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is_ppt

Checks if a quantum state violates the PPT criterion.

is_ppt 

is_ppt(
    mat: ndarray,
    sys: int = 2,
    dim: int | list[int] | ndarray | None = None,
    tol: float | None = None,
) -> bool

Determine whether or not a matrix has positive partial transpose 1.

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.

Parameters:

  • mat (ndarray) –

    A square matrix.

  • sys (int, default: 2 ) –

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

  • dim (int | list[int] | ndarray | None, default: None ) –

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

  • tol (float | None, default: None ) –

    Tolerance with which to check whether mat is PPT.

Returns:

  • bool –

    Returns True if mat is PPT and False if not.

Examples:

Consider the following matrix

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

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)
print(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
print(is_ppt(rho))

False

References

1 Wikipedia. Peres-Horodecki criterion. link.

Source code in toqito/state_props/is_ppt.py 
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def is_ppt(
    mat: np.ndarray, sys: int = 2, dim: int | list[int] | np.ndarray | None = None, tol: float | None = None
) -> bool:
    r"""Determine whether or not a matrix has positive partial transpose [@wikipediapereshorodecki].

    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.

    Args:
        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.

    Examples:
        Consider the following matrix

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

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

        ```python exec="1" source="above" result="text"
        from toqito.state_props import is_ppt
        import numpy as np
        mat = np.identity(9)
        print(is_ppt(mat))
        ```

        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.

        ```python exec="1" source="above" result="text"
        from toqito.states import bell
        from toqito.state_props import is_ppt
        rho = bell(2) @ bell(2).conj().T
        print(is_ppt(rho))
        ```

    """
    eps = np.finfo(float).eps

    sqrt_rho_dims = np.round(np.sqrt(list(mat.shape)))
    sqrt_rho_dims = sqrt_rho_dims.astype(int)

    if dim is None:
        dim = [
            [sqrt_rho_dims[0], sqrt_rho_dims[0]],
            [sqrt_rho_dims[1], sqrt_rho_dims[1]],
        ]
    if tol is None:
        tol = np.sqrt(eps)
    return is_positive_semidefinite(partial_transpose(mat, [sys - 1], dim), tol)