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Proceedings of the Samahang Pisika ng Pilipinas

Positive operator measure associated with homodyne tomography using the generalized Wigner function

Francisco Miguel Noora Balagot, Herbert B. Domingo

Published May 24, 2018• 0 citations
OpenAlex

Abstract

Recent studies have found a positive operator measure associated with quantum homodyne tomography which, with the state of a system, leads to the probability distribution phtT. Here, we introduce the general integral kernel for the state T, which arises from considering the General Wigner function, and show that phtT can still be expressed by the matrix elements of T when the general integral kernel is used, giving the same results as the standard integral kernel.

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Recent studies have found a positive operator measure associated with quantum homodyne tomography which, with the state of a system, leads to the probability distribution phtT. Here, we introduce the general integral kernel for the state T, which arises from considering the General Wigner function, and show that phtT can still be expressed by the matrix elements of T when the general integral kernel is used, giving the same results as the standard integral kernel.
In the framework of noisy quantum homodyne tomography with efficiency parameter 0 < η ⩽ 1, we propose two estimators of a quantum state whose density matrix elements ρ m , n decrease as , for fixed known B > 0 and 0 < r ⩽ 2. The first procedure estimates the matrix coefficients by a projection method on the pattern functions (that we introduce here for 0 < η ⩽ 1/2), the second procedure is a kernel estimator of the associated Wigner function. We compute the convergence rates of these estimators, in risk.
We report measurements of the Wigner distribution and the density matrix of an electromagnetic field mode, for both vacuum and quadrature-squeezed states. As proposed by Vogel and Risken,1 we obtained the Wigner distribution by tomographic inversion of a set of probability distributions of field-quadrature amplitudes measured by using balanced homodyne detection. A Fourier transform of the Wiener function yields the density matrix, which, according to the standard interpretation of quantum mechanics, contains all knowable information about a given quantum system.
The relation of theWigner function with the fair probability distribution called tomographic distribution or quantum tomogram associated with the quantum state is reviewed. The connection of the tomographic picture of quantum mechanics with the integral Radon transform of the Wigner quasidistribution is discussed. The Wigner–Moyal equation for the Wigner function is presented in the form of kinetic equation for the tomographic probability distribution both in quantum mechanics and in the classical limit of the Liouville equation. The calculation of moments of physical observables in terms of integrals with the state tomographic probability distributions is constructed having a standard form of averaging in the probability theory. New uncertainty relations for the position and momentum are written in terms of optical tomograms suitable for directexperimental check. Some recent experiments on checking the uncertainty relations including the entropic uncertainty relations are discussed.