A SIMPLE NEW PROOF OF THE KOCHEN-SPECKER PARADOX

Jan-Åke Larsson

Peter Heywood, Michael L. G. Redhead

John L. Bell

Jan-Åke Larsson, Luigi Accardi, Guillaume Adenier +5 more

The Kochen‐Specker paradox has recently been subject to experimental interest, and in this situation the number of steps in the proof in question is important. The fewer number of steps there are in the proof, the more imperfections can be tolerated in the experimental setup. In the spin‐1 version of the Kochen‐Specker paradox, when the settings used are directions in three‐dimensional space, the proofs can be easily visualized and the steps can easily be counted. In particular, the original Kochen‐Specker paradox makes use of so‐called great‐circle descents. Here, we will examine such descents in detail and also some other versions of the proof for spin‐1 systems. We will see that, perhaps contrary to intuition, the proofs that use a small number of steps do not in general use only great‐circle descents, and examine the reasons for this and possible extensions. At least one new proof will also be presented for the spin‐1 case.

Ravishankar Ramanathan, Yuan Liu, Paweł Horodecki

Abstract It is of interest to study how contextual quantum mechanics is, in terms of the violation of Kochen Specker state-independent and state-dependent non-contextuality inequalities. We present state-independent non-contextuality inequalities with large violations, in particular, we exploit a connection between Kochen–Specker proofs and pseudo-telepathy games to show KS proofs in Hilbert spaces of dimension d ⩾ 217 with the ratio of quantum value to classical bias being O ( d / log d ) . We study the properties of this KS set and show applications of the large violation. It has been recently shown that Kochen–Specker proofs always consist of substructures of state-dependent contextuality proofs called 01-gadgets. We show a one-to-one connection between 01-gadgets in C d and Hardy paradoxes for the maximally entangled state in C d ⊗ C d . We use this connection to construct large violation 01-gadgets between arbitrary vectors in C d , as well as novel Hardy paradoxes for the maximally entangled state in C d ⊗ C d , and give applications of these constructions. As a technical result, we show that the minimum dimension of the faithful orthogonal representation of a graph in R d is not a graph monotone, a result that may be of independent interest.