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Apress eBooks

Game Theory: With Quantum Mechanics, Odds Are Always in Your Favor

Vladimir Silva

Published Jan 1, 2018• 0 citations
Publisher

Abstract

This chapter explores two game puzzles that show the remarkable power of quantum algorithms over their classical counterparts:
DOI
10.1007/978-1-4842-4218-6_7

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Quantum mechanics, like relativity, is a pillar of contemporary physics, with remarkable, puzzling consequences. In these chapters some very simple ‘games’ – distributing prizes of different kinds – are used to explain the implications for ‘identity’. Classical particles have ‘Maxwell‐Boltzmann’ statistics, according to which swapping properties makes a difference; similarly for people, as it makes a difference if we swap bank accounts! Quantum particles with ‘Bose‐Einstein’ statistics (called bosons) or ‘Fermi‐Dirac’ statistics (fermions) are like dollars in bank accounts: if we swap $1 it makes no difference. Chapter 17 shows how Schrödinger used the prize games to argue that – like the dollars – bosons and fermions are not ‘identifiable things’. Chapter 18 extends the models to other possible quantum particles, arguing that quantum mechanics also restricts identity by restricting what states particles can possess together – for instance, Fermi's ‘exclusion principle’, that fermions must be in different states.
Quantum mechanics, like relativity, is a pillar of contemporary physics, with remarkable, puzzling consequences. In these chapters some very simple ‘games’ – distributing prizes of different kinds – are used to explain the implications for ‘identity’. Classical particles have ‘Maxwell‐Boltzmann’ statistics, according to which swapping properties makes a difference; similarly for people, as it makes a difference if we swap bank accounts! Quantum particles with ‘Bose‐Einstein’ statistics (called bosons) or ‘Fermi‐Dirac’ statistics (fermions) are like dollars in bank accounts: if we swap $1 it makes no difference. Chapter 17 shows how Schrödinger used the prize games to argue that – like the dollars – bosons and fermions are not ‘identifiable things’. Chapter 18 extends the models to other possible quantum particles, arguing that quantum mechanics also restricts identity by restricting what states particles can possess together – for instance, Fermi's ‘exclusion principle’, that fermions must be in different states.
We study quantum entanglement and some of its applications in graph theory and zero-error information theory. In Chapter 1 we introduce entanglement and other fundamental concepts of quantum theory. In Chapter 2 we address the question of how much quantum correlations generated by entanglement can deviate from classical predictions. We focus on non-local games: experiments in which two players are separated and forbidden to communicate, and have to collaborate to accomplish a task. We give two games exhibiting quantum-classical separations close to optimal. Remarkably, our results in theoretical physics are inspired by theoretical computer science. Chapter 3 is dedicated to the study of quantum graph parameters. Well-known quantities such as the chromatic number and the independence number of a graph can be interpreted as parameters for non-local games. A definition of quantum graph parameters follows from this fact. We contribute to the field in a number of ways. Among other results, we find a surprising characterization of the quantum chromatic number that relates to the Kochen-Specker theorem, a result in the foundations of quantum mechanics. In Chapter 4, we move to zero-error information theory. We study the zero error capacity of a classical noisy channel when the sender and the receiver can use quantum entanglement. We initiate the study of the source problem and source-channel problem with entanglement and we find channels and sources that exhibit a strong divergence in quantum and classical behaviours. To do that, we use results in combinatorics, linear algebra, optimization and number theory.