<p>We studied two topics: i) how much physical resources are needed to store information and ii) decoherent histories theory applied to Grover search. Given a system consisting of <i>d</i> degrees of freedom each of mass <i>m</i> to store an amount <i>S</i> of information, we find that its average energy, &lang;<i>H</i>&rang;, or size, &lang;<i>r</i>²&rang;, can be made arbitrarily small individually, but its product &lang;P&rang; = &lang;<i>H</i>&rang;&lang;<i> r</i>²&rang; is bounded below by (exp{<i>S/d</i>} &minus; 1)²<i>d</i>²/<i>m.</i> This result is obtained in a nonrelativistic, quantum mechanical setting, and it is independent of earlier thermodynamical results such as the Bekenstein bound on the entropy of black holes. </p><p> The second topic is decoherent histories applied to the Grover search problem. The theory of decoherent histories is an attempt to derive classical physics from positing only quantum laws at the fundamental level without notions of a classical apparatus or collapse of the wave-function. Searching for a marked target in a list of <i>N</i> items requires &Omega;(<i> N</i>) oracle queries when using a classical computer, while a quantum computer can accomplish the same task in <i>O</i>([special characters omitted]) queries using Grover's quantum algorithm. We study a closed quantum system executing Grover algorithm in the framework of decoherent histories and find it to be an exactly solvable model, thus yielding an alternate derivation of Grover's famous result. We also subject the Grover-executing computer to a generic external influence without needing to know the specifics of the Hamiltonian insofar as the histories decohere. Depending on the amount of decoherence, which is captured in our model by a single parameter related to the amount of information obtained by the environment, the search time can range from quantum to classical. Thus, we identify a key effect induced by the environment that can adversely affect a quantum computer's performance and demonstrate exactly how classical computing can emerge from quantum laws. </p>

<p> Bayesian networks and their accompanying graphical models are widely used for prediction and analysis across many disciplines. We will reformulate these in terms of linear maps. This reformulation will suggest a natural extension, which we will show is equivalent to standard textbook quantum mechanics. Therefore, this extension will be termed <i>quantum.</i> However, the term <i> quantum</i> should not be taken to imply this extension is necessarily only of utility in situations traditionally thought of as in the domain of quantum mechanics. In principle, it may be employed in any modelling situation, say forecasting the weather or the stock market&mdash;it is up to experiment to determine if this extension is useful in practice. Even restricting to the domain of quantum mechanics, with this new formulation the advantages of Bayesian networks can be maintained for models incorporating quantum and mixed classical-quantum behavior. The use of these will be illustrated by various basic examples. </p><p> Parrondo's paradox refers to the situation where two, multi-round games with a fixed winning criteria, both with probability greater than one-half for one player to win, are combined. Using a possibly biased coin to determine the rule to employ for each round, paradoxically, the previously losing player now wins the combined game with probabilitygreater than one-half. Using the extended Bayesian networks, we will formulate and analyze classical observed, classical hidden, and quantum versions of a game that displays this paradox, finding bounds for the discrepancy from naive expectations for the occurrence of the paradox. A quantum paradox inspired by Parrondo's paradox will also be analyzed. We will prove a bound for the discrepancy from naive expectations for this paradox as well. Games involving quantum walks that achieve this bound will be presented.</p>