Detection of every pure entangled qubit–qutrit state from incomplete tomography

Prove that the reconstructed density matrix obtained from the 23 experimentally accessible parameters detects every pure entangled state in a spin-1/2 times spin-1 (qubit–qutrit) system analyzed through a self-analyzing weak hyperon decay and a parity-conserving vector-meson two-body decay.

Background

The paper studies incomplete quantum-state tomography for $\Lambda V$ systems, where $V=\phi$ or $K^{\ast0}$. The decay data determine 23 of the 35 independent parameters of the underlying qubit–qutrit density matrix; the remaining 12 parameters are the vector-meson dipole polarization $P_i$ and the dipole–dipole correlations $C_{ij}$. The authors define $\rho_{\rm meas}$ by retaining the accessible parameters and setting the inaccessible ones to zero.

Theorem 2 establishes that entanglement is certifiable from the measured data exactly when $\rho_{\rm meas}$ has a negative eigenvalue. Numerical tests over 30,000 randomly generated pure entangled states found a strictly negative minimum eigenvalue in every case, but the paper does not provide an analytic proof covering all Schmidt bases and Schmidt coefficients. The conjecture is therefore that no pure entangled qubit–qutrit state shares all 23 accessible parameters with a separable state.