Limitations on testing quantum theory

How much of quantum theory can be experimentally tested? We investigate this question in an extreme scenario, where the experimenter cannot make any assumptions on the quantum description of a device or state. Instead, she can only directly manipulate and interpret classical information. We prove that in this scenario, there is almost nothing that she can ascertain about the internal state of a given quantum system. More precisely, for any initial state of a multi-partite system, there exists a different internal state so that any experiment performed on the system will produce identical statistics under these different states. In particular, we cannot certify if two subsystems are entangled. Our result implies that asserting the identity of a quantum state requires assumptions that cannot be experimentally tested through classical information alone.