## Codes Entanglement Mixed

The function performs the following decomposition for the input density matrix: The function F takes as

Input: an 8x8 Hermitian, positive matrix, of unit trace expressed in the computational basis

This matrix represents a mixed state of three qubits for which you would like to analyze its entanglement properties. Please do not enter rank 1! density matrices or else pure states.

The function F gives as

Output: an 8x8 Hermitian, positive matrix, of unit trace.

This matrix is essentially entangled part link of the input density matrix.

• the purity of the state

• whether the state is has a positive partial transpose (PPT) or negative (NPT)

• the weight (1-w) over the essentially entangled part of the density matrix. This can be used as an overall estimation of the entanglement content of the input state.

#### What you can do with the output:

The rank essentially entangled component according to the theorem in this link should be ≤5. Most of the times though is 1 or there is a principal eigenvector with an eigenvalue much greater than all the others. One can perform analysis of the entanglement of the principal vector using the habitually used measures of entanglement for pure states: 3-tangle, concurrence. Alternatively, this can be analyzed using the tanglemeter coefficients .

#### Accuracy:

The function finds the weight of the best separable approximation with accuracy . In other words, the weight w over the separable part ( or 1-w over the essentially entangled part) is a real number in the range [0, ] and this is calculated consistently up to third decimal number. One can in principle change the accuracy but this is not a straightforward task (other parameters needs to be adjusted along) so it is not recommended by us you to do so. If you run the function twice for the same input you may confirm this approximation on the weight. To be consistent with the accuracy, states with w > 0.999 are considered as separable states.

#### Extensions:

We have different versions of this function for different dimensions N of the Hilbert space. On the other hand this slows down considerably with the dimension (running time scales approximately as. So for instance for N = 12 the function concludes after one hour on a desktop computer. Please contact us if you are interested to learn more on the limits and possible extensions of this algorithmic solution for entanglement analysis.

#### If you obtain useful results with this function, please cite:

V. M. Akulin, G. A. Kabatyanski, A. Mandilara, Phys. Rev. A 92, 042322 (2015).  3qubitsEssentEntang.nb