Generalized Greenberger–Horne–Zeilinger Arguments from Quantum Logical Analysis

Abstract

The Greenberger–Horne–Zeilinger (GHZ) argument against noncontextual local hidden variables is recast in quantum logical terms of fundamental propositions, states and probabilities. Unlike Kochen–Specker- and Hardy-like configurations, this operator based argument proceeds within four nonintertwining contexts. The nonclassical performance of the GHZ argument is due to the choice or filtering of observables with respect to a particular state. We study the varieties of GHZ games one could play in these four contexts, depending on the chosen state of the GHZ basis.

Generalized Householder Transformations

“…As with original forms of expectation or operator based arguments such as Greenberger-Horne-Zeilinger [10,11] or Householder-based state-independent contextuality [3] those arguments use complementary and thus counterfactual observables. Additionally, additivity arguments use violations of admissibility [5], in particular, exclusivity and completeness.…”

“…Further methods to obtain such configurations based on parity proofs are discussed by Waegell, Aravind, Megill, and Pavičić [7][8][9]. The Greenberger-Horne-Zeilinger operator theorem is based on a similar argument [10,11].…”

“…with k ∈ N, so that 0 ≤ k = 9 + n 4 ≤ 18, and n = −36 + 4k. (11) This results in n arithmetically progressing from −36 in steps of 4, that is k ∈ {0, 1, . .…”

Generalized Householder transformations

“…As with original forms of expectation or operator based arguments such as Greenberger-Horne-Zeilinger [10,11] or Householder-based state-independent contextuality [3] those arguments use complementary and thus counterfactual observables. Additionally, additivity arguments use violations of admissibility [5], in particular, exclusivity and completeness.…”

“…Further methods to obtain such configurations based on parity proofs are discussed by Waegell, Aravind, Megill, and Pavičić [7][8][9]. The Greenberger-Horne-Zeilinger operator theorem is based on a similar argument [10,11].…”