Quantum Information Theory and Free Semialgebraic Geometry: One Wonderland Through Two Looking Glasses

Abstract

We illustrate how quantum information theory and free (i.e. noncommutative) semialgebraic geometry often study similar objects from different perspectives. We give examples in the context of positivity and separability, quantum magic squares, quantum correlations in non-local games, and positivity in tensor networks, and we show the benefits of combining the two perspectives. This paper is an invitation to consider the intersection of the two fields, and should be accessible for researchers from either field.

Figures (3)

• Figure 1: Free semialgebraic geometry and quantum information often look at similar landscapes from different perspectives, as in the fantastic world of this woodcut by M. C. Escher.
• Figure 2: Some three-dimensional spectrahedra taken from NP. Spectrahedra are convex sets described by a linear matrix inequality, and polyhedra are particular cases of spectrahedra.
• Figure 3: (Left) The magic square on the façade of the Sagrada Família in Barcelona, where every row and column adds to 33. (Right) The magic square in Albrecht Dürer's lithograph Melencolia I, where every row and column adds to 34.

Theorems & Definitions (6)

• Theorem 1: Faw19
• Theorem 2: D19Car
• proof
• Theorem 3: BN1
• Theorem 4: Naimark's Dilation Theorem
• Theorem 5: D20