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I am new to quantum computing and saw this argument on this site but I don't understand it.

First of all, I don't understand what is exactly meant by 'reversible'. Because even if you had a unitary matrix operation that doesn't mean you would be able to tell what were the inputs. For example the XOR gate which is like CNOT but without the control output, is actually unitary but that doesn't mean I can reverse the process. I can sometimes somewhat say what were the inputs but can I always tell 0,1 from 1,0 as inputs?

Suppose now, you have a 4x4 unitary matrix acting on two qubits as a tensor product of two operations. How can you reverse the process? I am not seeing how just being 'unitary' helps you? Can you tell what came in? Another thing I don't understand: if all the point of control gates is because of unitary matrices have to be reversible, then again why would we need control in first place?

Granted that all of your operations are Unitarian then also compositions and multiplications would be Unitarian. Then why do you need a control to guarantee reversibility if you claim reversibility follows from the unitary property?

Sure I am missing something here. But what is it? What's the point of control gates? Everyone says it's for reversibility. Why we need reversibility? People say due to unitarity. Unitarity is needed to keep the normal size of the probability vector. But then again, if all of them are unitary they are already reversible (if they aren't then again why is reversibility needed) so why then use controlled gates?

If you are correct then, take the XOR gate which is unitary, provided you have an output can you say what were the inputs? – bilanush – 2019-02-07T23:38:43.810

If I may ask two related questions regarding controlled gates. 1) your answer implying that a universal system must include an entanglement? Is this a correct phrasing? Since you say controlled gates are for entanglement and it's a gate inside universal set. 2) you didn't address or explain so why we need the control output? I mean, I can understand the gate is entangling two qubits together. But whats the control output I thought it's for reversability, but you say it's reversable regardless. Thanks! – bilanush – 2019-02-07T23:54:32.503

@bilanush edited the answer, since i like to answer with good formatting! – Hemant – 2019-02-08T00:53:19.943

Your answer is perfect so I accepted it even though it became late- and last. Now, one question, the trick of adding output qubits and thus rendering the gates reversable can't be done on other classical gates besides xor? – bilanush – 2019-03-02T01:49:24.433