Source code for thewalrus.quantum.entanglement

# Copyright 2022 Xanadu Quantum Technologies Inc.

# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at

#     http://www.apache.org/licenses/LICENSE-2.0

# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""
Entanglement measures for pure and mixed Gaussian states.
"""

import numpy as np
from ..decompositions import symplectic_eigenvals
from .gaussian_checks import is_valid_cov, is_pure_cov
from .conversions import reduced_gaussian


def get_partition(modes_A, split, M):
    r"""Helper function to validate and, if necessary, pre-prepare the inputs
    of ``vonneumann_entropy()`` and ``log_negativity()``.

    Args:
        modes_A (iterable or int): the subset of modes used for the bipartition
        split (int): the index of the mode separating the two partitions
            (alternative to ``modes_A``)
        M (int): total number of modes

    Returns:
        (list): mode indices of substate A
    """
    if modes_A is not None:
        if not isinstance(modes_A, (int, range, list, tuple, np.ndarray)):
            raise TypeError("``modes_A`` must be either integer, range, tuple or np.ndarray.")
        if isinstance(modes_A, int):
            modes_A = [modes_A]
        for mode in modes_A:
            if not isinstance(mode, int) or mode < 0 or mode > M - 1:
                raise ValueError(
                    f"Every element of 'modes_A' must be an integer between 0 and {M - 1}"
                )

    if modes_A is None and split is not None:
        if not isinstance(split, int) or split > M - 1:
            raise ValueError("'split' must be an integer smaller than the number of modes.")
        modes_A = range(split)

    if modes_A is None and split is None:
        raise TypeError("A bipartition must be defined, either by 'modes_A' or 'split'.")

    return list(modes_A)


def vonneumann_entropy(cov, hbar=2):
    r"""Returns the vonNeumann entropy of a covariance matrix.

    Args:
        cov (array): a covariance matrix
        hbar (float): the value of :math:`\hbar` in the commutation
            relation :math:`[\x,\p]=i\hbar`

    Returns:
        (float): vonNeumann entropy
    """
    if not is_valid_cov(cov, hbar=hbar):
        raise ValueError("Input is not a valid covariance matrix.")

    cov = cov / (hbar / 2)

    nus = symplectic_eigenvals(cov)

    S = 0
    for nu in nus:
        if not np.isclose(nu, 1):
            g = (nu + 1) / 2 * np.log((nu + 1) / 2) - (nu - 1) / 2 * np.log((nu - 1) / 2)
            S += g

    return S


def entanglement_entropy(cov, modes_A=None, split=None, hbar=2):
    r"""Returns the entanglement entropy of a covariance matrix under a given
    bipartition.

    Args:
        cov (array): a covariance matrix
        modes_A (iterable or int): the subset of modes used for the bipartition
        split (int): the index of the mode separating the two partitions
            (alternative to ``modes_A``)
        hbar (float): the value of :math:`\hbar` in the commutation
            relation :math:`[\x,\p]=i\hbar`
    Returns:
        (float): logarithmic negativity
    """
    if not is_pure_cov(cov, hbar=hbar):
        raise ValueError("Input is not a pure covariance matrix.")

    cov = cov / (hbar / 2)

    M = int(len(cov) / 2)

    modes_A = get_partition(modes_A, split, M)

    _, cov_A = reduced_gaussian(np.zeros(2 * M), cov, modes_A)

    E = vonneumann_entropy(cov_A)

    return E


def log_negativity(cov, modes_A=None, split=None, hbar=2):
    r"""Returns the logarithmic negativity of a covariance matrix under a given
    bipartition.

    Args:
        cov (array): a covariance matrix
        modes_A (iterable or int): the subset of modes used for the bipartition
        split (int): the index of the mode separating the two partitions
            (alternative to ``modes_A``)

    Returns:
        (float): entanglement entropy
    """
    if not is_valid_cov(cov, hbar=hbar):
        raise ValueError("Input is not a valid covariance matrix.")

    cov = cov / (hbar / 2)

    M = int(len(cov) / 2)

    modes_A = get_partition(modes_A, split, M)

    X = np.ones(M)
    P = np.ones(M)
    P[modes_A] = -1

    S = np.diag(np.concatenate((X, P)))

    cov_tilde = S @ cov @ S

    nus = symplectic_eigenvals(cov_tilde)
    E = np.sum([-np.log(nu) for nu in nus if nu < 1])

    return E

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