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Institute of Physics eBooks

Case studies: summaries, conclusions and outlook

Paul Hockett

Published Dec 1, 2023• 0 citations
Publisher

Abstract

In this chapter, the three case studies are summarised, compared and contrasted. General conclusions on the state of matrix element retrieval using the generalised bootstrap protocol are made (herein explored up to 38 complex matrix elements or, correspondingly, a 38 x 38 density matrix, in the largest case study). Future routes of exploration for these cases and improvements for the bootstrap methodology beyond those tested herein are discussed, along with an outlook and general perspective on the topic.

DOI
10.1088/978-0-7503-5022-8ch11

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In this chapter, the three case studies are summarised, compared and contrasted. General conclusions on the state of matrix element retrieval using the generalised bootstrap protocol are made (herein explored up to 38 complex matrix elements or, correspondingly, a 38 x 38 density matrix, in the largest case study). Future routes of exploration for these cases and improvements for the bootstrap methodology beyond those tested herein are discussed, along with an outlook and general perspective on the topic.
In this chapter, numerical methodologies/protocols for the complete quantum metrology of a photoionization event (a.k.a. a ‘complete’ photoionization experiment) are outlined. Following the theory outlined in chapter 3, this can be characterized as recovery of the photoionization matrix elements (including phase information) from the experimental measurements or, equivalently, the continuum density matrix. In general, such a recovery or reconstruction may be possible provided the channel functions are known, and the information content of the measurements is sufficient. These methods, specifically the generalised bootstrap retrieval protocol, are developed and illustrated, and they form the basis for the numerical reconstruction work for realistic cases presented in Part II (chapters 5–11).
As was repeatedly seen in the last chapter, for computing most of the usual matrix elements only a few of the many coordinates of an N -particle wavefunction are used, where due to the symmetry properties it is irrelevant which ones. Density matrices are a tool to extract the relevant information out of such monstrous constructs as are N -particle wavefunctions. (For reviews, see [Erdahl and Smith, 1987, Coleman, 1963, McWeeny, 1960]; in the text below we try to keep the notation as canonical as possible.) These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In this final chapter, we systematically examine and synthesize the key conclusions derived from the previous parts, while also proposing several important and interesting directions for future research that could extend our current topics. Specifically, we highlight (1) generalizations and variants of the historicized ontology, including the coevent formalism or quantum measure theory, the superdensity operators formalism, and the pseudo-density matrix formalism; (2) the physical extension of the application of quantum histories formalism, such as open quantum systems, infinite-dimensional cases, and classical/quantum relativity, and some interpretational and philosophical reflections. Finally, we give some remarks from the methodological perspectives for future foundational research of quantum mechanics.
The Lindblad (GKLS) master equation, which represents the mathematical form for the general evolution of a density matrix, is a versatile and widely-used tool in open quantum systems. In contrast with the typical approach of imposing additional conditions on the system, such as weak coupling or energy conservation, we explore the structure of the equation with no assumptions. We demonstrate that general quantum dynamics can be expressed through a combination of free evolution, exchanges of some physical quantities (generalised charges), not necessarily commuting with the Hamiltonian, between the system and the bath, and pure dephasing. This result comprises a novel perspective on quantum master equations, employing physical processes as elemental parts. We use it to explore the dynamics and stationary states of a two-level system and show that strong coupling, particle exchange, and non-Abelian effects all share the same physical origin. Moreover, we demonstrate that the generalised Gibbs state for all three cases contains a non-commutation term, which has not been previously considered.