A complete characterization of all-versus-nothing arguments for stabilizer states

Area of Science:

• Quantum Foundations
• Quantum Information Theory
• Quantum Computation

Background:

• All-versus-nothing (AvN) proofs are crucial for demonstrating quantum contextuality.
• Generalizing Mermin's construction, AvN proofs challenge local hidden variable theories.
• Understanding AvN arguments in stabilizer state systems is key to quantum foundations.

Purpose of the Study:

• To provide a general formulation and complete characterization of all-versus-nothing (AvN) arguments for stabilizer states.
• To reduce complex AvN arguments for n-qubit stabilizer states to a simpler, fundamental case.
• To develop a computational method for generating AvN arguments in quantum systems.

Main Methods:

• Developed a general formulation for all-versus-nothing (AvN) arguments.
• Utilized graph state theory to prove the 'AvN triple theorem', a combinatorial characterization.
• Established a reduction of n-qubit stabilizer state AvN arguments to a three-qubit Greenberger-Horne-Zeilinger state.

Main Results:

• A complete characterization of all-versus-nothing (AvN) arguments arising from stabilizer states is presented.
• Every AvN argument for an n-qubit stabilizer state is shown to be reducible to a three-qubit state (local Clifford-equivalent to the tripartite Greenberger-Horne-Zeilinger state).
• A computational method for generating AvN arguments on n-qubit stabilizer states is enabled by the AvN triple theorem.

Conclusions:

• The study significantly simplifies the understanding and generation of all-versus-nothing (AvN) arguments in quantum foundations.
• The findings provide new insights into the stabilizer formalism and its logical connections.
• This work contributes to the foundational questions within the 'Second quantum revolution'.