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purity

Calcultes the purity of a quantum state.

purity 

purity(rho: ndarray) -> float

Compute the purity of a quantum state 1.

The negativity of a subsystem can be defined in terms of a density matrix \(\rho\): The purity of a quantum state \(\rho\) is defined as

\[ \text{Tr}(\rho^2), \]

where \(\text{Tr}\) is the trace function.

Examples:

Consider the following scaled state defined as the scaled identity matrix

\[ \rho = \frac{1}{4} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \in \text{D}(\mathcal{X}). \]

Calculating the purity of \(\rho\) yields \(\frac{1}{4}\). This can be observed using |toqito⟩ as follows.

from toqito.state_props import purity
import numpy as np
print(purity(np.identity(4) / 4))

0.25

Calculate the purity of the Werner state:

from toqito.states import werner
rho = werner(2, 1 / 4)
print(purity(rho))

0.26530612244897955

Raises:

  • ValueError –

    If matrix is not density operator.

Parameters:

  • rho (ndarray) –

    A density matrix of a pure state vector.

Returns:

  • float –

    A value between 0 and 1 that corresponds to the purity of \(\rho\).

References

1 Wikipedia. Purity (quantum mechanics). link.

Source code in toqito/state_props/purity.py 
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def purity(rho: np.ndarray) -> float:
    r"""Compute the purity of a quantum state [@WikiPurity].

    The negativity of a subsystem can be defined in terms of a density matrix \(\rho\): The
    purity of a quantum state \(\rho\) is defined as

    \[
        \text{Tr}(\rho^2),
    \]

    where \(\text{Tr}\) is the trace function.

    Examples:
        Consider the following scaled state defined as the scaled identity matrix

        \[
            \rho = \frac{1}{4} \begin{pmatrix}
                             1 & 0 & 0 & 0 \\
                             0 & 1 & 0 & 0 \\
                             0 & 0 & 1 & 0 \\
                             0 & 0 & 0 & 1
                           \end{pmatrix} \in \text{D}(\mathcal{X}).
        \]

        Calculating the purity of \(\rho\) yields \(\frac{1}{4}\). This can be observed using
        `|toqito⟩` as follows.

        ```python exec="1" source="above" session="purity_example"
        from toqito.state_props import purity
        import numpy as np
        print(purity(np.identity(4) / 4))
        ```


        Calculate the purity of the Werner state:

        ```python exec="1" source="above" session="purity_example"
        from toqito.states import werner
        rho = werner(2, 1 / 4)
        print(purity(rho))
        ```

    Raises:
        ValueError: If matrix is not density operator.

    Args:
        rho: A density matrix of a pure state vector.

    Returns:
        A value between 0 and 1 that corresponds to the purity of \(\rho\).

    """
    if not is_density(rho):
        raise ValueError("Purity is only defined for density operators.")
    # "np.real" get rid of the close-to-0 imaginary part.
    return np.real(np.trace(np.linalg.matrix_power(rho, 2)))