diff --git a/qiskit/circuit/annotated_operation.py b/qiskit/circuit/annotated_operation.py
index 6006e68f58d..c1093d8fc5b 100644
--- a/qiskit/circuit/annotated_operation.py
+++ b/qiskit/circuit/annotated_operation.py
@@ -133,7 +133,7 @@ def __eq__(self, other) -> bool:
 
     def copy(self) -> "AnnotatedOperation":
         """Return a copy of the :class:`~.AnnotatedOperation`."""
-        return AnnotatedOperation(base_op=self.base_op, modifiers=self.modifiers.copy())
+        return AnnotatedOperation(base_op=self.base_op.copy(), modifiers=self.modifiers.copy())
 
     def to_matrix(self):
         """Return a matrix representation (allowing to construct Operator)."""
diff --git a/qiskit/primitives/backend_estimator.py b/qiskit/primitives/backend_estimator.py
index f6a53c02ea9..722ee900cee 100644
--- a/qiskit/primitives/backend_estimator.py
+++ b/qiskit/primitives/backend_estimator.py
@@ -36,7 +36,7 @@
 )
 from qiskit.utils.deprecation import deprecate_func
 
-from .base import BaseEstimatorV1, EstimatorResult
+from .base import BaseEstimator, EstimatorResult
 from .primitive_job import PrimitiveJob
 from .utils import _circuit_key, _observable_key, init_observable
 
@@ -88,17 +88,18 @@ def _prepare_counts(results: list[Result]):
     return counts
 
 
-class BackendEstimator(BaseEstimatorV1[PrimitiveJob[EstimatorResult]]):
+class BackendEstimator(BaseEstimator[PrimitiveJob[EstimatorResult]]):
     """Evaluates expectation value using Pauli rotation gates.
 
     The :class:`~.BackendEstimator` class is a generic implementation of the
-    :class:`~.BaseEstimatorV1` interface that is used to wrap a :class:`~.BackendV2`
-    (or :class:`~.BackendV1`) object in the :class:`~.BaseEstimatorV1` API. It
+    :class:`~.BaseEstimator` (V1) interface that is used to wrap a :class:`~.BackendV2`
+    (or :class:`~.BackendV1`) object in the :class:`~.BaseEstimator` V1 API. It
     facilitates using backends that do not provide a native
-    :class:`~.BaseEstimatorV1` implementation in places that work with
-    :class:`~.BaseEstimatorV1`.
+    :class:`~.BaseEstimator` V1 implementation in places that work with
+    :class:`~.BaseEstimator` V1.
     However, if you're using a provider that has a native implementation of
-    :class:`~.BaseEstimatorV1` or :class:`~.BaseEstimatorV2`, it is a better
+    :class:`~.BaseEstimatorV1` ( :class:`~.BaseEstimator`) or
+    :class:`~.BaseEstimatorV2`, it is a better
     choice to leverage that native implementation as it will likely include
     additional optimizations and be a more efficient implementation.
     The generic nature of this class precludes doing any provider- or
diff --git a/qiskit/primitives/backend_sampler.py b/qiskit/primitives/backend_sampler.py
index 213b9701a55..905fa237154 100644
--- a/qiskit/primitives/backend_sampler.py
+++ b/qiskit/primitives/backend_sampler.py
@@ -26,22 +26,23 @@
 from qiskit.utils.deprecation import deprecate_func
 
 from .backend_estimator import _prepare_counts, _run_circuits
-from .base import BaseSamplerV1, SamplerResult
+from .base import BaseSampler, SamplerResult
 from .primitive_job import PrimitiveJob
 from .utils import _circuit_key
 
 
-class BackendSampler(BaseSamplerV1[PrimitiveJob[SamplerResult]]):
-    """A :class:`~.BaseSamplerV1` implementation that provides a wrapper for
+class BackendSampler(BaseSampler[PrimitiveJob[SamplerResult]]):
+    """A :class:`~.BaseSampler` (V1) implementation that provides a wrapper for
     leveraging the Sampler V1 interface from any backend.
 
     This class provides a sampler interface from any backend and doesn't do
     any measurement mitigation, it just computes the probability distribution
     from the counts. It facilitates using backends that do not provide a
-    native :class:`~.BaseSamplerV1` implementation in places that work with
-    :class:`~.BaseSamplerV1`.
+    native :class:`~.BaseSampler` V1 implementation in places that work with
+    :class:`~.BaseSampler` V1.
     However, if you're using a provider that has a native implementation of
-    :class:`~.BaseSamplerV1` or :class:`~.BaseESamplerV2`, it is a better
+    :class:`~.BaseSamplerV1` ( :class:`~.BaseSampler`) or
+    :class:`~.BaseESamplerV2`, it is a better
     choice to leverage that native implementation as it will likely include
     additional optimizations and be a more efficient implementation.
     The generic nature of this class precludes doing any provider- or
diff --git a/qiskit/primitives/estimator.py b/qiskit/primitives/estimator.py
index 1ca1529852c..6ae50ee7f0f 100644
--- a/qiskit/primitives/estimator.py
+++ b/qiskit/primitives/estimator.py
@@ -26,7 +26,7 @@
 from qiskit.quantum_info.operators.base_operator import BaseOperator
 from qiskit.utils.deprecation import deprecate_func
 
-from .base import BaseEstimatorV1, EstimatorResult
+from .base import BaseEstimator, EstimatorResult
 from .primitive_job import PrimitiveJob
 from .utils import (
     _circuit_key,
@@ -36,9 +36,9 @@
 )
 
 
-class Estimator(BaseEstimatorV1[PrimitiveJob[EstimatorResult]]):
+class Estimator(BaseEstimator[PrimitiveJob[EstimatorResult]]):
     """
-    Reference implementation of :class:`BaseEstimatorV1`.
+    Reference implementation of :class:`BaseEstimator` (V1).
 
     :Run Options:
 
diff --git a/qiskit/primitives/sampler.py b/qiskit/primitives/sampler.py
index d93db4f0411..9155e50ad2c 100644
--- a/qiskit/primitives/sampler.py
+++ b/qiskit/primitives/sampler.py
@@ -26,7 +26,7 @@
 from qiskit.result import QuasiDistribution
 from qiskit.utils.deprecation import deprecate_func
 
-from .base import BaseSamplerV1, SamplerResult
+from .base import BaseSampler, SamplerResult
 from .primitive_job import PrimitiveJob
 from .utils import (
     _circuit_key,
@@ -36,11 +36,11 @@
 )
 
 
-class Sampler(BaseSamplerV1[PrimitiveJob[SamplerResult]]):
+class Sampler(BaseSampler[PrimitiveJob[SamplerResult]]):
     """
     Sampler V1 class.
 
-    :class:`~Sampler` is a reference implementation of :class:`~BaseSamplerV1`.
+    :class:`~Sampler` is a reference implementation of :class:`~BaseSampler` (V1).
 
     :Run Options:
 
diff --git a/test/python/circuit/test_annotated_operation.py b/test/python/circuit/test_annotated_operation.py
index e2ca9f4af9a..375c19fcf3a 100644
--- a/test/python/circuit/test_annotated_operation.py
+++ b/test/python/circuit/test_annotated_operation.py
@@ -204,6 +204,17 @@ def test_invalid_params_access(self):
             with self.assertRaises(AttributeError):
                 _ = annotated.validate_parameter(1.2)
 
+    def test_invariant_under_assign(self):
+        """Test the annotated operation is not changed by assigning."""
+        p = Parameter("p")
+        annotated = RXGate(p).control(2, annotated=True)
+        circuit = QuantumCircuit(annotated.num_qubits)
+        circuit.append(annotated, circuit.qubits)
+        bound = circuit.assign_parameters([1.23])
+
+        self.assertEqual(list(circuit.parameters), [p])
+        self.assertEqual(len(bound.parameters), 0)
+
 
 if __name__ == "__main__":
     unittest.main()