Qubit bi-negativity

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Cite as: http://qig.itp.uni-hannover.de/qiproblems/18

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Problem

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

[math] | \sigma^{T_2} |^{T_2} \geq 0 [/math]

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

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.

Literature

  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)