arXiv:1205.5015 · Journal of Physics A Mathematical and Theoretical

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

Mordecai Waegell, P. K. Aravind

Published Sep 18, 2012• 44 citations

Abstract

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.

DOI: 10.1088/1751-8113/45/40/405301

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arXiv:1205.5015 · Journal of Physics A Mathematical and Theoretical

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

Mordecai Waegell, P. K. Aravind

Published Sep 18, 2012• 44 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1751-8113/45/40/405301

Related papers

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Aug 20, 20120 citations

We present a number of observables-based proofs of the Kochen-Specker (KS) theorem based on the

N

-qubit Pauli group for $N\ensuremath{\ge}4$, 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

observables, 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 2 up, and show how it can be used to generate projectors-based KS proofs involving only 9 bases (or experimental contexts) in any dimension of the form

{2}^{N}

for $N\ensuremath{\ge}2$. Some implications of our results are discussed.

We investigate multiple qubit Pauli groups and the quantum states/rays arising from their maximal bases. Remarkably, the real rays are carried by a Barnes-Wall lattice

BW_n

(

n=2^m

). We focus on the smallest subsets of rays allowing a state proof of the Bell-Kochen-Specker theorem (BKS). BKS theorem rules out realistic non-contextual theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases. We investigate the geometrical structure of small BKS-proofs

v-l

involving

v

rays and

l

2n

-dimensional bases of

n

-qubits. Specifically, we look at the classes of parity proofs 18-9 with two qubits (A. Cabello, 1996), 36-11 with three qubits (M. Kernaghan & A. Peres, 1995) and related classes. One finds characteristic signatures of the distances among the bases, that carry various symmetries in their graphs.

We discuss two approaches to producing generalized parity proofs of the Kochen-Specker theorem. Such proofs use contexts of observables whose product is

I

-I

; we call them constraints. In the first approach, one starts with a fixed set of constraints and methods of linear algebra are used to produce subsets that are generalized parity proofs. Coding theory methods are used for enumeration of the proofs by size. In the second approach, one starts with the combinatorial structure of the set of constraints and one looks for ways to suitably populate this structure with observables. As well, we are able to show that many combinatorial structures can not produce parity proofs.

We investigate multiple qubit Pauli groups and the quantum states/rays arising from their maximal bases. Remarkably, the real rays are carried by a Barnes-Wall lattice

BW_n

(

n=2^m

). We focus on the smallest subsets of rays allowing a state proof of the Bell-Kochen-Specker theorem (BKS). BKS theorem rules out realistic {\it non-contextual} theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases. We investigate the geometrical structure of small BKS-proofs

v-l

involving

v

rays and

l

2n

-dimensional bases of

n

-qubits. Specifically, we look at the classes of parity proofs

18-9

with two qubits (A. Cabello, 1996),

36-11

with three qubits (M. Kernaghan \& A. Peres, 1995) and related classes. One finds characteristic signatures of the distances among the bases, that carry various symmetries in their graphs.