qinfo module deprecations and removals (#5911)

**Context:**

As part of our v0.39 deprecation cycle, `qinfo` will be deprecated. All
PRs for deprecating each function can be merged to here,

**Description of the Change:**

Deprecates `qinfo`. Specifically:

- [x] deprecate `qinfo.mutual_info`:
#5917
- [x] deprecate `qinfo.reduced_dm`:
#5915
- [x] deprecate `qinfo.purity`:
#5916
- [x] deprecate `qinfo.vn_entropy`:
#5912
- [x] deprecate `qinfo.vn_entanglement_entropy`
#5914
- [x] deprecate `qinfo.fidelity`:
#5915
- [x] deprecate `qinfo.relative_entropy`:
#5915
- [x] deprecate `qinfo.trace_distance`
- [x] remove `qinfo.classical_fisher`
- [x] remove `qinfo.quantum_fisher`

**Benefits:**

Desired UI and less redundancies.

**Possible Drawbacks:**

None

**Related GitHub Issues:**

[sc-67217] [sc-67216] [sc-66716] [sc-66715] [sc-66714] [sc-66713]
[sc-67664] [sc-67665] [sc-67663] [sc-67446]

---------

Co-authored-by: Mudit Pandey <mudit.pandey@xanadu.ai>
Co-authored-by: Astral Cai <astral.cai@xanadu.ai>
Co-authored-by: Ahmed Darwish <exclass9.24@gmail.com>
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Co-authored-by: ringo-but-quantum <github-ringo-but-quantum@xanadu.ai>
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Co-authored-by: Pietropaolo Frisoni <pietropfrisoni@gmail.com>

doc/code/qml_qinfo.rst

@@ -4,8 +4,12 @@ qml.qinfo
 Overview
 --------
 
-.. warning:: 
-    The `qinfo` module is deprecated and scheduled to be removed in v0.40.
+.. warning::
+
+    The ``qinfo`` module is deprecated and scheduled to be removed in v0.40. Most quantum information transforms
+    are available as measurement processes (see the :mod:`~pennylane.measurements` module for more details).
+    Additionally, the transforms are also available as standalone functions in the :mod:`~pennylane.math` and
+    :mod:`~pennylane.gradients` modules.
 
 This module provides a collection of methods to return quantum information quantities from :class:`~.QNode`
 returning :func:`~pennylane.state`.

doc/development/deprecations.rst

@@ -9,12 +9,18 @@ deprecations are listed below.
 Pending deprecations
 --------------------
 
+* The ``qml.qinfo`` module has been deprecated. Please see the respective functions in the ``qml.math`` and ``qml.measurements``
+  modules instead.
+
+  - Deprecated in v0.39
+  - Will be removed in v0.40
+
 * ``Device``, ``QubitDevice``, and ``QutritDevice`` will no longer be imported top level in v0.40.  They instead
   we be available as ``qml.devices.LegacyDevice``, ``qml.devices.QubitDevice``, and ``qml.devices.QutritDevice``
   respectively.
 
   - Deprecated top level access in v0.39
-  - Top level access removed in v0.40
+  - Top level access will be removed in v0.40
 
 * `QNode.gradient_fn` is deprecated. Please use `QNode.diff_method` instead. `QNode.get_gradient_fn` can also be used to
   process the diff method.
@@ -83,6 +89,12 @@ Other deprecations
 Completed deprecation cycles
 ----------------------------
 
+* The functions ``qml.qinfo.classical_fisher`` and ``qml.qinfo.quantum_fisher`` have been removed and migrated to the ``qml.gradients``
+  module. Therefore, ``qml.gradients.classical_fisher`` and ``qml.gradients.quantum_fisher`` should be used instead.
+
+  - Deprecated in v0.38
+  - Removed in v0.39
+
 * All of the legacy devices (any with the name ``default.qubit.{autograd,torch,tf,jax,legacy}``) are removed. Use ``default.qubit`` instead,
   as it supports backpropagation for the many backends the legacy devices support.
 

doc/releases/changelog-dev.md

@@ -101,6 +101,10 @@
 
 <h3>Deprecations 👋</h3>
 
+* The `qml.qinfo` module has been deprecated. Please see the respective functions in the `qml.math` and
+  `qml.measurements` modules instead.
+  [(#5911)](https://github.com/PennyLaneAI/pennylane/pull/5911)
+
 * The ``QubitStateVector`` template is deprecated.
    Instead, use ``StatePrep``.
    [(#6172)](https://github.com/PennyLaneAI/pennylane/pull/6172)

pennylane/measurements/mutual_info.py

@@ -74,7 +74,7 @@ def circuit_mutual(x):
         using the classical backpropagation differentiation method (``diff_method="backprop"``)
         with a compatible device and finite differences (``diff_method="finite-diff"``).
 
-    .. seealso:: :func:`~.vn_entropy`, :func:`pennylane.qinfo.transforms.mutual_info` and :func:`pennylane.math.mutual_info`
+    .. seealso:: :func:`~pennylane.vn_entropy`, :func:`pennylane.math.mutual_info`
     """
     wires0 = qml.wires.Wires(wires0)
     wires1 = qml.wires.Wires(wires1)

pennylane/measurements/purity.py

@@ -57,7 +57,7 @@ def circuit_purity(p):
     >>> circuit_purity(0.1)
     array(0.7048)
 
-    .. seealso:: :func:`pennylane.qinfo.transforms.purity` and :func:`pennylane.math.purity`
+    .. seealso:: :func:`pennylane.math.purity`
     """
     wires = Wires(wires)
     return PurityMP(wires=wires)

pennylane/measurements/vn_entropy.py

@@ -65,7 +65,7 @@ def circuit_entropy(x):
         using the classical backpropagation differentiation method (``diff_method="backprop"``)
         with a compatible device and finite differences (``diff_method="finite-diff"``).
 
-    .. seealso:: :func:`pennylane.qinfo.transforms.vn_entropy` and :func:`pennylane.math.vn_entropy`
+    .. seealso:: :func:`pennylane.math.vn_entropy`
     """
     wires = Wires(wires)
     return VnEntropyMP(wires=wires, log_base=log_base)