This is a short example note. I would use this format for a small calculation, a useful physics idea, or a thought that is not yet a full paper.

Setup

Suppose a quantum state \(\rho_{ABC}\) is defined on three regions. A useful quantity is the conditional mutual information

\[ I(A:C|B)_\rho = S(AB)_\rho + S(BC)_\rho - S(B)_\rho - S(ABC)_\rho . \]

It measures how much correlation remains between \(A\) and \(C\) once the region \(B\) is known.

Simple observation

If \(I(A:C|B)_\rho = 0\), then the state has a quantum Markov structure. Informally, all correlations between the two outer regions can be mediated through the middle region. One way to remember this is:

\[ A \longleftrightarrow B \longleftrightarrow C . \]

This is the kind of small fact that can be useful to keep in a note.

How I might use this template

  • Write the main idea in the first paragraph.
  • Put the important equation near the top.
  • Add a short derivation or explanation.
  • End with references or links.

References

  • Hayden, Jozsa, Petz, and Winter, Structure of states which satisfy strong subadditivity of quantum entropy with equality.
  • Fawzi and Renner, Quantum conditional mutual information and approximate Markov chains.