Source code for cvxpy.atoms.von_neumann_entr

"""
Copyright 2022, 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.special import entr

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



[docs]
class von_neumann_entr(Atom):
    """
    Represents the von Neumann Entropy of the positive-definite matrix :math:`X,`

        .. math::
            -\\operatorname{tr}(X \\log X).

    Mathematically, this is equivalent to

        .. math::
            \\texttt{von_neumann_entr}(X) = -\\textstyle\\sum_{i=1}^n \\lambda_i \\log \\lambda_i

    where :math:`\\lambda_{i}` are the eigenvalues of :math:`X.`

    Parameters
    ----------
    X : Expression or numeric
        A PSD matrix

    quad_approx : Tuple[int,...]
        This is either an empty tuple (default) or a 2-tuple. If a 2-tuple, then this atom
        is approximately canonicalized. The approximations replace ExpCone constraints with SOC
        constraints based on a quadrature scheme from https://arxiv.org/abs/1705.00812;
        quad_approx[0] is the number of quadrature nodes and quad_approx[1] is the number of
        scaling points in the quadrature scheme.

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

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

    def numeric(self, values):
        N = values[0]
        if hasattr(N, 'value'):
            N = N.value  # assume this is an ndarray
        w = LA.eigvalsh(N)
        val = np.sum(entr(w))
        return val

    def validate_arguments(self) -> None:
        """Verify that the argument is Hermitian."""
        if not self.args[0].is_hermitian():
            raise ValueError(
                f"""
                The argument {self.args[0].name()} to von_neumann_entr must be a Hermitian matrix.
                If you know for a fact that the input is Hermitian, wrap it with the hermitian_wrap
                atom before calling von_neumann_entr.
                """
            )
        
    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 False

    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 True

    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.
        """
        # N = values[0]
        # L = cholesky(N)
        # derivative = 2*(L + L * logm(np.dot(L.T, L)))
        # TODO: have to wrap derivative around scipy CSC sparse matrices
        #  compare to log_det atom.
        raise NotImplementedError()

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