Maybe I’ll give it a try!

If we are talking about the distance between quantum states, we would refer to the fidelity.

So what would it be for the case of the set of unitary operators earlier? Would it describe a physical interpretation of what type of unitary channel? Or is it the distance between the initial state and unitary transformed state?

If the uncertainty is zero, the state corresponds to either spin-up or spin-down; this is a trivial case.

Say we have σ_z. The observable σ_z corresponds to the measurement of spin along the z-axis. A non-zero uncertainty in σ_z indicates that the state is not in a definite spin-up or spin-down eigenstate, but is instead in a superposition of both.

So if I have a state rho, and I made measurements, i.e. by using POVM. That particular measurement will produce a set of outcomes with certainty probability. The uncertainty of each of the outcomes is measured by using Shannon entropy. Hence the uncertainty of the results is classical.

Corrected paper: arxiv.org/abs/2506.19232

The consequences may not be immediate, but in five to ten years, the overreliance on LLMs could reshape the landscape of academic writing, especially in STEM, where originality, novelty, and rigor are required.

The dependency on LLMs could dull students’ skills in problem solving, critical thinking, and the ability to create accurate, precise, and compelling narratives in research.

I’m not advocating to put on stop to LLM use, but when students outsource their entire writing process, they miss the opportunity to develop critical thinking and articulate their own thoughts on the research and findings.

As a result, most submissions contained generic answers, with identical sentence structures and repetitive phrasing, and the worst part was the lack of depth on the assigned topics. It feels hollow.

This issue isn’t limited to research. While teaching a pre-university course to Gen Z students, I noticed that many of them relied heavily on LLMs to complete their assignments.

Another time, I caught the LLM misapplying properties of POVMs, and I pointed out how it was wrong, and it actually agreed with me.