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bell

Bell states represent the simplest examples of quantum entanglement of two qubits.

Also known as EPR pairs, Bell states comprise of four quantum states in a superposition of 0 and 1.

bell 

bell(idx: int) -> ndarray

Produce a Bell state 1.

Returns one of the following four Bell states depending on the value of idx:

\[ \begin{equation} \begin{aligned} u_0 = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right), & \qquad & u_1 = \frac{1}{\sqrt{2}} \left( |00 \rangle - |11 \rangle \right), \\ u_2 = \frac{1}{\sqrt{2}} \left( |01 \rangle + |10 \rangle \right), & \qquad & u_3 = \frac{1}{\sqrt{2}} \left( |01 \rangle - |10 \rangle \right). \end{aligned} \end{equation} \]

Parameters:

  • idx (int) –

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

Returns:

  • ndarray –

    Bell state with index idx.

Raises:

  • ValueError –

    If idx is not an integer.

Examples:

When idx = 0, this produces the following Bell state:

\[ u_0 = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right). \]

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

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

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

References

1 Wikipedia. Bell State. link.

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

    Returns one of the following four Bell states depending on the value of `idx`:

    \[
        \begin{equation}
            \begin{aligned}
                u_0 = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right), &
                \qquad &
                u_1 = \frac{1}{\sqrt{2}} \left( |00 \rangle - |11 \rangle \right), \\
                u_2 = \frac{1}{\sqrt{2}} \left( |01 \rangle + |10 \rangle \right), &
                \qquad &
                u_3 = \frac{1}{\sqrt{2}} \left( |01 \rangle - |10 \rangle \right).
            \end{aligned}
        \end{equation}
    \]

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

    Returns:
        Bell state with index `idx`.

    Raises:
        ValueError: If `idx` is not an integer.

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

        \[
            u_0 = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right).
        \]

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

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

    """
    match idx:
        case 0:
            return 1 / np.sqrt(2) * np.array([[1], [0], [0], [1]])
        case 1:
            return 1 / np.sqrt(2) * np.array([[1], [0], [0], [-1]])
        case 2:
            return 1 / np.sqrt(2) * np.array([[0], [1], [1], [0]])
        case 3:
            return 1 / np.sqrt(2) * np.array([[0], [1], [-1], [0]])
    raise ValueError("Invalid integer value for Bell state.")