Quantum tomography of helicity states for general scattering processes
Alexander Bernal
Abstract
Quantum tomography has become an indispensable tool in order to compute the density matrix <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>ρ</a:mi></a:math> of quantum systems in physics. Recently, it has further gained importance as a basic step to test entanglement and violation of Bell inequalities in high-energy particle physics. In this work, we present the theoretical framework for reconstructing the helicity quantum initial state of a general scattering process. In particular, we perform an expansion of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>ρ</c:mi></c:math> over the irreducible tensor operators <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mo stretchy="false">{</e:mo><e:msubsup><e:mi>T</e:mi><e:mi>M</e:mi><e:mi>L</e:mi></e:msubsup><e:mo stretchy="false">}</e:mo></e:math> and compute the corresponding coefficients uniquely by averaging, under properly chosen Wigner D-matrices weights, the angular distribution data of the final particles. Besides, we provide the explicit angular dependence of a novel generalization of the production matrix <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi mathvariant="normal">Γ</i:mi></i:math> and of the normalized differential cross section of the scattering. Finally, we rederive all our previous results from a quantum-information perspective using the Weyl-Wigner-Moyal formalism and we obtain, in addition, simple analytical expressions for the Wigner <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>P</l:mi></l:math> and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>Q</n:mi></n:math> symbols. Published by the American Physical Society 2024