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is_ensemble

Checks if a set of quantum states form an ensemble.

is_ensemble 

is_ensemble(states: list[ndarray]) -> bool

Determine if a set of states constitute an ensemble.

For more info, see Section: Ensemble Of Quantum States from 1.

An ensemble of quantum states is defined by a function

\[ \eta : \Gamma \rightarrow \text{Pos}(\mathcal{X}) \]

that satisfies

\[ \text{Tr}\left( \sum_{a \in \Gamma} \eta(a) \right) = 1. \]

Parameters:

  • states (list[ndarray]) –

    The list of states to check.

Returns:

  • bool –

    True if states form an ensemble and False otherwise.

Examples:

Consider the following set of matrices

\[ \eta = \left\{ \rho_0, \rho_1 \right\} \]

where

\[ \rho_0 = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad \rho_1 = \frac{1}{2} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}. \]

The set \(\eta\) constitutes a valid ensemble.

from toqito.state_props import is_ensemble
import numpy as np
rho_0 = np.array([[0.5, 0], [0, 0]])
rho_1 = np.array([[0, 0], [0, 0.5]])
states = [rho_0, rho_1]
print(is_ensemble(states))

True

References

1 Watrous, John. The Theory of Quantum Information. (2018). doi:10.1017/9781316848142.

Source code in toqito/state_props/is_ensemble.py 
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def is_ensemble(states: list[np.ndarray]) -> bool:
    r"""Determine if a set of states constitute an ensemble.

    For more info, see Section: Ensemble Of Quantum States from [@watrous2018theory].

    An ensemble of quantum states is defined by a function

    \[
        \eta : \Gamma \rightarrow \text{Pos}(\mathcal{X})
    \]

    that satisfies

    \[
        \text{Tr}\left( \sum_{a \in \Gamma} \eta(a) \right) = 1.
    \]

    Args:
        states: The list of states to check.

    Returns:
        `True` if states form an ensemble and `False` otherwise.

    Examples:
        Consider the following set of matrices

        \[
            \eta = \left\{ \rho_0, \rho_1 \right\}
        \]

        where

        \[
            \rho_0 = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad
            \rho_1 = \frac{1}{2} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.
        \]

        The set \(\eta\) constitutes a valid ensemble.

        ```python exec="1" source="above" result="text"
        from toqito.state_props import is_ensemble
        import numpy as np
        rho_0 = np.array([[0.5, 0], [0, 0]])
        rho_1 = np.array([[0, 0], [0, 0.5]])
        states = [rho_0, rho_1]
        print(is_ensemble(states))
        ```

    """
    trace_sum = 0
    for state in states:
        trace_sum += np.trace(state)
        # Constraint: All states in ensemble must be positive semidefinite.
        if not is_positive_semidefinite(state):
            return False
    # Constraint: The sum of the traces of all states within the ensemble must
    # be equal to 1.
    return np.allclose(trace_sum, 1)