Anti- (Conjugate) Linearity

TL;DR

This paper introduces the theory of antilinear operators, highlighting their differences from linear operators, and explores their applications in quantum systems and mathematical structures like the Lagrangian Grassmannian.

Contribution

It provides an elementary yet comprehensive exposition of antilinear operators, including classification, spectral theory, and applications in quantum information and geometry.

Findings

Antilinear operators have unique spectral and structural properties.

Applications include representing quantum entanglement and teleportation.

Rich structure of antilinear operator spaces is analyzed.

Abstract

This is an introduction to antilinear operators. In following E.P.Wigner the terminus "antilinear" is used as it is standard in Physics. Mathematicians prefer to say "conjugate linear". By restricting to finite-dimensional complex-linear spaces, the exposition becomes elementary in the functional analytic sense. Nevertheless it shows the amazing differences to the linear case. Basics of antilinearity is explained in sections 2, 3,4, 7 and in subsection 1.2: Spectrum, canonical Hermitian form, antilinear rank one and two operators, the Hermitian adjoint, classification of antilinear normal operators, (skew) conjugations, involutions, and acq-lines, i.e. the antilinear counterparts of 1-parameter operator groups. Applications include the representation of the Lagrangian Grassmannian by conjugations, its covering by acq-lines, as well as results on equivalence relations. After remembering elementary Tomita-Takesaki theory, antilinear maps, assiciated associated to a two-partite quantum system, are defined. By allowing to write modular objects as twisted products of pairs of them, they open some new ways to express EPR and teleportation tasks. The appendix presents a look onto the rich structure of antilinear operator spaces.