At first slaves were purely objects, the private property of their master who could use them as they wanted and kill them at will. As I told in a past answer, there are records of masters known for clearly abusing and killing their slaves but there was no repercussion other than on their reputation.

However this evolved slowly and progressively with time under the Empire, and the State imposed more and more protection and rights for the slaves, allowing them to go to court for instance. Finally it's under Antoninus Pius in the second half of the 2nd century that it became illegal for a master to kill his slaves without trial.

Corollary question. In class today we discussed that the punishment for a slave murdering their owner was the execution of every slave living in the household, including their families.

Is there any truth to this? Or is it an exaggeration?