  A conceptually simpler proof of the separability criterion for two-qubit
systems, which is referred to as "Hefei inequality" in literature, is
presented. This inequality gives a necessary and sufficient separability
criterion for any mixed two-qubit system unlike the Bell-CHSH inequality that
cannot test the mixed-states such as the Werner state when regarded as a
separability criterion. The original derivation of this inequality emphasized
the uncertainty relation of complementary observables, but we show that the
uncertainty relation does not play any role in the actual derivation and the
Peres-Hodrodecki condition is solely responsible for the inequality. Our
derivation, which contains technically novel aspects such as an analogy to the
Dirac equation, sheds light on this inequality and on the fundamental issue to
what extent the uncertainty relation can provide a test of entanglement. This
separability criterion is illustrated for an exact treatment of the Werner
state.
