Pauli sentence bug (#5407)

**Context:**
Earlier `PauliSentence.to_mat(wire_order)` failed with exception when it
had `only empty PauliWords`.
This has been fixed under this PR.

**Description of the Change:**
Earlier when `PauliWord` didn't have wires the function `_pw_wires`
defaulted to `PauliSentence` wires. Now rather than `PauliSentence`
itself, it defaults to `wire_order`

**Benefits:**

**Possible Drawbacks:**

**Related GitHub Issues:**
Fixes #5354

---------

Co-authored-by: Korbinian Kottmann <43949391+Qottmann@users.noreply.github.com>
Co-authored-by: Christina Lee <christina@xanadu.ai>

doc/releases/changelog-dev.md

@@ -264,9 +264,13 @@
 * We no longer perform unwanted dtype promotion in the `pauli_rep` of `SProd` instances when using tensorflow.
   [(#5246)](https://github.com/PennyLaneAI/pennylane/pull/5246)
 
-* Fixed `TestQubitIntegration.test_counts` in `tests/interfaces/test_jax_qnode.py` to always produce counts for all outcomes.
+* Fixed `TestQubitIntegration.test_counts` in `tests/interfaces/test_jax_qnode.py` to always produce counts for all
+  outcomes.
   [(#5336)](https://github.com/PennyLaneAI/pennylane/pull/5336)
 
+* Fixed `PauliSentence.to_mat(wire_order)` to support identities with wires.
+  [(#5407)](https://github.com/PennyLaneAI/pennylane/pull/5407)
+
 <h3>Contributors ✍️</h3>
 
 This release contains contributions from (in alphabetical order):

pennylane/pauli/pauli_arithmetic.py

@@ -832,7 +832,16 @@ def _pw_wires(w: Iterable) -> Wires:
             the PauliWord is empty ({}), choose any arbitrary wire from the
             PauliSentence it is composed in.
             """
-            return w or Wires(self.wires[0]) if self.wires else self.wires
+            if w:
+                # PauliWord is not empty, so we can use its wires
+                return Wires(w)
+
+            if wire_order:
+                # PauliWord is empty, treat it as Identity operator on any wire
+                # Pick any arbitrary wire from wire_order
+                return Wires(wire_order[0])
+
+            return wire_order
 
         if len(self) == 0:
             n = len(wire_order) if wire_order is not None else 0

tests/pauli/test_pauli_arithmetic.py

@@ -23,7 +23,6 @@
 from pennylane import numpy as np
 from pennylane.pauli.pauli_arithmetic import PauliWord, PauliSentence, I, X, Y, Z
 
-
 matI = np.eye(2)
 matX = np.array([[0, 1], [1, 0]])
 matY = np.array([[0, -1j], [1j, 0]])
@@ -877,6 +876,11 @@ def test_to_mat_empty(self, ps, true_res):
         assert sparse.issparse(res_sparse)
         assert qml.math.allclose(res_sparse.todense(), true_res)
 
+    def test_empty_pauli_to_mat_with_wire_order(self):
+        """Test the to_mat method with an empty PauliSentence and PauliWord and an external wire order."""
+        actual = PauliSentence({PauliWord({}): 1.5}).to_mat([0, 1])
+        assert np.allclose(actual, 1.5 * np.eye(4))
+
     ps_wire_order = ((ps1, []), (ps1, [0, 1, 2, "a", "b"]), (ps3, [0, 1, "c"]))
 
     @pytest.mark.parametrize("ps, wire_order", ps_wire_order)