Topological proofs of contextuality in quantum mechanics

Abstract

We provide a cohomological framework for contextuality of quantum mechanics that is suited to describing contextuality as a resource in measurement-based quantum computation. This framework applies to the parity proofs first discussed by Mermin, as well as a different type of contextuality proofs based on symmetry transformations. The topological arguments presented can be used in the state-dependent and the state-independent case.

Hidden Variable Model for Universal Quantum Computation with Magic States on Qubits

“…Note that the phase point operators of the form eq. ( 19) only exist when the local Hilbert space dimension d is odd since noncontextual value assignments on E exist only when d is odd [41]. The proof of this statement requires the following lemma.…”

“…The functions Φ and β have a cohomological interpretation elucidated in Ref. [41] (also see Ref. [42]).…”

Commutative d-torsion K-theory and its applications

“…Part (1) follows from the definition of the chain complex; see also Ref. 6. Part (2) follows from the observation that γG is the image of the identity map in…”

Cohomology and the Algebraic Structure of Contextuality in Measurement Based Quantum Computation

“…More recently Okay et al described an obstruction for contextuality in measurement based quantum computation (MBQC) that is based on group cohomology [15]. While the Čech cohomology obstruction is well defined for any set of quantum measurements, their obstruction exploits the algebraic structure of the Pauli measurements used in MBQC.…”