Pure Maps between Euclidean Jordan Algebras
arXiv:1805.11496 · doi:10.4204/EPTCS.287.19
Abstract
We propose a definition of purity for positive linear maps between Euclidean Jordan Algebras (EJA) that generalizes the notion of purity for quantum systems. We show that this definition of purity is closed under composition and taking adjoints and thus that the pure maps form a dagger category (which sets it apart from other possible definitions.) In fact, from the results presented in this paper, it follows that the category of EJAs with positive contractive linear maps is a dagger-effectus, a type of structure originally defined to study von Neumann algebras in an abstract categorical setting. In combination with previous work this characterizes EJAs as the most general systems allowed in a generalized probabilistic theory that is simultaneously a dagger-effectus. Using the dagger structure we get a notion of dagger-positive maps of the form f = g*g. We give a complete characterization of the pure dagger-positive maps and show that these correspond precisely to the Jordan algebraic version of the sequential product that maps (a,b) to sqrt(a) b sqrt(a). The notion of dagger-positivity therefore characterizes the sequential product.
arXiv commentIn Proceedings QPL 2018, arXiv:1901.09476
References in corpus (18)
- The Resource Theory of Quantum States Out of Thermal Equilibrium
- Informational derivation of Quantum Theory
- Probabilistic theories with purification
- Higher-order interference and single-system postulates characterizing quantum theory
- Distillation of local purity from quantum states
- Characterization of the Sequential Product on Quantum Effects
- An effect-theoretic reconstruction of quantum theory
- Sequential Product Spaces are Jordan Algebras
- Paschke Dilations
- Quotient-Comprehension Chains
- The Category of Von Neumann Algebras
- A universal property for sequential measurement
- Dagger and Dilation in the Category of Von Neumann algebras
- A really simple elementary proof of the uniform boundedness theorem
- Purity through Factorisation
- A Royal Road to Quantum Theory (or Thereabouts), Extended Abstract
- Three characterisations of the sequential product
- An algebraic semi-automated proof of the fundamental identity of Jordan algebras