User:Aschimmo/sandbox/Quantum Contextuality Sandbox

Quantum Contextuality is a foundational concept in quantum mechanics stating that the outcome one gets 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.

Contextuality was first proposed in the Bell-Kochen-Specker theorem, which revealed that hidden variables and non-contextuality are incompatible with quantum mechanics. That is, the state of a quantum system cannot be described either deterministically or independent of the experimental setup. Since then, contextuality has developed under several mathematical frameworks, including the Sheaf Theoretic, Spekkens' operational contextuality, and the graph theoretic. The Sheaf Theoretic proposed by Samson Abramsky and Adam Brandenburger employs sheaf theory to generalize contextuality to all forms of measurement, not just measurements in quantum mechanics. Meanwhile, Spekkens defines and expands upon contextuality as it applies to quantum information and experimentation, and the graph theoretic explains contextuality using the mathematical formalism present in graph theory. There also exists contextuality by default, which has recently been applied in psychology and sociology to test for contextual effects in behavioral and social experiments.

Recently, quantum contextuality has been explored as a potential means for quantum computing. In 2013, Robert Raussendorf showed that in general, any Mermin-like, inequality-free proof for quantum contextuality (such as the Kochen-Specker theorem) can be turned into a measurement-based quantum computation. More formally, "under quite natural assumptions for multi-qubit systems, MBQCs (measurement based quantum computations) which compute a non-linear Boolean function with sufficiently high success probability are contextual." Then in 2014, Mark Howard, et. al. showed that applying contextuality to magic state distillation (MSD) provides a means for arriving at universal, fault-tolerant quantum computing. MSD is a process by which a single qubit's polarization is increased along one of several "magic" directions and its state is considered "magic" once its polarization reaches a certain level. In particular, the group proved that states are non-contextual if and only if they cannot be used as inputs for magic state distillation. This provides a selection criteria for proper input states needed in MSD, and by extension aids in the efficient creation of magic states that, when coupled with fault-tolerant operators, allows for universal quantum computing. In both cases, contextuality has a potential advantage over other quantum computational techniques since contextuality itself can be thought of as a theory of information that is built directly into quantum mechanics. Comparatively, more traditional quantum phenomena such as entanglement and interference use delicate physical states which are incredibly susceptible to noise and very difficult to manipulate experimentally.