P. K. Aravind

Proofs of the Kochen–Specker theorem based on a system of three qubits

A number of new proofs of the Kochen–Specker theorem are given based on the observables of the three-qubit Pauli group. Each proof is presented in the form of a diagram from which it is obvious by inspection. Each of our observable-based proofs leads to a system of projectors and bases that generally yields a large number of ‘parity proofs’ of the Kochen–Specker theorem. Some examples of such proofs are given and some of their applications are discussed.

Quantum mysteries revisited again

This paper describes a device, consisting of a central source and two widely separated detectors with six switch settings each, that provides a simple gedanken demonstration of the nonclassical correlations that are the subject of Bell’s theorem without relying on either statistical effects or the occurrence of rare events. The mechanism underlying the operation of the device is revealed for readers with a knowledge of quantum mechanics.

Bell’s Theorem Without Inequalities and Only Two Distant Observers

A proof of Bell’s theorem without inequalities and involving only two observers is given by suitably extending a proof of the Bell-Kochen-Specker theorem due to Mermin. This proof is generalized to obtain an inequality-free proof of Bell’s theorem for a set of n Bell states (with n odd) shared between two distant observers. A generalized CHSH inequality is formulated for n Bell states shared symmetrically between two observers and it is shown that quantum mechanics violates this inequality by an amount that grows exponentially with increasing n.

Parity proofs of the Kochen?Specker theorem based on 60 complex rays in four dimensions

It is pointed out that the 60 complex rays in four dimensions associated with a system of two qubits yield over 109 critical parity proofs of the Kochen–Specker theorem. The geometrical properties of the rays are described, an overview of the parity proofs contained in them is given and examples of some of the proofs are exhibited.

Solution to the King's Problem in prime power dimensions

It is shown how to ascertain the values of a complete set of mutually complementary observables of a prime power degree of freedom by generalizing the solution in prime dimensions given by Englert and Aharonov [Phys. Lett. A284, 1-5 (2001)].It is shown how to ascertain the values of a complete set of mutually complementary observables of a prime power degree of freedom by generalizing the solution in prime dimensions given by Englert and Aharonov [Phys. Lett. A284, 1 - 5 (2001)].

IMPOSSIBLE COLORINGS AND BELL'S THEOREM

Abstract An argument due to Zimba and Penrose is generalized to show how all known non-coloring proofs of the Bell–Kochen–Specker (BKS) theorem can be converted into inequality-free proofs of Bells nonlocality theorem. A compilation of many such inequality-free proofs is given.

HOW REYE'S CONFIGURATION HELPS IN PROVING THE BELL–KOCHEN–SPECKER THEOREM: A CURIOUS GEOMETRICAL TALE

It is shown that the 24 quantum states or “rays” used by Peres (J. Phys. A24, 174-8 (1991)) to give a proof of the Bell–Kochen–Specker (BKS) theorem have a close connection with Reyes configuration, a system of twelve points and sixteen lines known to projective geometers for over a century. The interest of this observation stems from the fact that it provides a ready explanation for many of the regularities exhibited by the Peres rays and also permits a systematic construction of all possible non-coloring proofs of the BKS theorem based on these rays. An elementary exposition of the connection between the Peres rays and Reyes configuration is given, following which its applications to the BKS theorem are discussed.

Parity Proofs of the Bell-Kochen-Specker Theorem Based on the 600-cell

It is shown how the 300 rays associated with the antipodal pairs of vertices of a 120-cell (a four-dimensional regular polytope) can be used to give numerous “parity proofs” of the Kochen–Specker theorem ruling out the existence of noncontextual hidden variables theories. The symmetries of the 120-cell are exploited to give a simple construction of its Kochen–Specker diagram, which is exhibited in the form of a “basis table” showing all the orthogonalities between its rays. The basis table consists of 675 bases (a basis being a set of four mutually orthogonal rays), but all the bases can be written down from the few listed in this paper using some simple rules. The basis table is shown to contain a wide variety of parity proofs, ranging from 19 bases (or contexts) at the low end to 41 bases at the high end. Some explicit examples of these proofs are given, and their implications are discussed.

Parity Proofs of the Kochen-Specker Theorem Based on the 24 Rays of Peres

A diagrammatic representation is given of the 24 rays of Peres that makes it easy to pick out all the 512 parity proofs of the Kochen-Specker theorem contained in them. The origin of this representation in the four-dimensional geometry of the rays is pointed out.

Proofs of the Kochen-Specker theorem based on the N-qubit Pauli group

We present a number of observables-based proofs of the Kochen- Specker (KS) theorem based on the N-qubit Pauli group for N � 4, thus adding to the proofs that have been presented earlier for the 2- and 3-qubit groups. These proofs have the attractive feature that they can be presented in the form of diagrams from which they are obvious by inspection. They are also irreducible in the sense that they cannot be reduced to smaller proofs by ignoring some subset of qubits and/or observables in them. A simple algorithm is given for transforming any observables-based KS proof into a large number of projectors-based KS proofs; if the observables-based proof has O observ- ables, with each observable occurring in exactly two commuting sets and any two commuting sets having at most one observable in common, the number of associated projectors-based parity proofs is 2 O. We introduce symbols for the observables- and projectors-based KS proofs that capture their important features and also convey a feeling for the enormous variety of both these types of proofs within the N-qubit Pauli group. We discuss an infinite family of observables-based proofs whose members include all numbers of qubits from two up, and show how it can be used to generate projectors-based KS proofs involving only nine bases (or experimental contexts) in any dimension of the form 2 N for N � 2. Some implications of our results are discussed.