Shapes of Quantum States and Their Applications

Konrad Szymański ()
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Konrad Szymański: Institute of Physics, Slovak Academy of Sciences, Research Centre for Quantum Information

A chapter in Handbook of Visual, Experimental and Computational Mathematics, 2026, pp 543-571 from Springer

Abstract: Abstract In this chapter a geometric perspective on the structure of quantum information is presented, helping with understanding of many unintuitive properties of the quantum theory. Such a viewpoint arises from the statistical nature of the theory, with probabilities being its cornerstone – the considered sets are exactly a result of probabilistic constraints. A short introduction to quantum information is provided, with special attention to geometric interpretations and methods. In particular, emphasis is put on the sets of mixed quantum states, also known as density operators, and their images under linear maps and their properties. This topic is expanded in the subsequent section, showing more detailed properties of the sets of states through their projections – called numerical ranges – and low-dimensional cuts. The description of the sets through the lens of semidefinite, convex, and algebraic geometry simplifies reasoning in quantum information significantly. This point is exemplified in the last section of the chapter with two solutions to problems in quantum information – finding uncertainty relations and entanglement detection – along with visualization of the fundamental mathematical structures they are based on.

Keywords: Quantum information; Convex geometry; Semidefinite geometry; Algebraic geometry; Density matrices; Numerical ranges; Spectrahedra; Uncertainty relations; Quantum entanglement (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-16368-4_53

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DOI: 10.1007/978-3-032-16368-4_53

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