Why does the formalism of QM represent reversible changes (eg the time evolution operator, quantum gates, etc) with unitary operators?
To put it another way, can it be shown that unitary transformations preserve entropy?
Answer
The fact that evolutions of quantum mechanics are unitary after finite periods of time can be proven from the Schrödinger equation, and hinges on the characterization of unitary operators as those linear operators which are norm-preserving.
Recall the Schrödinger equation: $$ \frac{\mathrm d}{\mathrm d t} |\psi\rangle \;=\; -i H |\psi\rangle \;,$$ where $H$ may or may not be time-dependent, but in any case has real eigenvalues, so that $H = H^\dagger$. As a result, the way in which $|\psi\rangle$ changes instantaneously with time is in such a way that its magnitude, as a vector in Hilbert space, does not increase. We can see this by simply computing: $$ \frac{\mathrm d}{\mathrm dt} \langle\psi|\psi\rangle \;=\; \left[\frac{\mathrm d}{\mathrm dt} \langle\psi|\right]|\psi\rangle + \langle\psi|\left[\frac{\mathrm d}{\mathrm dt}|\psi\rangle\right] \;=\; i\langle\psi|H^\dagger|\psi\rangle - i\langle\psi|H|\psi\rangle \;=\; 0.$$ because $H = H^\dagger$. Then at all times the state vector remains the same length. That is to say, the norm is preserved under finite-time evolution.
Because unitary operators are exactly those ones which preserve the norm, it follows that the finite-time evolution will be unitary.
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