Indicate database migration in /api/core/state response (#122445)

* Indicate database migration in /api/core/state response

* Change API response according to review comment

* Adjust API response

* Update test

* Add test

tests/components/api/test_init.py

@@ -770,4 +770,43 @@ async def test_api_core_state(hass: HomeAssistant, mock_api_client: TestClient)
     resp = await mock_api_client.get("/api/core/state")
     assert resp.status == HTTPStatus.OK
     json = await resp.json()
-    assert json["state"] == "RUNNING"
+    assert json == {
+        "state": "RUNNING",
+        "recorder_state": {"migration_in_progress": False, "migration_is_live": False},
+    }
+
+
+@pytest.mark.parametrize(
+    ("migration_in_progress", "migration_is_live"),
+    [
+        (False, False),
+        (False, True),
+        (True, False),
+        (True, True),
+    ],
+)
+async def test_api_core_state_recorder_migrating(
+    hass: HomeAssistant,
+    mock_api_client: TestClient,
+    migration_in_progress: bool,
+    migration_is_live: bool,
+) -> None:
+    """Test getting core status."""
+    with (
+        patch(
+            "homeassistant.helpers.recorder.async_migration_in_progress",
+            return_value=migration_in_progress,
+        ),
+        patch(
+            "homeassistant.helpers.recorder.async_migration_is_live",
+            return_value=migration_is_live,
+        ),
+    ):
+        resp = await mock_api_client.get("/api/core/state")
+    assert resp.status == HTTPStatus.OK
+    json = await resp.json()
+    expected_recorder_state = {
+        "migration_in_progress": migration_in_progress,
+        "migration_is_live": migration_is_live,
+    }
+    assert json == {"state": "RUNNING", "recorder_state": expected_recorder_state}