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tile

Tile state.

tile 

tile(idx: int) -> ndarray

Produce a Tile state 1.

The Tile states constitute five states on 3-by-3 dimensional space that form a UPB (unextendible product basis).

Returns one of the following five tile states depending on the value of idx:

\[ \begin{equation} \begin{aligned} |\psi_0 \rangle = \frac{1}{\sqrt{2}} |0 \rangle \left(|0\rangle - |1\rangle \right), \qquad & |\psi_1\rangle = \frac{1}{\sqrt{2}} \left(|0\rangle - |1\rangle \right) |2\rangle, \\ |\psi_2\rangle = \frac{1}{\sqrt{2}} |2\rangle \left(|1\rangle - |2\rangle \right), \qquad & |\psi_3\rangle = \frac{1}{\sqrt{2}} \left(|1\rangle - |2\rangle \right) |0\rangle, \\ \qquad & |\psi_4\rangle = \frac{1}{3} \left(|0\rangle + |1\rangle + |2\rangle)\right) \left(|0\rangle + |1\rangle + |2\rangle \right). \end{aligned} \end{equation} \]

Parameters:

  • idx (int) –

    A parameter in [0, 1, 2, 3, 4]

Returns:

  • ndarray –

    Tile state.

Raises:

  • ValueError –

    Invalid value for idx.

Examples:

When idx = 0, this produces the following tile state

\[ \frac{1}{\sqrt{2}} |0\rangle \left( |0\rangle - |1\rangle \right). \]

Using |toqito⟩, we can see that this yields the proper state.

from toqito.states import tile
import numpy as np
print(tile(0))

[[ 0.70710678]
 [-0.70710678]
 [ 0.        ]
 [ 0.        ]
 [ 0.        ]
 [ 0.        ]
 [ 0.        ]
 [ 0.        ]
 [ 0.        ]]

References

1 Bennett, Charles and DiVincenzo, David and Mor, Tal and Shor, Peter and Smolin, John and Terhal, Barbara. Unextendible Product Bases and Bound Entanglement. Physical Review Letters. vol. 82(26). (1999). doi:10.1103/physrevlett.82.5385.

Source code in toqito/states/tile.py 
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def tile(idx: int) -> np.ndarray:
    r"""Produce a Tile state [@bennett1999unextendible].

    The Tile states constitute five states on 3-by-3 dimensional space that form a UPB (unextendible product basis).

    Returns one of the following five tile states depending on the value of `idx`:

    \[
        \begin{equation}
            \begin{aligned}
                |\psi_0 \rangle = \frac{1}{\sqrt{2}} |0 \rangle
                \left(|0\rangle - |1\rangle \right),
                \qquad &
                |\psi_1\rangle = \frac{1}{\sqrt{2}}
                \left(|0\rangle - |1\rangle \right) |2\rangle, \\
                |\psi_2\rangle = \frac{1}{\sqrt{2}} |2\rangle
                \left(|1\rangle - |2\rangle \right),
                \qquad &
                |\psi_3\rangle = \frac{1}{\sqrt{2}}
                \left(|1\rangle - |2\rangle \right) |0\rangle, \\
                \qquad &
                |\psi_4\rangle = \frac{1}{3}
                \left(|0\rangle + |1\rangle + |2\rangle)\right)
                \left(|0\rangle + |1\rangle + |2\rangle \right).
            \end{aligned}
        \end{equation}
    \]

    Args:
        idx: A parameter in [0, 1, 2, 3, 4]

    Returns:
        Tile state.

    Raises:
        ValueError: Invalid value for `idx`.

    Examples:
        When `idx = 0`, this produces the following tile state

        \[
            \frac{1}{\sqrt{2}} |0\rangle \left( |0\rangle - |1\rangle \right).
        \]

        Using `|toqito⟩`, we can see that this yields the proper state.

        ```python exec="1" source="above" result="text"
        from toqito.states import tile
        import numpy as np
        print(tile(0))
        ```

    """
    e_0, e_1, e_2 = basis(3, 0), basis(3, 1), basis(3, 2)
    match idx:
        case 0:
            return 1 / np.sqrt(2) * np.kron(e_0, (e_0 - e_1))
        case 1:
            return 1 / np.sqrt(2) * np.kron((e_0 - e_1), e_2)
        case 2:
            return 1 / np.sqrt(2) * np.kron(e_2, (e_1 - e_2))
        case 3:
            return 1 / np.sqrt(2) * np.kron((e_1 - e_2), e_0)
        case 4:
            return 1 / 3 * np.kron((e_0 + e_1 + e_2), (e_0 + e_1 + e_2))
    raise ValueError("Invalid integer value for Tile state.")