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 \(H\rightarrow W W^*\) and \(H\rightarrow Z Z^*\), 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 \(H\rightarrow Z Z^*\) 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.

对撞机产生的弱相互作用规范玻色子的量子纠缠可以通过计算相应的极化密度矩阵来探索。为此，我们考虑希格斯玻色子衰变\(H\rightarrow WW^*\)和\(H\rightarrow ZZ^*\)，其中\(W^*\)和\(Z^*\)是壳外状态，以及WW、WZ和ZZ质子碰撞中双玻色子的产生。双玻色子态的极化密度矩阵由产生过程的振幅决定，并且可以根据规范玻色子衰变成的最终态动量的角分布通过实验重建。我们证明了贝尔不等式的一个合适实例在\(H\rightarrow ZZ^*\)中被违反的程度可以在大型强子对撞机上用未来的数据进行测试。对于 900 GeV 以上的不变质量和接近\(\pi /2\) 的散射角，在WW和ZZ玻色子对的产生中违反了相同的贝尔不等式在质量框架的中心。在这种情况下，大型强子对撞机的数据不足以证明贝尔不等式的违反。我们还分析了在未来\(e^+e^-\)和 μ 子对撞机中检测双玻色子最终状态中贝尔不等式违规的前景。对于每个检查的过程，计算了进一步的可观测量，它提供了双玻色子系统中偏振纠缠量的下限。极化密度矩阵的解析表达式在附录中完整给出。我们还提供测试贝尔不等式所需的优化过程所需的酉矩阵。