Two distinct puzzles, which are both known as Gibbs' paradox, have interested physicists since they were first identified in the 1870s. They each have significance for the foundations of statistical mechanics and have led to lively discussions with a wide variety of suggested resolutions. Most proposed resolutions had involved quantum mechanics, although the original puzzles were entirely classical and were posed before quantum mechanics was invented. In this paper, I show that contrary to what has often been suggested, quantum mechanics is not essential for resolving the paradoxes. I present a resolution of the paradoxes that does not depend on quantum mechanics and includes the case of colloidal solutions, for which quantum mechanics is not relevant.
Two losing games can be played in a certain manner to produce a winning outcome-a phenomenon known as Parrondo's paradox. Of particular interest is the emergence of quantum game theory and the attempt to model known Parrondo's games through quantum computation notation. In this article, we investigate whether flipping four-sided quantum coins will result in the emergence of Parrondo's paradox. We discover that by playing two losing games A and B in a sequential order, a winning scenario can be derived. Furthermore, four-sided quantum coin is the first instance where the ratcheting effect from the classical Parrondo's game is necessary. Crucially, our study is designed with quantum protocols as its basis and does not have a direct classical counterpart.
Based on the axioms of quantum theory, we identify a class of topological singularities that encode a fundamental difference between classic and quantum probability, and explain quantum theory’s puzzles and phenomena in simple mathematical terms so they are no longer ‘quantum paradoxes’. The singularities provide also new experimental insights and predictions that are presented in this article and establish a surprising new connection between the physical and social sciences. The key is the topology of spaces of quantum events and of the frameworks postulated by these axioms. These are quite different from their counterparts in classic probability and explain mathematically the interference between quantum experiments and the existence of several frameworks or ‘violation of unicity’ that characterizes quantum physics. They also explain entanglement, the Heisenberg uncertainty principle, order dependence of observations, the conjunction fallacy and geometric phenomena such as Pancharatnam–Berry phases. Somewhat surprisingly, we find that the same topological singularities explain the impossibility of selecting a social preference among different individual preferences: which is Arrow’s social choice paradox: the foundations of social choice and of quantum theory are therefore mathematically equivalent. We identify necessary and sufficient conditions on how to restrict experiments to avoid these singularities and recover unicity, avoiding possible interference between experiments and also quantum paradoxes; the same topological restriction is shown to provide a resolution to the social choice impossibility theorem of Chichilnisky.