Quantifying Entanglement of Maximal Dimension in Bipartite Mixed States

The Schmidt coefficients capture all entanglement properties of a pure bipartite state and therefore determine its usefulness for quantum information processing. While the quantification of the corresponding properties in mixed states is important both from a theoretical and a practical point of view, it is considerably more difficult, and methods beyond estimates for the concurrence are elusive. In particular this holds for a quantitative assessment of the most valuable resource, the forms of entanglement that can only exist in high-dimensional systems. We derive a framework for lower bounding the appropriate measure of entanglement, the so-called $G$-concurrence, through few local measurements. Moreover, we show that these bounds have relevant applications also for multipartite states.

Figure 1

The $G$-concurrence (red) and the 2-concurrence (light blue) for $4\times 4$ axisymmetric states. The 2-concurrence (for pure states) is defined here as $C_2=\sqrt{(d/d-1)(1-\mathrm{Tr}{\rho }_A^2)}$, where ${\rho }_A$ denotes the reduced state of party $A$. In the plane we show the axisymmetric states and the borders between the Schmidt-number classes (red solid lines) which, for $x>0$ are lines of constant fidelity $F=\mathrm{Tr}({\rho }^{\text{axi}}|{\mathrm{\Phi }}_4⟩⟨{\mathrm{\Phi }}_4|)$. Here, $x$ and $y$ are the appropriate coordinates to parametrize the axisymmetric states in a geometry that corresponds to the Hilbert-Schmidt metric [20, 22].