How to build deck steps down to patio?
As long as the steps are not attached to anything and the patio has a good mortar base and a packed gravel underbed then it should support the steps.

During the winter ground moves up and down due to freezing/thawing, so the steps cannot be attached to anything, because the whole patio will be moving up and down during the winter.

If the patio is old and has been been through several winters and shows no signs of cracking, then it is probably good to support stone steps. If it has cracks already, the steps will likely cause more cracking and settling. If it is a new patio, then it is hard to tell, because it depends on how well the patio was made and the soil.

Therefore, if it is a new patio, it would be better to use a separate foundation for the steps. If the steps are disconnected from the house, then you can use a gravel footing. Make sure it is drained with a pipe going to the storm runoff. If it is attached to the house, then it should have pillar footings that go below the frost line (about 4-5 feet deep).