# How does quantum contextuality relate to mutually commuting observables?

I am trying to get a better understanding of what is the idea behind quantum contextuality. Quoting from the wikipedia page (emphasis mine):

Quantum contextuality is a foundational concept in quantum mechanics stating that the outcome one observes in a measurement is dependent upon what other measurements one is trying to make. More formally, the measurement result of a quantum observable is dependent upon which other commuting observables are within the same measurement set.

I am a bit confused by the phrasing here. If I have two commuting observables, $$[A,B]=0$$, that means that they do not interfere with one another, or, quoting from the relevant Wikipedia page, that "the measurement of one observable has no effect on the result of measuring another observable in the set".

Is this just bad phrasing (or a typo) on Wikipedia's side, or am I missing something?

• – Sanchayan Dutta May 28 '19 at 19:01
• @SanchayanDutta ah, that makes sense. So they mean that one is considering a set of observables $\{A_k\}_k$, some of which can be commuting, but which is not a set of mutually commuting observables. – glS May 28 '19 at 19:01

It does sound like bad phrasing. The idea here is that our set of observables is not necessarily mutually (pairwise) commuting. If you have three observables $$A$$, $$B$$ and $$C$$, and $$[A,B]=0$$ and $$[B,C]=0$$, then you're right that the measurement of $$A$$ will have no effect on $$B$$ and the measurement of $$B$$ will have no effect on the measurement of $$C$$. But the measurement statistics of $$B$$ will still be context-dependent if you have a two observable set i.e. it'll depend on whether you have $$A$$ or $$C$$ in your set.
Now if you have all three, then there's some uncertainty in the joint measurement statistics. In other words, given well-defined marginal distributions $$P(a,b|A,B)$$ and $$P(b,c|B,C)$$, you cannot have a well-defined joint probability distribution $$P(a,b,c|A,B,C)$$ in this case (although interestingly, the uncertainty can be lower bounded in many cases).