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The experimental confirmation of Landauer principle and the emerging concept of a computational universe make it more and more crucial to understand the physical sense of information, as it has an intrinsic relation with observer knowledge that is often rejected as subjective. In this paper we propose an objective definition of phase information in a purely classical computational universe with quantized phase states, this quantization being imposed by our fundamental hypothesis that physical information has a finite and limited density, responsible for the irreversibility. We use a statistical study of results obtained by numerical simulations of a billiard to highlight an excessive and paradoxical loss of phase information that we solve by involving a classical to quantum transition. After discussing the pertinence of such a transition to clarify some problematic aspects of statistical physics, we conclude that a computational universe should automatically lose phase information through time in an irreversible way, which could be compensated by a gain of physical information due to observation and decoherence., Comment: 28 pages, 8 figures