TIM PALMER: Three compelling reasons for discretising Hilbert Space

As the recent Nature survey confirms, we do not seem to be getting any closer to a consensus understanding of quantum mechanics (QM). I will argue that this is because QM’s principal assumption, that the state of a quantum system is a vector in Hilbert Space, is incorrect. More specifically, I will give three compelling reasons why the continuum nature of Hilbert Space is merely an approximation for something discrete.
Firstly, as John Wheeler asserted, the continuum nature of Hilbert Space conceals the information-theoretic nature of the wavefunction: this nature is revealed by discretisation. Secondly, discrete Hilbert Space allows a novel EPR/Bell-local interpretation of the violation of Bell’s inequality. Here we focus on the axiomatic role that counterfactual definiteness plays in Bell’s theorem, a role emphasised by Anton Zeilinger. Thirdly, discretisation provides a novel solution to the long-standing measurement problem: as a reduction in the information content of the wavefunction until the classical limit is reached.
If we assume discretisation scales are set by gravity, a testable prediction can be made. Even when qubits are perfectly shielded from their environment, the exponential advantage of Shor’s algorithm will have saturated at 1,000 qubits. At this limit, there is not enough information in the quantum state to allocate even 1 bit to each dimension of Hilbert Space. Hence, insofar as classical computers will never be able to factor 2,048-bit RSA integers, neither will quantum computers. This potential breakdown of QM can be verified or falsified in 5 years, if quantum-tech roadmaps are to be believed.