Does ignorance of the whole imply ignorance of the parts? — Large violations of non-contextuality in quantum theory Thomas Vidick1 and Stephanie Wehner2 Computer Science division, UC Berkeley, USA∗ Center for Quantum Technologies, National University of Singapore, 2 Science Drive 3, 117543 Singapore† (Dated: May 5, 2011) 1
arXiv:1011.6448v2 [quant-ph] 4 May 2011
2
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 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. Our inequality has interesting implications for quantum cryptography.
In this note we examine the following seemingly innocent question: does one’s ignorance about the whole necessarily imply ignorance about at least one of its parts? Given just a moments thought, the initial reaction is generally to give a positive answer. Surely, if one cannot know the whole, then one should be able to point to an unknown part. Classically, and more generally for any deterministic non-contextual hidden variable model, our intuition turns out to be correct: ignorance about the whole does indeed imply the existence of a specific part which is unknown, so that one can point to the source of one’s ignorance. However, we will show that in a quantum world this intuition is flawed. THE PROBLEM
Let us first explain our problem more formally. Consider two dits y0 and y1 ∈ {0, . . . , d − 1}, where the string y = y0 y1 plays the role of the whole, and y0 , y1 are the individual parts. Let ρy denote an encoding of the string y into a classical or quantum state. In quantum theory, ρy is simply a density operator, and in a NC-HV model it is a preparation Py described by a probability distribution over hidden variables λ ∈ Λ. Let PY be a probability distribution over {0, . . . , d − 1}2 , and imagine that with probability PY (y) we are given the state ρy . The optimum probability of guessing y given its encoding ρy , which lies in a register E, can be written as X Pguess (Y |E) = max PY (y) p(y|M, Py ) , (1) {M}
y∈{0,...,d−1}2
where p(y|M, Py ) is the probability of obtaining outcome y when measuring the preparation Py with M,
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and the maximization is taken over all d2 -outcome measurements