I have asked two similar questions 1, 2 but have not received a definite answer, so here I have narrowed the issue down to one hopefully more easily addressed aspect of the big question. Unfortunately I do not have a quantum computation background, and don't yet understand the usual notation used in the field. A link to an article that provides an answer that I can gradually decipher would be sufficient.
Given an elementary entangled state of N qubits, is there a quantum operation that produces the complement of the entangled state?
By "elementary entangled state of N qubits", I mean a state in which all of the N qubits are entangled with each other such that measuring the value of any one qbit will determine the values of all the other qubits.
By "complement", I mean a superposition of all states that are orthogonal to the elementary entangled state, where the "weights" of all the independent components of the superposition are the same.
The closest I've been able to come on my own to answering the question is to see that the hypothetical operation would not lose any information and would be reversible. And, it seems that simply adding an ancillary bit to identify a state as the complement of another state would almost be good enough (except that the components of that state wouldn't be very easily accessible for downstream operations).