Karl Svozil

Quantum field theory on fractal spacetime: a new regularisation method

Spacetime is modelled as a fractal subset of Rn. Analysis on homogeneous sets with non-integer Hausdorff dimensions is applied to the low-order perturbative renormalisation of quantum electrodynamics. This new regularisation method implements the Dirac matrices and tensors in R4 without difficulties, is gauge invariant, covariant and differs from dimensional regularisation in some aspects.

Optimal tests of quantum nonlocality

We present a general method for obtaining all Bell inequalities for a given experimental setup. Although the algorithm runs slowly, we apply it to two cases. First, the Greenberger-Horne-Zeilinger setup with three observers each performing one of two possible measurements. Second, the case of two observers each performing one of three possible experiments. In both cases we obtain hundreds of inequalities. Since this is the set of all inequalities, the one that is maximally violated in a given quantum state must be among them. We demonstrate this fact with a few examples. We also note the deep connection between the inequalities and classical logic, and their violation with quantum logic.

Quantum randomness and value indefiniteness

As computability implies value definiteness, certain sequences of quantum outcomes cannot be computable.

Greechie diagrams, nonexistence of measures in quantum logics, and Kochen–Specker‐type constructions

We use Greechie diagrams to construct finite orthomodular lattices ‘‘realizable’’ in the orthomodular lattice of subspaces in a three‐dimensional Hilbert space such that the set of two‐valued states is not ‘‘large’’ (i.e., full, separating, unital, nonempty, resp.). We discuss the number of elements of such orthomodular lattices, of their sets of (ortho)generators and of their subsets that do not admit a ‘‘large’’ set of two‐valued states. We show connections with other results of this type.

The quantum coin toss-testing microphysical undecidability

Abstract A critical review of randomness criteria shows that no-go theorems severely restrict the validity of actual “proofs” of undecidability. It is suggested to test microphysical undecidability by physical processes with low extrinsic complexity, such as polarized laser light. The publication and distribution of a sequence of pointer readings generated by such methods is proposed. Unlike any pseudorandom sequence generated by finite deterministic automata, the postulate of microscopic randomness implies that this sequence can be safely applied for all purposes requiring stochasticity and high complexity.

Experimental Evidence of Quantum Randomness Incomputability

In contrast with software-generated randomness (called pseudo-randomness), quantum randomness can be proven incomputable; that is, it is not exactly reproducible by any algorithm. We provide experimental evidence of incomputability--an asymptotic property--of quantum randomness by performing finite tests of randomness inspired by algorithmic information theory.

Strong Kochen-Specker theorem and incomputability of quantum randomness

The Kochen-Specker theorem shows the impossibility for a hidden variable theory to consistently assign values to certain (finite) sets of observables in a way that is non-contextual and consistent with quantum mechanics. If we require non-contextuality, the consequence is that many observables must not have pre-existing definite values. However, the Kochen-Specker theorem does not allow one to determine which observables must be value indefinite. In this paper we present an improvement on the Kochen-Specker theorem which allows one to actually locate observables which are provably value indefinite. Various technical and subtle aspects relating to this formal proof and its connection to quantum mechanics are discussed. This result is then utilized for the proposal and certification of a dichotomic quantum random number generator operating in a three-dimensional Hilbert space.

Physical versus computational complementarity. I

The dichotomy between endophysical/intrinsic and exophysical/extrinsic perception concerns how a model—mathematical, logical, computational—universe is perceived, from inside or from outside. This paper, the first in a proposed series, discusses some limitations and tradeoffs between endophysical/intrinsic and exophysical/extrinsic perceptions in both physical and computational contexts. We build our work on E. F. Moores Gedanken-experiments in which the universe is modeled by a finite deterministic automaton. A new type of computational complementarity, which mimics the state of quantum entanglement, is introduced and contrasted with Moores computational complementarity. Computer simulations of both types of computational complementarity are developed for fourstate Moore automata.

CONSISTENT USE OF PARADOXES IN DERIVING CONSTRAINTS ON THE DYNAMICS OF PHYSICAL SYSTEMS AND OF NO-GO-THEOREMS

The classical methods used by recursion theory and formal logic to block paradoxes do not work in quantum information theory. since quantum information can exist as a coherent superposition of the classical “yes” and “no” states, certain tasks which are not conceivable in the classical setting can be performed in the quantum setting. Classical logical inconsistencies do not arise, since there exist fixed point states of the diagnonalization operator. In particular, closed timelike curves need not be eliminated in the quantum setting, since they need not lead to the classical antinomies. Quantum information theory can also be subjected to the treatment of inconsistent information in databases and expert systems. It is suggested that any two pieces of contradicting information are stored and processed as coherent superposition. In order to be tractable, this strategy requires quantum computation.We investigate the orthoalgebras of certain non-Boolean models which have a classical realization. Our particular concern will be the partition logics arising from the investigation of the empirical propositional structure of Moore and Mealy type automata.

Logical Equivalence Between Generalized Urn Models and Finite Automata

To every generalized urn model there exists a finite (Mealy) automaton with identical propositional calculus. The converse is true as well.