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Cambridge University Press eBooks

Density Operator

Lilia M. Woods, Pablo Rodriguez-López

Published Jun 6, 2024• 0 citations
Publisher

Abstract

Problems involving calculations of various properties associated with the density operator and entropies and their relations to more general situations in physics are included.
DOI
10.1017/9781009355414.004

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Abstract In various contexts in mathematical physics, such as out-of-equilibrium physics and the asymptotic information theory of many-body quantum systems, one needs to compute the logarithm of a positive unbounded operator. Examples include the von Neumann entropy of a density matrix and the flow of operators with the modular Hamiltonian in the Tomita-Takesaki theory. Often, one encounters the situation where the operator under consideration, which we denote by ∆, can be related by a perturbative series to another operator ∆0, whose logarithm is known. We set up a perturbation theory for the logarithm log ∆. It turns out that the terms in the series possess a remarkable algebraic structure, which enables us to write them in the form of nested commutators plus some “contact terms”.
Abstract Following arXiv:2210.12963 [hep-th], we investigate aspects of the time evolution operator regarded as a density operator and associated entanglement-like structures in various quantum systems. These involve timelike separations and generically lead to complex-valued entropy, although there are interesting real subfamilies. There are many parallels and close relations with reduced transition matrices and pseudo-entropy, which we discuss and clarify. For instance, a related quantity involves the time evolution operator along with a projection onto some initial state, which amounts to analysing pseudo-entropy for the initial state and its time-evolved final state.
Heavy use of matrices is made in the discussions of density operators and rotation operators. Thus, the general properties of matrices are presented and, since they represent a non-Abelian group, some of their peculiarities are included. Their addition, subtraction and various kinds of multiplications are discussed. Determinants are developed from matrices and used in the solution of simultaneous equations. Various matrix transformations are discussed with a view towards their later use in density matrix calculations and finally, the exponentials of matrices are discussed.