Positive phase space transformation incompatible with classical physics.

Area of Science:

• Quantum mechanics
• Quantum information theory
• Foundations of physics

Background:

• Bell's conjecture posits that a positive Wigner function precludes violations of local hidden variable theories.
• This conjecture typically assumes phase space measurements are performed.
• Local hidden variable theories represent a classical view of reality, incompatible with quantum mechanics.

Purpose of the Study:

• To investigate whether Bell's conjecture holds under broader conditions, specifically including operationally local transformations.
• To introduce and utilize the concept of operationally local transformations, independent of specific theoretical frameworks.
• To demonstrate a scenario where Bell's inequality is violated despite a positive Wigner function.

Main Methods:

• Introduction of "operationally local transformations" as changes to local measurement apparatus settings.
• Theoretical analysis of composite quantum systems with positive Wigner functions.
• Utilizing entangled ancillae to perform operationally local transformations that preserve Wigner function positivity.

Main Results:

• Demonstration that Bell's inequality can be violated even when parties perform only phase space measurements.
• Confirmation that operationally local transformations, preserving Wigner function positivity, can lead to violations.
• Entangled ancillae are shown to be crucial for realizing these transformative operations.

Conclusions:

• Bell's conjecture, in its strictest sense, may not hold universally when operationally local transformations are considered.
• The study highlights the subtle interplay between Wigner function positivity, measurement settings, and Bell inequality violations.
• Quantum correlations can be more profound than predicted by local hidden variable theories, even under specific operational constraints.