurements instead of the model of standard projection measurements makes it possible to suppress systematic errors of quantum tomography even when using high-order wave plates. It turns out that the fuzzy measurement model can give a significant increase in the reconstruction accuracy compared to the standard measurement model.
The wave plate is a basic device for transforming and measuring the polarization states of light. It is known that the transformation of light by means of two wave plates makes it possible to measure the state of polarization in an arbitrary basis. The finite spectral width of the light, however, leads to a chromatic aberration of the polarization quantum transformation caused by the parasitic dispersion of the birefringence of the plate material. This causes systematic errors in the tomography of quantum polarization states and significantly reduces its accuracy. This study is a development of our work1, in which an adequate model for quantum measurements of polarization qubits under chromatic aberration was first formulated. This work includes a generalization of the results obtained earlier for the cases of two-qubit states. Along with examples of random states those uniformly distributed over the Haar measure are considered. Using a matrix of complete information, it is quantitatively traced how the presence of chromatic aberrations under conditions of a finite spectral width of light leads to the loss of information in quantum measurements. It is shown that the use of the developed model of fuzzy measurements instead of the model of standard projection measurements makes it possible to suppress systematic errors of quantum tomography even when using high-order wave plates. It turns out that the fuzzy measurement model can give a significant increase in the reconstruction accuracy compared to the standard measurement model.
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.
Systematic errors are inevitable in most measurements performed in real life because of imperfect measurement devices. Reducing systematic errors is crucial to ensuring the accuracy and reliability of measurement results. To this end, delicate error-compensation designs are often necessary in addition to device calibration to reduce the dependence of the systematic error on the imperfection of the devices. The art of error-compensation designs is well appreciated in nuclear magnetic resonance systems by using composite pulses. In contrast, there are few works on reducing systematic errors in quantum optical systems. Here we propose an error-compensation design applicable to reducing the systematic error in projective measurements on ensembles of both single and multiqubit systems. This design can significantly decrease the systematic error due to dominant error sources in typical optical experiments. In particular, it can reduce the systematic error to the second order of the phase errors of both the half-wave plate (HWP) and the quarter-wave plate as well as the angle error of the HWP. Its power in reducing the systematic error was verified experimentally on qubit state tomography and numerically on two-qubit state tomography. Our study may find applications in high-precision tasks in polarization optics and quantum optics.
We propose a technique for performing quantum state tomography of photonic polarization-encoded multiqubit states. Our method uses a single rotating wave plate, a polarizing beam splitter, and two photon-counting detectors per photon mode. As the wave plate rotates, the photon counters measure a pseudocontinuous signal which is then Fourier transformed. The density matrix of the state is reconstructed using the relationship between the Fourier coefficients of the signal and the Stokes' parameters that represent the state. The experimental complexity, i.e., different wave plate rotation frequencies, scales linearly with the number of qubits.