Fix output of `DensityMatrix.partial_transpose` to match input dimensions (#10163)

* (fix) change output of partial_transpose to return DensityMatrix matching input dimensions

* (test) check output dims match input dims when using DensityMatrix.partial_transpose

* (docs) add bugfix reno

* lint changes

* Fix up release note

---------

Co-authored-by: Jake Lishman <jake.lishman@ibm.com>

qiskit/quantum_info/states/densitymatrix.py

@@ -832,4 +832,4 @@ class DensityMatrix(QuantumState, TolerancesMixin):
             lst[i], lst[i + n] = lst[i + n], lst[i]
         rho = np.transpose(arr, lst)
         rho = np.reshape(rho, self._op_shape.shape)
-        return DensityMatrix(rho)
+        return DensityMatrix(rho, dims=self.dims())

releasenotes/notes/fix-partial-transpose-output-dims-3082fcf4147055dc.yaml

@@ -0,0 +1,5 @@
+---
+fixes:
+  - |
+    Fixed the dimensions of the output density matrix from :meth:`.DensityMatrix.partial_transpose`
+    so they match the dimensions of the corresponding input density matrix.

test/python/quantum_info/states/test_densitymatrix.py

@@ -1219,6 +1219,15 @@ class TestDensityMatrix(QiskitTestCase):
             self.assertEqual(rho.partial_transpose([0]), DensityMatrix(rho1))
             self.assertEqual(rho.partial_transpose([1]), DensityMatrix(rho1))
 
+        with self.subTest(msg="dims(3,3)"):
+            mat = np.zeros((9, 9))
+            mat1 = np.zeros((9, 9))
+            mat[8, 0] = 1
+            mat1[0, 8] = 1
+            rho = DensityMatrix(mat, dims=(3, 3))
+            rho1 = DensityMatrix(mat1, dims=(3, 3))
+            self.assertEqual(rho.partial_transpose([0, 1]), rho1)
+
     def test_clip_probabilities(self):
         """Test probabilities are clipped to [0, 1]."""
         dm = DensityMatrix([[1.1, 0], [0, 0]])