Building on a new definition and characterization of probabilistic event structures, a general definition of distributed probabilistic strategies is proposed. Probabilistic strategies are shown to compose, with probabilistic copy-cat strategies as identities. A higher-order probabilistic process language reminiscent of Milnerʼs CCS is interpretable within probabilistic strategies. W.r.t. a new definition of quantum event structure, it is shown how consistent parts of a quantum event structure are automatically probabilistic event structures, and so possess a probability measure. This gives a non-traditional take on the consistent-histories approach to quantum theory. It leads to an extension to quantum strategies. Probabilistic games extend to games with payoff, symmetry and games of imperfect information.
Building on a new definition and characterization of probabilistic event structures, a general definition of distributed probabilistic strategies is proposed. Probabilistic strategies are shown to compose, with probabilistic copy-cat strategies as identities. A higher-order probabilistic process language reminiscent of Milnerʼs CCS is interpretable within probabilistic strategies. W.r.t. a new definition of quantum event structure, it is shown how consistent parts of a quantum event structure are automatically probabilistic event structures, and so possess a probability measure. This gives a non-traditional take on the consistent-histories approach to quantum theory. It leads to an extension to quantum strategies. Probabilistic games extend to games with payoff, symmetry and games of imperfect information.
In the context of quantum information theory, “quantization” of various mathematical constructions is said to occur upon the replacement, at various points in the construction, of the classical randomization notion of probability distribution with higher order randomization notions from quantum mechanics such as quantum superposition with measurement. For this to be done “properly”, a faithful copy of the original construction is required to exist within the new “quantum” one, just as is required when a function is extended to a larger domain. Here, Markov processes that serve as mathematical models of history dependent Parrondo games are quantized properly.
Abstract We study properties of quantum strategies, which are complete specifications of a givenparty's actions in any multiple-round interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantumstrategies that generalizes the Choi-Jamiol/kowski representation of quantum operations. This new representation associates with each strategy a positive semidefinite operator acting onlyon the tensor product of its input and output spaces. Various facts about such representations are established, and two applications are discussed: the first is a new and conceptually simpleproof of Kitaev's lower bound for strong coin-flipping, and the second is a proof of the exact characterization QRG = EXP of the class of problems having quantum refereed games. 1 Introduction The theory of games provides a general structure within which both cooperation and competitionamong independent entities may be modeled, and provides powerful tools for analyzing these models. Applications of this theory have fundamental importance in many areas of science.This paper considers games in which the players may exchange and process quantum information. We focus on competitive games, and within this context the types of games we consider arevery general. For instance, they allow multiple rounds of interaction among the players involved, and place no restrictions on players ' strategies beyond those imposed by the theory of quantuminformation. While classical games can be viewed as a special case of quantum games, it is important tostress that there are fundamental differences between general quantum games and classical games. For example, the two most standard representations of classical games, namely the normal formand extensive form representations, are not directly applicable to general quantum games. This is due to the nature of quantum information, which admits a continuum of pure (meaning extremal)
This thesis explores both theoretical and practical aspects of concurrent programming languages that integrate probabilistic and/or quantum computations. On the theoretical side, we investigate how event structures, a model for concurrency, can be used to furnish concurrent languages with denotational semantics. Event structures are studied in three distinct settings: non-deterministic, probabilistic, and quantum computation. Based on these three realms, we successively extend a core imperative-like concurrent language with non-deterministic choice, probabilistic choice, and standard quantum operations. For each extension, we define both operational and denotational semantics and establish their equivalence through soundness and adequacy results. Soundness ensures that every behavior obtainable through the operational semantics is reflected denotationally, showing that the denotational semantics correctly abstracts the computations. Conversely, adequacy guarantees that whenever the denotational semantics yields a value, this value can be produced by the operational semantics, demonstrating that the operational semantics explicitly realises what the denotational semantics predicts. On the practical side, motivated by the growing interplay between classical and quantum computers, we integrate non-deterministic choice, probabilistic choice, and quantum operations into a single language, further extended with a synchronization primitive. We develop an interpreter for this language and test it on several illustrative examples, including three versions of the Dining Philosophers problem: concurrentonly, probabilistic, and quantum. Using our tool, we demonstrate that a result stated in the literature about the quantum Dining Philosophers problem does not hold, leaving open the question of whether an efficient algorithm for solving it exists.
We consider game theory from the perspective of quantum algorithms. Strategies in classical game theory are either pure (deterministic) or mixed (probabilistic). While not every two-person zero-sum finite game has an equilibrium in the set of pure strategies, von Neumann showed that there is always an equilibrium at which each player follows a mixed strategy. A mixed strategy deviating from the equilibrium strategy cannot increase a player's expected payoff. We show by example, however, that a player who implements a quantum strategy can increase his expected payoff, and explain the relation to efficient quantum algorithms.
Quantum games extend classical game theory into the quantum realm by incorporating quantum players and quantum strategies. Two-player quantum games have been widely studied and applied, among other areas, in optimization and decision-making. In this work, we introduce, formalize and analyze a novel class of three-player quantum games, referred to as Proxy Quantum Games (PQGs). In this setting, one player, the Controller, influences or governs the strategic operations executed by a second player, the Proxy, who, in turn, competes directly against a third player, the Opponent. The strategic influence is implemented via controlled unitary operations, whereby the Proxy’s available strategies are conditioned on the Controller’s quantum state. We further generalize the PQG framework by incorporating multiple Controllers acting on a common Proxy, thereby enabling the study of higher-order strategic interactions and multi-agent influence structures. The objective under consideration is the maximization of the Controller’s expected payoff, under various configurations of strategic control and entanglement. Comparative analyses are conducted between the PQG formulation and conventional multiplayer quantum games, highlighting key differences in strategic behavior and payoff distributions.
Abstract Despite predictive success, population dynamics and evolutionary game theory still pose fundamental problems. Violation of the competitive exclusion principle in plankton communities provides an example. A promising solution of this ‘paradox of the plankton’ comes from theories involving cyclic competition, an evolutionary analogue of the classical rock-paper-scissors (RPS) game. However, modeling probabilistic RPS structures, one encounters a fundamental difficulty: the pairs rock-scissors, scissors-paper, and paper-rock possess representations in separate Kolmogorovian probability spaces, but a single global probability space for entire triplets does not exist. Populations that take part in cyclic competition should therefore involve probabilistic incompatibilities, analogous to those occurring in quantum mechanics. Here, using experimental data collected from 1990 to 2011 on the RPS cycles of lizards, we show that the incompatibilities are indeed unavoidable, and the data cannot be reconstructed from a single Kolmogorovian probability space. We then prove that the effect is genuinely quantum probabilistic, i.e. all the probabilities can be formulated in terms of a single density matrix and a set of non-commuting projectors. This formal quantum structure is dormant in games where probabilities of strategies do not entangle with probabilities of payoffs, and thus could be overlooked. In more realistic scenarios, involving games ‘with ace in a sleeve’, the non-Kolmogorovian structure can be activated. Surprisingly, lizards occasionally do play such games. In consequence, the formalism of evolutionary games, similarly to quantum mechanics, should begin with density matrix equations. Implications of our finding extend beyond lizard communities, given that RPS games are common in nature and higher dimensional RPS games may be even more common in ecosystems.