code: def max_entropy(
      state, indices, base=None, check_state=False, c_dtype="complex128"
  ):
      r"""Compute the maximum entropy of a density matrix on a given subsystem. It supports all
      interfaces (Numpy, Autograd, Torch, Tensorflow and Jax).
  
      .. math::
          S_{\text{max}}( \rho ) = \log( \text{rank} ( \rho ))
  
      Args:
          state (tensor_like): Density matrix of shape ``(2**N, 2**N)`` or ``(batch_dim, 2**N, 2**N)``.
          indices (list(int)): List of indices in the considered subsystem.
          base (float): Base for the logarithm. If None, the natural logarithm is used.
          check_state (bool): If True, the function will check the state validity (shape and norm).
          c_dtype (str): Complex floating point precision type.
  
      Returns:
          float: The maximum entropy of the considered subsystem.
  
      **Example**
  
      The maximum entropy of a subsystem for any state vector can be obtained by first calling
      :func:`~.math.dm_from_state_vector` on the input. Here is an example for the
      maximally entangled state, where the subsystem entropy is maximal (default base for log is exponential).
  
      >>> x = [1, 0, 0, 1] / np.sqrt(2)
      >>> x = dm_from_state_vector(x)
      >>> max_entropy(x, indices=[0])
      0.6931472
  
      The logarithm base can be changed. For example:
  
      >>> max_entropy(x, indices=[0], base=2)
      1.0
  
      The maximum entropy can be obtained by providing a quantum state as a density matrix. For example:
  
      >>> y = [[1/2, 0, 0, 1/2], [0, 0, 0, 0], [0, 0, 0, 0], [1/2, 0, 0, 1/2]]
      >>> max_entropy(y, indices=[0])
      0.6931472
  
      The maximum entropy is always greater or equal to the Von Neumann entropy. In this maximally
      entangled example, they are equal:
  
      >>> vn_entropy(x, indices=[0])
      0.6931472
  
      However, in general, the Von Neumann entropy is lower:
  
      >>> x = [np.cos(np.pi/8), 0, 0, -1j*np.sin(np.pi/8)]
      >>> x = dm_from_state_vector(x)
      >>> vn_entropy(x, indices=[1])
      0.4164955
      >>> max_entropy(x, indices=[1])
      0.6931472
  
      """
      density_matrix = reduce_dm(state, indices, check_state, c_dtype)
      maximum_entropy = _compute_max_entropy(density_matrix, base)
      return maximum_entropy
  
imports: import itertools
  import functools
  from string import ascii_letters
  from autoray import numpy
  from numpy import float64
  import pennylane
  from .None import single_dispatch
  from .multi_dispatch import diag, dot, scatter_element_add, einsum, get_interface
  from .utils import is_abstract, allclose, cast, convert_like, cast_like
  from .matrix_manipulation import _permute_dense_matrix
label_desc: Use image node_id 19 for calling a global function with example usage: max_entropy(state, indices, base, check_state, c_dtype) and returns: maximum_entropy