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breuer

Breuer states represent the Breuer bound entangled states.

These states are based on the Breuer-Hall criterion.

breuer 

breuer(dim: int, lam: float) -> ndarray

Produce a Breuer state 1.

Gives a Breuer bound entangled state for two qudits of local dimension dim, with the lam parameter describing the weight of the singlet component as described in 1.

This function was adapted from the QETLAB package.

Parameters:

  • dim (int) –

    Dimension of the Breuer state.

  • lam (float) –

    The weight of the singlet component.

Returns:

  • ndarray –

    Breuer state of dimension dim with weight lam.

Raises:

  • ValueError –

    Dimension must be greater than or equal to 1.

Examples:

We can generate a Breuer state of dimension \(4\) with weight \(0.1\). For any weight above \(0\), the state will be bound entangled, that is, it will satisfy the PPT criterion, but it will be entangled.

from toqito.states import breuer
print(breuer(2, 0.1))

[[0.3 0.  0.  0. ]
 [0.  0.2 0.1 0. ]
 [0.  0.1 0.2 0. ]
 [0.  0.  0.  0.3]]

References

1 Breuer, Heinz-Peter. Optimal Entanglement Criterion for Mixed Quantum States. Physical Review Letters. vol. 97(8). (2006). doi:10.1103/physrevlett.97.080501.

Source code in toqito/states/breuer.py 
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def breuer(dim: int, lam: float) -> np.ndarray:
    r"""Produce a Breuer state [@breuer2006optimal].

    Gives a Breuer bound entangled state for two qudits of local dimension `dim`, with the
    `lam` parameter describing the weight of the singlet component as described in
    [@breuer2006optimal].

    This function was adapted from the QETLAB package.

    Args:
        dim: Dimension of the Breuer state.
        lam: The weight of the singlet component.

    Returns:
        Breuer state of dimension `dim` with weight `lam`.

    Raises:
        ValueError: Dimension must be greater than or equal to 1.

    Examples:
        We can generate a Breuer state of dimension \(4\) with weight \(0.1\). For any weight above \(0\), the
        state will be bound entangled, that is, it will satisfy the PPT criterion, but it will be entangled.

        ```python exec="1" source="above" result="text"
        from toqito.states import breuer
        print(breuer(2, 0.1))
        ```

    """
    if dim % 2 == 1 or dim <= 0:
        raise ValueError(f"The value {dim} must be an even positive integer.")

    v_mat = np.fliplr(np.diag((-1) ** np.mod(np.arange(1, dim + 1), 2)))
    max_entangled(dim)
    psi = np.dot(np.kron(np.identity(dim), v_mat), max_entangled(dim))

    return lam * (psi * psi.conj().T) + (1 - lam) * 2 * symmetric_projection(dim) / (dim * (dim + 1))