Concurrence of Mixed Bipartite Quantum States in Arbitrary Dimensions

We derive a lower bound for the concurrence of mixed bipartite quantum states, valid in arbitrary dimensions. As a corollary, a weaker, purely algebraic estimate is found, which detects mixed entangled states with a positive partial transpose.

Figure 1

Lower bound (full line) of the concurrence $c$ of the family of two spin-1 states ${\varrho }_a$ [16], together with its purely algebraic approximation $c_l^{(1)}={\lambda }_1^{(1)}-\sum _{i>1}{\lambda }_i^{(1)}$ (dashed line), over the entire parameter range $a=0\dots 1$. While ${\varrho }_a$ has positive partial transpose and is therefore not detected via the negative partial transpose criterion for entanglement, already the algebraic approximation of our lower bound indentifies the state as nonseparable, without need for numerical optimization, for all $a$. Optimal lower bound and optimal upper bound [dots, obtained by minimizing $\mathcal{C}$ in (4) [17]] confine the actual value of concurrence to an interval with relative uncertainty of the order of approximately $10\%$.