Quantum state tomography, entanglement detection and Bell violation prospects in weak decays of massive particles

Abstract

A rather general method for determining the spin density matrix of a multi-particle system from angular decay data is presented. The method is based on a Bloch parameterisation of the $d$-dimensional generalised Gell-Mann representation of $ρ$ and exploits the associated Wigner- and Weyl-transforms on the sphere. Each parameter of a (possibly multipartite) spin density matrix can be measured from a simple average over an appropriate set of experimental angular decay distributions. The general procedures for both projective and non-projective decays are described, and the Wigner $P$ and $Q$ symbols calculated for the cases of spin-half, spin-one, and spin-3/2 systems. The methods are used to examine Monte Carlo simulations of $pp$ collisions for bipartite systems: $pp\rightarrow W^+W^-$, $pp\rightarrow ZZ$, $pp\rightarrow ZW^+$, $pp\rightarrow W^+\bar{t}$, $t\bar{t}$, and those from the Higgs boson decays $H\rightarrow WW^{*}$ and $H\rightarrow ZZ^*$. Measurements are proposed for entanglement detection and Bell inequality violation in bipartite systems.

Figures (10)

• Figure 1: Plots showing the form of the eight Wigner $Q$ symbols $\Phi^{Q+}_j$ corresponding to each of the Gell-Mann operators $\lambda^{(3)}_j$. These functions show the contribution \ref{['eq:pGMbasis']} of the corresponding density matrix parameters to the angular probability density function for lepton emission in a $W^+$ boson decay.
• Figure 2: Plots showing the form of the eight Wigner $P$ symbols $\Phi^{P+}_i$ corresponding to each of the Gell-Mann operators $\lambda^{(3)}_i$. These functions are used in recovering the density matrix parameters from the lepton angular probability distribution according to \ref{['eq:extractGMW']}.
• Figure 3: Cartoon of the right-handed coordinate axes used for the quantum state tomography of bipartite systems such as $W^+W^-$. The axes are aligned as indicated (with ${{\hat{\bf n}}}=\hat{\bf x}$ pointing out of the page), and are each defined in the respective bosons' rest frames. For tomography of other bipartite systems the axes are similarly defined, with the $\hat{z}$ axis parallel to the direction of the first of the two named particles in the bipartite centre-of-mass frame.
• Figure 4: Reconstructed Gell-Mann parameters obtained from quantum state tomography of pairs of simulated $W^\pm$ bosons obtained from (a) $W^+W^-\rightarrow \ell^+\nu \ell^- \bar{\nu}$ (b) $H\rightarrow WW^{(*)} \rightarrow \ell^+\nu \ell^- \bar{\nu}$, (c) and a 200 GeV scalar decaying to $WW$ then to a state with a 30 GeV '$\tau$' lepton. We also show (d) the results expected for an ideal singlet state \ref{['eq:singletstate']} of two qutrits. The bottom row of each plot contains the $a_i$ parameters for a $W^+$ boson, the leftmost column the $b_j$ parameters for the $W^-$ boson and the rows and columns 1-8 the $c_{ij}$ parameters. The (0,0) element has no meaning.
• Figure 5: Reconstructed expectation values of the complete set of ${\mathsf{j}}=1$ spin operators for pairs of $W^\pm$ bosons obtained from (a) $pp\rightarrow W^+W^-\rightarrow \ell^+\nu \ell^- \bar{\nu}$ (b) $pp\rightarrow H\rightarrow WW^{(*)} \rightarrow \ell^+\nu \ell^- \bar{\nu}$, (c) a 200 GeV Higgs boson decaying to $WW$ involving a 30 GeV '$\tau$' lepton, and (d) an ideal singlet state \ref{['eq:singletstate']} of two qutrits. In (a)--(c) the bottom row contains the operators for the $W^+$ boson, the leftmost column the $b_j$ parameters for the $W^-$ boson and others rows and columns the product operators for those bosons. The bottom-left-hand bin of each plot has no meaning.