  Quantum entanglement of weak interaction gauge bosons produced at colliders
can be explored by computing the corresponding polarization density matrix. To
this end, we consider the Higgs boson decays $H\to W W^*$ and $H\to Z Z^*$, in
which $W^*$ and $Z^*$ are off-shell states, and the $WW$, $WZ$ and $ZZ$
di-boson production in proton collisions. The polarization density matrix of
the di-boson state is determined by the amplitude of the production process and
can be experimentally reconstructed from the angular distribution of the
momenta of the final states into which the gauge bosons decay. We show that a
suitable instance of the Bell inequality is violated in $H\to Z Z^*$ to a
degree that can be tested at the LHC with future data. The same Bell inequality
is violated in the production of $WW$ and $ZZ$ boson pairs for invariant masses
above 900 GeV and scattering angles close to $\pi/2$ in the center of mass
frame. LHC data in this case are not sufficient to establish the violation of
the Bell inequality. We also analyze the prospects for detecting Bell
inequality violations in di-boson final states at future $e^+e^-$ and muon
colliders. A further observable that provides a lower bound on the amount of
polarization entanglement in the di-boson system is computed for each of the
examined processes. The analytic expressions for the polarization density
matrices are presented in full in an Appendix. We also provide the unitary
matrices required in the optimization procedure necessary in testing the Bell
inequalities.
