Maximal Beable Subalgebras of Quantum-Mechanical Observables

Beables in Algebraic Quantum Mechanics

John Bell once argued that one ought to select, out of the 'observables' of quantum theory, some subset of 'beables' that can be consistently ascribed determinate values. Moreover, this subset should be selected so as to guarantee (among other things) that we can dispense with the orthodox interpretation's loose talk about 'measurement values': "...the probability of a beable being a particular value would be calculated just as was formerly calculated the probability of observing that value". Working in the framework of C*-algebras (in particular, Segal algebras), I propose an algebraic charac…

基于量子集合论的上下文隐变量理论的逻辑刻画

Logical Characterization of Contextual Hidden-Variable Theories based on Quantum Set Theory

本研究通过量子集合论证明，集合论宇宙与beable子代数关联当且仅当其为ZFC可满足，并给出唯一极大隐式可定义子宇宙，从而严格重构玻尔的经典语言概念。

Orthogonal maximal abelian *-subalgebras of the 6x6 matrices

We construct new pairs of orthogonal maximal abelian $*$-subalgebras of $M_6(\mathbb C)$, by classifying all self-adjoint complex Hadamard matrices of order 6. In particular, we exhibit a non-affine one-parameter family of non-equivalent Hadamard matrices of order 6. In the last part of the paper we present other previously unknown examples of complex Hadamard matrices of higher orders.

Entropy and optimal decompositions of states relative to a maximal commutative subalgebra

To calculate the entropy of a subalgebra or of a channel with respect to a state, one has to solve an intriguing optimalization problem. The latter is also the key part in the entanglement of formation concept, in which case the subalgebra is a subfactor. I consider some general properties, valid for these definitions in finite dimensions, and apply them to a maximal commutative subalgebra of a full matrix algebra. The main method is an interplay between convexity and symmetry. A collection of helpful tools from convex analysis for the problems in question is collected in an appendix.

Quantum coherence generating power, maximally abelian subalgebras, and Grassmannian Geometry

We establish a direct connection between the power of a unitary map in $d$-dimensions ($d<\infty$) to generate quantum coherence and the geometry of the set ${\cal M}_d$ of maximally abelian subalgebras (of the quantum system full operator algebra). This set can be seen as a topologically non-trivial subset of the Grassmannian over linear operators. The natural distance over the Grassmannian induces a metric structure on ${\cal M}_d$ which quantifies the lack of commutativity between the pairs of subalgebras. Given a maximally abelian subalgebra one can define, on physical grounds, an associat…