How contextuality and antidistinguishability are related

Abstract

Contextuality is a key characteristic that separates quantum from classical phenomena and an important tool in understanding the potential advantage of quantum computation. However, when assessing the quantum resources available for quantum information processing, there is no formalism to determine whether a set of states can exhibit contextuality and whether such proofs of contextuality indicate anything about the resourcefulness of that set. Introducing a well-motivated notion of what it means for a set of states to be contextual, we establish a relationship between contextuality and antidistinguishability of sets of states. We go beyond the traditional notions of contextuality and antidistinguishability and treat both properties as resources, demonstrating that the degree of contextuality within a set of states has a direct connection to its level of antidistinguishability. If a set of states is contextual, then it must be weakly antidistinguishable and vice-versa. However, critical contextuality emerges as a stronger property than traditional antidistinguishability.

Figures (2)

• Figure 1: A contextual scenario with 18 vectors and 9 contexts in dimension $d=4$Cabello_1996. The vertices are projectors (pure states) where $(a,b,c,d)$ is identified with the projector $\ket{a,b,c,d}\bra{a,b,c,d}$ where $\ket{a,b,c,d}=a\ket{00}+b\ket{01}+c\ket{10}+d\ket{11}$. The hyperedges form contexts of mutually orthogonal sets, where for simplicity the hyperedges are illustrated as straight lines and ovals. The scenario is generated by projectors indicated by orange triangles, along with its SA and A context given by the orange, dotted line. The projectors indicated by blue squares do not generate the entire graph -- all but the $(1,0,0,0)$ state -- and therefore demonstrate only a contextual instance with the associated (strongly) antidistinguishing context given by the blue, dash-dotted line.
• Figure 2: (a) The Lisoněk state-independent proof of contextuality PhysRevA.89.042101 with 21 projectors and 7 contexts in dimension $d=6$ is the smallest KS set in terms of contexts. The contexts here are given by straight lines and the labelling $10ab10$ denotes the projector associated with the vector $(1,0,a,b,1,0)$, where $a=e^{2\pi i/3}$ and $b=e^{4\pi i/3}$. The three projectors indicated by orange triangles generate the scenario with the associated antidistinguishing context given by the orange, dotted line. (b) Yu and Oh's set 10.1103/physrevlett.108.030402 is the simplest state independent proof of contextuality with the smallest number of projectors in dimension $d=3$. See section \ref{['sec:yu_oh']} specifying the projectors. Contexts in this figure are illustrated as circles. The orange projectors generate a contextual instance with the SA measurement given by the orange, dotted line. However, the scenario is noncontextual by our definition of contextuality (Def. 3) as it admits a consistent value assignment highlighted by the blue squares.

Theorems & Definitions (19)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Theorem 1
• proof
• Definition 7
• Lemma 1