How does the Planck scale affect qubits?

Abstract

Gedanken experiments in quantum gravity motivate generalised uncertainty relations (GURs) implying deviations from the standard quantum statistics close to the Planck scale. These deviations have been extensively investigated for the non-spin part of the wave function but existing models tacitly assume that spin states remain unaffected by the quantisation of the background in which the quantum matter propagates. Here, we explore a new model of nonlocal geometry in which the Planck-scale smearing of classical points generates GURs for angular momentum. These, in turn, imply an analogous generalisation of the spin uncertainty relations. The new relations correspond to a novel representation of {\rm SU(2)} that acts nontrivially on both subspaces of the composite state describing matter-geometry interactions. For single particles each spin matrix has four independent eigenvectors, corresponding to two $2$-fold degenerate eigenvalues $\pm (\hslash +\beta )/2$, where $\beta$ is a small correction to the effective Planck's constant. These represent the spin states of a quantum particle immersed in a quantum background geometry and the correction by $\beta$ emerges as a direct result of the interaction terms. In addition to the canonical qubits states, $|0⟩=|\uparrow ⟩$ and $|1⟩=|\downarrow ⟩$, there exist two new eigenstates in which the spin of the particle becomes entangled with the spin sector of the fluctuating spacetime. We explore ways to empirically distinguish the resulting `geometric' qubits, $|0^{\prime }⟩$ and $|1^{\prime }⟩$, from their canonical counterparts.

25 pages of main text, 9 pages of appendices, 2 pages of references. No figures, no tables. Invited contribution to the Quantum Reports special issue "Relevance of Information Geometry in Quantum Information Science", C. Cafaro (ed.). Comments welcome. Published version (v2)

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• Generalized uncertainty principle and burning stars
• Problems with Modified Commutators
• Fractal properties of particle paths due to generalised uncertainty relations
• Towards a Quantum Erlangen Program