Quaternions, octonions and Bell-type inequalities

Two-qubit analogs of octonions and projective planes

A quantum spin-1/2, and its associated su(2) algebra of Pauli spin matrices are familiarly linked to Clifford algebra and quaternions. Somewhat more loosely, we develop connections between the su(4) algebra of two spins and of its sub-algebras, which are important throughout the field of quantum information, with octonions, the projective plane of seven elements, and entities in projective geometry and design theory called partial Steiner systems of 10 and 15 elements. Although not the exact correspondence between a single spin and quaternions, the close analogy between these geometrical objec…

Non-associativity, supersymmetry and hidden variables

It is shown that the supersymmetric quantum mechanics has an octonionic generalization. The generalization is based on the inclusion of quaternions into octonions. The elements from the coset octonions/quaternions are unobservables bacause they cannot be considered as quantum operators as a consequence of their non-associative properties. The idea that the octonionic generalization of the supersymmetric quantum mechanics describes an observable particle formed with unobservable ``particles'' is presented.

通过 Clifford 代数中的中心荷反常研究八元数与量子引力

Octonions and Quantum Gravity through the Central Charge Anomaly in the Clifford Algebra

本文基于超代数、复化八元数与 Clifford 代数构建含 AdS$_3$ 等距/量子比特对偶的量子引力理论，利用中心荷反常得到全息面积律并提出黑洞信息悖论的解决机制。

Bell inequalities from multilinear contractions

We provide a framework for Bell inequalities which is based on multilinear contractions. The derivation of the inequalities allows for an intuitive geometric depiction and their violation within quantum mechanics can be seen as a direct consequence of non-vanishing commutators. The approach is motivated by generalizing recent work on non-linear inequalities which was based on the moduli of complex numbers, quaternions and octonions. We extend results on Peres conjecture about the validity of Bell inequalities for quantum states with positive partial transposes. Moreover, we show the possibilit…