The modern tools of quantum mechanics (A tutorial on quantum states, measurements, and operations)
Matteo G. A. Paris

TL;DR
This tutorial reviews modern quantum mechanics tools for describing states, measurements, and operations in open systems, emphasizing generalized measurements, CP maps, and their physical implementations.
Contribution
It provides a comprehensive overview of generalized measurement theory, including the Naimark and Stinespring theorems, connecting abstract concepts with physical realizations.
Findings
Introduction of POVMs and their physical implementation
Explanation of the Naimark theorem and joint measurement limitations
Discussion of CP maps and the Kraus operator-sum representation
Abstract
This tutorial is devoted to review the modern tools of quantum mechanics, which are suitable to describe states, measurements, and operations of realistic, not isolated, systems in interaction with their environment, and with any kind of measuring and processing devices. We underline the central role of the Born rule and and illustrate how the notion of density operator naturally emerges, together the concept of purification of a mixed state. In reexamining the postulates of standard quantum measurement theory, we investigate how they may formally generalized, going beyond the description in terms of selfadjoint operators and projective measurements, and how this leads to the introduction of generalized measurements, probability operator-valued measures (POVM) and detection operators. We then state and prove the Naimark theorem, which elucidates the connections between generalized and…
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