A recent theoretical paper by Leonard Mlodinow and Todd Brun suggests that the functioning of physical records or memories is never accompanied by a decrease in entropy, meaning that all memories align with the thermodynamic arrow of time. In this thesis, we characterize a class of physical systems as memories in terms of inferences that can be made about the state of the world, given certain information about these systems. Tools from Bayesian probability theory are used to quantify the informativeness and reliability associated with such inferences. Based on consideration of two model systems, one classical and one quantum, we argue in favor of Mlodinow and Brun\u27s claim that the functioning of memory systems is conditioned by thermodynamic constraints. For the classical model, we show that a memory which operates against the thermodynamic arrow, and thus remembers a relatively high-entropy state, is much less informative than a similar memory which aligns with the thermodynamic arrow. Our analysis of the quantum model, expressed in the density matrix formalism of quantum mechanics, allows us to consider the inferences that can be made when a quantum system is coupled to a simple type of quantum memory system. We ultimately show that these inferences can be expressed in terms of a probabilistic matrix completion problem
