Probabilistic and Deterministic Description

C. H. Woo

Between the quantum domain and the classical world of facts there is an intermediate domain for which a ‘‘classical’’ probabilistic description is appropriate. As a result of decoherence, such a description obeys the classical additivity of probabilities. In contrast to the usual interpretation in classical probability theory, however, the ‘‘classical’’ probability which emerges from an underlying quantum theory is meant to be applicable to a single system, such as the universe.It is argued that small inhomogeneities and anisotropies in the very early universe should be described probabilistically. If, furthermore, the transition between this probabilistic description and a later deterministic description happens as late as after the decoupling of photons, then the smoothness of the microwave background receives a different interpretation: it is not because the fluctuations at decoupling were extremely small, but because the contributions from different fluctuations add to smooth things out.

Kierner, Kaleb

Quantum programming introduces both probabilistic and non-deterministic behavior, making formal verification a challenging task. This work develops a framework for modeling non-deterministic quantum programs using Markov Decision Processes (MDPs). Each program configuration is represented as a combination of the remaining program and its quantum state, while transitions capture both quantum evolution and non-deterministic branching. We define operational semantics for quantum programs in terms of MDPs and prove their equivalence to established denotational semantics. This equivalence ensures consistency between high-level mathematical descriptions and executable models. The approach enables the application of model-checking techniques, originally designed for classical probabilistic systems, to quantum programs. Our results show that MDPs provide a foundation for verifying properties of non-deterministic quantum programs.

Roger Balian

Zeng Zhu-shi

The development of physics is investigated in the light of methodology. Two revelations are derived : The deterministic description and probabilistic description co-exist side by side in Newton Mechanics,Quantum Mechanics and Statistic Physics. It is faced with the possibility of a third great synthetization.