Does Ignorance of the Whole Imply Ignorance of the Parts?

PRL 107, 030402 (2011)
PHYSICAL    REVIEW    LETTERS
week ending 15 JULY 2011
Does Ignorance of the Whole Imply Ignorance of the Parts? Large Violations of Noncontextuality in Quantum Theory
Thomas Vidick1 and Stephanie Wehner2
1Computer Science Division, University of California at Berkeley, Berkeley, California 94720, USA* 2Center for Quantum Technologies, National University of Singapore, 2 Science Drive 3, Singapore 117543 Singapore† (Received 19 December 2010; published 13 July 2011)
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 noncontextual (NC) hidden-variable model (HV). Curiously, however, such a conjecture is 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 arbitrarily violated by quantum mechanics.