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arXiv:1209.5176 · WORLD SCIENTIFIC eBooks

QUANTUM STATES ARISING FROM THE PAULI GROUPS: SYMMETRIES AND PARADOXES

Michel R. P. Planat

Published Aug 14, 2013• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/9789814518550_0038

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QUANTUM STATES ARISING FROM THE PAULI GROUPS, SYMMETRIES AND PARADOXES

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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.

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Physics Letters A•Oct 15, 2010•3 citations
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 BWnBW_nBWn​ (n=2mn=2^mn=2m). 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−lv-lv−l involving vvv rays and lll 2n2n2n-dimensional bases of nnn-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 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 BWnBW_nBWn​ (n=2mn=2^mn=2m). 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−lv-lv−l involving vvv rays and lll 2n2n2n-dimensional bases of nnn-qubits. Specifically, we look at the classes of parity proofs 18−918-918−9 with two qubits (A. Cabello, 1996), 36−1136-1136−11 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 present a number of observables-based proofs of the Kochen-Specker (KS) theorem based on the NNN-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 OOO 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 2O{}^{O}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 NNN-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 2N{2}^{N}2N for $N\ensuremath{\ge}2$. Some implications of our results are discussed.
The 60 complex rays and 105 orthonormal bases arising from the two-qubit Pauli group form the Kochen-Specker system of Waegell and Aravind (2011), recently named by Aravind, Burton, Nunez Ponasso and Richter (2026) as a natural target for symmetry-based analysis. This note determines exactly the largest number of the 105 bases satisfiable by a single {0,1} ray assignment: the answer is 93, so at least twelve bases must fail. The optimum has clean structure: an attaining assignment selects exactly one state from each of the fifteen maximal abelian subgroups of the Pauli group, so all fifteen stabilizer bases are satisfied and all twelve failures are hybrid bases. The bound is certified by two independent integer-programming solvers (CBC proven optimum; HiGHS optimality and infeasibility of 94), and the attaining assignment is re-verified in exact Gaussian-integer arithmetic. The tight inequality is 93 versus the quantum value 105 (ratio 35/27 in the +/-1 normalization). This is the exact two-qubit analog of the value 65 of 75 determined by the author for the 600-cell system. Verification: run python3 verify2q.py with rays60c.txt in the same directory; it rebuilds the system from the fixed data and re-verifies the anchors (60 rays, 105 bases, seven per ray), the certificate count of 93, the transversal property, and the twelve failing bases in seconds, without any solver.
Structures that demonstrate nonclassicality are of foundational interest in quantum mechanics, and can also be seen as resources for numerous applications in quantum information processing - particularly in the Hilbert space of NNN qubits. The theory of entanglement, quantum contextuality, and quantum nonlocality within the NNN-qubit Pauli group is further developed in this thesis. The Strong Kochen-Specker theorem and the structures that prove it are introduced and explored in detail. The pattern of connections between structures that show entanglement, contextuality, and nonlocality is explained. Computational search algorithms and related tools were developed and used to perform complete searches for minimal nonclassical structures within the NNN-qubit Pauli group up to values of NNN limited by our computational resources. Our results are surveyed and prescriptions are given for using the elementary nonclassical structures we have found to construct more complex types of such structures. Families of nonclassical structures are presented for all values of NNN, including the most compact family of projector-based parity proofs of the Kochen-Specker theorem yet discovered in all dimensions of the form 2N2^N2N, where N≥2N\geq2N≥2. The applications of our results and their connection with other work is also discussed.