Quantum theory allows for absolute maximal contextuality

Contextuality is a fundamental feature of quantum theory and a necessary resource for quantum computation and communication. It is therefore important to investigate how large contextuality can be in quantum theory. Linear contextuality witnesses can be expressed as a sum $S$ of $n$ probabilities, and the independence number $\alpha$ and the Tsirelson-like number $\vartheta$ of the corresponding exclusivity graph are, respectively, the maximum of $S$ for noncontextual theories and for the theory under consideration. A theory allows for absolute maximal contextuality if it has scenarios in which $\vartheta /\alpha$ approaches $n$. Here we show that quantum theory allows for absolute maximal contextuality despite what is suggested by the examination of the quantum violations of Bell and noncontextuality inequalities considered in the past. Our proof is not constructive and does not single out explicit scenarios. Nevertheless, we identify scenarios in which quantum theory allows for almost-absolute-maximal contextuality.

Figure 1

A physical system in state $\rho$ enters the box. The player presses a button $i$ (with $i=1,...,5$) and a light ($r=0,1$) flashes. The player can also press a second button $j$ such that $i$ and $j$ are adjacent in the graph drawn at the top of the box. Two important properties can be checked: (i) adjacent buttons correspond to tests without mutual disturbance, i.e., $P(i=r|ij)=P(i=r\left|ik\right)$ for any $j,k$ adjacent to $i$; (ii) exclusiveness relation: no adjacent vertices can “happen” together, i.e., $P(i=1,j=1|ij)=0$, for all $i,j$ adjacent.