toqito.state_props.is_ppt

toqito.state_props.is_ppt(mat, sys=2, dim=None, tol=None)[source]

Determine whether or not a matrix has positive partial transpose [WikPPT].

Yields either True or False, indicating that mat does or does not have positive partial transpose (within numerical error). The variable mat is assumed to act on bipartite space.

For shared systems of \(2 \otimes 2\) or \(2 \otimes 3\), the PPT criterion serves as a method to determine whether a given state is entangled or separable. Therefore, for systems of this size, the return value True would indicate that the state is separable and a value of False would indicate the state is entangled.

Examples

Consider the following matrix

\[\begin{split}X = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{pmatrix}.\end{split}\]

This matrix trivially satisfies the PPT criterion as can be seen using the toqito package.

>>> from toqito.state_props import is_ppt
>>> import numpy as np
>>> mat = np.identity(9)
>>> is_ppt(mat)
True

Consider the following Bell state:

\[u = \frac{1}{\sqrt{2}}\left( |01 \rangle + |10 \rangle \right).\]

For the density matrix \(\rho = u u^*\), as this is an entangled state of dimension \(2\), it will violate the PPT criterion, which can be seen using the toqito package.

>>> from toqito.states import bell
>>> from toqito.state_props import is_ppt
>>> rho = bell(2) * bell(2).conj().T
>>> is_ppt(rho)
False

References

[WikPPT]

Quantiki: Positive partial transpose https://www.quantiki.org/wiki/positive-partial-transpose

Parameters:

  • mat – A square matrix.

  • sys – Scalar or vector indicating which subsystems the transpose should be applied on.

  • dim – The dimension is a vector containing the dimensions of the subsystems on which mat acts.

  • tol – Tolerance with which to check whether mat is PPT.

Returns:

Returns True if mat is PPT and False if not.