Can you measure something about something without adding or removing energy from it?
This comment has got me wondering if there is a way to measure a property or a quantity of something without some kind of energy exchange or energy conversion. It asks me for an example, and I can't think of any right now.
So the answer here needs some, any, well though out example if the answer is yes.
Answer
Yes. The Aharonov-Bohm effect is an example where something can be measured without exchanging energy.
A long solenoid inside the two paths of an electron two-slit experiment induces a phase shift $\Delta \phi=q \Phi_B / \hbar$, even if there is no external electromagnetic field. The only thing that matters is that the integrated magnetic potential around the path is nonzero [1]. So here we can measure magnetic flux without adding or removing energy to it.
I think interaction-free measurements in quantum mechanics also fits [2]. The Elitzur–Vaidman bomb-testing problem is likely the most famous example [3]. Here measurements are using counterfactual interactions, where information is gained even when there is no energy exchange (and clever setups can make it arbitrarily unlikely to happen).
[1] https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect
[2] Elitzur, A. C., & Vaidman, L. (1993). Quantum mechanical interaction-free measurements. Foundations of Physics, 23(7), 987-997. https://arxiv.org/abs/hep-th/9305002
[3] https://en.wikipedia.org/wiki/Elitzur%E2%80%93Vaidman_bomb_tester
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