Does ignorance of the whole imply ignorance of the parts? - Large violations of non-contextuality in quantum theory

Abstract

A central question in our understanding of the physical world is how our knowledge of the whole relates to our knowledge of the individual parts. One aspect of this question is the following: to what extent does ignorance about a whole preclude knowledge of at least one of its parts? Relying purely on classical intuition, one would certainly be inclined to conjecture that a strong ignorance of the whole cannot come without significant ignorance of at least one of its parts. Indeed, we show that this reasoning holds in any non-contextual hidden variable model (NC-HV). Curiously, however, such a conjecture is \emph{false} in quantum theory: we provide an explicit example where a large ignorance about the whole can coexist with an almost perfect knowledge of each of its parts. More specifically, we provide a simple information-theoretic inequality satisfied in any NC-HV, but which can be \emph{arbitrarily} violated by quantum mechanics. Our inequality has interesting implications for quantum cryptography.

v2: Removed claim about most general NC-HV models, improved explanations

References in corpus (4)

• The operational meaning of min- and max-entropy
• A simple test for hidden variables in spin-1 system
• The uncertainty principle determines the non-locality of quantum mechanics
• Preparation contextuality powers parity-oblivious multiplexing

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• Large violations in Kochen Specker contextuality and their applications
• Multi-Faced Entanglement
• Twenty years of quantum contextuality at USTC
• Optimal counterfeiting attacks and generalizations for Wiesner's quantum money
• Quantum Measurements Are Noncontextual
• Convexity of noncontextual wirings and how they order the set of correlations