Minimum distance of the boundary of the set of PPT states from the maximally mixed state using the geometry of the positive semidefinite cone

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.

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

$$\begin{aligned} \rho _{w_{3}}= pP_{\psi _{3}}+(1-p)\frac{\mathbb {I}}{8}, \end{aligned}$$

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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,

$$\begin{aligned} \rho _{w_{3}}=\left( \begin{array}{cccccccc} \frac{1-p}{8}+\frac{p}{2} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{p}{2} \\ 0 &{}\quad \frac{1-p}{8} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \frac{1-p}{8} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1-p}{8} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1-p}{8} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1-p}{8} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1-p}{8} &{}\quad 0 \\ \frac{p}{2} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1-p}{8}+\frac{p}{2} \\ \end{array} \right) \end{aligned}$$

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Carrying out the partial transposition operation, we have

$$\begin{aligned} \rho _{w_{3}}^{\mathrm{T}_B}=\left( \begin{array}{cccccccc} \frac{1-p}{8}+\frac{p}{2} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad \frac{1-p}{8} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \frac{1-p}{8} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1-p}{8} &{}\quad \frac{p}{2} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{p}{2} &{}\quad \frac{1-p}{8} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1-p}{8} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1-p}{8} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1-p}{8}+\frac{p}{2} \\ \end{array} \right) \end{aligned}$$

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Eigenvalues of $\rho _{w_{3}}^{\mathrm{T}_B}$ are found to be,

$$\begin{aligned} \frac{1}{8} (1-5 p),\frac{1-p}{8},\frac{1-p}{8},\frac{1-p}{8},\frac{1-p}{8},\frac{1}{8} (3 p+1),\frac{1}{8} (3 p+1),\frac{1}{8} (3 p+1) \end{aligned}$$

Hence, the maximum value of p such that the partially transposed Werner state is positive semidefinite is $\frac{1}{5}$.