Qubit bi-negativity – Open Quantum Problems

Cite this problem as Problem 18.

Problem

A little problem introduced in [1] is the bi-negativity on two qubits: Prove that

$| \sigma^{T_2} |^{T_2} \geq 0$

holds for every two-qubit state $\sigma$ . Here, $T_2$ denotes the partial transpose with respect to the second system (see also problem 2) and $|.|$ is the operator absolute value, $|x| = \sqrt{x \cdot x}$ .

Solution

The problem has been solved by S. Ishizaka in [2] where it is proven that the bi-negativity is indeed positive for all two-qubit states.

References

[1] K. Audenaert, B. De Moor, K. G. H. Vollbrecht, and R. F. Werner, »Asymptotic Relative Entropy of Entanglement for Orthogonally Invariant States«, Phys. Rev. A 66, 032310 (2002) and quant-ph/0204143 (2002).

[2] S. Ishizaka, »Binegativity and geometry of entangled states in two qubits«, Phys. Rev. A 69, 020301(R) (2004) and quant-ph/0308056 (2003).