"""
Copyright 2023, the CVXPY authors

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.
"""
from typing import List, Tuple

import numpy as np
from scipy import linalg as LA
from scipy.stats import entropy

from cvxpy import settings
from cvxpy.atoms.atom import Atom
from cvxpy.constraints.constraint import Constraint


class quantum_rel_entr(Atom):
    """
    An approximation of the quantum relative entropy between systems with (possibly un-normalized) 
    density matrices :math:`X` and :math`Y:`
     
    .. math::
        \\operatorname{tr}\\left( X ( \\log X - \\log Y ) \\right).
      
    The approximation uses a quadrature scheme described in https://arxiv.org/abs/1705.00812.

    Parameters
    ----------
    X : Expression or numeric
        A PSD matrix
    Y : Expression or numeric
        A PSD matrix
    quad_approx : Tuple[int, int]
        quad_approx[0] is the number of quadrature nodes and quad_approx[1] is the number of scaling
        points in the quadrature scheme from https://arxiv.org/abs/1705.00812.

    Notes
    -----
    This function does not assume :math:`\\operatorname{tr}(X)=\\operatorname{tr}(Y)=1,` which
    would be required for most uses of this function in the context of quantum information.
    """

    EVAL_TOL = min(settings.ATOM_EVAL_TOL, 1e-6)

    def __init__(self, X, Y, quad_approx: Tuple[int, int] = (3, 3)) -> None:
        self.quad_approx = quad_approx
        super(quantum_rel_entr, self).__init__(X, Y)

    def numeric(self, values):
        X, Y = values
        if hasattr(X, 'value') and hasattr(Y, 'value'):
            X = X.value
            Y = Y.value
        # Symmetrize A and B
        X = (X + X.conj().T) / 2
        Y = (Y + Y.conj().T) / 2
        w1, V = LA.eigh(X)
        w2, W = LA.eigh(Y)
        u = w1.T @ np.abs(V.conj().T @ W) ** 2
        if np.any(w1 < - self.EVAL_TOL) or np.any(w2 < -self.EVAL_TOL):
            return np.inf
        else:
            w1[w1 < 0] = 0
            w2[w2 < 0] = 0
        r1 = -entropy(w1)
        r2 = u @ np.log(w2)
        return (r1 - r2)

    def validate_arguments(self) -> None:
        if not (self.args[0].is_hermitian() and self.args[1].is_hermitian()):
            raise ValueError(
                "The arguments to quantum_rel_entr must both be hermitian."
            )
        
    def sign_from_args(self) -> Tuple[bool, bool]:
        """Returns sign (is positive, is negative) of the expression.
        """
        return (False, False)

    def is_atom_convex(self) -> bool:
        """Is the atom convex?
        """
        return True

    def shape_from_args(self) -> Tuple[int, ...]:
        """Returns the shape of the expression.
        """
        return tuple()

    def is_atom_concave(self) -> bool:
        """Is the atom concave?
        """
        return False

    def is_incr(self, idx) -> bool:
        """Is the composition non-decreasing in argument idx?
        """
        return False

    def is_decr(self, idx) -> bool:
        """Is the composition non-increasing in argument idx?
        """
        return False

    def get_data(self):
        return [self.quad_approx]

    def _grad(self, values):
        """Gives the (sub/super)gradient of the atom w.r.t. each argument.

        Matrix expressions are vectorized, so the gradient is a matrix.

        Args:
            values: A list of numeric values for the arguments.

        Returns:
            A list of SciPy CSC sparse matrices or None.
        """
        raise NotImplementedError()

    def _domain(self) -> List[Constraint]:
        """Returns constraints describing the domain of the node.
        """
        return [self.args[0] >> 0, self.args[1] >> 0]