Quantum Communications pp 183-249 | Cite as

# Quantum Decision Theory: Analysis and Optimization

## Abstract

Decision is the heart of any quantum digital communication system and is concerned with the formulation of how a measurement must be taken to argue on the transmitted message in the presence of uncertainties (due to the nature of any quantum measurement). The state of the quantum system is considered as assigned through a constellation of pure states or of density operators, whereas the measurement operators must be found with the goal of achieving the “best decision” (*optimization*), usually obtained by minimizing the error probability. The chapter, after a mathematical formulation of decision, moves on to optimization, where the guidelines are given by Holevo’s and Kennedy’s theorems. These theorems determine the conditions that must be fulfilled by an optimal system of measurement operators, but do not provide any clue on how to identify it. In any case, the problem of optimization is very difficult, and exact solutions (nonnumerical) are only known in few cases, mainly in the binary systems and in general when the state constellation has a symmetry. The specific symmetry that simplifies detection and optimization is called *geometrically uniform symmetry* (GUS).

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