Derivation of Quantum Mechanical Observables and Dynamics from II₁ Factor Operator Algebras

Holger Friedrich Hofmann

Siendong Huang

Quantum states with perfect correlation of continuous variable systems are investigated. On the basis of the property of perfect correlation, the local observable algebras are assumed to be type II1 factor von Neumann algebras. Consequently, finite dimensional matrices may be represented using elements of local observable algebras. Therefore, the dense coding and teleportation processes are identical in performance to those in the finite dimensional cases.

Werner Timmermann

Roberto Isai Crotone

We present a mathematically rigorous W*-algebraic theory of the Eigenstate Thermalization Hypothesis (ETH) on quantum spin lattices, establishing operational thermalization as a continuous algebraic contraction. Constructing the physical framework as the inductive limit of local C*-algebras on Zd, we represent the thermodynamic system via the GNS construction as a hyperfinite II1 von Neumann factor M equipped with a faithful normal tracial state τ . The observer’s computational limits are strictly encoded by a local subalgebra N defined on a finite spatial support Λ ⊂ Zd . Utilizing Lieb-Robinson causal bounds and the non-commutative ergodic properties of asymptotically abelian systems, we prove that the conditional expectation onto N contracts to the tracial state in the operator norm. Furthermore, we derive the classical Srednicki ansatz directly from the concentration of measure on energy shells of the quasi-local net. Lastly, we analyze the mathematical obstruction of this algebraic contraction in integrable systems and Many-Body Localized (MBL) phases.

Pöttinger, Jürgen

Marie Choda

This is an expository survey article based on a talk for the Spe- cial Session (Operator Algebra) in AMC2009. For two subalgebras A and B of a finite von Neumann algebra M, we discuss a modified version h(A|B) of the Connes-Stormer relative entropy H(A|B). As one of the most elementary examples, we pick two maximal abelian *-subalgebras A, B of the n × n com- plex matrices Mn(C), and show relations between h(A|B) and the entropy for unistochastic matrices. Also we show related results in the case of a pair of subfactors of a type II1 factor.