Impossibility Theorem for Extending Contextuality to Disturbing Systems

Recently there has been much interest and progress in extending the definition of contextuality to systems with disturbance. We prove that such an endeavor cannot simultaneously satisfy the following principles: (1) any deterministic system is noncontextual; (2) discarding information cannot turn a noncontextual system into a contextual one; (3) classical post-processing cannot create contextuality; (4) the joint realization of two statistically independent noncontextual systems is noncontextual. We also prove the same result without principle 4, under a stronger version of principle 1.