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gen_bell

Generalized Bell state represents a bigger set of Bell states.

This set includes the standard bell states and other higher dimensional bell states as well. Generalized Bell states are the basis of multidimensional bipartite states having maximum entanglement.

gen_bell 

gen_bell(k_1: int, k_2: int, dim: int) -> ndarray

Produce a generalized Bell state 1.

Produces a generalized Bell state. Note that the standard Bell states can be recovered as:

bell(0) : gen_bell(0, 0, 2)

bell(1) : gen_bell(0, 1, 2)

bell(2) : gen_bell(1, 0, 2)

bell(3) : gen_bell(1, 1, 2)

Examples:

For \(d = 2\) and \(k_1 = k_2 = 0\), this generates the following matrix

\[ G = \frac{1}{2} \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{pmatrix} \]

which is equivalent to \(|\phi_0 \rangle \langle \phi_0 |\) where

\[ |\phi_0\rangle = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right) \]

is one of the four standard Bell states. This can be computed via |toqito⟩ as follows.

from toqito.states import gen_bell
dim = 2
k_1 = 0
k_2 = 0
print(gen_bell(k_1, k_2, dim))

[[0.5+0.j 0. +0.j 0. +0.j 0.5+0.j] [0. +0.j 0. +0.j 0. +0.j 0. +0.j] [0. +0.j 0. +0.j 0. +0.j 0. +0.j] [0.5+0.j 0. +0.j 0. +0.j 0.5+0.j]]

It is possible for us to consider higher dimensional Bell states. For instance, we can consider the \(3\)-dimensional Bell state for \(k_1 = k_2 = 0\) as follows.

from toqito.states import gen_bell
dim = 3
k_1 = 0
k_2 = 0
print(gen_bell(k_1, k_2, dim))

[[0.33333333+0.j 0. +0.j 0. +0.j 0. +0.j 0.33333333+0.j 0. +0.j 0. +0.j 0. +0.j 0.33333333+0.j] [0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j] [0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j] [0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j] [0.33333333+0.j 0. +0.j 0. +0.j 0. +0.j 0.33333333+0.j 0. +0.j 0. +0.j 0. +0.j 0.33333333+0.j] [0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j] [0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j] [0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j] [0.33333333+0.j 0. +0.j 0. +0.j 0. +0.j 0.33333333+0.j 0. +0.j 0. +0.j 0. +0.j 0.33333333+0.j]]

Parameters:

  • k_1 (int) –

    An integer 0 <= k_1 <= n.

  • k_2 (int) –

    An integer 0 <= k_2 <= n.

  • dim (int) –

    The dimension of the generalized Bell state.

References

1 Sych, Denis and Leuchs, Gerd. A complete basis of generalized Bell states. New Journal of Physics. vol. 11(1). (2009). doi:10.1088/1367-2630/11/1/013006.

Source code in toqito/states/gen_bell.py 
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def gen_bell(k_1: int, k_2: int, dim: int) -> np.ndarray:
    r"""Produce a generalized Bell state [@Sych_2009_AComplete].

    Produces a generalized Bell state. Note that the standard Bell states can be recovered as:

        bell(0) : gen_bell(0, 0, 2)

        bell(1) : gen_bell(0, 1, 2)

        bell(2) : gen_bell(1, 0, 2)

        bell(3) : gen_bell(1, 1, 2)


    Examples:
        For \(d = 2\) and \(k_1 = k_2 = 0\), this generates the following matrix

        \[
            G = \frac{1}{2} \begin{pmatrix}
                            1 & 0 & 0 & 1 \\
                            0 & 0 & 0 & 0 \\
                            0 & 0 & 0 & 0 \\
                            1 & 0 & 0 & 1
                        \end{pmatrix}
        \]

        which is equivalent to \(|\phi_0 \rangle \langle \phi_0 |\) where

        \[
            |\phi_0\rangle = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right)
        \]

        is one of the four standard Bell states. This can be computed via `|toqito⟩` as follows.

        ```python exec="1" source="above"
        from toqito.states import gen_bell
        dim = 2
        k_1 = 0
        k_2 = 0
        print(gen_bell(k_1, k_2, dim))
        ```

        It is possible for us to consider higher dimensional Bell states. For instance, we can consider
        the \(3\)-dimensional Bell state for \(k_1 = k_2 = 0\) as follows.

        ```python exec="1" source="above"
        from toqito.states import gen_bell
        dim = 3
        k_1 = 0
        k_2 = 0
        print(gen_bell(k_1, k_2, dim))
        ```

    Args:
        k_1: An integer 0 <= k_1 <= n.
        k_2: An integer 0 <= k_2 <= n.
        dim: The dimension of the generalized Bell state.

    """
    gen_pauli_w = gen_pauli(k_1, k_2, dim)
    return 1 / dim * vec(gen_pauli_w) @ vec(gen_pauli_w).conj().T