How Does the Planck Scale Affect Qubits?
Abstract
:1. Introduction
2. The Smeared Space Model of Nonlocal Geometry
3. Spin in Smeared-Space Quantum Mechanics
3.1. One-Particle Systems in Smeared Space
3.2. Two-Particle Systems in Smeared Space
3.3. Bell States in Smeared Space
4. The Smeared-Space Representation of SU(2)
5. Discussion
5.1. Conclusions
5.2. Future Work
- In [10], we argued that a smeared spatial “point” can be viewed as a quantum reference frame (QRF). However, there is a fundamental difference between a QRF embedded as a material quantum system in a classical background geometry, as proposed in [14], and one generated by the nonlocality of the geometry itself [10,13]. Realistic observers are embedded as material quantum systems in quantum geometries, so that the smeared-space and canonical QRF formalisms should be combined to describe this scenario.
- The model should be extended to the relativistic regime via an appropriate smearing of the time coordinate and we expect this to generate a generalisation of the canonical Poincaré algebra. The generalisation should support an appropriate subalgebra structure that classifies the symmetries of nonlocal Minkowski space. Generalisations of the canonical Dirac and Klein–Gordon equations can then be obtained, which are compatible with the minimum length and minimum momentum phenomenology implied by the EGUP.
- If the subalgebra structure of the Poincaré group, which classifies the symmetries of nonlocal spacetime, can be consistently constructed, this has immediate consequences for the standard model of particle physics. The seminal realisation, by Eugene Wigner, that canonical quantum particles are “unitary representations of the inhomogeneous Lorentz group” [19], is conventionally applied to particles propagating in classical spacetimes. We aim to construct representations that act nontrivially on both subspaces of a tensor product Hilbert space describing matter-geometry interactions. From these representations, we can expect to obtain minimum length and minimum momentum phenomenology that remains consistent with the symmetries of special relativity.
- Finally, the existence of qubits that are entangled with the spacetime background may have profound implications for the black hole information loss paradox [17]. Consider, for example, two black holes with identical mass, charge, and “spin”. (In the common terminology, the “spin” of a black hole refers to its orbital angular momentum [20] and must not be confused with genuine quantum mechanical spin.) Let us imagine that the first black hole emits a particle of Hawking radiation, say, an electron, in the spin “up” eigenstate , whereas the second emits an electron in the spin “up” eigenstate . Due to recoil, both particles are entangled with their respective black holes, and, according to the smeared space theory, the states of the two black-hole-plus-electron systems are indistinguishable, via simultaneous measurements of and . Nonetheless, the electron in state possesses additional entanglement, with the nonlocal spacetime background, and, hence, additional entanglement entropy. Such "geometric" qubits may radiate more entropy away from the black hole than their canonical counterparts, and, therefore, more information. According to our previous arguments, this information should be encoded in additional noncanonical correlations between the subsystems of multiparticle states, due to their mutual interaction with the nonlocal background geometry. However, existing models of black hole evaporation do not account for this type of entanglement [17], which is explicitly generated by the delocalisation of spatial “points” over regions comparable to the Planck volume.