*Robert Alicki and Mark Fannes*

- Published in print:
- 2001
- Published Online:
- February 2010
- ISBN:
- 9780198504009
- eISBN:
- 9780191708503
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504009.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces quantum spin chains for two different purposes: prototypes of infinite quantum systems discretized in space, and general models of stochastic processes in discrete time on a ...
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This chapter introduces quantum spin chains for two different purposes: prototypes of infinite quantum systems discretized in space, and general models of stochastic processes in discrete time on a discrete quantum state space. The algebra of local observables is constructed. The probability functionals, i.e. the states, are defined in terms of reduced density matrices with their compatibility relations. Several examples of shift-invariant states are discussed: product states, classical states, and limiting Gibbs states. The chapter concludes with the construction of nearest-neighbour dynamics and the need for an appropriate algebra of quasi-local observables.Less

This chapter introduces quantum spin chains for two different purposes: prototypes of infinite quantum systems discretized in space, and general models of stochastic processes in discrete time on a discrete quantum state space. The algebra of local observables is constructed. The probability functionals, i.e. the states, are defined in terms of reduced density matrices with their compatibility relations. Several examples of shift-invariant states are discussed: product states, classical states, and limiting Gibbs states. The chapter concludes with the construction of nearest-neighbour dynamics and the need for an appropriate algebra of quasi-local observables.

*Robert Alicki and Mark Fannes*

- Published in print:
- 2001
- Published Online:
- February 2010
- ISBN:
- 9780198504009
- eISBN:
- 9780191708503
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504009.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter begins with the discussion of various meanings of the entropy: Boltzmann entropy, Shannon entropy, von Neumann entropy. It then presents the basic properties of the von Neumann entropy ...
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This chapter begins with the discussion of various meanings of the entropy: Boltzmann entropy, Shannon entropy, von Neumann entropy. It then presents the basic properties of the von Neumann entropy such as concavity, sub-additivity, strong sub-additivity, and continuity. It proves the existence of mean entropy for shift-invariant states on quantum spin chains, and then derives the expression for the mean entropy for quasi-free Fermions on a chain. The chapter defines the quantum relative entropy and presents its basic properties, including behaviour with respect to completely positive maps. The maximum entropy principle defines thermal equilibrium states (Gibbs states). This variational principle is illustrated by the Hartree–Fock approximation for a model of interacting Fermions. The entropy of an equilibrium state for a free quantum particle on a compact Riemannian manifold is also estimated. Finally, the notion of relative entropy is formulated in the algebraic setting using the relative modular operator.Less

This chapter begins with the discussion of various meanings of the entropy: Boltzmann entropy, Shannon entropy, von Neumann entropy. It then presents the basic properties of the von Neumann entropy such as concavity, sub-additivity, strong sub-additivity, and continuity. It proves the existence of mean entropy for shift-invariant states on quantum spin chains, and then derives the expression for the mean entropy for quasi-free Fermions on a chain. The chapter defines the quantum relative entropy and presents its basic properties, including behaviour with respect to completely positive maps. The maximum entropy principle defines thermal equilibrium states (Gibbs states). This variational principle is illustrated by the Hartree–Fock approximation for a model of interacting Fermions. The entropy of an equilibrium state for a free quantum particle on a compact Riemannian manifold is also estimated. Finally, the notion of relative entropy is formulated in the algebraic setting using the relative modular operator.

*Jochen Rau*

- Published in print:
- 2017
- Published Online:
- November 2017
- ISBN:
- 9780199595068
- eISBN:
- 9780191844300
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199595068.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

The limited data available about a macroscopic system may come in various forms: sharp constraints, expectation values, or control parameters. While these data impose constraints on the state, they ...
More

The limited data available about a macroscopic system may come in various forms: sharp constraints, expectation values, or control parameters. While these data impose constraints on the state, they do not specify it uniquely; a further principle—the maximum entropy principle—must be invoked to construct it. This chapter discusses basic notions of information theory and why entropy may be regarded as a measure of ignorance. It shows how the state—called a Gibbs state—is constructed using the maximum entropy principle, and elucidates its generic properties, which are conveniently summarized in a thermodynamic square. The chapter further discusses the second law and how it is linked to the reproducibility of macroscopic processes. It introduces the concepts of equilibrium and temperature, as well as pressure and chemical potential. Finally, this chapter considers statistical fluctuations of the energy and of other observables in case these are given as expectation values.Less

The limited data available about a macroscopic system may come in various forms: sharp constraints, expectation values, or control parameters. While these data impose constraints on the state, they do not specify it uniquely; a further principle—the maximum entropy principle—must be invoked to construct it. This chapter discusses basic notions of information theory and why entropy may be regarded as a measure of ignorance. It shows how the state—called a Gibbs state—is constructed using the maximum entropy principle, and elucidates its generic properties, which are conveniently summarized in a thermodynamic square. The chapter further discusses the second law and how it is linked to the reproducibility of macroscopic processes. It introduces the concepts of equilibrium and temperature, as well as pressure and chemical potential. Finally, this chapter considers statistical fluctuations of the energy and of other observables in case these are given as expectation values.