code: def relative_entropy(
      state0, state1, base=None, check_state=False, c_dtype="complex128"
  ):
      r"""
      Compute the quantum relative entropy of one state with respect to another.
  
      .. math::
          S(\rho\,\|\,\sigma)=-\text{Tr}(\rho\log\sigma)-S(\rho)=\text{Tr}(\rho\log\rho)-\text{Tr}(\rho\log\sigma)
          =\text{Tr}(\rho(\log\rho-\log\sigma))
  
      Roughly speaking, quantum relative entropy is a measure of distinguishability between two
      quantum states. It is the quantum mechanical analog of relative entropy.
  
      Each state must be given as a density matrix. To find the relative entropy given
      a pure state, call :func:`~.math.dm_from_state_vector` first.
  
      Args:
          state0 (tensor_like): ``(2**N, 2**N)`` or ``(batch_dim, 2**N, 2**N)`` density matrix.
          state1 (tensor_like): ``(2**N, 2**N)`` or ``(batch_dim, 2**N, 2**N)`` density matrix.
          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: Quantum relative entropy of state0 with respect to state1
  
      **Examples**
  
      The relative entropy between two equal states is always zero:
  
      >>> x = np.array([1, 0])
      >>> x = qml.math.dm_from_state_vector(x)
      >>> qml.math.relative_entropy(x, x)
      0.0
  
      and the relative entropy between two non-equal pure states is always infinity:
  
      >>> y = np.array([1, 1]) / np.sqrt(2)
      >>> y = qml.math.dm_from_state_vector(y)
      >>> qml.math.relative_entropy(x, y)
      inf
  
      The quantum states can be provided as density matrices, allowing for computation
      of relative entropy between mixed states:
  
      >>> rho = np.array([[0.3, 0], [0, 0.7]])
      >>> sigma = np.array([[0.5, 0], [0, 0.5]])
      >>> qml.math.relative_entropy(rho, sigma)
      tensor(0.08228288, requires_grad=True)
  
      It is also possible to change the log base:
  
      >>> qml.math.relative_entropy(rho, sigma, base=2)
      tensor(0.1187091, requires_grad=True)
  
      .. seealso:: :func:`pennylane.qinfo.transforms.relative_entropy`
      """
      # Cast as a c_dtype array
      state0 = cast(state0, dtype=c_dtype)
  
      # Cannot be cast_like if jit
      if not is_abstract(state0):
          state1 = cast_like(state1, state0)
  
      if check_state:
          # pylint: disable=expression-not-assigned
          _check_density_matrix(state0)
          _check_density_matrix(state1)
  
      # Compare the number of wires on both subsystems
      if qml.math.shape(state0)[-1] != qml.math.shape(state1)[-1]:
          raise qml.QuantumFunctionError(
              "The two states must have the same number of wires."
          )
  
      return _compute_relative_entropy(state0, state1, base=base)
  
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 16 for calling a global function with example usage: relative_entropy(state0, state1, base, check_state, c_dtype) and returns: _compute_relative_entropy