# Why can time-translation invariant quantum operations never increase coherence between energy eigenspaces?

Set $$\hbar =1$$. Let $$U(t) = e^{-itH}$$ be evolution under a Hamiltonian $$H$$ (for convenience let's assume $$H$$ is not degenerate). A time-translation invariant quantum operation $$\mathcal{E}$$ is one that effectively commutes with evolution under $$H$$, i.e. for any state $$\rho$$,

$$\mathcal{E}(U(t) \rho U^\dagger(t)) = U(t)\mathcal{E}(\rho)U^\dagger(t).$$

In my lecture notes I came across the statement that “time-translation invariant operations cannot generate any coherence between energy eigenspaces”. I took this to mean that these operations do not increase the coherence of quantum states.

For a qubit state, I thought a good measure of its coherence would be the magnitude of either of its off-diagonal elements, e.g. $$|\rho_{0,1}|$$. So my question is: how can I prove that $$|\rho_{0,1}|$$ is never increased by $$\mathcal{E}$$ in the qubit case?

I can show that $$\mathcal{E}$$ cannot change a qubit state $$\rho$$ with $$|\rho_{0,1}| = 0$$ to a state $$\mathcal{E}(\rho)$$ with $$|\mathcal{E}(\rho)_{0,1}| \neq 0$$ (in this case $$\mathcal{E}(\rho)$$ commutes with $$H$$, and so must be a completely incoherent state). But I haven't been able to go further than this.

Do I have a bad definition of coherence? Or have I misinterpreted the statement from my lecture notes?