Why people believe this
A bit is a bit — measuring in different ways should give the same information. Classical information does not depend on how you read it.
The correction
In quantum mechanics, the choice of measurement basis fundamentally changes what information you extract. A qubit in state |+> = (|0> + |1>)/sqrt(2) gives completely random results in the computational basis, but gives a definite result in the Hadamard basis. Quantum information is basis-dependent. This is why quantum key distribution works — measuring in the wrong basis destroys the information rather than revealing it.
Try it in the simulator
What to do
Place H on q0, then Z, then H. Run it — you get |1> with 100% probability. Now remove the middle Z and run again — you get |0>. Same initial state, same final measurement, completely different result because of what happened in between. The basis transformation changed everything.
Research notes
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