Geometry of joint reality: Device-independent steering and operational completeness

Abstract

We look at what type of arguments can rule out the joint reality (or value definiteness) of two observables of a physical system, such as a qubit, and give several strong yet simple no-go results based on assumptions typically weaker than those considered previously. The first result uses simple geometry combined with a locality assumption to derive device-independent steering inequalities. These may also be regarded as "conditional" Bell inequalities, are simpler in principle to test than standard Bell inequalities, and for two-qubit systems are related to properties of the quantum steering ellipsoid. We also derive a Bell inequality from locality and a one-sided reality assumption, and demonstrate a close connection between device-independent steering and Bell nonlocality. Moreover, we obtain a no-go result without the use of locality or noncontextuality assumptions, based on similar geometry and an assumption that we call "operational completeness". The latter is related to, but strictly weaker than, preparation noncontextuality. All arguments are given for finite statistics, without requiring any assumption that joint relative frequencies converge to some (unobservable) joint probability distribution. We also generalise a recent strong result of Pusey, for preparation noncontextuality, to the scenarios of device-independent steering and operational completeness.

Device-independent verification of Einstein-Podolsky-Rosen steering

“…Moreover, the resource ecient method used in [44] could improve the success probability of the partial BSM, and the self-testing could be more noise robust by adopting other techniques [10]. Finally, it is interesting to follow an alternate framework [45] to verify quantum steering.…”

Experimentally ruling out joint reality based on operational completeness

“…Here, we address this question by experimentally demonstrating that an assumption of operational completeness is sufficient to rule out the joint reality of two incompatible observables, even for a single qubit. This assumption is strictly weaker than preparation noncontextuality, and requires that if two ensembles are operationally equivalent then the joint relative frequencies of observables with predetermined real values are the same (up to statistical errors) [29]. Using a single trapped 171 Yb + ion, we perform two device-independent (DI) no-go tests in the form of linear and nonlinear inequalities [29].…”

“…This assumption is strictly weaker than preparation noncontextuality, and requires that if two ensembles are operationally equivalent then the joint relative frequencies of observables with predetermined real values are the same (up to statistical errors) [29]. Using a single trapped 171 Yb + ion, we perform two device-independent (DI) no-go tests in the form of linear and nonlinear inequalities [29]. With the techniques of microwave operation and highly distinguishable state-dependent fluorescence detection, we achieve quantum gates with very high fidelity (99.97%) and low measurement error (2.08%).…”

“…These constraints on AB are illustrated as shaded regions in Fig. 1(a), for the choice c = 0 (which is optimal for orthogonal qubit observables [29]).…”

“…Further, if the mixtures ε and ε at the intersection point are operationally equivalent (e.g., via direct test, or because the ensembles ε ± , ε ± lie in an operational plane of A and B so that any two convex combinations with the same values of A and B are operationally equivalent [29]), operational completeness then implies the predetermined real values of A and B have equal relative frequencies up to statistical errors, i.e., N(α, β|ε)/N ≈ N(α, β|ε )/N. Thus, AB ε ≈ AB ε , which contradicts the strict inequality above.…”