Abstract
Using a geometric measure of entanglement quantification based on Euclidean distance of the Hermitian matrices (Patel and Panigrahi in Geometric measure of entanglement based on local measurement, 2016. arXiv:1608.06145), we obtain the minimum distance between the set of bipartite n-qudit density matrices with a positive partial transpose and the maximally mixed state. This minimum distance is obtained as \(\frac{1}{\sqrt{d^n(d^n-1)}}\), which is also the minimum distance within which all quantum states are separable. An idea of the interior of the set of all positive semidefinite matrices has also been provided. A particular class of Werner states has been identified for which the PPT criterion is necessary and sufficient for separability in dimensions greater than six.
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The authors want to acknowledge valuable inputs from Prof. Somshubhro Bandyopadhyay (Bose Institute, Kolkata).
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Appendix A
Appendix A
1.1 Calculation for 3-qubit Werner state to be PPT
The 3-qubit Werner state with \(1 \otimes 2\) bi-partition is given by
where \(N=2^n\), \(p \in [0, 1]\); \(P_\psi \) is 3-qubit GHZ state and \(\frac{\mathbb {I}}{8}\) is the normalized identity matrix of order 8. Hence,
Carrying out the partial transposition operation, we have
Eigenvalues of \( \rho _{w_{3}}^{\mathrm{T}_B}\) are found to be,
Hence, the maximum value of p such that the partially transposed Werner state is positive semidefinite is \(\frac{1}{5}\).
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Banerjee, S., Patel, A.A. & Panigrahi, P.K. Minimum distance of the boundary of the set of PPT states from the maximally mixed state using the geometry of the positive semidefinite cone. Quantum Inf Process 18, 296 (2019). https://doi.org/10.1007/s11128-019-2411-6
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DOI: https://doi.org/10.1007/s11128-019-2411-6