How typical is contextuality?

TL;DR

This work analyzes how commonly contextuality arises under generalized noncontextuality by testing randomly sampled prepare-and-measure scenarios with a simplex-embedding linear program. By defining typicality $t_{(n,m,d;N)}$ as the fraction of trials with a positive robustness $r$, the authors show contextuality is highly typical, often approaching 1 as the number of states $n$ or measurements $m$ increases, including in the presence of realistic noise. They map how purity and measurement sharpness affect typicality and demonstrate that high typicality does not automatically translate to large quantum advantage, using parity-oblivious multiplexing as a case study. An open-source toolbox accompanies the study, enabling practitioners to predict typicality under experimental constraints and to guide the design of contextuality-based experiments and protocols.

Abstract

Identifying when observed statistics cannot be explained by any reasonable classical model is a central problem in quantum foundations. A principled and universally applicable approach to defining and identifying nonclassicality is given by the notion of generalized noncontextuality. Here, we study the typicality of contextuality -- namely, the likelihood that randomly chosen quantum preparations and measurements produce nonclassical statistics. Using numerical linear programs to test for the existence of a generalized-noncontextual model, we find that contextuality is fairly common: even in experiments with only a modest number of random preparations and measurements, contextuality arises with probability over 99%. We also show that while typicality of contextuality decreases as the purity (sharpness) of the preparations (measurements) decreases, this dependence is not especially pronounced, so contextuality is fairly typical even in settings with realistic noise. Finally, we show that although nonzero contextuality is quite typical, quantitatively high degrees of contextuality are not as typical, and so large quantum advantages (like for parity-oblivious multiplexing, which we take as a case study) are not as typical. We provide an open-source toolbox that outputs the typicality of contextuality as a function of tunable parameters (such as lower and upper bounds on purity and other constraints on states and measurements). This toolbox can inform the design of experiments that achieve the desired typicality of contextuality for specified experimental constraints.

Figures (9)

• Figure 1: Visualization of the effects $\ket{j,k}$ from Eq. \ref{['90effects']} (blue dots) and their antipodal counterparts (purple dots) for $k=0$, showing only the ZX-hemisphere of the Bloch sphere, for clarity. The full set of effects is obtained by additionally rotating this polygon 4 times in steps of $\frac{\pi}{5}$ around the Z axis of the Bloch sphere.
• Figure 2: Typicality of contextuality for different numbers of preparations and measurements for randomly-sampled (a) pure states and projective measurements, and (b) mixed states and POVMs.
• Figure 3: Typicality of contextuality for 92 dichotomic projective measurements over a qubit (that approximate the set of all projective effects), parametrized by Eq. \ref{['90effects']}, and $n$ random (pure in violet, mixed in blue) states.
• Figure 4: Examples of $n=5$ (a) pure states and (b) mixed states and their convex hulls (violet), measured by the same 92 projective measurements provided by Eq. \ref{['90effects']} and their convex hull (blue, excluding the null and the unit effects).
• Figure 5: Minimal number of states $n$ needed to observe typicality $t_{(n,m=20,d=2;N=10^6)} >99\%$ for $m=20$ measurements as a function of the lower bound on purity of the states with (a) sharp measurements and (b) unsharp measurements, for two different upper bounds on their purity. In (b), the bounds on sharpness are the same as the purity of the states.

Theorems & Definitions (9)

• Lemma 1: Typicality zero for scenarios $\mathbf{(n<d^2,m,d)}$
• proof
• Proposition 1
• Proposition 1
• Corollary 2
• Proposition 2
• proof
• Proposition 2
• proof