  A tunnel barrier in a degenerate electron gas was recently discovered as a
source of entangled electron-hole pairs. Here, we investigate the loss of
entanglement by dephasing. We calculate both the maximal violation E_max of the
Bell inequality and the degree of entanglement (concurrence) C. If the
initially maximally entangled electron-hole pair is in a Bell state, then the
Bell inequality is violated for arbitrary strong dephasing. The same relation
E_max=2\sqrt{1+C^{2}} then holds as in the absence of dephasing. More
generally, for a maximally entangled superposition of Bell states, the Bell
inequality is satisfied for a finite dephasing strength and the entanglement
vanishes for somewhat stronger (but still finite) dephasing strength. There is
then no one-to-one relation between E_max and C.
