Observable Entanglement Measure for Mixed Quantum States

We quantify an unknown mixed quantum state’s entanglement by suitable, local parity measurements on its twofold copy. The associated observable qualifies as a generalized entanglement witness.

Figure 1

Squared mixed-state concurrence $c(\varrho {)}^2$, approximated by its measurable lower bound $\mathrm{Tr}[\varrho \otimes \varrho (V_1+V_2)/2]$ as given by (2), vs its lower bound in quasipure approximation, for random states of a $2\times 5$ dimensional ($a$, $b$, and $c$), and of a $3\times 3$ dimensional system ($d$, $e$, $f$). The different panels represent states with different degrees of mixing: $a$ and $d$ display the case of weakly mixed states ($0.2\le \sqrt{1-\mathrm{Tr}{\varrho }^2}\le 0.21$), $b$ and $e$ correspond to the regime of intermediate mixing ($0.4\le \sqrt{1-\mathrm{Tr}{\varrho }^2}\le 0.405$), and $c$ and $f$ show strongly mixed states ($0.529\le \sqrt{1-\mathrm{Tr}{\varrho }^2}\le 0.533$). The dashed lines indicate equality of both bounds. In particular, for highly entangled states and for states with little mixing the bound is very good; but it also yields a surprisingly good characterization of rather strongly mixed states.