In superdense coding, you can use one qubit to control the Hilbert space of two qubits and steer it into 4 mutually orthogonal states, so that measurement of both qubits together will not have a probabilistic outcome.
I think the idea is cool. I want to understand a more general question that arises from this result.
What if I have 100 qubits (or N qubits, if you like), entangled by a neutral party, and 99 are sent to my friend and 1 is sent to me. How much control do I have over the composite quantum state? How many mutually orthogonal states can one steer the global state into?
Thank you for your interest in this question. I can't seem to figure out who is right, so let's consider a slightly different question.
There are a total of 5 qubits. I have 2, and my friend has 3. Clearly the upper bound of classical bits I can send to my friend is 5. But is it possible to find 5 unitary operations I can perform on my two qubits to steer the state into 5 orthogonal states.
If you can answer that, then how does this generalize to N and M qubits?
Hopefully this comes closer to the real question I want to ask.