Probabilistic Description

Roger Balian

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.

Randa Herzallah, Abdessamad Belfakir

In this paper, we propose a novel probabilistic control framework for efficiently controlling an ensemble of quantum systems that can also compensate for the interaction of the systems with the external environment. The main challenge in this problem is to simultaneously steer an ensemble of systems with variation in their internal parameters from an initial state to a desired final state. The minimisation of the discrepancy between the probabilistic description of the dynamics of a quantum ensemble and a predefined desired probabilistic description is the key step in the proposed framework. With this objective, the derived solution will not only allow the transitioning of the ensemble from one state to another, but will generally allow steering an initial distribution of the ensemble to a final distribution. Numerical results are presented, demonstrating the effectiveness of the proposed probabilistic control framework.

Abdessamad Belfakir, Randa Herzallah

In this paper, we propose a novel probabilistic control framework for efficiently controlling an ensemble of quantum systems that can also compensate for the interaction of the systems with the external environment. The main challenge in this problem is to simultaneously steer an ensemble of systems with variation in their internal parameters from an initial state to a desired final state. The minimisation of the discrepancy between the probabilistic description of the dynamics of a quantum ensemble and a predefined desired probabilistic description is the key step in the proposed framework. With this objective, the derived solution will not only allow the transitioning of the ensemble from one state to another, but will generally allow steering an initial distribution of the ensemble to a final distribution. Numerical results are presented, demonstrating the effectiveness of the proposed probabilistic control framework.