code: def cov_matrix(prob, obs, wires=None, diag_approx=False):
      """Calculate the covariance matrix of a list of commuting observables, given
      the joint probability distribution of the system in the shared eigenbasis.
  
      .. note::
          This method only works for **commuting observables.**
          If the probability distribution is the result of a quantum circuit,
          the quantum state must be rotated into the shared
          eigenbasis of the list of observables before measurement.
  
      Args:
          prob (tensor_like): probability distribution
          obs (list[.Observable]): a list of observables for which
              to compute the covariance matrix
          diag_approx (bool): if True, return the diagonal approximation
          wires (.Wires): The wire register of the system. If not provided,
              it is assumed that the wires are labelled with consecutive integers.
  
      Returns:
          tensor_like: the covariance matrix of size ``(len(obs), len(obs))``
  
      **Example**
  
      Consider the following ansatz and observable list:
  
      >>> obs_list = [qml.PauliX(0) @ qml.PauliZ(1), qml.PauliY(2)]
      >>> ansatz = qml.templates.StronglyEntanglingLayers
  
      We can construct a QNode to output the probability distribution in the shared eigenbasis of the
      observables:
  
      .. code-block:: python
  
          dev = qml.device("default.qubit", wires=3)
  
          @qml.qnode(dev, interface="autograd")
          def circuit(weights):
              ansatz(weights, wires=[0, 1, 2])
              # rotate into the basis of the observables
              for o in obs_list:
                  o.diagonalizing_gates()
              return qml.probs(wires=[0, 1, 2])
  
      We can now compute the covariance matrix:
  
      >>> shape = qml.templates.StronglyEntanglingLayers.shape(n_layers=2, n_wires=3)
      >>> weights = np.random.random(shape, requires_grad=True)
      >>> cov = qml.math.cov_matrix(circuit(weights), obs_list)
      >>> cov
      tensor([[0.9275379 , 0.05233832], [0.05233832, 0.99335545]], requires_grad=True)
  
      Autodifferentiation is fully supported using all interfaces.
      Here we use autograd:
  
      >>> cost_fn = lambda weights: qml.math.cov_matrix(circuit(weights), obs_list)[0, 1]
      >>> qml.grad(cost_fn)(weights)
      array([[[ 4.94240914e-17, -2.33786398e-01, -1.54193959e-01],
              [-3.05414996e-17,  8.40072236e-04,  5.57884080e-04],
              [ 3.01859411e-17,  8.60411436e-03,  6.15745204e-04]],
             [[ 6.80309533e-04, -1.23162742e-03,  1.08729813e-03],
              [-1.53863193e-01, -1.38700657e-02, -1.36243323e-01],
              [-1.54665054e-01, -1.89018172e-02, -1.56415558e-01]]])
      """
      variances = []
  
      # diagonal variances
      for i, o in enumerate(obs):
          eigvals = cast(o.eigvals(), dtype=float64)
          w = (
              o.wires.labels
              if wires is None
              else wires.indices(o.wires)
          )
          p = marginal_prob(prob, w)
  
          res = dot(eigvals**2, p) - (dot(eigvals, p)) ** 2
          variances.append(res)
  
      cov = diag(variances)
  
      if diag_approx:
          return cov
  
      for i, j in itertools.combinations(range(len(obs)), r=2):
          o1 = obs[i]
          o2 = obs[j]
  
          o1wires = (
              o1.wires.labels
              if wires is None
              else wires.indices(o1.wires)
          )
          o2wires = (
              o2.wires.labels
              if wires is None
              else wires.indices(o2.wires)
          )
          shared_wires = set(o1wires + o2wires)
  
          l1 = cast(o1.eigvals(), dtype=float64)
          l2 = cast(o2.eigvals(), dtype=float64)
          l12 = cast(np.kron(l1, l2), dtype=float64)
  
          p1 = marginal_prob(prob, o1wires)
          p2 = marginal_prob(prob, o2wires)
          p12 = marginal_prob(prob, shared_wires)
  
          res = dot(l12, p12) - dot(l1, p1) * dot(l2, p2)
  
          cov = scatter_element_add(cov, [i, j], res)
          cov = scatter_element_add(cov, [j, i], res)
  
      return cov
  
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 1 for calling a global function with example usage: cov_matrix(prob, obs, wires, diag_approx) and returns: cov, cov